When I was a child in primary school, a maths teacher asked me this question: “What is heavier, 1000kg of bricks or 1000kg of feathers?” I would always answer this question incorrectly because I wrongly assumed that a brick is always heavier than a feather. This assumption clouded my logical reasoning and my understanding of the question asked of me wasn’t comprehensive. So both a metric ton of bricks and a metric ton of feathers “weigh the same” .
Or do they?
Both sets of bricks and feathers have identical mass, but they don’t necessarily weigh the same. We were taught to assume that the gravity is the same at every point on Earth, however satellite images depict an interesting phenomenon.
On these colourful maps, the warm colours (red, orange, yellow) represent areas with strong gravity, while cool colours (green, blue) represent areas with weak gravity. Since the strength of the gravitational force is dependent on the mass (Fg = m*g), we observe variance in Earth’s gravity on the surface with mountains (higher masses) having stronger gravitational forces, while trenches (lower masses) having weaker gravitational forces.
Thoughty2 video on time:
Vsauce3 video ft. Neil DeGrasse Tyson on time:
https://www.youtube.com/watch?v=AORsw8NpN4E
Aperture video on time:
https://www.youtube.com/watch?v=7P3Ous2IjiQ
Ted-Ed video on time:
https://www.youtube.com/watch?v=AORsw8NpN4E
Aperture video on time:
https://www.youtube.com/watch?v=7P3Ous2IjiQ
Ted-Ed video on time:
What is time?
In physics, time is a measurement reading on your modern-day clock. However, before the invention of clocks, humans measured time according to the following physical processes understandable to each epoch of civilisation:
— The first appearance of Sirius (heliacal rising) to mark the flooding of the Nile each year.
— The period succession of night and day, which seemed eternal.
— The position on the horizon of the first appearance of the sun at dawn
— The position of the sun in the sky.
— The marking of the moment of noontime during the day
— The length of the shadow case by a gnomon.
Eventually humans began using operational definitions and instrumentation to characterise the passage of time, which coincided with the evolution of our conception of time.
How did humans conceptualise time?
The International System of Units (SI) declared the unit of time is the second (s), which was mathematically defined as (1/24)*(1/60)*(1/60) = 1/86400th of a day until 1967. Since then, the duration of a second is defined as “9,192,631,770 [cycles] of the radiation corresponding to the transition between the 2 hyperfine levels of the ground state of the Caesium 133 atom”, which operates the caesium atomic clock.
The Coordinated Universal Time (UTC) timestamp in use worldwide is an atomic time standard because it is the most relatively accurate time standard in order of 10-15 of a second (i.e. an error of 1 second in approximately 30 million years). The smallest time step considered theoretically observable is approximately 5.391x10-44 seconds, known as the Planck time. After 1950, advances in electronics enabled reliable measurement of the microwave frequencies being generated, which provided the caesium atomic clock practical use. As timekeeping technology further advanced, atomic clock research progressed to larger frequencies, leading to higher accuracies and greater precision. Although clocks have been invented based on these techniques, they aren’t yet in use s primary reference standards.
Before the 20th century, many people including Galileo and Newton thought the perception of time was identical for everyone everywhere on Earth. One second passed for me, one second passed for you and one second passed for everyone else. According to Neil DeGrasse Tyson, this belief is known as linear time, which became the basis for timelines, where time is a parameter. Nowadays, the modern understanding of time evolved with Einstein’s theory of relativity, which states that time run at different rates depending on relative motion, along with the emerging concept of spacetime, where the universe operates on a world line rather than a timeline. If you and I are living in spacetime, time itself is a coordinate.
Describe the regularities in nature
In the past, time was measured from historical recording of the number of occurrences (events) of some periodic phenomenon. Before the laws of physics were invented, humans noted and tabulated the regular recurrences of Earth’s seasons, sunrises and sunsets, the motion of the moon and stars for millennia. Hose & McDougall (1912) stated that early humans thought the sun was the arbiter of the flow of time, hence time back then was known only to the hour for millennia. This is why you see gnomons (part of a sundial) placed across the globe, especially Eurasia, and at least as far southward as the jungles of Southeast Asia. When curious humans began peering upwards at the night sky, they maintained their astronomical observatories for religious purposes since they demonstrated sufficient accuracy to ascertain the regular motions of the stars and nearby planets.
Originally, timekeeping was performed manually by priests, and then for commerce with watchmen tasked to note time. As time progressed, the development of the equinoxes, the sandglass, and the water clock improved in terms of accuracy and reliability.
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This is a photo of a gnomon in Taganrog, Rostov Oblost, Russia, which has a triangular blade shape.
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Around 1330, Richard of Wallingford (1292–1336), abbot of St. Alban's abbey, famously constructed a mechanical clock as an astronomical orrery. During his time, the application of ratchets and gears enabled the towns of Europe to develop mechanisms to display the time on their respective town clocks. At the time of the scientific revolution, the clocks was adequately scaled down for families to share a personal clock, or own a pocket watch. Prior to the commercialisation of clocks, royalty including kings and queens were the only humans who could afford clocks. Throughout the 18th and 19th centuries, pendulum clocks were used worldwide. As technology further advanced into the 20th and 21st centuries, pendulum clocks were superseded by quartz and digital clocks, as well as atomic clocks, for both standards and scientific use.
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This is a photo of the 14th century Wallingford clock at St Albans Cathedral, England.
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How did Galileo describe the flow of time?
In 1583, Galileo Galilei (1564–1642) swayed a lamp in harmonic motion at mass at the cathedral of Pisa, and used his pulse to time its motion. He discovered that a pendulum’s harmonic motion has a constant period. In his 1638 work “Two New Sciences”, Galileo used a water clock to measure the time taken for a bronze ball to roll a known distance down an inclined plane. He describes his clock as “a large vessel of water placed in an elevated position. A thin pipe was soldered to the bottom of this vessel to allow a thin jet of water through it, which was collected in a small glass during the time of each descent (from the vessel’s top or partially along the vessel). After each descent, the water collected was then weighed on the most accurate balance at the time. The differences and ratios of these weights yielded the differences and ratios of the times measured, which showed no substantial errors”.
His experimental setup measured the literal flow of time, in order to describe the motion of a ball. This preceded Newton’s statement in his Principia: “I do not define time, space, place and motion, as being well known to all”.
How did Newton describe linear time?
Around 1665, Isaac Newton (1643–1727) derived the motion of objects falling under gravity, which was the first clear formulation for mathematical physics of linear time, which was conceived as a universal clock. In his Principia, he stated:
“Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.”
The water clock mechanism described by Galileo was designed to provide laminar flow of the water, which allowed a constant flow of water throughout the experiments. Newton embodied this phenomenon as “duration”. Back then, time was regarded as a parameter that served as an index to the behaviour of the physical system being considered. Since Newton's fluents treat a linear flow of time (mathematical time), time was viewed as a linearly varying parameter, a concept of the hours marching along the clock face. Therefore, calendars and ship's logs were used to map the march of the hours, days, months, years and centuries.
What is thermodynamics & and the paradox of irreversibility?
— By 1798, Benjamin Thompson (1753–1814) had discovered that work could be transformed to heat without limit, a precursor of the conservation of energy or the 1st law of themodynamics.
— In 1824, Sadi Carnot (1796-1832) analysed the steam engine with his abstract engine, namely Carnot engine.
— Rudolf Clausius (1822–1888) mentioned a measure of disorder, or entropy, that affected the continually decreasing amount of free energy available for the Carnot engine to use. This is described by the 2nd law of themodynamics.
Hence the continual march of a thermodynamic system, from small to large entropy, at any given temperature, defines an arrow of time. In 1996, Stephen Hawking identified 3 arrows of time:
— Psychological arrow of time = Our perception of an inexorable flow.
— Thermodynamic arrow of time = Distinguished by the growth of entropy
— Cosmological arrow of time = Distinguished by the expansion of the universe.
In an isolated thermodynamic system, entropy reaches a maximal level, and increases. In contrast, Erwin Schrödinger (1887–1961) indicated that life depends on a "negative entropy flow” in 1945. Ilya Prigogine (1917–2003) asserted other thermodynamic systems, which aren’t close to equilibrium, can also exhibit stable spatio-temporal structures. In 1989, the Belousov-Zhabotinsky equations were claimed to demonstrate oscillating colors in a chemical solution. In 1996, Prigogine found that these nonequilibrium thermodynamic branches approach a bifurcation point that is unable, and another thermodynamic branch becomes stable in its stead.
I’ll discuss the laws of thermodynamics in another post.
What is electromagnetism and the speed of light?
— In 1864, James Clerk Maxwell (1831–1879) presented a combined theory of electricity and magnetism by combining all the laws then known relating to those 2 phenomenon into 4 equations. These vector calculus equations are known as Maxwell’s equations for electromagnetism. In free space (i.e. space not containing electric charges), the equations take the form:
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— ∇ (del operator)
— ε0 = Electric permittivity of free space
— μ0 = Magnetic permittivity of free space
— c = 1/ (ε0*μ0)0.5 : Speed of light in free space
— E = Electric field
— B = Magnetic field
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These equations yielded solutions in the form of electromagnetic waves.
How did Einstein describe spacetime?
In 1905, Albert Einstein challenged the notion of absolute time with his theory of special relativity, however he only defined how clocks synchronise to mark a linear flow of time.
If a clock is situated at point A in space, an observer at point A can determine the time values of events in the immediate proximity of point A by simultaneously observing the positions of the clock’s hands. If another clock is situated at point B in space with respect to point A, then it’s possible for an observer at point B to determine the time values of events in the immediate surroundings of point B. However, it’s not possible to compare the events at both points A and B, in respect to time, without further assumption. So far we only defined “a time at point A”, and “a time at point B”.
Since we haven’t defined a common “time” for observers at points A and B, the latter cannot be defined at all unless the “time required by light to travel from A to B” equals the "time it requires to travel from B to A” is clearly defined. For instance, a ray of light starts moving at time (tA) from point A to point B. It arrives at point B at time (tB) before being reflected at point B in the direction of A. Finally, it arrives again at point A at time (t’A).
If we accord with the definitions, the 2 clocks synchronise if:
tB - tA = t’A - tB
Einstein assumed this definition of synchronism is free from contradictions, and possible for any number of points. This means the the following relations are universally valid:
(1) If the clock at B synchronises with the clock at A, the clock at A synchronises with the clock at B.
(2) If the clock at A synchronises with the clock at B and also with the clock at C, the clocks at B and C also synchronise with each other.
Einstein proved that if the speed of light remains constant between reference frames, space and time must be changing so that the moving observer measures the same speed of light as the stationary observer. Recall that velocity is defined by space and time:
v = dr/dt
— r = Position
— t = Time
Indeed, the Lorentz transformation (for 2 reference frames in relative motion, whose x-axis is directed in the direction of the relative velocity) mixes space and time analogous to a Euclidean rotation around the z axis that integrates x and y coordinates.
Consequently, this leads to the relativity of simultaneity.
More specifically, the Lorentz transformation is a hyperbolic rotation:
This is a change of coordinates in the 4-dimensional Minkowski space, a dimension of which is ct.
In Euclidean space, an ordinary rotation (expressed below) is the corresponding change of coordinates.
The speed of light (c) acts as a conversion factor for the measurement of the dimensions of spacetime in different units, which is measured as 299 792 458 m/s. A similar factor in Euclidean space is needed for the measurement of width in nautical miles and depth in feet.
Einstein showed time in a "moving" reference frame passed by slower than in a "stationary" reference frame according to the following equation (derived by the Lorentz transformation by putting ∆x′ = 0, ∆τ = ∆t′):
Δt = Δτ / (1 - v2/c2)0.5
— Δτ = Time between 2 events as measured in the moving reference frame in which they occur at the same place (e.g. two ticks on a moving clock). This is called the proper time between the 2 events.
— Δt = Time between these same 2 events, but as measured in the stationary reference frame.
— v = Speed of the moving reference frame relative to the stationary one.
— c = Speed of light
When moving objects demonstrate a slower passage of time, it is described as undergoing time dilation. These transformations are only valid for 2 frames at constant relative velocity. If we naively applied this concept to other situations, this fosters such paradoxes, such as the twin paradox. Nevertheless, the twin paradox can be resolved with Einstein’s general theory of relativity, as it uses Riemannian geometry (in accelerated, non-inertial reference frames). If we employ the metric tensor describing the Minkowski space:
[(dx1)2 + (dx2)2 + (dx3)2 - c*(dt)2]
then we yield a geometric solution n to Lorentz's transformation that preserves Maxwell’s equations, as Einstein proved. Maxwell’s field equations described an exact relationship between the measurements of space and time in a given region of spacetime and the energy density of that region.
Nonetheless, Einstein’s equations predicted that the gravitational fields adjust time:
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— T = Gravitational time dilation of an object at a distance of r.
— dt = Change in coordinate time, or the interval of coordinate time.
— G = Gravitational constant
— M = Mass generating the field
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The above surd describes the change in proper time (dτ), or the interval of proper time. A simpler approximation can be used:
This means the strength of the gravitational field (as well as the acceleration) are inversely proportional to the passing of time. Particle acceleration experiments and cosmic ray findings supported theories of time dilation, in which moving particles decay slower than their less energetic counterparts. Because gravitational time dilation causes gravitational red-shift and Shapiro signal travel time delays near massive objects such as the sun, the Global Positioning System (GPS) has to adjust its signal to account for this effect.
Einstein's general theory of relativity states that a freely moving particle outlines a history in spacetime that maximises its proper time. Taylor and Wheeler (2000) described this phenomenon as the principle of external ageing, which they defined as “The path a free object takes between two events in spacetime is the path for which the time lapse between these events, recorded on the object's wristwatch, is an extremum.”
Assuming every point in the universe is treated as a ‘centre’, this corresponds with the physics acting the same in all reference frames. This supports Einstein’s theory that time is relative to an inertial frame of reference. Contrast to Newton’s first law, it has its own local geometry in an inertial frame, hence its own measurements of space and time, meaning there is no 'universal clock’
Describe time in quantum mechanics
There is a time parameter in the equations of quantum mechanics. The Schrödinger equation is:
H(t)|ψ(t)) = i*h*(d/dt) |ψ(t))
One solution can be:
|ψ(t)) = e-i*H*t/h |ψe(0))
— e-i*H*t/h = The ‘time evolution operator’
— H = Hamiltonian
However, the Schrödinger picture shown above is equivalent to the Heisenberg picture, which is similar to the Poisson brackets of classical mechanics. The Poisson brackets are superseded by a non-zero communicator, e.g. [H,A] for observable A, and Hamiltonian (H).
d/dt(A) = (i*h)-1 [A, H] + (dA/dt)classical
The above equation denotes an uncertainty relation in quantum physics. For instance, with time (the observable A), the energy E (from the Hamiltonian H) gives:
ΔΕ*ΔT > h/2
— ΔΕ = Uncertainty in energy
— ΔT = Uncertainty in time
— h = Planck’s constant
If the precision of measurement of the duration of a sequence of events increases, the precision of the energy associated with that sequence decreases, and vice versa. Because time is not an operator in quantum mechanics, this equation is different from the standard uncertainty principle.
Corresponding commutator relations also work for momentum p and position q, which are conjugate variables of each other. Moreover, a corresponding uncertainty principle in momentum and position also work for the same variables, similar to the energy and time relation above.
I’ll discuss quantum mechanics in another post.
Describe time in dynamical systems
In dynamical systems, time is a parameterisation of a dynamical system that manifests and operates on the geometry of the system. Physicists suggested that time is an implicit consequence of chaos (i.e. non-linearity / irreversibility) known as the characteristic time, or the rate of information entropy production, of a system.
What is signalling in terms of time?
One application of the electromagnetic waves described above is signalling, which is part of communication between parties and places. e.g. A yellow ribbon tied to a tree, or the ringing of a church bell. A signal can be part of a conversation, which includes a protocol. Moreover, a signal can be the position of the hour hand on a town clock or a railway station.
If you observe, you can still signal different parties and places as long as you live within their past light cone. However, you can’t receive signals from those parties and places outside our past light cone.
I’ll delve into the details of signalling and communications in another post.
Describe time in cosmology
During the early 20th century, there has been debate between physicists including Einstein and Lemaître regarding the primordial nature of the universe. In 1929, Edwin Hubble used his telescopic observations of the stars to show the universe was expanding. Moreover, the Lambda-CDM model computed a positive cosmological constant, which physicists interpreted the universe was expanding at an accelerating rate.
If the universe were expanding, then it was minuscule and significantly hotter and denser billions of years ago, known as a singularity. George Gamow hypothesised that the abundance of chemical elements humans have discovered, named and organised into a modern Periodic Table may have been products of nuclear reactions in a scorching, dense universe. Fred Hoyle trivialised this hypothesis with his ‘Big Bang’ theory. Meanwhile, Fermi and others argued the “big bang” process would have ceased after only the light elements were created such as Hydrogen and Helium, hence it failed to account for the abundance of heavier elements.
Gamow predicted the black-body radiation in the universe cooled to 5-10 Kelvin during its expansion after the Big Bang. Subsequent experiments managed to cool matter down to as low as 2.7 Kelvins, which corresponds to an approximate age of the universe of 13.8 billion years after the Big Band.
This raised a few questions:
— What happened between the singularity of the Big Bang and the Planck time, the smallest observable time?
— What did time separate from the spacetime foam?
General relativity have provided a modern concept of the expanding universe that began from the Big Bang. Relativity and quantum theory have lead to approximate reconstructions of the history of the observable universe. In our epoch, we can see distant stars in our night sky, since electromagnetic waves can propagate without disturbance from conductors or charges. Prior to this epoch, before the universe cooled enough for electrons and nuclei to combine into atoms about 377,000 years after the Big Bang, starlight would not have been visible over large distances.
What is mass?
Every time you stand on a scale (depending on the country you live in), you see your weight measured in kg or lb. e.g. I weigh about 80kg on Earth, and the Earth weighs 80kg on me. The former is caused by the Earth’s gravitational force on you, and the latter is caused by your gravitational force on the Earth. Since both you and Earth are made of atoms, you and Earth exert a gravitational force against one another. If you weighed yourself on the same scale on the moon’s surface, you would see a number that is about 1/6 of the previous value. e.g. I weigh about 13.33 kg on the Moon’s surface. So if you’re desperate to lose about 83% of your weight quickly, perhaps organise an expensive holiday to the Moon. This is because the Moon has a weaker gravitational force than Earth on you, and it has less mass than Earth too.
Mass is defined as a property of the physical body, as well as a measure of its resistance to acceleration (a change in state of motion) after an application of net force. It also determines an object’s strength of its gravitational attraction to other bodies.
According to Newton’s 2nd law of motion, a body of fixed mass (m) accelerates after being subject to a net force (F), which is calculated by a = F/m. Furthermore, a body’s mass determines how its motion changes or is affected by the Earth’s gravitational field. For instance, if I place a tennis ball of mass (mA) at a distance (r, centre of mass to centre of mass) from another tennis ball of mass (mB), each tennis ball will experience an attractive force Fg = (G*mA*mB)/r2, where G = 6.67×10−11 N kg−2 m2 is the “universal gravitational constant”.
What are the units of mass?
The standard International System of Units (SI) unit of mass is the kilogram (kg), which is equivalent to 1000 grams. This unit was first defined in 1795 as 1 dm3 of water at the melting point of ice. However, this definition was scrapped because it was difficult to precisely measure 1 dm3 of water at the proper temperature and pressure. Therefore, in 1889, the kilogram was redefined as the mass of the international prototype of the kilogram of cast iron, hence becoming independent of the metre and the properties of water.
However, the mass of the international prototype and its supposedly identical national copies were discovered to vary over time. Following a final vote by the General Conference on Weights and Measures (CGPM) in November 2018, the kilogram and several other units were redefined on May 20, 2019 according to only invariant quantities of nature: the speed of light, the caesium hyperfine frequency, and the Planck constant.
Other units are accepted for use in SI:
— The tonne (t) or (“metric ton”) is equal to 1000 kg.
— The electronvolt (eV) is a unit of energy that can be converted to a unit of mass thanks to the mass-energy equivalence. Its unit of measurement is (e*V)/c2.
— The atomic mass unit (u) is measured as 1/12 of the mass of a Carbon-12 atom, approximately 1.66×10−27 kg, which helps express the masses of atoms and molecules.
Outside the SI system, other units of mass include:
— The slug (sl) is an Imperial unit of mass that is approximately 14.6 kg.
— The pound (lb) is a unit of both mass and force, used mainly in the United States (about 0.45 kg or 4.5 N).
— The Planck mass (mP) is the maximum mass of point particles (about 2.18×10−8 kg).
— The solar mass (M☉) is the mass of the Sun (≈1.99×1030 kg), which astrophysicists use to compare large masses such as stars or galaxies.
— The mass of a tiny particle may be identified by its inverse Compton wavelength (1 cm−1 ≈ 3.52×10−41 kg).
— The mass of an enormous star or black hole may be identified with its Schwarzchild radius (1 cm ≈ 6.73×1024 kg).
What are the different definitions of mass?
In physical science, Rindler (2006) identified at least 7 different aspects of mass, or physical notions that involve the concept of mass. Every experiment to date has demonstrated these 7 values to be proportional, and equal in some experiments, which lead to the abstract concept of mass.
a. Inertial Mass = A measure of an object’s resistance to acceleration when a force is applied. When we apply a force to an object, we can measure its acceleration resulting from that force to determine its mass. An object with small inertial mass tends to accelerate more than an object with large inertial mass when acted upon by the same force.
b. Active Gravitational Mass = A measure of the strength of an object's gravitational flux (equal to the surface integral of gravitational field over an enclosing surface). We can measure the gravitational field by dropping a small "test object” freely and measuring its free-fall acceleration. For example, if I drop a bowling ball on the Moon, it experiences a smaller gravitational field, therefore accelerates more slowly. Meanwhile, I drop the same bowling ball on the Earth, it experiences a greater gravitational field, therefore accelerates more quickly. Since the Moon’s gravitational field is weaker than Earth’s, we can conclude that the Moon has less active gravitational mass.
c. Passive Gravitational Mass = A measure of the strength of an object's interaction with a gravitational field. If we divide an object's weight by its free-fall acceleration, we can evaluate its passive gravitational mass. For example, 2 objects within the same gravitational field will experience the same acceleration. However, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.
d. Energy = This contains mass according to the principle of mass-energy equivalence. This equivalence is demonstrated in many physical processes including pair production, nuclear fusion, and the gravitational bending of light. e.g. Pair production and nuclear fusion involve the conversion of mass into energy, or vice versa. The gravitational bending of light demonstrates behaviour of photons of pure energy that is analogous to passive gravitational mass.
e. Curvature of spacetime = This is a relativistic manifestation of the existence of mass, which is significantly weak and difficult to measure. Hence, the concept of curvature wasn’t known until Einstein proposed his theory of general relativity. For example, extremely precise atomic clocks placed on the Earth’s surface were discovered to measure less time (run slower) when compared to similar clocks in space. This difference in elapsed time is a form of curvature called ‘gravitational time dilation’.
f. Quantum mass = This is defined as a difference between an object's quantum frequency and its wave number. Various forms of spectroscopy calculates the quantum mass of an electron i.e. the Compton wavelength, which associates with the Rydberg constant, the Bohr radius, and the classical electron radius. A Kibble balance is used to measure the quantum mass of larger objects. In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.
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This figure illustrates the relationship between the properties of mass and their associated physical constants. Every massive object is believed to exhibit all 5 properties. However, due to extremely large or extremely small constants, it’s unfeasible to verify more than 2 or 3 properties for any object.
— rs = Schwarzschild radius: The ability of mass to cause curvature in space and time.
— μ = Standard gravitational parameter: The ability of a massive body to exert Newtonian gravitational forces on other bodies.
— m = Inertial mass: The Newtonian response of mass to forces.
— E0 = Rest energy: The ability of mass to be converted into other forms of energy.
— λ = Compton wavelength: The quantum response of mass to local geometry.
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Do you often use the terms ‘mass’ and ‘weight’ interchangeably for the same concept? For example, do you often say “I weigh X kg” or “My mass is X kg”? In scientific terms, mass and weight are different concepts and quantities that often get confused and tend to be used incorrectly. In a scientific context, mass is the amount of “matter” that comprise an object, whereas weight is the force exerted on an object by gravity. So, for the sake of scientific correctness, you weigh X Newtons, and your mass is Y kg. Your weight changes on the surface of different planets because they have different gravitational field strengths, however your mass doesn’t change at all. Another way to differentiate mass and weight is conceptualising “mass” as an intrinsic property of an object, and “weight” as an object’s resistance to deviating from its natural course of free fall.
But why do so many people get weight and mass mixed up?
In everyday layperson language, all everyday objects are viewed to have both mass and weight, with weight almost exactly proportional to mass. However, the use of “weight” to describe both properties of force and matter depends on the context. For example, in retail commerce, the "net weight" of products actually refers to mass, expressed in units of mass such as grams or ounces. On the contrary, the load index rating on automobile tires refers to weight, because it specifies the maximum structural load for a tire in kilograms. Before the late 20th century, the distinction between the 2 terms was not strictly applied in technical writing, so expressions such as "molecular weight” are still used.
How do we convert units of mass to equivalent forces on Earth?
When an object’s weight (its gravitational force) is expressed in kilograms, it refers to the kilogram-force (kgf or kg-f), also known as the kilopond (kp). Note that all objects on the Earth’s subject are subject to an average gravitational acceleration of approximately 9.8 m/s2. The General Conference on Weights and Measures fixed the value of standard gravity at precisely 9.80665 m/s2 because disciplines such as metrology required a standard value for converting units of defined mass into defined forces and pressures. Therefore the kilogram-force is defined as precisely 9.80665 newtons. In real life, gravitational acceleration at any point on Earth’s surface varies with latitude, elevation and sub-surface density by about (n/10)*0.01.
Engineers in disciplines that involve weight loading (force on a structure due to gravity), such as structural engineering, often convert the mass of objects like steel and vehicles to a force in newtons. This conversion involves the multiplication by a factor around 9.8, and 2 significant figures is usually sufficient for such calculations) to derive the load of the object.
How is buoyancy related to weight?
The relationship between mass and weight on Earth is highly proportional most of the time. For example, a 1000 kg object would have 10 times more weight force than a 100 kg object. However, some objects violate this mass / weight proportionality.
Consider a balloon filled with helium, instead of falling towards the ground, it floats upwards. This phenomenon is buoyancy, a force that opposes gravity. If you partially deflate that balloon, it becomes neutrally buoyant above the ground. If you attempted to weigh a neutrally buoyant balloon on a scale, it would give a reading of 0 kg, because it is perfectly weightless. Even if both the material comprising of the balloon and the gaseous particles inside the balloon have mass, it’s difficult to measure them on everyday scales.
Buoyancy doesn’t make a portion of an object's weight vanish, instead the missing weight is being borne by the ground, meaning less force (weight) is being applied to any scale theoretically placed underneath the object in question. If you try to weigh a swimming pool and then with a person entering and floating in, the entire weight of that human is being borne by the pool and, ultimately, , the scale underneath the pool. If you attempted to weight a buoyant object on a working scale for buoyant objects, it would weigh less, however, the object/fluid system becomes heavier by the additional object's full mass. Since air has fluid properties, it can support the the weight a body loses through mid-air buoyancy.
Both liquids and gases act as fluids in the physical science realm, meaning all macro‑size objects larger than dust particles are immersed in fluids on Earth, in which they experience buoyancy. Consider a swimmer floating in a pool or a balloon floating in air, buoyancy fully counters the gravitational weight of the object being weighed. However, if an object is supported by a sling or cable, it is fundamentally no different from being supported by a fluid, because the weight has transferred to another location.
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Regardless of the fluid in which an object is immersed (gas or liquid), the buoyant force on an object is equal to the weight of the fluid it displaces.
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Although no effort is required to counter the weight of airborne hot air balloons, the inertia associated with their considerable mass can knock fully grown men off their feet when the balloon's basket is moving horizontally over the ground.
I’ll delve into the details of buoyancy and Archimedes’ Principle in another post.
How does the air buoyancy affect on measurement?
Usually, the effect of air buoyancy on objects of normal density is negligible and insignificant in day-to-day activities. For example, the buoyancy’s diminishing effect upon your body weight is about 1/860 that of Earth’s gravity. Moreover, changes in barometric pressure hardly affect a person's weight more than ±1 part in 30,000. In metrology (the science of measurement), air density is accounted for to compensate for buoyancy effects in order to achieve accurate precision mass standards for calibrating laboratory scales and balances. High-quality "working" standards are made of special stainless steel alloys with densities of about 8,000 kg/m3, while platinum-iridium alloys have densities of about 21,550 kg/m3. According to the International Organisation of Legal Metrology, a standard value of buoyancy relative to stainless steel was developed for metrology work, which lead to the coining of “conventional mass”. This term is defined as “the mass of a reference standard of density 8,000 kg/m3 which it balances in air with a density of 1.2 kg/m3 at 20 °C”.
High-precision scales (or balances) used in laboratories are calibrated to conventional mass uses stainless steel standards, i.e. true mass minus 150 ppm of buoyancy. If 2 objects have identical mass, they may have different densities. This means they displace different volumes and therefore have different buoyancies and weights. Hence, any object measured on this scale (compared to a stainless steel mass standard) has its conventional mass measured, i.e. its true mass minus an unknown degree of buoyancy.
What are the different types of scales?
When you stand on a balance-beam-type scale at the doctor’s office, your mass is being measured directly. Balances (or "dual-pan" mass comparators) are designed to compare the gravitational force exerted on the person on the platform with that on the sliding counterweights on the beams. Gravity is the force that diverges the needle from the "balanced" (null) point. If you move these balances from Earth's equator to the poles, they may give identical measurements. However they don’t indicate that the doctor's patient became heavier. This means the balances aren’t affected by the gravity-countering centrifugal force due to Earth's rotation about its axis.
If you step onto spring-based or digital load cell-based scales (single-pan devices), the number you see on the screen is your weight (gravitational force) measurement. This means variations in the strength of the gravitational field lead to variations of the readings. Therefore you notice such scales often used in commerce or hospitals because they require a highly accurate measurement of mass.
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This is a photo of a balance-type weighing scale, which is unaffected by the strength of gravity.
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This is a photo of a spring-based hanging balance.
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This is a photo of a load cell-based balance scale.
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Inertial vs. gravitational mass
So far, no experiment has unambiguously demonstrated any difference between inertial mass, passive and active gravitational masses. In classical mechanics, Newton’s 3rd law implies that active and passive gravitational mass must always be identical (or at least proportional). However, the classical theory doesn’t provide a compelling reason that explains why the gravitational mass has to equal the inertial mass.
Albert Einstein developed his general theory of relativity under the assumption that the intentionality of correspondence between inertial and passive gravitational mass. No experiment at that time was able to detect a difference between them, in essence the equivalence principle. This equivalence is often described as the "Galilean equivalence principle" or the “weak equivalence principle”, which is an important outcome for free-falling objects. Consider an object has inertial mass (m) and gravitational mass (M). If the only force acting on the object originates from a gravitational field g, the force on the object is F = M*g. Given this force, the acceleration of the object can be determined by Newton's 2nd law: F = m*a. Combining these formulae gives the gravitational acceleration: a = M*g/m
This equation describes the ratio of gravitational to inertial mass of any object is equal to some constant K if and only if all objects fall at the same rate in a given gravitational field. This phenomenon is called the "universality of free-fall”. Moreover, the constant K can be taken as 1 by defining our units appropriately.
According to scientific folklore, the first experiments that claimed to demonstrate the universality of free-fall were performed by Galileo when he dropped objects from the Leaning Tower of Pisa. Historians argued this story is most likely apocryphal, instead they claimed Galileo likely performed his experiments with balls rolling down nearly frictionless inclined planes to slow the motion and increase the timing accuracy. In 1889, Eötvös, Pekár & Fekete (1922) used the torsion balance pendulum to perform increasingly precise experiments. As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been discovered, at least to the precision 10−12.
Since the universality of free-fall only applies to systems in which gravity is the only acting force, all other forces, such as friction and air resistance, are ignored or absent. For example, if I drop a hammer and a feather from the same height through the air on Earth, the feather will take much longer to reach the ground due to upwards air resistance force. This means the feather is technically not in free-fall. If I drop the same hammer and feather from the same height in a perfect vacuum, in which air resistance is absent, the hammer and the feather should land on the ground at exactly the same time. We assume the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible.
A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, is expounded by the “general theory of relativity”. Einstein's equivalence principle states that it is impossible to distinguish between a uniform acceleration and a uniform gravitational field within sufficiently small regions of space-time. Hence, the theory proposes that the force acting on a massive object caused by a gravitational field is caused by the object's tendency to move in a straight line. Thus, it should be be a function of its inertial mass and the strength of the gravitational field.
What generates mass?
In theoretical physics, a mass generation mechanism is a theory that attempts to explain the origin of mass from the most fundamental laws of physics. Although many models have been proposed by physicists, however, it is difficult to explain the notion of mass being strongly related to the gravitational interaction. 2 types of mass generation models have been proposed so far: gravity-free models and models that involve gravity.
— The Higgs mechanism is based on a symmetry-breaking scalar field potential, such as the quartic.
— The Standard Model uses this mechanism as part of the Gashow-Weinberg-Salam model to unify electromagnetic and weak interactions. It was one of several models that predicted the existence of the scalar Higgs boson. — Gravity-free models, such as the Standard Model, suggests that the gravitational interaction is either not involved or does not play a crucial role.
Other theories include:
- Unparticle physics and the unhiggs models
- UV-Completion by Classicalisation = Unitarisation of the WW scattering occurs by creation of classical configurations.
- Technicolour model
- Coleman-Weinberg mechanism
- Symmetry breaking driven by non-equilibrium dynamics of quantum fields above the electroweak scale
- Asymptotically safe weak interactions based on some nonlinear sigma models
- Models of composite of W and Z vector bosons
- Top quark condensate
Gravitational models include:
- Extra-dimensional Higgsless models = This concerns the 5th component of the gauge fields in place of the Higgs fields. Csaki et al. (2004) suggested that imposing certain boundaries conditions on the extra dimensional fields can produce electroweak symmetry breaking, which theoretically increase the unitarity breakdown scale up to the energy scale of the extra dimension. Calmet et al. (2009) implied that this model can be related to technicolor models and to UnHiggs models through the AdS/QCD correspondence, in which the Higgs field is of unparticle nature.
- Unitary Weyl gauge = If we add a suitable gravitational term to the standard model action with gravitational coupling, this theory becomes locally scale-invariant (i.e. Weyl-invariant) in the unitary gauge for the local SU(2). Pawlowski & Raczka (1994) proposed that Weyl transformations can act multiplicatively on the Higgs field, which meant the Higgs scalar needs to be a constant to fix the Weyl gauge.
- Preon-inspired models = Such models include the Ribbon model of Standard Model particles by Sundance Bilson-Thompson, based in braid theory and compatible with loop quantum gravity and other similar theories. It not only explains the origin of mass, but also interprets electric charge as a topological quantity (its twists carried on the individual ribbons), and colour charge as modes of twisting.
- Superfluid vacuum theory = Avdeenkov & Zloshchastiev (2011) suggested that masses of elementary particles arise from interaction with a physical vacuum, analogous to the gap generation mechanism in superfluids. Zloshchastiev & Dzhunushaliev (2013) suggested the low-energy limit of the theory implicated an effective potential for the Higgs sector contrasting to the Standard Model’s, which yields the mass generation nonetheless. Under certain conditions, this potential gives rise to an elementary particle with a role and characteristics similar to the Higgs boson.
Discuss the Pre-Newtonian concepts of mass
i. Weight as an amount
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This painting illustrates the early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, c. 1285 BCE). The scene shows Anubis weighing the heart of Hunefer.
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The concept of “amount” is quite ancient, predating recorded history. At a particular early era, humans realised the weight of a collection of similar objects was directly proportional to the number of objects in the collection:
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— W = Weight of the collection of similar objects
— n = Number of objects in the collection
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This proportionality indicates that 2 values have a constant ratio:
Wn/n = Wm/m, or equivalently Wn/Wm = n/m
An early use of this relationship is a balance scale, which balances the force of one object's weight against the force of another object's weight. When the 2 sides of the balance scale are on the same level, the objects experience similar gravitational fields. Therefore, if have similar masses then their weights will also be similar. Scales were then routinely used to compare weights or masses.
As a consequence, historical weight standards were often defined in terms of amounts. For instance, the Romans used the carob seed (carat or siliqua) as a standard of measurement. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If the object's weight was equivalent to 144 carob seeds, then the object was said to weigh one Roman ounce (uncia). This shows the carob seed was used as the common mass standard for defining both the Roman pound and ounce.
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This is a picture of carob seeds.
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The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:
ounce / pound = W144 / W1728 = 144/1728 = 1/12
ii. Planetary motion
In 1600 AD, Johannes Kepler sought employment with Tycho Brahe, who possessed some of the most precise astronomical data available. Using Brahe's precise observations of the planet Mars, Kepler spent a number of years conjuring his own method for characterising planetary motion. In 1609, he published his 3 laws of planetary motion in an attempt to explain the planets’ orbital motion around the Sun. Kepler's final planetary model illustrated the planetary orbits as elliptical paths with the Sun at a focal point of the ellipse. He also discovered that the square of the orbital period of each planet is directly proportional to the cube of the semi-major axis of its orbit. In order words, the ratio of these 2 values is constant for all planets in the Solar System.
On 25 August 1609, Galileo Galilei showcased his first telescope to a group of Venetian merchants. Then in early January 1610, Galileo observed 4 dim objects near Jupiter, which he mistook for stars. A few days later, Galileo observed those objects were orbiting Jupiter, which were the first celestial bodies observed to orbit a celestial body other than the Earth or Sun. Nowadays, we know these objects as the moons of Jupiter, they were later named the Galilean moons in his honour.
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Painting of Galileo Galilei (1636)
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iii. How did Galileo describe free fall?
Prior to 1683, Galileo become interested in the phenomenon of objects in free fall, and attempted to denote their motion. A biographical entry by Galileo’s pupil Vincenzo Viviani stated that “Galileo had dropped balls of some material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass”. Galileo then provided the following theoretical argument: “If 2 bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?” In 1632, Galileo believed that answering this question requires all bodies falling at the same rate.
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This diagram illustrates the distance travelled by a freely falling ball is proportional to the square of the elapsed time.
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In 1638, Galileo published a later experiment in his Two New Sciences. Imagine a fictional character named Salviati, who described an experiment using a bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick” with a straight, smooth, polished groove. The groove was lined with “parchment, also smooth and polished as possible”. He then placed "a hard, smooth and very round bronze ball” into this groove. The ramp was inclined at various angles to adequately reduce the ball’s acceleration in order to measure the elapsed time. Subsequently, the ball rolled a known distance down the ramp, and the time taken for the ball to move the known distance was measured. Galileo used a water clock to measure time elapsed.
Galileo found that the distance a falling object traverses s always proportional to the square of the elapsed time:
He demonstrated that free falling objects under the influence of the Earth's gravitational field have a constant acceleration. Furthermore, his contemporary, Johannes Kepler, demonstrated that the planets traced elliptical paths under the influence of the Sun's gravitational mass. However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.
What is Newtonian mass?
In 1674, Robert Hooke published his concept of gravitational forces, which states that all celestial bodies have an attraction or gravitating power towards their own centres, and also attract all the other celestial bodies that are within the sphere of their activity. Furthermore, he stated that gravitational attraction increases by the body’s distance from its own centre. In 1679-80, Hooke conjectured that gravitational forces are inversely proportional to double of the distance between the 2 bodies, which corresponded with Newton’s hypothesis. There were accounts of Hooke persuading Newton to check the mathematical details of orbits described by Kepler to verify his hypothesis. Although Newton’s own investigations verified Hooke’s hypothesis, Newton chose not to reveal this finding to Hooke due to personal differences. In 1684, Newton told his friend Edmond Halley about his solution to the problem of gravitational orbits, but somehow misplaced the solution in his office. After a persuasion from Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Newton forwarded a document to Halley that may have been titled De motu corporum in gyrum (Latin for "On the motion of bodies in an orbit”). Halley presented Newton's findings to the Royal Society of London, promising a complete presentation would follow. Later, Newton recorded his ideas in a 3 book set, entitlted Philosophiæ Naturalis Principia Mathematica (Latin: Mathematical Principles of Natural Philosophy). They were received by the Royal Society in order on the following dates: 28 April 1685–86, 2 March 1686–87 and 6 April 1686–87, which were then published in May 1686-87.
Newton’s discovery of the relationship between gravity and radius helped bridge the gap between Kepler's gravitational mass and Galileo's gravitational acceleration.
g = -μ*(R^/ |R|2)
— g = Apparent acceleration of a body as it passes through a region of space where gravitational fields exist
— μ = Gravitational mass (standard gravitational parameter) of the body causing gravitational fields
— R = Radial radial coordinate, i.e. the distance between the centres of the two bodies.
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Painting of Edmond Halley
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Newton provided an additional method for evaluating gravitational mass by developing the relationship between a body's gravitational mass and its gravitational field. There are 2 solutions for calculating the mass of the Earth. One is Kepler’s method (based on the orbit of Earth's Moon), and the other is through measurement of the gravitational acceleration on the Earth's surface before multiplying it by the square of the Earth's radius. It’s estimated the mass of the Earth is approximately 3/1,000,000th of the mass of the sun. As of 2003, Cuk (2003) noted that no other accurate method for measuring gravitational mass has been discovered.
Describe Newton’s cannonball
https://en.wikipedia.org/wiki/Newton%27s_cannonball
In his book ‘A Treatise of the System of the World’, Isaac Newton outlined a thought experiment to conjecture the idea of the force of gravity being universal, which was the key force behind planetary motion.
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This diagram illustrates a cannon on top of a very high mountain shoots a cannonball horizontally. If the speed is low, the cannonball quickly falls back to Earth (A,B). If the speed is intermediate, the cannonball will revolve around Earth along an elliptical orbit (C,D). If the speed is sufficiently high, the cannonball would theoretically leave the Earth entirely (E).
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Imagine a cannon placed on top of a colossal mountain, and it fires a cannonball. In the absence of the forces of gravitation or air resistance, the cannonball would follow a straight line away from Earth, tangential to Earth’s orbital path at the point of firing, in the direction of firing. If the force of gravitation exists and acts on the cannonball, the cannon’s path would head towards the Earth’s surface, depending on the initial velocity of the cannonball.
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If the initial speed of the cannonball is 0 m/s horizontally from Newton’s hypothetically tall mountain, it would fall down to the Earth’s surface in a straight line.
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If the initial speed of the cannonball is 6000 m/s (low speed) horizontally from Newton’s hypothetically tall mountain, it would follow a curved trajectory and fall towards the Earth’s surface.
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If the initial speed of the cannonball is about 7300 m/s (orbital speed) horizontally from Newton’s hypothetically tall mountain, it would circle around the Earth along a fixed circular orbit, just like the Moon and International Space Station.
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If the initial speed of the cannonball is about 8000 m/s horizontally from Newton’s hypothetically tall mountain (faster than orbital speed but lower than escape velocity), it would follow an elliptical orbit as it revolves around Earth.
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If the initial speed of the cannonball is exactly escape velocity (around 11,200 m/s) horizontally from Newton’s hypothetically tall mountain, it would follow a parabolic or hyperbolic trajectory away from Earth.
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This is a photograph of page 6 from Newton’s Philosophiæ Naturalis Principia Mathematica Volume 3, De mundi systemate (On the system of the world). It is part of the image collection on the Voyager Golden Record that is being carried into deep space aboard the Voyager 1 and 2 spacecraft.
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Discuss universal gravitational mass
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This diagram illustrates an apple experiencing gravitational fields directed towards every part of the Earth. Nevertheless, the sum total of these many fields produces a single gravitational field directed towards the Earth's centre.
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In the 1680s, Newton introduced the concept of universal gravitational mass that states that “every object has gravitational mass, and therefore, every object generates a gravitational field”. He assumed that the strength of each object's gravitational field decreases with the the square of the distance to that object. If a large collection of tiny objects combined into a giant spherical body such as the Earth or Sun, it would generate a gravitational field proportional to the total mass of the body, and inversely proportional to the square of the distance to the body's centre.
For example, consider a carob seed that produces its own gravitational field. If I were to gather copious amounts of carob seeds and combine them into a gargantuan sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Newton believed it was theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun. Using the method of unit conversion on any traditional mass unit can theoretically evaluate the gravitational mass.
However, measuring gravitational mass based on Newton’s theory is difficult practically. Smaller objects have significantly weak gravitational fields, hence difficult to measure. It wasn’t until the 1797 Cavendish experiment, which was the first to successfully measure Earth's mass in terms of traditional mass units. Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water.
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This is a vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside.
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If 2 objects A and B, of masses MA & MB, separated by a displacement RAB, then each object exerts a gravitational force on the other, of magnitude FAB = -G*MA*MB*(R^AB)/(|RAB|2).
— G = Universal gravitational constant
Discuss inertial mass
Proposed by Ernst Mach and developed into the notion of operationalism by Percy W. Bridgman, the definition of an “inertial mass” is the mass of an object measured by its resistance to acceleration. Although the simple classical mechanics definition of mass varies slightly than that in the theory of special relativity, the essential meaning is identical.
Belkind (2012) argued that Mach’s definition of mass fails to account for the potential energy (or binding energy) required to bring 2 masses sufficiently close to one another to perform the measurement of mass. If we compared the mass of the proton in the nucleus of deuterium with the mass of the proton in free space, it would be greater by about 0.239% due to the binding energy of deuterium. For instance, if we take the reference mass (m2) as the mass of the neutron in free space, and compute the relative accelerations for the proton and neutron in deuterium, then the above formula over-estimates the mass m1 or the proton in deuterium by 0.239%. So Mach’s formula can only be used to obtain ratios of masses, i.e. m1 / m2 = |a2| / |a1|. Henri Poincaré highlighted the impossibility of measuring instantaneous acceleration with a single measurement, and asserted multiple measurements (of position, time, etc.) are required to compute the acceleration. He described this as an "insurmountable flaw" in the Mach definition of mass.
What is atomic mass?
https://en.wikipedia.org/wiki/Dalton_(unit)
The mass of all objects was formerly measured relative to that of the kilogram, which was defined as the mass of the international prototype of the kilogram (IPK). The IPK was a platinum alloy cylinder stored in an environmentally-monitored safe secured in a vault at the International Bureau of Weights and Measures in France. However, the IPK demonstrated inconvenience in measuring the masses of atoms and particles of similar scale, as it contains quintillions of atoms, because it lost or gained a little mass over time in spite of the best efforts to prevent these variations. In order to precisely compare an atom’s mass to that of another atom, scientists developed the atomic mass unit (u) or Dalton (Da). They defined 1u as exactly 1/12 of the mass of a Carbon-12 atom, and by extension a carbon-12 atom has a mass of exactly 12 u. Studies stated that this definition is subject to change by the proposed redefinition of SI base units, which will leave the Dalton close to 1, but no longer exactly equal to it.
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This is a photo of the international Prototype Kilogram (IPK).
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Veritasium’s videos on the kilogram and its redefinition:
https://www.youtube.com/watch?v=c_e1wITe_ig
I’ll discuss the changing definitions of the kilogram in another post.
What is weight?
https://en.wikipedia.org/wiki/Weight
i. Gravitational definition
In introductory physics textbooks, weight is commonly defined as the force exerted on a body by gravity. It is expressed n the formula W = mg, where W is the weight, m is the mass of the object, and g is the gravitational acceleration. In 1901, the 3rd General Conference on Weights and Measures (CGPM) established the official definition of weight:
“The word weight denotes a quantity of the same nature as a force: the weight of a body is the product of its mass and the acceleration due to gravity.”
— Resolution 2 of the 3rd General Conference on Weights and Measures
This resolution defines weight as a vector, since force is a vector quantity. However, some textbooks define weight as a scalar. The 2007 Fundamentals of Physics. 1 (8th ed.) defined the “weight W of a body is equal to the magnitude Fg of the gravitational force on the body.” A 1979 study defined weight as the m kilogram weight (which term is abbreviated to kg-wt.
ii. Operational definition
The operational definition states that the weight of an object is the force measured by the operation of weighing it, which is the force it exerts on its support. Since W is the downward force on the body by the centre of Earth and there is zero acceleration in the body, there is an opposite and equal force by the support on the body. Furthermore, weight is equal to the force exerted by the body on its support because action and reaction have same numerical value and opposite direction. For instance, if an object in free fall exerts little if any force on its support, this object experiences “weightlessness”.
However, this definition doesn’t explicitly exclude the effects of buoyancy, which decreases the measured weight of an object when immersed in a fluid such as air or water. This suggests a floating balloon or an object floating in water might have zero weight.
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Left: A spring scale measures weight, by seeing how much the object pushes on a spring (inside the device). If we weighed an object on the Moon, would give a lower reading.
Right: A balance scale indirectly measures mass, by comparing an object to references. If we measured an object on this scale on the Moon, it would give the same reading, because the object and references would both become lighter.
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iii. ISO definition
In the ISO International standard ISO 80000-4:2006, which describes the basic physical quantities and units in mechanics as a part of the International standard ISO/IEC 80000, the definition of weight is provided as:
Fg = m*g
— m = mass
— g = local acceleration of free fall.
Remarks include:
- When the reference frame is Earth, this quantity comprises not only the local gravitational force, but also the local centrifugal force due to the rotation of the Earth, a force which varies with latitude.
- The effect of atmospheric buoyancy is excluded in the weight.
- In common parlance, the name "weight" continues to be used where "mass" is meant, but this practice is deprecated. — ISO 80000-4 (2006)
The definition relies on the selected frame of reference. If the chosen frame is co-moving with the object of concern, then it supports the operational definition. However, if the specified frame is the surface of the Earth, the weight according to the ISO and gravitational definitions differ only by the centrifugal effects due to the rotation of the Earth.
What is apparent weight?
In real world situations, the act of weighing may produce a result that differs from the ideal value provided by the definition used. This is known as “apparent weight”. For example, when an object is immersed in a fluid the displacement of the fluid will cause an upward force on the object, which makes it appear lighter when weighed on a scale due to the effects of buoyancy. Levitation and mechanical suspension both similarly affect the apparent weight of an object. Galili (1993) asserted that, if the gravitational definition of weight is used, the apparent weight of an object would be the operational weight measured by an accelerating scale.
In physics, apparent is defined as the property of objects that corresponds to how heavy an object is. Whenever the force of gravity acting on the object is not balanced by an equal but opposite normal force, the weight of that object will not match the apparent weight of the object. This means a “weightless” astronaut in low Earth orbit, with an apparent weight of zero, weighs the same as another astronaut standing on the ground. This is because the force of gravity in low Earth orbit and on the ground are virtually the same.
When an object rests on the ground, it experiences a normal force exerted by the ground, which acts only on the boundary of the object that is in contact with the ground. While the normal is transferred into the body, the force of gravity on every part of the body is balanced by stress forces acting on that part. In outer space, an astronaut is not subject to such stress forces, making them feel “weightless”. If we defined the apparent weight of an object in terms of normal forces, we can understand this effect of the stress forces. Beiser (2004) preferred to define the weight force as "the force the body exerts on whatever it rests on.”
Consider a person riding in an elevator. When the elevator rises, they begin to exert a force in the downward direction. When that person stands on a scale, their weight would be heavier due to the extra downward force, which changes their apparent weight.
Vsauce video on gravity:
What are statics?
https://en.wikipedia.org/wiki/Statics
It is a branch of mechanics that analyses the effects of loads (force and torque, “moment”) on physical systems that don’t experience an acceleration (a = 0), but rather, are in static equilibrium with their environment. The pioneering work in statics was first conducted by Archimedes (c.287 - c.212 BC), which was later developed by Thebit.
In the analysis of structures, e.g. in architectural and structural engineering, static equilibrium is applied to understand the strength of materials. At rest, the body’s centre of gravity is a key concept because it represents an imaginary point at which all the mass of a body resides. A body’s stability in response to external forces is determined by the position of its point relative to the foundations on which a body lies on.
— If the centre of gravity shifts outside the foundations, then the body is determined to be unstable, leading to an active torque. This means any disturbance, no matter how minuscule, will cause the body to fall or topple.
— If the centre of gravity is within the foundations, the body is stabilised, meaning no net torque acts on the body.
— If the centre of gravity coincides with the foundations, then the body is metastable.
Fluid statics, also known as hydrostatics, studies fluids at rest (i.e. in static equilibrium). When a force exerts on any particle of the fluid, it is the same at all points at the same depth (or altitude) within the fluid. If the net force is is greater than zero the fluid will move in the direction of the resulting force. This concept was first formulated in an extended form by Blaise Pascal in 1647, famously known as Pascal’s Law. I’ll discuss hydrostatics in another post.
What are dynamics?
https://en.wikipedia.org/wiki/Dynamics_(mechanics)
It is a branch of classical mechanics that studies forces and its effects on motion. The fundamental physical laws that govern dynamics in physics were defined by the Isaac Newton. Physicists involved in dynamics study how a physical system might develop or alter over time and study the causes of those changes. Understanding the dynamics of a system of mechanics requires the application of Newton’s 3 laws of motion, especially the 2nd law.
The study of dynamics are categorised into linear and rotational dynamics.
(a) Linear dynamics concerns the motion of objects in a straight line and involves such quantities as:
— Force
— Mass/inertia
— Displacement (in units of distance)
— Velocity (distance per unit time)
— Acceleration (distance per unit of time squared)
— Momentum (mass times unit of velocity).
(b) Rotational dynamics concerns the motion of objects in a rotating or along a curved path and involves such quantities as:
— Torque
— Moment of inertia / rotational inertia
— Angular displacement (in radians or less often, degrees)
— Angular velocity (radians per unit time)
— Angular acceleration (radians per unit of time squared)
— Angular momentum (moment of inertia multiplied by unit of angular velocity).
What is force?
https://en.wikipedia.org/wiki/Force
i. Describe bodies with spatial extent
https://en.wikipedia.org/wiki/Gravitational_field
This equation shows the gravitational field on mass (mj) as the sum of all gravitational fields due to all other masses (mi) except the mass (mj) itself. The unit vector (R^ij) is in the direction of Ri — Rj.
https://en.wikipedia.org/wiki/Free_fall
https://www.youtube.com/watch?v=KDp1tiUsZw8
https://en.wikipedia.org/wiki/Coulomb%27s_law
https://en.wikipedia.org/wiki/Hooke%27s_law
https://en.wikipedia.org/wiki/Drag_(physics)
https://en.wikipedia.org/wiki/Virtual_work
https://en.wikipedia.org/wiki/Work_(physics)
https://en.wikipedia.org/wiki/Mechanical_energy
https://en.wikipedia.org/wiki/Energy
In physics, a force is defined as any (unopposed) interaction that changes the object’s motion. It causes an object with mass to change its velocity, leading to acceleration. Since it has both magnitude and direction, force is a vector quantity.
What are its units of measurement?
Its SI units is Newtons or kg*ms-1, represented by the symbol F. Forces can be described as a push or pull on an object due to phenomena such as gravity, magnetism, or anything that can cause a mass to accelerate.
As with other physical concepts such as temperature, the precise operational definitions are used to quantify intuitive understanding of forces. These definitions need to be consistent with direct observations and compared to a standard measurement scale. Laboratory measurements of force are determined through experimentation to demonstrate consistency with the conceptual definition of force offered by Newtonian mechanics.
Forces act in a particular direction and their magnitudes depend on the strength of the pulling or pushing action. Thus, these characteristics classify forces as “vector quantities”. This means forces would follow a different set of mathematical rules than physical quantities that lack direction (i.e. scalar quantities). For instance, to determine the outcome of 2 forces acting on the same object, we require information on both the magnitude and the direction of both forces. If both of these pieces of information are unknown for each force, then the situation is ambiguous and a result is virtually impossible to predict. For example, consider 2 people are pulling on the same rope with known magnitudes of pulling force in a tug of war. Without information on the exact direction of either person’s pulling force, it is impossible to determine the exact acceleration of the rope. Therefore, knowing the directions of all forces, represented by vectors, are essential in determining the net force.
The first instances of quantitative investigation of force were in situations where several forces canceled each other out leading to static equilibrium. Sears experimentally demonstrated that forces can be treated as additive vector quantities because they have both magnitude and direction. When 2 forces act on a point particle, the resultant force (or net force) is determined using the parallelogram rule of vector addition. Kleppner & Kolenkow (2010) explained this rule as the addition of 2 vectors representing sides of a parallelogram, which yields an an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The difference between the magnitudes of the 2 forces and the angle between their lines of action influences the magnitude of the resultant net force. However, if the forces are exerted on an extended body, we need to specify their respective lines of application in order to account for their effects on the body’s motion.
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These are free body diagrams of a block on a flat surface and an inclined plane. Forces are resolved and added together to determine their magnitudes and the net force.
(i) N = m*g*cos(θ)
m*g*cos(θ) + (m*g*sin(θ) - μk*N) = m*g
m*g*cos(θ) + (m*g*sin(θ) - μk*(m*g*cos(θ)) = m*g
m*g*(cos(θ) + sin(θ) - μk*cos(θ)) = m*g
cos(θ)*(1 - μk) + sin(θ) = 1
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(ii) Normal force = Weight force (m*g)
m*g is also considered to the weight of an object.
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Free body diagrams conveniently illustrate the forces acting on a system. These diagrams are ideally drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be performed to determine the net force. In addition to vector addition, forces can also be segregated into independent components at right angles with each other. For example, a horizontal force directed southwestwards can split into 2 forces, 1 southwards, and 1 westwards. Combining these component forces together using vector addition would compute the original force. In the case of orthogonal components, the components of the vector sum are evaluated by the scalar addition of the components of the individual vectors. Since orthogonal components act independently of each other, forces acting perpendicularly to each other have no effect on the magnitude or direction of the other. Consider selecting a set of orthogonal basis vectors in order to make the mathematical working most convenient. Moreover, consider selecting a basis vector that points in the same direction as one of the forces, since that force would have only 1 non-zero component. A few papers stated orthogonal force vectors can be 3-dimensional with the 3rd component being at right angles to the other 2.
What is mechanical equilibrium?
https://en.wikipedia.org/wiki/Mechanical_equilibrium
— Static
In classical mechanics, a particle in mechanical equilibrium has a net force of zero on it. Therefore, a physical system is in mechanical equilibrium if all of its individual components has a net force of zero on each of them.
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This schematic shows an object resting on a surface and the corresponding free body diagram indicating the forces acting on the object. The normal force (N) s equal, opposite, and collinear to the gravitational force (mg), so the net force and moment is zero. Consequently, the object is in a state of static mechanical equilibrium.
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There are alternative definitions for mechanical equilibrium that are all mathematically equivalent. The system can be in equilibrium if the momentum or velocity is constant, the angular momentum is conserved and net torque is zero. A conservative system is in equilibrium if a point in configuration space is zero where the gradient of the potential energy with respect to the generalised coordinates.
A particle in static equilibrium has zero velocity. Therefore, if all particles in equilibrium have constant velocity, then we can find an inertial reference frame in which the particle is stationary with respect to the frame
What is stability?
Consider a function that describes the system’s potential energy, we can use calculus to determine the system's equilibria. It should be at mechanical equilibrium at the function’s critical points that describe the system's potential energy. These points can be located when the derivative of the function is zero at these points. To determine the stability of the system, the second derivative test is applied:
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i. 2nd derivative < 0
The potential is at a local maximum, meaning the system is in an unstable equilibrium state. We can interpret that an arbitrarily small displacement of the system from equilibrium state would allow the forces of the system to shift it even farther away.
The diagram shows the ball in an unstable equilibrium.
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ii. 2nd derivative > 0
The potential energy is at a local minimum, meaning the system is a stable equilibrium. We can determine that a small displacement from equilibrium state causes the forces to restore that equilibrium. If more than 1 stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states.
This diagram shows the ball placed in a stable equilibrium.
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iii. 2nd derivative = 0 or Does not exist
The state of the system is neutral to the lowest order and virtually remains in equilibrium unless a force causes a small displacement. To understand the precise stability of the system, we need to examine higher-order derivatives. If the lowest non-zero derivative is is of odd order or has a negative value, then the state is unstable. On the other hand, if the lowest non-zero derivative is both of even order and has a positive value, then the state is stable. however if all higher order derivatives are zero, then the state is neutral. So, in a truly neutral state, the energy doesn’t vary and the state of equilibrium has a finite width. It can be referred to as a marginally stable state or an indifferent state.
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If the system exists in more than 1 dimension, this yields different results in different directions. For example, a system may be stable with respect to displacements in the x-direction but unstable in the y-direction. This is known as a “saddle point”. If a system is stable in all directions, then it can safely be described as in stable equilibrium.
If there is not enough information about the forces acting on a body to determine if it is in equilibrium or not, then it makes the system statically indeterminate.
Examples of static equilibrium include:
— A paperweight on a desk
— A rock balance sculpture
— A stack of blocks in the game of Jenga (as long as the sculpture or stack of blocks is not in the state of collapsing)
— An object on a level surface being pulled or attracted downward toward the center of the Earth by the force of gravity. Simultaneously, the Earth’s surface applies an equal upwards force that resists the downward force, also known as the normal force. This leads to zero net force and hence no acceleration.
Examples of moving object in mechanical (but not static) equilibrium include:
— A child sliding down a slide at constant speed.
— A person pressing a spring to a defined point. They can push it to an arbitrary point and maintain its position there, at which point the compressive load and the spring reaction are equal. When the compressive force is relieved, then the spring returns to its original state.
— An object suspended on a vertical spring scale experiences the downwards force of gravity acting on the object balanced by a force applied by the upwards "spring reaction force", which equals the object's weight. This lead to the discovery of a few quantitative force laws as such that “the force of gravity is proportional to volume for objects of constant density). Those laws include:
- Archimedes principle for buoyancy
- Archimedes’ analysis of the lever
- Boyle’s law for gas pressure
- Hooke’s law for springs
When one pushes against an object resting on a frictional surface, this results in the object not moving because the applied force is opposed by static friction, generated between the object and the surface. If the static friction force exactly balances the applied force, the object will not move, meaning no acceleration. Young & Zemansky stated that the static friction increase or decrease in response to the applied force up to an upper limit, which is dependent on the characteristics of the contact between the surface and the object.
— Dynamic
Any object travelling at a constant velocity must be subject to zero net force (resultant force), meaning they are experiencing dynamic equilibrium.
Consider an object travelling at constant velocity across a surface with kinetic friction. A kinetic friction force exactly opposes the applied force in the direction of motion, resulting in zero net force. Since the object began with a non-zero velocity, it continues to move with a non-zero velocity.
Describe forces in quantum mechanics
In quantum mechanics, the definition of force is maintained, however it has to account for operators as described by the ‘Schrödinger equation’. This results in measurements becoming “quantised” i.e. appear in discrete portions, which is difficult to imagine in the context of “forces”. Nevertheless, the potentials V(x,y,z) or fields, from which the forces are derived, are treated similarly to classical position variables:
This outline is different only in quantum field theory, where these fields are also quantised.
One “caveat” in quantum mechanics is the particles that act onto each other lacking the spatial variable. Instead they have a discrete intrinsic angular momentum-like variable called the “spin”, and the relationship between the space and the spin variables are denoted by ‘Pauli exclusion principle’. Depending on the spin’s value, identical particles can be divided into 2 different classes: ‘fermions’ and ‘bosons’. If 2 identical fermions (e.g. electrons) have a symmetric spin function (e.g. parallel spins), that means the spatial variables must be antisymmetric (i.e. they exclude each other from their places much as if there was a repulsive force), and vice versa. Generally speaking, if 2 identical fermions have antiparallel spins, their position variables must be symmetric (i.e. the apparent force must be attractive). Therefore, in 2 fermions scenarios, there is strictly a negative correlation between spatial and spin variables. On the other hand, in 2 bosons, (e.g. quanta of electromagnetic waves, photons) the correlation is strictly positive. This shows how force loses part of its meaning. I’ll delve more into quantum mechanics into another post.
What are Feynman diagrams?
https://en.wikipedia.org/wiki/Feynman_diagram
In modern particle physics, forces and the acceleration of particles are described by a mathematical by-product of exchange of momentum-carrying gauge bosons. The introduction of quantum field theory and general relativity determined force as a redundant concept that arose from the conservation of momentum (or 4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). Since the conservation of momentum is derived from the homogeneity or symmetry of space, physicists considered it as more fundamental than the concept of force. Weinberg (1994) described the currently known fundamental forces as more accurately “fundamental interactions”.
When particle A emits (creates) or absorbs (annihilates) virtual particle B, a conservation of momentum results in recoil of particle A impresses repulsion or attraction between particles A A' exchanging by B. Feynman diagrams are a conceptually simple way to describe such fundamental interactions in combination with sophisticated mathematical descriptions. Each matter particle is depicted as a straight line travelling through time, which normally increases up or to the right in the diagram. Besides their direction of propagation through the Feynman diagram, matter and anti-matter particles are considered identical. World lines of particles intersect at interaction vertices, and any force arising from an interaction is situated at the vertex with an associated instantaneous change in the direction of the particle world lines. Shifman (1999) discovered gauge bosons discharged away from the vertex as wavy lines, and can be absorbed at an adjacent vertex during vertical particle exchange.
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This Feynmann diagram illustrates the decay of a neutron into a proton. The W boson is situated between 2 vertices indicating a repulsion.
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Feynmann diagrams can succinctly describe the decay of neutrons into electrons, protons and neutrinos, which is an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
I’ll discuss Feynmann diagrams in another post.
What are the fundamental forces?
In physics, the fundamental forces or interactions are defined as interactions that can be reduced to more basic interactions. There are 4 known fundamental interactions:
i. Gravitational Forces / Gravity
https://en.wikipedia.org/wiki/Gravity
Gravity, or gravitation, is a natural phenomenon that brings all things with mass or energy, such as planets, stars, galaxies, humans and even light, towards one another. On Earth, gravity gives weight to all physical objects, while the Moon’s gravity causes the ocean tides. Physicists theorised the gravitational attraction was the main contributor to the coalescing of gaseous matter in the Universe and formation of stars, which lead to the amalgamation of stars to form galaxies. Although its effects are increasingly weaker as objects move further away, gravity can affect objects from an infinite range.
The earliest theories on gravity were postulated by the ancient Greek philosopher Archimedes, which discovered the centre of gravity using a triangle. Modern work on gravitational theory were published by Galileo Galilei in the late 16th and early 17th centuries. His famous experiments involved dropping balls from the Tower of Pisa, and rolling balls down inclines. He calculated that the ball’s gravitational acceleration was the same in both scenarios. He also postulated air resistance contributed to the slow falling of masses in an atmosphere. His works set the foundation for the formulation of Newton’s theory of gravity.
Describe Newton’s law of universal gravitation
https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
In modern language, the law states the following:
“Every point mass attracts every single other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the 2 masses and inversely proportional to the square of the distance between them.”
F = (G*(m1*m2))/r2
— F = Force between the masses
— G = Gravitational constant (6.674 x 10-11 m3*kg-1s-2)
— m1 = First mass
— m2 = Second mass
— r = Distance between the centres of the masses
The SI units of the following parameters:
— F = Newtons (N)
— m1 & m2 = Kilograms (kg)
— r = metres (m)
— Constant G = 6.67430(15)×10−11 m3⋅kg−1⋅s−2.
The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, notwithstanding the fact that Cavendish didn’t calculate the numerical value for G himself. It also tested Newton’s theory of gravity between masses in the laboratory for the first time, which occurred 111 years after the publication of Newton's Principia and 71 years after Newton's death.
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This is a error plot showing experimental values for big G.
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If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them can be evaluated through summation of contributions of the notional point masses that constitute the bodies. As the component point masses become “infinitely small”, the limit entails integration of the force over the extents of 2 bodies.
This demonstrates how an object with a spherically symmetric distribution of mass exerts the same gravitational attraction on external bodies, assuming all the object’s mass were concentrated at a point at its centre. Note this is not generally true for non-spherically-symmetrical bodies.
For points inside a spherically symmetric distribution of matter, Newton’s shell theorem can help evaluate the gravitational force. It states how different parts of the mass distribution affect the gravitational force measured at a point located a distance (r0) from the centre of the mass distribution:
— The portion of the mass located at radii r < r0 causes the same force at the radius r0, assuming all of the mass enclosed within a sphere of radius r0 was concentrated at the centre of the mass distribution.
— The portion of the mass located at radii r > r0 exerts no net gravitational force at the radius r0 from the centre. This means the individual gravitational forces exerted on a point at r0 by the elements of the mass outside the radius r0 cancel each other.
Consequently, there is no net gravitational acceleration anywhere within the hollow sphere within a shell of uniform thickness and density. Moreover, the gravity increases linearly with the distance from the centre within a uniform sphere. The additional mass leads to a 1.5 times increase in gravitational force, while the increasing distance from the centre leads to a decrease. Therefore, if a spherically symmetric body has a uniform core and a uniform mantle with a density lower than 2/3 of the core’s density, then the gravity initially decreases outwardly beyond the boundary. However, if the sphere is large enough, the gravity increases further outward again, eventually exceeding the gravity of the core/mantle boundary. It’s suggested Earth’s gravity is highest at the core / mantle boundary.
ii. Vector Form
To account for the direction of the gravitational force as well as its magnitude, the vector equation for Newton’s law of universal gravitation expressed quantities in bold represent vectors.
F2-1 = -G*[(m1*m2) / |r1-2|2]*r^1-2
— F2-1 = Force applied on object 2 exerted by object 1
— G = Gravitational constant
— m1 & m2 = Masses of the objects 1 and 2, respectively
— |r12| = |r2 − r1| : Distance between objects 1 and 2
— r^1-2 = (r2 - r1)/ (|r2 - r1|) : Unit vector from object 1 to 2
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This diagram illustrates the gravity field surrounding Earth from a macroscopic perspective.
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iii. Gravitational Field
https://en.wikipedia.org/wiki/Gravitational_field
A gravitational field is a model that attempts to explain the influence that a massive body extends into the space around itself, which then produces a force on another massive body. It is used to illustrate gravitational phenomena, which is measured in Newtons per kilogram (N/kg).
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This graph shows the gravitational field strength within the Earth.
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Originally, gravity was conceptualised by Isaac Newton as a force between point masses. Then Pierre-Simon Laplace attempted to model gravity as a kind of radiation field or fluid.
Since the 19th century, the field model gradually became accepted in the scientific community. The model states that particles distort spacetime via their mass, meaning the force is measured as the perceived distortion. Geroch (1981) explained that this model suggests that matter moves in certain ways in response to the curvature of spacetime. This inferred there is either no gravitational force, or that gravity is a fictitious force.
In classical mechanics, Richard Feynman (1971) described the gravitational field as a physical quantity. The gravitational field (g) around a single particle of mass (M) is a vector field that consists of every point of a vector pointing directly towards the particle. Application of the universal law leads to the calculation of the magnitude of the field at every point, which represents the force per unit mass on any object at that point in space. Since the force field is conservative, there is a scalar potential energy per unit mass (φ) at each point in space associated with the force fields, known as a gravitational potential. The gravitational field equation is given as:
g = F / m = d2R / dt2 = -G*M*(R^/ |R|2) = -∇*φ
— F = Gravitational force
— m = Mass of the test particle
— R = Position of the test particle
— R^ = A unit vector in the radial direction of R
— t = Time
— G = Gravitational constant
— ∇ = Del operator
Note that the equation includes Newton’s law of the universal gravitation, and the association between gravitational potential and field acceleration. Note that d2R / dt2 and F/m are both equal to the gravitational acceleration (g). The negative signs indicate the forces acting antiparallel to the displacement. The equivalent field equation in terms of mass density (ρ) of the attracting mass:
-∇*g = φ*∇2 = 4*π*G*ρ
— This equation includes Gauss’s law for gravity, and Poisson’s equation for gravity.
The field around multiple particles is the vector sum of the fields around each individual particle. An object in such a field experiences a force equal to the vector sum of the forces experienced in these individual fields.
In general relativity, the Christoffel symbols represents the gravitational force field and the metric tensor represents the gravitational potential. The gravitational field is determined by solving the Einstein field equations:
G = T*(8*π*G)/c4
— T = Stress-energy tensor
— G = Einstein tensor
— c = Speed of light
These equations depend on the distribution of matter energy in a region of space. The fields in general relativity represent the curvature of spacetime. General relativity suggests that an object being in a region of curved space is equivalent to accelerating up the gradient of the field. According to Newton’s 2nd law, if the object is held stationary with respect to the field, it experiences a fictitious force. So if you stand still on Earth’s surface, you feel yourself being pulled down by the force of gravity. This means the gravitational fields predicted by general relativity differ in their effects only slightly from those predicted by classical mechanics, with a few differences. One of the easily verifiable differences is the bending of light in such fields.
A gravitational field can be generalised in vector form for interactions involving more than 2 objects (e.g. An asteroid and Earth). For 2 objects, we define the gravitational field g(r) as:
g(r) = -G*(m1/|r|2)*r^
This can be written as:
F(r) = m*g(r)
The SI units of a gravitational field is m/s2.
Since gravitational fields are conservative, the done by gravity from one position to another is path-independent. Consequently, the gravitational potential field,V(r), exists.
g(r) = -∇*V(r)
If m1 is a point mass or the mass of a sphere with homogeneous mass distribution, then force field g(r) outside the sphere is considered isotropic. This means, it depends only on the distance r from the centre of the sphere.
V(r) = -G*(m1/r)
— Note the gravitational field is on, inside and outside of symmetric masses.
According to Gauss’s law, the field in a symmetric body can be evaluated by the mathematical equation:
— dV = Closed surface
— Menc = Mass enclosed by the surface
Therefore, for a hollow sphere of radius R and total mass M.
For a uniform solid sphere of radius R and total mass M,
iv. What are the problematic aspects?
Deviations from Newton’s description of gravity are small when the dimensionless quantities φ/c2 and (v/c)2 are both less than 1.
— φ = Gravitational potential
— v = Velocity of the objects being studied
— c = Speed of light in vacuum
— rorbit = Radius of the Earth's orbit around the Sun
If either dimensionless parameter is large, then general relativity needs to be applied to describe the system. General relativity can be simplified to Newtonian gravity in the limit of small potential and low velocities.
— Discuss the theoretical concerns with Newton’s expression
“There is no immediate prospect of identifying the mediator of gravity”. Despite considerable progress being made over the past 50 years, physicists still haven’t resolved the relationship between the gravitational force and other known fundamental forces. Newton himself believed that the concept of an inexplicable action at a distance was insufficient, but he was limited in his discovering capabilities at that time.
“Newton’s theory of gravitation requires the instantaneous transmission of gravitational force”. Before the development of general relativity, physicists used the classical assumptions of the nature of space and time. They couldn’t fully explain why a significant propagation delay in gravity leads to unstable planetary and stellar orbits.
— What observations conflict with Newton’s formula?
- Newton’s theory doesn’t fully explain the precession of the perihelion of the orbits of the planets, especially Mercury’s. Comparing the Newtonian calculation and observations from advanced telescopes during the 19th century, there is a 43 arcsecond per century discrepancy.
- The predicted angular deflection of light rats by gravity evaluated by Newton’s Theory only confirmed one-half of the deflection actually observed by astronomers. On the other hand, calculations based on the theory of general relativity closely agree with the astronomical observations.
- In spiral galaxies, the orbiting of stars around their centres seemed to disobey Newton's law of universal gravitation. Astrophysicists speculated that the presence of large amounts of dark matter may explain this spectacular phenomenon in the framework of Newton's laws.
— What was Einstein’s solution?
Einstein’s theory of general relativity described gravitation as an attribute of curved spacetime rather than caused by a force propagated between bodies. Furthermore, the theory states energy and momentum distort spacetime in their vicinity, and other particles move in trajectories that are determined by the geometry of spacetime. This lead to a description of the motions of light and mass that accurately matched all available observations. Due to the curvature of spacetime, gravitational force is considered a fictitious force. Moreover, due to its world line being a geodesic of spacetime, the gravitational acceleration of a body is in free fall.
v. Extensions
Newton was the first to consider in his works an extended expression of his law of gravity including an inverse-cube term of the form. This equation attempted to explain the Moon’s apsidal motion.
— B is a constant.
Other extensions were proposed by Laplace (around 1790) and Decombes (1913):
In recent years, Greene and Gudkov (2007) attempted to use neutron interferometry to add non-inverse square terms in the law of gravity.
vi. Describe solutions to Newton’s law of universal gravitation
The n-body problem attempts to predict the individual motions of a group of celestial objects interacting with each other gravitationally. The desire to understand the motions of the Sun, planets and the visible stars lead to attempts in solving this problem. In the 20th century, solving the n-body problem required understanding of the dynamics of globular cluster star systems. The n-body problem in general relativity is quite difficult to solve using current mathematical concepts.
The classical physical problem informally states that “given the quasi-steady orbital properties (instantaneous position, velocity and time) of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times”.
How strong is Earth’s gravity?
Every planetary body (including Earth) is surrounded by its own gravitational field, which Newtonian physics conceptualised as an exertion of an attractive force on all objects. If we assume a spherically symmetrical planet, the strength of its gravitational field at any given point above the surface is proportional to the planetary body's mass and inversely proportional to the square of the distance from the body’s centre.
Cantor et al. (2006) stated that the strength of the gravitational field is numerically equal to the acceleration of objects under its influence. The rate at which falling objects accelerate near the Earth's surface varies slightly depending on latitude, surface features such as mountains and ridges, and unusually high or low sub-surface densities. For purposes of weights and measures, the International Bureau of Weights and Measures defined a value of standard gravity, under the International System of Units (SI). Denoted g, that value is g = 9.80665 m/s2 (32.1740 ft/s2). In meteorology, this value of gravity is applied precisely to latitude of 45°32’33”.
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If an object with comparable mass to that of the Earth were to fall towards it, then the corresponding acceleration of the Earth would be observable.
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According to Newton’s 3rd law, the Earth itself experiences a force equal in magnitude and opposite in direction to a force exerted on a falling object. In theory, the Earth accelerates towards the object until they collide. Because the mass of the Earth is substantially larger than the object, the acceleration experienced by the Earth exerted by the opposite force is negligible in comparison to the object's. If the object doesn’t bounce after its collision with the Earth, each of them then exerts a repulsive contact force on the other. This effectively balances the attractive force of gravity and prevents further acceleration.
Hofmann-Wellenhof & Horitz (2006) stated the force of gravity on Earth is the resultant (vector sum) of 2 forces:
(a) The gravitational attraction dictated by the Newton's universal law of gravitation, &
(b) The centrifugal force, resulting from the choice of an earthbound, rotating frame of reference.
The force of gravity is weakest at the equator because of the centrifugal force caused by the Earth's rotation and points on the equator are furthest from the Earth’s centre. Moreover, the force of gravity varies with latitude and increases from about 9.780 m/s2 at the Equator to about 9.832 m/s2 at the poles.
I’ll delve into the details of Earth’s gravity in another post.
— Discuss equations for a falling body near the surface of the Earth
Galileo was the first to demonstrate and then formulate a set of equations that describe the resultant trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions.
Near the surface of the Earth, the acceleration due to gravity g = 9.807 m/s2 or 32.18 ft/s2 approximately. In all cases, the object is assumed to be initially at rest, and air resistance is ignored. Nonetheless, in real life, air resistance induces a drag force on any body that falls through any atmosphere other than a perfect vacuum. This drag force increases with velocity until it equals the gravitational force, when the object then falls at a constant terminal velocity.
— Gravity in astronomy
Newton’s law of gravity has helped astrophysicists understand more about the planets in the Solar System, the mass of the Sun, quasars, and dark matter. It allowed them to accurately calculate the masses of the planets and the Sun without the need to travel there and weigh them on a scale. In space, an object maintains its orbit because of the force of gravity acting upon it. For example, planets orbits stars, stars orbit galactic centres, galaxies orbit a centre of mass in clusters, and clusters orbits in superclusters. Hence, the force of gravity exerted on one object by another is directly proportional to the product of those objects' masses and inversely proportional to the square of the distance between them.
A 2016 study theorised that the earliest gravity (in the form of quantum gravity, supergravity, or a gravitational singularity), along with ordinary space and time, developed during the Planck epoch (up to 1*10−43 seconds after the birth of the Universe), possibly from a primeval state (such as a false vacuum, quantum vacuum, or virtual particle). However the cause is currently unknown.
— Gravitational radiation (wave)
https://en.wikipedia.org/wiki/Gravitational_wave
Einstein’s theory of general relativity predicts that energy can be emitted from a system in the form of gravitational radiation. Any accelerating matter traces curvatures in the space-time metric, which is how the gravitational radiation is transported away from the system. Examples of co-orbiting objects generating curvatures in space-time include the Earth-Sun system, pairs of neutron stars, and pairs of black holes. Exploding supernovae are another astrophysical system predicted to lose energy in the form of gravitational radiation, but this needs evidence.
In 1973, the first indirect evidence for gravitational radiation was through measurements of the Hulse-Taylor binary. This binary system contains a pulsar and neutron star in orbit around one another. Since its initial discovery, its orbital period has shortened due to the energy loss, which is consistent for the amount of energy loss due to gravitational radiation. This discovery helped those physicists win the Physics Nobel Prize in 1993.
The first direct evidence for gravitational radiation was measured on 14 September 2015 by the LIGO detectors, located in the LIGO Hanford Observatory in Washington, US. The physicists in the observatory measured the gravitational waves emitted during the collision of two black holes 1.3 billion-light years from Earth. Their findings confirmed the theoretical predictions of Einstein and others that such waves exist. A 2016 report hyped the discovery as a portal to practical observation and understanding of the nature of gravity and events in the Universe including the Big Bang. Abbott et al. (2017) discovered detectable amounts of gravitational radiation were emitted during the formation of neutron stars and black holes.
As of 2020, the gravitational radiation emitted by the Solar System is too minuscule to be detected and measured with current technology.
— What is the speed of gravity?
In December 2012, a Chinese research team measured the phase lag of Earth tides during full and new moons. The findings supported the theory that the speed of gravity is equal to the speed of light. So, if the Sun suddenly disappeared, the Earth would continue orbiting along the original path for 8 minutes, which is the time light takes to travel that distance.
In October 2017, the LIGO and Virgo detectors picked up gravitational wave signals within 2 seconds of gamma ray satellites and optical telescopes detecting signals from the same direction. This discovery confirmed that the speed of gravitational waves was the same as the speed of light.
I’ll discuss the speed of gravity in more detail in another post.
List anomalies and discrepancies
A list of observations haven’t been sufficiently accounted for, which may point to the need for better theories of gravity or perhaps be explained in other ways.
— Extra-fast stars = Stars in galaxies follow a distribution of velocities where stars on the outskirts move faster than they should according to the observed distributions of normal matter. Galaxies within galaxy clusters follow a similar trajectory. Dark matter, which interact through gravitational forces but not electromagnetic forces, would account for the discrepancy. Physicists have proposed various modifications to Newtonian dynamics.
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This graph shows a rotation curve of a typical spiral galaxy: predicted (A) and observed (B). The discrepancy between the curves is attributed to dark matter.
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— Flyby anomaly = Various spacecraft have experienced greater acceleration than expected during gravity assist manoeuvres.
— Accelerating expansion = The metric expansion of space is observed to be accelerating, leading to the theory of the existence of dark energy. This concept is still debated.
— Anomalous increase of the astronomical unit = Recent measurements indicate that planetary orbits are widening faster than anticipated. It was assumed that it may have been due to the Sun losing mass through the energy it radiates.
— Extra energetic photons = Photons usually gain energy when they travel through galaxy clusters and then radiate that energy after they exit those clusters. Studies suggested the accelerating expansion of the Universe should theoretically prevent the photons from expending all that energy. If we take into this account, photons from the cosmic microwave background radiation would gain twice as much energy as expected. Chown (2009) suggested this phenomenon indicated that gravity falls off faster than inverse-squared at certain distance scales.
— Extra massive hydrogen clouds = The spectral lines of the Lyman-alpha forest suggest that hydrogen clouds tend to clump together at certain scales than expected. Chown (2009) suggested this indicated gravity falling off slower than inverse-squared at certain distance scales, analogous to dark flow.
What are alternative theories to general relativity?
— Historical alternative theories
- Aristotelian theory of gravity
- Le Sage’s theory of gravitation (1784): This is also called LeSage gravity, proposed by Georges-Louis Le Sage, which is based on a fluid-based explanation where a light gas fills the entire Universe.
- Ritz’s theory of gravitation (1908)
- Nordström’s theory of gravitation (1912,13)
- Kaluza Klein theory (1921)
- Whitehead’s theory of gravitation (1922)
— Modern alternative theories
- Brans-Dicke theory of gravity (1961)
- Induced gravity (1967): Proposed by Andrei Sakharov according to which general relativity might arise from quantum field theories of matter.
- String theory (late 1960s)
- f(R) gravity (1970)
- Horndeski theory (1974)
- Supergravity (1976)
- In the modified Newtonian dynamics (MOND) (1981), Mordehai Milgrom proposes a modification of Newton’s 2nd law of motion for small accelerations.
- The self-creation cosmology theory of gravity (1982) by G.A. Barber in which the Brans-Dicke theory is modified to allow mass creation.
- Loop quantum gravity (1988) by Carlo Rovelli, Lee Smolin and Abhay Ashtekar.
- Non-symmetric gravitational theory (NGT) (1994) by John Moffat
- Tensor-vector-scalar gravity (TeVeS) (2004), relativistic modification of MOND by Jacob Bekenstein
- Chameleon theory (2004) by Justin Khoury and Amanda Weltman
- Pressuron theory (2003) by Olivier Minazzoli and Aurélien Hees
- Conformal gravity
- Gravity as an entropic force, arising as an emergent phenomenon from the thermodynamic concept of entropy.
- In the superfluid vacuum theory, the gravity and curved space-time arise as a collective excitation mode of non-relativistic background superfluid.
What is free fall?
https://en.wikipedia.org/wiki/Terminal_velocityhttps://en.wikipedia.org/wiki/Free_fall
If gravity is the only force acting upon an object in space, it is experiencing ‘free fall’ in Newtonian physics. Examples of objects in free fall include:
— A spacecraft (in space) with no propulsion (e.g. in a continuous orbit, or on a suborbital trajectory ascending and then descending.
— An object dropped at top of a drop tube.
— An object thrown upward or a person jumping off the ground at low speed (Note: Air resistance is negligible in comparison to weight.)
Furthermore, if you throw a ball upwards, notice that it continues moving upwards up to a certain point before changing direction and begins to fall downwards at an accelerating rate. Technically, the ball is in free fall during the upward motion and instantaneous rest at the top of its motion. If the only acting force is gravity, then the acceleration is always directed downwards and has the same magnitude for all bodies. Since all objects fall at the same rate in the absence of other forces regardless of their velocity, objects and people will experience weightlessness in these situations.
Examples of objects that aren’t in free fall include:
— Aircraft flying in the air due to lift and thrust.
— Standing on the ground due to the upwards normal force from the ground against the gravitational force.
— Skydiving using a parachute due to the aerodynamic drag force and air resistance balancing the force of gravity. Some parachutes provide an additional lift force.
If an object is in free fall in a vacuum near the Earth’s surface, it will accelerate at approximately 9.8 m/s2, independent of its mass. When we account for air resistance, the falling object reaches terminal velocity, that is, around 53 m/s (195 km/h or 122 mph) for a human skydiver. The terminal velocity depends on many factors including mass, drag coefficient, and relative surface area. However it can only be achieved if the object is falling from a sufficient altitude from the ground. A 2010 study measured a typical skydiver falling in a spread-eagle position reaches terminal velocity after about 12 seconds, during which he has fallen around 450 m (1,500 ft).
https://www.youtube.com/watch?v=KDp1tiUsZw8
On August 2, 1971, astronaut David Scott demonstrated the phenomenon of free fall on the moon. He held a hammer in one hand and a feather in the other hand at the same height above the moon’s surface. When he releases both the hammer and the feather, which falls to the ground first? If you guessed the hammer hit the ground, you’ll be in for a surprise. In fact, both the hammer and feather fell at the same rate and landed on the ground simultaneously. This proved Galileo’s theory that all objects experience the same downwards acceleration due to gravity in the absence of air resistance. In the case of the Moon’s gravity, the gravitational acceleration is approximately 1.63 m/s2, or only about 1⁄6 that on Earth. I wonder what Galileo’s reaction would be had he lived and witnessed his theory being proven on camera?
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Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of (2h/g)0.5.
— h = Height
— g = Free-fall acceleration due to gravity
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(A) Describe free fall in Newtonian mechanics
I. Uniform gravitational field without air resistance
v(t) = v0 - g*t
y(t) = v0*t + y0 - 0.5*g*t2
— v0 = Initial velocity (m/s)
— v(t) = Vertical velocity with respect to time (m/s)
— y0 = Initial altitude (m)
— y(t) = Altitude with respect to time (m)
— t = Time elapsed (s)
— g = Acceleration due to gravity (9.81 m/s2 near the surface of the earth)
These formulae describe the vertical motion of an object falling a small distance close to the surface of a planet. It closely approximates its fall in air, assuming the force of gravity on the object overwhelms the force of air resistance. In other words, the object's velocity is always much less than the terminal velocity.
II. Uniform gravitational field with air resistance
m*(dv/dt) = m*g - 0.5*ρ*CD*A*v2
— m = Mass
— A = Cross-sectional area
— ρ = Air density
— v = Fall velocity
— CD = Drag coefficient (assumed to be constant, depending on the Reynolds number)
This equation describes the motion of skydivers, parachutists or any body of mass, with Reynolds number above the critical Reynolds number, so that the air resistance is proportional to the square of the fall velocity.
Terminal speed is:
v∞ = [(2*m*g) / (ρ*CD*A)]0.5
The object’s speed with respect to time can be integrated over time to find the vertical position as a function of time:
y = y0 - (v∞2/g) * ln*cosh(g*t/v∞)
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This graph depicts the acceleration of a small meteoroid when entering the Earth's atmosphere at different initial velocities.
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III. Inverse-square law gravitational field
When 2 objects in space orbit each other in the absence of other forces, they are also in free fall of each other, e.g. the Moon or an artificial satellite "falls around" the Earth, or a planet “falls around” the Sun. If we assume the object is perfectly spherical, the equation describing its motion includes Newton’s Law of Universal Gravitation. The solutions to the gravitational two-body problem illustrate elliptic orbits, which obeys Kepler’s laws of planetary motion. To understand this association between falling objects close to the Earth and orbiting objects, we’ll discuss the Newton’s cannonball thought experiment later on.
A special case of an elliptical orbit of eccentricity, e = 1 (radial elliptic trajectory), is the motion of 2 objects moving radially towards each other with no angular momentum. We can compute the free-fall time for 2 point objects on a radial path. The solution yields time as a function of separation:
t(y) = (y03 / 2μ)0.5 * [((y/y0)*(1 - y/y0))0.5 + arccos((y/y0)0.5)]
— t = Time after the start of the fall
— y = Distance between the centres of the bodies
— y0 = Initial value of y
— μ = G(m1 + m2): Standard gravitational parameter
If we substitute y = 0, we can calculate the free-fall time.
The separation as a function of time is evaluated as the inverse of the the equation, which is represented below exactly by the analytic power series:
If we evaluate this power series, we get:
y(t) = y0 * (x - x2/5 - 3*x3/175 - 23*x4/7875 - 1894*x5/3931875 - 3293*x6/21896875 - 2418092*x7/62077640625 - …)
— x = [1.5*(0.5π - t*(2μ/y03)0.5)]2/3
(B) Free fall in general relativity
In general relativity, a free falling object is subject to no net force and its inertial motion describes a geodesic. The Newtonian theory of free fall agrees with Einstein’s theory of general relativity if the object concerned is far away from any from any sources of space-time curvature, where spacetime is flat. In other cases, both theories demonstrate their distinguishing proposals. For instance, general relativity can explain the precession of orbits, the orbital decay or inspiral of compact binaries due to gravitational waves, and the relativity of direction (geodetic precession and frame dragging), whereas newton’s theory fails to explain them.
Modern experimental observations such as the Eötvös experiment have demonstrated both Galileo’s thought experiment and Newton’s theory regarding gravitational and inertial masses accelerating at the same rate when in free fall. This formulated the basis of the equivalence principle, from which form the basis for Einstein’s theory of general relativity. I’ll discuss the equivalence principle in detail later.
What is terminal velocity?
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This force diagram illustrates the downward force of gravity (Fg) equaling the restraining force of drag (Fd) plus the buoyancy. The net force on the object is zero, and the result is that the velocity of the object remains constant.
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Terminal velocity is defined as the maximum velocity attainable by an object as it falls through a fluid (e.g. air). This occurs when the sum of the drag force (Fd) and the buoyancy is equal to the downward force of gravity (Fg) acting on the object. This leads to a net force of zero on the object, which means it has zero acceleration. In fluid dynamics, Riazi & Türker (2019) defined an object moves at terminal velocity when its speed is constant due to the restraining force exerted by the fluid through which it is moving through.
Examples:
Imagine a skydiver falling in a belly-to-earth (i.e., face down) free fall position. Based on wind resistance, a skydiver’s terminal velocity in that position is about 195 km/h (120 mph, 54 m/s), which is the asymptotic limiting speed. If we graph this fall, we find that the skydiver reaches 50% of terminal velocity after 3 seconds, then 90% of terminal velocity after 8 seconds, then 99% of terminal velocity after 15 seconds and so on.
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This is a graph of velocity versus time of a skydiver reaching a terminal velocity.
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If a skydiver pulls in their limbs, they can increase their terminal velocity to about 320 km/h (200 mph or 90 m/s), which is close to the terminal velocity of the peregrine falcon diving down on its prey. Competition speed skydivers fly in a head-down position to reach faster speeds of 530 km/h (330 mph; 150 m/s). The current world record for the fastest terminal velocity by a human is 1,357.6 km/h (840 mph; 380 m/s), which was achieved by Felix Baumgartner after jumping from a height of 39,000 m (128,100 ft).
Discuss the physics of terminal velocity
Ignoring buoyancy, terminal velocity can be calculated using the following equation:
Vt = [(2*m*g) / (ρ*A*Cd)]0.5
— Vt = Terminal velocity
— m = Mass of the falling object
— g = Gravitational acceleration
— Cd = Drag coefficient
— ρ = Density of the fluid through which the object is falling
— A = Projected area of the object
If we take into account the effects of buoyancy, we have to apply Archimedes’ Principle to depict the upward force on the falling object by the surrounding fluid. It states that the mass (m) reduces by the displaced fluid mass (ρ*V), with V being the volume of the object. The equation is as follows: mr = m - ρ*V
The properties of the fluid, the mass of the object and its projected cross-sectional surface area are all factors that influence the terminal speed of an object. According to the barometric formula, as altitude decreases, air density increases, by 1% every 80 metres. For every 160 metres (520 ft) an object falls through the atmosphere, their terminal velocity decreases by 1%. After they reach local terminal velocity, their speed decreases to change with the local terminal speed as they continue to fall.
How is terminal velocity derived?
Using the drag equation, the net force acting on an object falling near the surface of Earth is:
Fnet = m*a = m*g - 0.5*ρ*v2*A*Cd
— v(t) = Velocity of the object as a function of time (t)
At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity limt→∞v(t) = Vt :
m*g - 0.5*ρ*v2*A*Cd = 0
If we solve for Vt, this yields:
[5] Vt = [(2*m*g) / (ρ*A*Cd)]0.5
Assuming ρ, g and Cd are constants, the drag equation is:
m*a = m*(dv/dt) = m*g - 0.5*ρ*v2*A*Cd
A more practical form of this equation can be obtained by substituting α2 = (ρ*A*Cd) / (2*m*g)
If we divide both sides of the equation by m, we obtain:
dv/dt = g*(1 - α2*v2)
dt = dv / g*(1 - α2*v2)
If we take the integral of both sides of the equation, we obtain:
This leads to:
This equation can be simplified into:
t = [(1/ (2*α*g))*ln((1 + α*v)/(1 - α*v))] = [arctanh(α*v)] / α*g
— arctanh = Inverse hyperbolic tangent function
Alternatively,
v = (1/α)*tanh(α*g*t)
— tanh = Hyperbolic tangent function
If we assume g is positive, and substitute α back in, then the terminal speed v becomes:
v = [(2*m*g) / (ρ*A*Cd)]0.5 * tanh[t*(g*ρ*A*Cd)/(2*m)]
As time tends towards infinity (t—> ∞), the hyperbolic tangent tends to 1. Therefore the resulting terminal speed is:
Describe terminal speed in a creeping flow
When fluids move sluggishly, their inertia forces are negligible (assuming the fluid is massless) compared to other force. Such forces are called “creeping flows”, and their viscosity is represented by Reynolds number, Re << 1. The equation for the motion of creeping flow is a simplified version of Navier-Stokes Equation.
∇p = μ*v*∇2
— v = Fluid velocity vector field
— p = Fluid pressure field
— μ = Liquid / fluid viscosity
In 1851, Sir George Gabriel Stokes worked out the analytical solution for the creeping flow around a sphere. His solution described the drag force acting on the sphere.
[6] D = 3*π*μ*d*V or Cd = 24/Re
— Reynolds Number: Re = (ρ*d*V)/μ
The above equation that expresses the drag force is called “Stokes’ Law”.
When the value of Cd is substituted into equation [5], it yields the expression for terminal speed of a spherical object moving under creeping flow conditions:
Vt = (g*d2/18μ)*(ρs - ρ)
— ρs = Density of the object
Stokes Law can assist in studying how sediments settle near the ocean bottom and moisture drops fall in the atmosphere. The principle can also be applied in the falling sphere viscometer, an experimental device used to measure the viscosity of highly viscous fluids, such as oil, paraffin, tar etc.
How is terminal velocity calculated in the presence of buoyancy force?
If we do take the effects of buoyancy into account, then an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the object reaches terminal velocity, its weight is exactly balanced by the upward buoyancy force and drag force.
[1] W = Fb + D
— W = Weight of the object
— Fb = Buoyancy force
— D = Drag force
If the falling object’s shape is spherical, then the equations for the 3 forces are:
[2] W = (π/6)*d3*ρs*g
[3] Fb = (π/6)*d3*ρ*g
[4] D = Cd*0.5*ρ*V2*A
— d = Diameter of the spherical object
— g = Gravitational acceleration
— ρ = Density of the fluid
— ρs = Density of the object
— A = (π/4)*d2 : Projected area of the sphere
— Cd = Drag coefficient
— V = Characteristic velocity (taken as terminal velocity, Vt)
If we substitute equations [2], [3] & [4] into [1] and solve for terminal velocity (Vt), then we obtain the following equation [5]:
In [1], the object is assumed to be denser than the fluid. If we ignore this assumption, then the sign of the drag force would be negative, since the object will be moving upwards, against gravity. For instance, bubbles forming at the bottom of a champagne glass and helium balloons move upwards. The terminal velocity in such cases will have a negative value, which corresponds to the rate of rising up.
ii. Electromagnetic Force
https://en.wikipedia.org/wiki/Electromagnetismhttps://en.wikipedia.org/wiki/Coulomb%27s_law
In 1784, Charles-Augustin de Coulomb first described the electrostatic force existing intrinsically between 2 charges. The properties of electrostatic force include:
— Varies as the inverse square law directed in the radial direction.
— Independent of the mass of the charged objects
— Follows the superposition principle
Coulomb’s law unifies all these observations into one succinct statement.
Subsequent mathematicians and physicists applied the construct of an electric field to evaluate the electrostatic force on an electric charge at any point in space. Richard Feynman used a hypothetical “test charge” anywhere in space to base the electric field and then applied Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as:
E—>= F—>/q
— q = Magnitude of the hypothetical test charge
Meanwhile, the Lorentz force of magnetism was discovered to exist between 2 electric currents. Similar to the electric field, the magnetic field can be applied to create a magnetic force on an electric current at any point in space. At any case, the magnitude of the magnetic field can be calculated using the following formula:
B = F/(I*L)
— I = Magnitude of the hypothetical test current
— L = Length of hypothetical wire through which the test current flows.
The magnetic field exerts a force on all magnets including, e.g. those used in compasses. Since the Earth’s magnetic field is aligned closely with the orientation of the Earth’s axis, it causes compass magnets to become oriented as the magnetic force pulls on the needle.
A combination of the definition of electric current as the time rate of change of electric charge lead to the rule of vector multiplication called Lorentz’s Law, which describes the force on a charge moving in a magnetic field. The link between electricity and magnetism lead to the description of a unified electromagnetic force that acts on a charge. This electromagnetic force is defined as the sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field:
F—>= q*(E—> + v—>*B—>)
— F—> = Electromagnetic force
— q = Magnitude of the charge of the particle
— E—>= Electric field
— v—>= Velocity of the particle
— B—>= Magnetic field
In 1864, James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations to explain the origin of electric and magnetic fields. Oliver Heaviside and Josiah Willard Gibbs then reformulated Maxwell’s equations into 4 vector equations. The Maxwell Equations mathematically describe the sources of the electric and magnetic fields as being stationary and moving charges, and the interactions of the fields themselves.This lead to the discovery of electric and magnetic fields being “self-generating” through a wave when travelling at the speed of light. Duffin (1980) united the nascent fields of electromagnetic theory with optics, which directly completed the description of the electromagnetic spectrum.
However, there is difficulty reconciling electromagnetic theory with the photoelectric effect and the nonexistence of the ultraviolet catastrophe. Leading theoretical physicists used quantum mechanics to develop a new theory of electromagnetism, which ultimately led to quantum electrodynamics (QED). QED describes all electromagnetic phenomena as being mediated by wave–particles known as photons, which are the fundamental exchange particle that describe all interactions relating to electromagnetism including the electromagnetic force.
I’ll delve into the details of electromagnetism in another post.
Nuclear Force:
There are 2 “nuclear forces” that take place in quantum theories of particle physics.
iii. Strong nuclear force
https://en.wikipedia.org/wiki/Strong_interaction
The strong nuclear force is involved in the structural integrity of atomic nuclei. Stevens (2003) stated that the strong force represents the interactions between quarks and gluons according to the theory of quantum chromodynamics (QCD). In summary, the strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. its (aptly named) strong interaction makes it the “strongest” of the 4 fundamental forces.
The strong force only acts directly upon elementary particles. Nevertheless, a residual of the force is observed between hadrons as the nuclear force. On the other hand, the strong force acts indirectly, transmitted as gluons, which form part of the virtual π (pi) and ρ (rho) mesons, which classically transmit the nuclear force. However, failure to ascertain free quarks has demonstrated that the elementary particles affected are not directly observable, such a phenomenon is called ‘colour confinement’.
iv. Weak nuclear force
https://en.wikipedia.org/wiki/Weak_interaction
The weak nuclear force is involved in the decay of certain nucleons into leptons and other types of hadrons. It is formed from the exchange of the heavy W and Z bosons. The most common effect of the weak force is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The weak force’s field strength is measured to be approximately 1013 times less than that of the strong force. In addition, the weak force is stronger than gravity over short distances. Physicists proposed an electroweak theory that states electromagnetic forces and the weak force are indistinguishable at temperatures hotter than approximately 1015 Kelvins. Modern particle accelerators have probed such scorching temperatures, and demonstrated the possible conditions of the universe in the early moments of the Big Bang.
I’ll discuss nuclear forces in detail in another post.
I’ll discuss the Standard Model and conceptual model of fundamental interactions in another post.
Force as potential energy
Rather than a force, the mathematically related concept of potential energy is used to understand motion. For example, , the gravitational force acting upon an object is acted upon the gravitational field present at the object's location. Using the definition of energy, a potential scalar field (U(r—>)) is defined as that field whose gradient is equal and opposite to the force produced at every point:
Forces can be classified as conservative or non-conservative. Studies distinguished conservative forces from non-conservative forces as being equivalent to the gradient of a potential.
What are conservative forces?
A conservative force exerts work on a particle between 2 points independent of the path chosen. It is considered as a force that conserves mechanical energy. Suppose a particle starts at point A, and a force F acts on it. Other forces impact on the particle’s motion, eventually returning to point A, traversing a closed path. If the net work done by F at this point is 0, then F passes the closed path test. Examples of conservative forces include gravitational force, spring force, magnetic force and electric force.
One of the caveats of the closed path test is the work done by a conservative force on a particle moving between any 2 points doesn’t depend on the path taken by the particle.
Since the gravitational force is conservative, the work done by the gravitational force on an object depends only on its change in height. Consider two paths 1 and 2, both traversing from point A to point B. The net change in energy for a particle moving along path 1 from A to B and then path 2 backwards from B to A is 0. Therefore the the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.
e.g. If a child slides down a frictionless slide, the work done by the gravitational force on the child from the beginning to the end of the slide is independent of the shape of the slide. This means the work done by the gravitational force on the child during the slide depends on the vertical displacement of the child.
Describe the mathematics of conservative forces
A force field (F), defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it satisfies any of the following 3 vector conditions:
[1] The curl of F is the zero vector:
∇ * F—> = 0—>
[2] There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place:
[3] The force can be written as the negative gradient of a potential (φ),
F—> = -∇*φ
How do we prove that these 3 conditions are equivalent when F is a force field?
(a) [1] implies [2]
Let C be any simple closed path (i.e. a path starting and ending at the same point with no self-intersections. Consider a surface S of which C is the boundary. Then Stokes’ theorem states that:
If the curl of F is zero, the left hand side is zero - therefore statement 2 is true.
(b) [2] implies [3]
Assuming statement 2 holds, let c be a simple curve from the origin to a point x and define a function.
This function is well-defined (independent of the choice of c), which follows from statement 2. Applying the fundamental theorem of calculus gives:
F—> = -∇*φ
That means statement 2 implies statement 3.
(c) [3] implies [1]
Let’s assume that the 3rd statement is true. A vector calculus identity states that the curl of the gradient of any function is 0. Therefore, if the 3rd statement is found to be true, the 1st statement must be true as well. This demonstrates [1] implies [2], [2] implies [3], and [3] implies [1]. Therefore, all 3 statements are equivalent, Q.E.D.. (The equivalence of [1] and [3] is also known as Helmholtz’s theorem.)
Depending on velocity, many forces aren’t actually force fields. This means the above 3 conditions aren’t mathematically equivalent. For instance, the magnetic force satisfies statement [2] because the work done by a magnetic field on a charged particle is always zero. However it fails statement [3], and statement [1] is not defined, since the force is not a vector field, so the curl cannot be evaluated. However there is still debate whether the magnetic force can be classified as conservative or not.
What are non-conservative forces?
For certain physical scenarios, some forces are impossible to model because of the gradient of potentials. This is caused by macrophysical considerations generating forces that arise from a macroscopic statistical average of microstates. For instance, friction is caused by the gradients of numerous electrostatic potentials between the atoms. However it demonstrates a force model independent of any macroscale position vector.
Examples of non-conservative forces include friction, and non-elastic material stress, as well as contact forces, tension, compression and drag. Kuypers (2005) claimed non-conservative forces can arise in classical physics due to neglected degrees of freedom or from time-dependent potentials in spite of conservation of total energy. For example, friction is treated through consideration of the motion of individual molecules without violation of the conservation of energy. Nevertheless, every molecule's motion must be considered instead of treating it through statistically. For macroscopic systems, the non-conservative approximation is more straightforward to deal with than millions of degrees of freedom.
The link between macroscopic non-conservative forces and microscopic conservative forces is explained by statistical mechanics. In macroscopic closed systems, non-conservative forces act to change the internal energies of the system, which associate with heat transfer. According to the 2nd Law of Thermodynamics, non-conservative forces lead to energy transformations within closed systems from ordered to more random conditions as entropy increases.
What are the non-fundamental forces?
i. Normal force
https://en.wikipedia.org/wiki/Normal_force |
FN = Normal force
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In mechanics, the normal force (FN) is the component of a contact force that is perpendicular to the surface that an object contacts. For instance, the surface of a floor or table that prevents an object from falling exerts a normal force.The normal force is on the object is equal but in opposite direction to the gravitational force applied on the object. i.e. N = m*g. In this case, the definition of ‘normal force’ is used in the geometric sense and means perpendicular. This normal force represents the force applied by the table against the object that prevents it from sinking through the table. Note that the table needs to be sturdy enough to deliver this normal force without breaking.
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Weight (W), the frictional force (Fr), and the normal force (Fn) acting on a block. Weight is the product of mass (m) and the acceleration of gravity (g).
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Where an object rests on an incline, the normal force is perpendicular to the plane the object rests on. Nonetheless, the normal force may be sufficiently large to prevent sinking through the surface, presuming the surface is adequately sturdy. The strength of the force can be calculated as:
N = m*g*cos(θ)
— N = Normal force
— m = Mass of the object
— g = Gravitational field strength
— θ = Angle of the inclined surface measured from the horizontal.
Note that this diagram hasn’t accounted for friction.
The magnitude of the normal force (N) is the projection of the net surface interaction force (T) in the normal direction (n). This means the normal force vector can be evaluated by scaling the normal direction by the net surface interaction force. This makes the surface interaction force equal to the dot product of the unit normal with the Caunchy stress tensor describing the stress state of the surface.
N = n*N = n*(T*n) = n*(n*τ*n)
or, in indicial notation,
Ni = ni*N = ni*Tj*nj = ni*nk*τjk*nj
The parallel shear component of the contact force is known as the ‘frictional force’ (Ff*r).
The state coefficient of friction for an object on incline plane is calculated by the formula:
μs = tan(θ)
— For an object on the point of sliding, θ is the angle between the slope and the horizontal.
Where does the normal force originate?
When the object and the constraint (e.g. a table) are pressed against each other, the van der Waals force increases substantially to oppose the object, meaning the 2 can’t penetrate each other. If the 2 objects are separated by a tiny distance, the normal force disappears. Bettini explains the intermolecular force instantly vanishes past the point of equilibrium.
How does the normal apply in real life?
When you’re in an elevator either stationary or moving at constant velocity, the normal force exerted on your feet balances your weight. If the elevator is accelerating downward, the normal force becomes less than your ground weight, hence your perceived weight decreases. If you were standing on a weighing scale as the elevator is moving vertically, the scale would read the normal force being delivered to your feet. If the elevator is accelerating vertically, the normal force reading would differ from your ground weight. So, if the elevator cab accelerates the weighing scale measures normal force, whereas if the elevator cab doesn’t accelerate the weighing scale measures gravitational force.
If we define up as the positive direction, then we can construct Newton’s 2nd law and solve for the normal force on a passenger to yield the following equation:
N = m*(g + a)
If you’re riding in a gravitron, the static friction caused by and perpendicular to the normal force acting on you against the walls suspends you above the floor as the ride rotates. This means the walls of the ride applies normal force to you in the direction towards the centre, which is an outcome of centripetal force applied to you as the ride rotates. This results in the static friction between you and the walls of the ride counteracting the pull of gravity on the passengers. This leads to suspension above ground of the passengers throughout the duration of the ride.
If we define the centre of the ride as the positive direction, then we solve for the normal force on a body that is suspended above ground with the following equation:
N = m*v2/r
— N = Normal force on the passenger
— m = Mass of the passenger
— v = Tangential velocity of the passenger
— r = Distance of the passenger from the centre of the ride.
If the normal force is known, then we can can solve for the static coefficient of friction required to maintain a net force of zero in the vertical direction:
μ = m*g/N
— μ = Static coefficient of friction
— g = Gravitational field strength.
ii. Friction
https://en.wikipedia.org/wiki/Friction
Friction is a force that resists the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Between the 15th to 18th centuries, early physicists discovered the elementary property of sliding (kinetic) friction, in which they were expressed in empirical laws:
— Amontons’ First Law =The force of friction is directly proportional to the applied load.
— Amontons’ Second Law = The force of friction is independent of the apparent area of contact.
— Coulomb’s Law of Friction = Kinetic friction is independent of the sliding velocity.
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This diagram simulates blocks with fractal rough surfaces, exhibiting static frictional interactions.
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There are several types of friction:
1. Dry friction
Dry friction resists relative lateral motion of 2 solid surfaces in contact. 2 types of dry friction include ‘static friction’ (stiction) between non-moving surfaces, and ‘kinetic friction’ (sliding friction or dynamic friction) between moving surfaces.
Named after Charles-Augustin de Coulomb, Coulomb friction is an approximate model used to calculate the force of dry friction. The model is expressed as:
Ff < μ*Fn
— Ff = Force of friction exerted by each surface on the other, which is parallel to the surface directed opposite to the net applied force.
— μ = Coefficient of friction, an empirical property of the contacting materials
— Fn = Normal force exerted by each surface on the other, directed perpendicular (normal) to the surface.
The Coulomb friction (Ff) can be any value from 0 to μ*Fn, and the direction of the frictional force against the surface is opposite to the motion that surface would experience in the absence of friction. Therefore, the frictional force would act to prevent the motion between the surfaces in the case of static motion. This balances the net force tending to cause such motion. In this case, the Coulomb approximation provides a threshold value for this force, above which motion would commence, with the maximum force known as ‘traction’.
The frictional force is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the 2 surfaces. For instance, a curling stone sliding along the ice experiences a kinetic force, which slows it down. Another example is the drive wheels of an accelerating car experiencing a frictional force directed forwards. If there was no frictional force, the wheels would spin, and the rubber would slide backwards along the pavement. Note that friction doesn’t oppose the direction of movement of the vehicle, but rather the direction of (potential) sliding between tire and road.
— Normal force
If the object is on a level surface and the force causing it to slide is directed horizontally, the normal force (N) between the object and the surface is equal to the object’s weight. This equals to its mass multiplied by the acceleration due to earth’s gravity (g).
If the object is on a tilted surface such as an inclined plane, there is reduced normal force because the gravitational force perpendicular to the face of the plane is reduced.
The normal and, ultimately the frictional force can be determined using vector analysis via a free body diagram. In certain situations, other forces may need to be taken into account when calculating the normal force.
— Coefficient of friction
The coefficient of friction (COF), symbolised μ, is a dimensionless scalar value that describes the ratio of the force of friction between 2 bodies and the force pressing them together. The COF depends on the materials involved, which is shown in the table below. COFs range from near 0 to greater than 1, values between 0 and 1 is considered “low”, while values larger than 1 is considered “high”. In 1921, the Air Brake Association proposed an axiom of the nature of friction between metal surfaces that it is greater between 2 surfaces of similar metals than between 2 surfaces of different metals. Therefore, brass has a higher coefficient of friction when moved against brass, but less if moved against steel or aluminium.
For surfaces at rest relative to each other μ = μs, where μs is the coefficient of static friction (COSF), which is usually larger than its kinetic counterpart. The COSF is exhibited by a pair of contacting surfaces, which depends upon the combined effects of material deformation characteristics and surface roughness. Both facts originate in the chemical bonding between atoms in each of the bulk materials and between the material surfaces and any adsorbed material. Hanaor, Gan & Einav (2016) stated the fractality of surfaces played an important role in determining the magnitude of the static friction.
For surfaces in relative motion μ = μk, where μk is the coefficient of kinetic friction (COKF). The Coulomb friction is equal to Ff, and the direction of frictional force on each surface is exerted opposite to its motion relative to the other surface.
The term and utility of the coefficient of friction was introduced by Arthur Morin to be used as an empirical measurement. Both COSF and COKF depend on the pair of surfaces in contact. For any given pair of surfaces, the COSF is usually larger than the COKF. However, both coefficients may be equal in some sets of surfaces such as teflon-on-teflon.
Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Other materials, such as teflon, have coefficient values as low as 0.04.
— A value of 0 indicates no friction at all.
— When rubber is in contact with other surfaces, it can yield friction coefficients from 1 to 2.
— A value greater than 1 indicates the force required to slide an object along the surface is greater than the normal force of the surface on the object. e.g. Silicone rubber or acrylic rubber-coated surfaces have a coefficient of friction that can be substantially larger than 1.
The COF for any 2 materials depends on system variables such as temperature, velocity, atmosphere, ageing and de-ageing times, and the materials’ surface structure.
Approximate coefficients of friction
— Static friction (μs)
Static friction occurs between 2 or more solid objects that aren’t moving relative to each other. For instance, static friction prevents an object from sliding down a sloped surface. They arise as a result of surface roughness features across multiple length-scales at solid surfaces. These features are called ‘asperities’, which are analysed at the nanoscale. A 2016 study found asperities result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area. Greenwood & Williamson (1966) implicated the linearity between applied load and true contact area that arose from asperity deformation lead to the linearity between static frictional force and normal force, found for typical Amonton-Coulomb type friction.
A sufficient force needs to be applied against the static friction force before the object can move. The maximum possible friction force between 2 surfaces before the object begins to slide can be evaluated as the product of the coefficient of static friction and the normal force: Fmax = μs*Fn. If the object doesn’t slide, the friction force ranges from 0 (zero) to Fmax.
— Any force smaller than Fmax that attempts to slide 1 surface over the other is opposed by a frictional force of equal magnitude and opposite direction.
— Any force larger than Fmax that overcomes the force of static friction would cause the object to slide.
— If the object instantly slides, then static friction is no longer applicable. Hence the friction between the two surfaces is then called kinetic friction.
An example of static friction at work is a car’s wheel slipping as it rolls on the ground. Although the wheel is in motion, the patch of the tire that contacts the ground is stationary relative to the ground, so it is static rather than kinetic friction.
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When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.
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— Kinetic friction (μk)
Also known as dynamic friction, or sliding friction, kinetic friction occurs when 2 objects move relative to each other and rub together (like a sled on the ground). The friction force between 2 surfaces after sliding begins is evaluated as the product of the coefficient of kinetic friction and the normal force: Fk = μk*Fn.
Persson & Volokitin (2002) proposed a model that demonstrated kinetic friction being greater than static friction. Beatty (2007) implied kinetic friction is caused by chemical bonding between the surfaces, rather than interlocking asperities. Persson & Volokitin (2002) argued there are cases where the effects of roughness were dominant e.g. rubber to road friction. Furthermore, Persson (2000) asserted that surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces.
Makkonen (2012) stated that thermodynamics could explain the origin of kinetic friction at a nanoscale. As the object slides, new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work is done to generate new surface, and heat energy is released to remove the old surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front.
— Angle of friction
For certain applications, it is useful to define static friction in terms of the maximum angle before an object begins to slide. This is called the angle of friction or friction angle.
tan(θ) = μs
- θ = Angle from horizontal
- μs = Static coefficient of friction between the objects
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This diagram shows the angle of friction, θ, when block just starts to slide.
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— Friction at the atomic level
One of the challenges of designing nanomachines is determining the forces required to move atoms past each other. For the first time in 2008, scientists developed a technique to move a single atom across a surface, and measure the forces required. A modified atomic force microscope used an ultrahigh vacuum and nearly zero temperature (5º K) to drag a cobalt atom, and a carbon monoxide molecule, across surfaces of copper and platinum.
— What are the limitations of the Coulomb model?
The Coulomb approximation assumes that surfaces are in atomically close contact only over a small fraction of their overall area. This contact area is proportional to the normal force (until saturation, during which all area is in atomic contact). This frictional force is proportional to the applied normal force, independently of the contact area. Although the relationship between normal force and frictional force is not exactly linear, the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.
Coulomb friction model becomes less accurate for conjoined surfaces e.g. adhesive tapes resists sliding even if the normal force is absent or negative. This suggests the frictional force depends on the area of contact, which may explain why drag racing tires demonstrate adhesion.
A 2012 study demonstrated the potential for an “effectively negative coefficient of friction in the low-load regime”, which suggested an inversely proportional relationship between the normal force and friction. This finding challenges the notion of a directly proportional relationship between the normal force and friction.
— Numerical simulation of the Coulomb model
https://www.sciencedirect.com/topics/engineering/coulomb-model
Although it is a simplified model of friction, it is useful in a multitude of numerical simulation applications such as multibody systems and granular material. Many studies discuss the idea of designing algorithms that can efficiently numerically integrate mechanical systems with Coulomb friction and bilateral or unilateral contact, including sticking and sliding. Acary & Brogliato (2008) highlighted Coulomb friction’s involvement in some non-linear effects, such as the Painlevé paradoxes.
— Instabilities
Bigoni (2012) found various types of instabilities are produced by dry friction in mechanical systems, which can be caused by:
- The friction force with an increasing velocity of sliding
- Material expansion due to heat generation during friction = The thermo-elastic instabilities
- Pure dynamic effects of sliding of two elastic materials = The Adams-Martins instabilities
Adams & Martins’ 1995 study on smooth surfaces discovered the latter, and Nosonovsky & Adams (2004) discovered the same effects on periodic rough surfaces. A few studies suggested friction-related dynamical instabilities accounted for brake squeals, the 'song' of a glass harp and stick and slip phenomena, which were modelled Rice & Ruina (1983) as a drop of friction coefficient with velocity. Nosonovsky (2013) discovered frictional instabilities induce the formation of new self-organised patterns at the sliding interface, such as in-situ formed tribofilms. Tribofilms’ main purpose is reducing friction and wear in self-lubricating materials.
2. Fluid friction
Fluid friction occurs between adjacent fluid layers that are moving relative to each other. This internal resistance to flow is called viscosity, which is defined as the “thickness” of a fluid. For example, water is “thin”, hence low viscosity, while honey is “thick”, thus high viscosity. The less viscous the fluid, the greater its ease of deformation or movement. I’ll delve into the viscosity of fluids in another post.
3. Lubricated friction
Lubricated friction involves a fluid separating 2 solid surfaces, which aims to reduce wear of 1 or both surfaces n close proximity moving relative to each another. A substance called a lubricant is interposed between the surfaces to lubricate them.
The applied load is carried by pressure generated within the fluid due to the frictional viscous resistance to motion of the lubricating fluid between the surfaces. If lubrication is sufficient, it yields smooth continuous operation of equipment, minimising wear, excessive stresses or seizures at bearings. Without lubrication, metal or other surfaces can rub destructively over each other, generating heat and possibly damage or failure.
4. Skin friction
Skin friction arises from the interaction between the fluid and the skin of the body. It is directly related to the area of the surface of the body that is in contact with the fluid. Skin friction is dictated by the drag equation, which increases with the square of the velocity.
Moreover, skin friction is caused by viscous drag in the boundary layer around the object. To decrease skin friction, one can shape the moving body to achieve smooth flow, such as an airfoil, or decrease the length and cross-section of the moving object as much as is practicable.
5. Internal friction
Internal friction is the force that resists motion between the elements making up a solid material while it undergoes deformation.
Plastic deformation in solids elicits irreversible changes in the internal molecular structure of an object caused by either (or both) an applied force or a change in temperature. This change of an object's shape is called strain, which is caused by a stress force.
On the other hand, elastic deformation in solids elicits reversible changes in the internal molecular structure of an object. During deformation, internal forces oppose the applied force. If the applied force is too weak to completely overcome the resistance of the opposing forces, the object can assume a new equilibrium state and return to its original shape upon removal of the applied force. This is the main property of elasticity.
6. Radiation friction
In 1909, Einstein predicted the existence of "radiation friction" that oppose the movement of matter as a consequence of light pressure. He claimed radiation exerts pressure on both sides of the plate. If the plate is at rest, the forces of pressure exerted are equal on both sides of the plate. However, if the plate is in motion, more radiation reflects on the surface ahead during the motion (front surface) than on the back surface. Since the force of pressure exerted on the front surface is acting backwards, it counteracts the motion of the plate, thence increases with the velocity of the plate.
7. Other types of friction
— Rolling resistance
This force resists the rolling of a wheel or other circular object along a surface due to deformations in the object or surface. Silliman (1871) implied the force of rolling resistance is less than that associated with kinetic friction. Butt, Graf & Kappi (2006) measured the values for the coefficient of rolling resistance as typically 0.001. One of the most common examples of rolling resistance as the movement of a motor vehicle’s tires on a road, which generates heat and sound.
— Braking friction
When a brake is applied to a wheel in order to slow or stop a vehicle or piece of rotating machinery, it generates a retarding force. Braking friction is different to rolling friction because the coefficient of friction for rolling friction is small whereas the coefficient of friction for braking friction is large depending on the materials the brake pads are composed of.
— Triboelectric effect
When dissimilar materials are rubbed against each other, this accumulates electrostatic charge. If flammable gases or vapours are in the vicinity, this would be hazardous. When the discharge of accumulated static ignites a flammable substance, it can cause an explosion.
— Belt friction
This physical property arises from the forces acting on a belt wrapped around a pulley, with one end of a belt being pulled. Since tension acts on both ends of the belt, it can be modelled by the belt friction equation.
For example, when designers construct rigs, they require the knowledge of the number of times the belt or rope need to be wrapped around the pulley to prevent it from slipping.
What is the energy of friction?
According to the law of conservation of energy, no energy is destroyed due to friction, but it can be transferred to the system of concern, such as thermal energy. For instance, a sliding hockey puck slows down to a halt as friction converts its kinetic energy into heat, which increases the thermal energy of the puck and the ice surface.
When an object is pushed along a surface tracing a path C, the energy converted to heat is calculated by the line integral:
- Ffric = Friction force
- Fn = Vector obtained by multiplying the magnitude of the normal force by a unit vector pointing against the object's motion
- μk = Coefficient of kinetic friction inside the integral indicates its variation according to the location
- x = Position of the object.
- The energy lost to a system due to friction is an example of thermodynamic irreversibility.
What is the work of friction?
Static friction does zero work in the reference frame of the interface between 2 surfaces, because there was zero displacement between the surfaces. Hartog (1961) explained that kinetic friction is always directed opposite the motion and does negative work in the same reference frame. In other cases, friction does positive work in a selective number of frames or reference. If you place a heavy box on a rug and then pull on the rug swiftly, the box slides backwards relative to the rug, but moves forward relative to the frame of reference in which the floor is stationary. Leonard (2000) implied that the kinetic friction between the box and rug accelerates the box in the same direction that the box moves, thus positive work is done.
The work done by friction can transform into deformation, wear, and heat, which can affect the contact surface properties. The work of friction can combine materials such as in the process of friction welding. However, if work due to frictional forces rise to substantial levels, it would cause excessive erosion or wear of combined sliding surfaces. In addition, harder corrosion particles situated between mating surfaces in relative motion (fretting) exacerbates wear of frictional forces. Bayer (2004) stated that the fit the surface finish of an object degrades due to its surfaces wearing by the work of friction to the point of non-functionality, such as, in the form of bearing seizure or failure.
What are the applications of friction?
i. Transportation
- Automobile brakes inherently rely on friction to slow a vehicle through conversion of kinetic energy into heat. Designers of brake systems have to consider how to safely disperse the sheer amount of heat. Disk brakes rely on friction between a disc and brake pads compressing transversely against the rotating disc. In drum brakes, brake shoes or pads are pressed outwards against a rotating cylinder (brake drum) to generate friction. Since braking discs can be more efficient than drums in terms of efficient cooling, disk brakes demonstrate superior stopping performance.
- Rail adhesion refers to the grip the wheels of a train have on the rails.
- Road slipperiness is an design and safety factor for automobiles:
— Split friction is a dangerous condition that arises from varying friction on either side of a car.
— Road texture affects the interaction of tires and the driving surface.
ii. Measurement
- A tribometer is an instrument that measures friction on a surface.
- A profilograph is a device used to measure pavement surface roughness.
iii. Household usage
- Friction can heat and ignite matchsticks, i.e. friction between the head of a matchstick and the rubbing surface of the match box.
- Sticky pads use friction (i.e. increase the friction coefficient between the surface and the object) to prevent objects from slipping off smooth surfaces.
When was the concept of friction first coined?
- The Greeks, including Aristotle, Vitruvius, and Pliny the Elder, were recorded to the first to be curious in the cause and mitigation of friction. In 350 AD, Themistius stated his awareness of the differences between static and kinetic friction after discovering the motion of a moving body is easier to change compared to a stationary body.
- In 1493, Leonardo da Vinci discovered the classic laws of sliding friction, but were not published and remained unknown.
- In 1699, Guillaume Amontons rediscovered da Vinci’s laws and made them known as Amonton’s 3 laws of dry friction. He presented the nature of friction in terms of surface irregularities and the force required to increase the weight pinning the surfaces together.
- Bernhard Forest de Bélidor and Leonhard Euler (1750) further elaborated Amontons’ view of friction.
- Euler derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction.
- John Theophilus Desaguliers (1750) first recognised the role of adhesion in friction. Since microscopic forces cause surfaces to stick together, he proposed that the frictional force could tear the adhering surfaces apart.
- Charles-Augustin de Coulomb (1785) further advanced the understanding of friction through his investigations of the influence of 4 main factors on friction. Those factors are (1) the nature of the materials in contact and their surface coatings, (2) the extent of the surface area, (3) the normal pressure (or load), and (4) the length of time that the surfaces remained in contact (time of repose). Moreover, he considered additional factors on friction including sliding velocity, temperature and humidity to further understand the complexity of friction.
- Although this distinction was already stated by Johann Andreas von Segner in 1758, Coulomb proposed his friction law to distinguish between static and dynamic friction.
- Pieter van Musschenbroek (1762) considered the surfaces of fibrous materials by meshing them together, which takes a finite time to increase friction, in order to explain the effect of the time of repose.
- John Leslie (1766 - 1832) pointed out a weakness in the theories proposed by Amontons and Coulomb. He speculated that if friction arises from a weight being dragged up the inclined plane of successive asperities, why then isn’t it balanced as the weight descends down the opposite slope? Dowson (1997) reported that Leslie also scrutinised the role of adhesion in friction proposed by Desaguliers, which lead to a tendency to accelerate rather than retard its motion. Leslie believed that friction was a time-dependent process of flattening, pressing down asperities, which generated new obstacles in addition to prior cavities.
- Arthur Jules Morin (1833) developed the concept of sliding versus rolling friction.
- Osborne Reynolds (1866) derived the equation of viscous flow, which Armstrong-Hélouvry (1991) reported to complete the classic empirical model of friction (static, kinetic, and fluid) commonly used today in engineering.
- In 1877, Fleeming Jenkin and J.A. Ewing investigated the continuity between static and kinetic friction.
- 20th century research focused on understanding of the physical mechanisms behind friction. In 1950, Frank Philip Bowden and David Tudor demonstrated the actual area of contact between surfaces as a microscopic fraction of the apparent area.
- Circa 1986, the invention of the atomic force microscope allows scientists to study friction at the atomic scale, which verified dry friction as the product of the inter-surface shear stress and the contact area. These 2 discoveries support Amonton's first law and the macroscopic proportionality between normal force and static frictional force between dry surfaces.
- In 2014, Sosnovskiy, Sherbahov & Komissarov indicated that the friction force is proportional to both the contact and the volumetric (tensile-compression, bending, torsion, etc.) load, provided that the volumetric load causes cyclic stresses (±σ) in the contact area.
iii. Tension
In physics, tension is a pulling force that transmits axially by a string, a cable, chain, or similar one-dimensional continuous object, or by each end of a rod, truss member, or similar three-dimensional object. The SI Unit of tension is measured in Newtons (or pounds-force).
At an atomic level, atoms or molecules are stretched apart and gain potential energy. This creates a kind of restoring force called tension. Each end of a string or rod under such tension pulls on on the object it is attached to, in order to restore the string/rod to its relaxed length.
In addition, tension can act as a transmitted force as part of an action-reaction pair of forces, or as a restoring force. The ends of a string or a separate object transmitting tension exerts forces on the objects to which the string or rod is attached, in the direction of the string at the point of attachment. These forces due to tension are also called "passive forces”.
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Imagine 2 people competing in a tug of war where they’re pulling on each of a rope. The rope in the drawing extends into a drawn illustration indicating adjacent segments of the rope. One segment is duplicated in a free body diagram showing a pair of action-reaction forces of magnitude T pulling the segment in opposite directions, where T is transmitted axially and is called the tension force. This end of the rope is pulling the tug of war team to the right. Each segment of the rope is pulled apart by the two neighbouring segments, stressing the segment in what is also called tension, which can change along the too football fields members.
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Tension described in:
(A) 1 Dimension
The tension in a string is scalar quantity, meaning it can not be negative. Zero tension indicates the string is slack. Physicists tend to treat a string or rope in one dimension, with it having length but being massless with zero cross section.
- If the string is not bent, due to vibrations or pulleys, then tension is a constant along the string, which is equal to the magnitude of the forces applied by the ends of the string.
- If the string curves around 1 or more pulleys, it still has constant tension along its length in the idealised situation of the pulleys being massless and frictionless.
- If a string is vibrating with a set of frequencies (derived from Newton’s laws of motion), it depends on the string’s tension.
Each microscopic segment of the string pulls on and is pulled upon by its neighbouring segments, with a force equal to the tension at that position along the string.
- If the string has curvature, then the 2 pulls on a segment by its 2 neighbours won’t add to zero. This generates a net force on that segment of the string, leading to an acceleration. This net force is a restoring force, and the motion of the string includes transverse waves, which can solve the equation central to Sturm-Liouville theory:
-d/dx [r(x)*[dρ(x)/dx]] + v(x)*ρ(x) = (ω2)*σ(x)*ρ(x)
— v(x) = Force constant per unit length [units force per area]
— ω2 = Eigenvalues for resonances of transverse displacement ρ(x) on the string
— There are solutions including the various harmonics on a stringed instruments.
(B) 3 Dimensions
The force exerted by the ends of a 3D continuous material such as a rod or truss member is labelled as tension. Both the amount a rod elongates and the loads that cause structural failure depends on the force per cross-sectional area rather than the force alone. Stress is represented as a 3x3 matrix called a tensor, and the σ11 element of the stress tensor is tensile force per area, or compression force per area. This element is denoted as a negative number when the rod compresses rather than elongates.
— System in equilibrium
When the sum of all forces is zero, a system is in equilibrium.
ΣF> = 0
For example, a system that contains an object being descended vertically by a string with tension (T) at a constant velocity. If the system is moving at constant velocity, it is in equilibrium because the tension in the string (pulling on the object) is equal to the weight force (pulling down on the object).
ΣF> = T> + m*g> = 0
— System under net force
When an unbalanced force is exerted on a system, it experiences a net force. Note that the sum of all forces is not zero, meaning acceleration and net force always co-exist.
ΣF> ≠ 0
For example, if the same system above involves an object descending at an accelerating rate downwards (positive acceleration), a net force exists in the system. In such case, negative acceleration would indicate that |m*g| > |T|.
ΣF> = T> - m*g> ≠ 0
Consider 2 bodies A and B having masses m1 & m2, respectively, attached to each other by an inextensible string over a frictionless pulley. There are 2 forces acting on the body A: its weight (w1 = m1*g), and the tension (T) in the string pulling up. Hence, the net force (F1) and body A is w1 - T, so m1*a = m1*g - T. In an extensible string, Hooke’s Law applies.
— Strings in modern physics
String-like objects in relativistic theories, such as the strings used in some models of interactions between quarks, and those used in the modern string theory, also experience tension. Physicists analyse these strings in terms of their world sheet, and the energy is then typically proportional to the length of the string. Therefore, the tension in such strings is independent of the amount of stretching.
Physics videos on pulley systems
The Organic Chemistry Tutor:
Khan Academy:
Q1. Imagine a pulley system that consists of 2 objects of masses m1 and m2. If m2 is accelerating downwards at a m/s2, derive the equation for acceleration of the system.
[1] T - m1*g = m1*a
[2] m2*g - T = m2*a
[1] + [2]: (T - m1*g) + (m2*g - T) = m2*a + m1*a
m2*g - m1*g = m2*a + m1*a
g*(m2 - m1) = a*(m2 + m1)
a = g*(m2 - m1) / (m2 + m1)
Q2. Imagine a pulley system that consists 1 hanging object (m2) and another object on a rough slope. If the system is accelerating to the left down the slope, then derive the equations that describe the motion of the system.
[1] T - m2*g = m2*a
[2] m1*g*sin(θ) - T - Fk = m1*a
[1] + [2]: (T - m2*g) + (m1*g*sin(θ) - T - Fk = m2*a + m1*a
m1*g*sin(θ) - m2*g - Fk = a*(m2 + m1)
a = [g*(m1*sin(θ) - m2) - Fk] / (m2 + m1)
Q3. Imagine an object of mass hanging from a ceiling from 2 strings attached to the same point each on an angle (θ) relative to the ceiling. If the mass is stable in this current setup, derive equations that evaluates the mass of the object.
[1] Fw = m*g
[2] FT = T1*sin(θ1) + Τ2*sin(θ2)
[1] = [2]:
Fw = FT
m*g = T1*sin(θ1) + Τ2*sin(θ2)
m = [T1*sin(θ1) + Τ2*sin(θ2)]/g
iv. Elastic force / Elasticity
https://en.wikipedia.org/wiki/Elasticity_(physics)https://en.wikipedia.org/wiki/Hooke%27s_law
When a spring is stretched to a certain length, an elastic force acts to the spring to its natural length. An ideal spring is assumed to be massless, frictionless, unbreakable, and infinitely stretchable. Nave (2013) explained such springs exert pushing forces as it contracts, and pulling forces as it extends, in proportion to the spring’s displacement from its equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom Hooke’s Law is named. If Δx is the displacement, the force exerted by an ideal spring equals:
F> = -k*Δx>
— k = Spring constant (or force constant), specific to the spring
— The minus sign accounts for the tendency of the force to act in opposition to the applied load.
I’ll delve into elasticity and Hooke’s Law in another post.
v. Continuum mechanics: Pressure, Drag, Stress
https://en.wikipedia.org/wiki/Pressurehttps://en.wikipedia.org/wiki/Drag_(physics)
Newton’s laws and Newtonian mechanics were first developed to describe how forces affect idealised point particles rather than 3D objects. However, in the real world, matter has complex structure and forces acting on 1 part of an object affects other parts of an object. The theories of continuum mechanics attempt to describe the way forces affect the material through studies of the lattice that holds the object’s atoms together in situations involving flow, contraction, expansion and shape alteration. For example, in extended fluids, pressure differences generate directed along the pressure gradients as follows:
F>/V = -∇>*P
— V = Volume of the object in the fluid
— P = Scalar function that describes the pressure at all locations in space.
Studies found pressure gradients and differentials generate buoyant forces for fluids suspended in gravitational fields, winds in atmospheric science, and the lift linked to aerodynamics and flight.
A force linked to dynamic pressure is known as fluid resistance, which resists the motion of an object through a fluid due to viscosity. In the case of “Stokes’ drag’, the force is approximately proportional to the velocity, but opposite in direction:
F>d = -b*v>
— b = A constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area).
— v> = Velocity of the object
Forces in continuum mechanics are fully described by a stress-tensor with terms that are roughly defined as:
σ = F/A
— A = Relevant cross-sectional area for the volume for which the stress-tensor is being calculated.
This formalism includes pressure terms linked to forces acting normal to the cross-sectional area (the matrix diagonals of the tensor) and shear terms linked to forces acting parallel to the cross-sectional area (the off-diagonal elements). Textbooks state the stress tensor accounts for forces that cause all strains (deformations) including tensile stresses and compressions. I’ll delve into continuum mechanics in another post.
vi. Fictitious forces
https://en.wikipedia.org/wiki/Fictitious_force
Fictitious forces depend on the frame of reference, which suggests they appear because of the adoption of non-Newtonian (or non-inertial) reference frames. Mallette (2008) listed examples of fictitious forces include the centrifugal force and the Coriolis force. Studies stated such forces don’t exist in frames of references that aren’t accelerating, which are referred to as “pseudo forces”.
In general relativity, gravity becomes a fictitious force in situations where space-time deviates from a flat geometry. Kaluza-Klein theory and string theory ascribe electromagnetism and the other fundamental forces respectively to the curvature of differently scaled dimensions.
vii. Torque
https://en.wikipedia.org/wiki/Torque
When forces push extended objects to rotation, it generates torque. Mathematically, the torque of a force (F>) is defined relative to an arbitrary reference point as the cross-product:
r> = r> *F>
— r> = The position vector of the force application point relative to the reference point.
Since torque is the rotation equivalent of force, there is rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. If we apply Newton’s 2nd law of motion, we can derive an analogous equation for the instantaneous angular acceleration of the rigid body:
r> = I*α>
— I = Moment of inertia of the body
— α> = Angular acceleration of the body
This equation defines the moment of inertia, which is the rotational equivalent for mass. If the rotation over a time interval is evaluated, the tensor need to substitute the moment of inertia in order to fully determine the characteristics of rotations including precession and nutation.
Nave (2013) used the differential form of Newton’s 2nd Law to derive an alternative definition of torque:
r> = dL>/dt
— L> = Angular momentum of the particle
Fitzpatrick (2007) highlighted that Newton’s 3rd Law requires all objects exerting torques themselves experience equal and opposite torques, which directly satisfy the conservation of angular momentum for closed systems experiencing rotations and revolutions through the action of internal torques.
viii. Centripetal force
https://en.wikipedia.org/wiki/Centripetal_force
When an object accelerates in circular motion, the unbalanced force acting on the object is equal to:
F> = -m*v2*r^/r
— m = Object’s mass
— v = Object’s velocity
— r = Radius of the circular path
— r^ = Unit vector pointing in the radial direction outwards from the centre
This means that the unbalanced centripetal force experienced by any object is always directed toward the centre of the curved path. Such forces act perpendicular to the velocity vector associated with the object’s motion, which changes the direction (but not the speed) of the object (magnitude of the velocity). The unbalanced force accelerating the object needs to be resolved into 2 components: one perpendicular to the curved path, and one tangential to the path. This obtains both the tangential force, and the radial (centripetal) force. The tangential force accelerates the force positively or negatively, and the radial force changes its direction.
I’ll discuss fictitious forces, torque and centripetal force in detail in the circular motion post.
What is the unit of force?
The SI Unit of force is the Newton (N), which is defined as the force required to accelerate a 1 kg mass at a rate of 1 m/s2 or kg*m*s-2. The corresponding CGs unit is the dyne (dyn), the force required to accelerate a 1 gram of mass by 1 cm/s2 or g*cm*s-2. Therefore, a Newton is equal to 100,000 dynes.
A gravitational foot-pound-second English unit of force is the pound-force (ld-f), which is defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m*s-2. Wandmacher & Johnson (1995) defined an alternative unit of mass to the pound-force called the slug, which is defined as the mass that can accelerate by 1 ft/s2 when acted on by 1 pound-force.
An alternative unit of force in a different foot-pound-second system, known as the absolute fps system, is called the poundal (pdl), which is defined as the force required to accelerate a 1-lb mass at a rate of 1 ft/s2. The units of slug and poundal are designed to avoid a constant of proportionality in Newton’s 2nd Law.
The pound-force’s metric counterpart is the kilogram-force (kgf), which is defined as the force exerted by standard gravity on 1 kg of mass. This leads to the metric slug (mug or hyl), which is the mass that can accelerate at 1 m/s2 when subjected to a force of 1 kgf. Although the kgf is not part of the modern SI system, it is occasionally used to express aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. Other arcane units of force include the sthène which is equivalent to 1000 N, and the kip, which is equivalent to 1000 lbf.
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units.
What is potential energy?
https://en.wikipedia.org/wiki/Potential_energy
In physics, potential energy is the energy obtained by an object relative to its relative to other objects, stresses within itself, its electric charge, or other factors.
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When the archer does work on the bow, drawing the string back, some of the chemical energy of the archer's body is transformed into elastic potential energy in the bent limb of the bow. When the string is released, the force between the string and the arrow does work on the arrow. The potential energy in the bow limbs is transformed into the kinetic energy of the arrow as it takes flight.
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There are various types of potential energy, each associated with a particular type of force. They include elastic, gravitational, electric, nuclear, intermolecular and chemical.
https://en.wikipedia.org/wiki/Virtual_work
https://en.wikipedia.org/wiki/Work_(physics)
https://en.wikipedia.org/wiki/Mechanical_energy
https://en.wikipedia.org/wiki/Energy
What is work and energy?
In physics, work is the product of force and displacement. A force is thought to do work if there is a displacement of the point of application in the direction of the force. This term was introduced in 1826 by the French mathematician Gaspard-Gustave Coriolis, who defined it as “weight lifted through a height”. This early definition was based on the early steam engines that lifted buckets of water out of flooded ore mines. The SI units of work is the joule (J) or Newton-metre (N*m), which is defined as the work expended by a force of 1 Newton (N) through a displacement of 1 metre. Non-SI units of work include the Newton-metre, erg, the foot-pound, the foot-poundal, the kilowatt-hour, the litre-atmosphere, and the horsepower-hour.
The work (W) done by a constant force of magnitude (F) on a point that traverses a displacement (s) in a straight line in the direction of the force is expressed as:
W = F*s
For example, if a force 5 Newtons (F = 5N) acts along a point that travels 10 metres (s = 10m), then the work done is (W = F*s = 5*10 = 50 J). This is approximately the work done lifting a 1kg object from ground level to over a person's head against the force of gravity. If the weight lifted the same distance is doubled or the same weight is lifted twice the distance, then the work is doubled.
The work-energy principle states that the increases in the rigid body’s kinetic energy is caused by an equal amount of positive work done on the body by the resultant force acting on that body, and vice versa.
From Newton’s 2nd law, work on a free (i.e. no fields), rigid (i.e. no internal degrees of freedom) body, is equal to the change in kinetic energy (EK) of the linear velocity and angular velocity of that body:
W = ΔEK
The work of forces generated by a potential function is known as ‘potential energy’ and the forces are conserved. Therefore, work on an object that is simply displaced in a conservative force field, without any change in velocity or rotation, is equal to minus the change of potential energy (EP) of the object:
W = -ΔEP
These formulas show that work is the energy associated with the action of a force, so it subsequently possesses the physical dimensions, and units, of energy.
What are constraint forces?
They limit the movement of components in a system, such as constraining an object to a surface. For example, when an object placed on a slope is stuck, its attachment to a taut string won’t make it move in an outwards direction to make the string any ‘tauter’. Since constraint forces restrict the velocity in the direction of the constraint to zero, they don’t perform work on the system.
Goldstein stated that constraint forces eliminate movement in directions that characterise the constraint in a mechanical system. Since the component of velocity along the constraint force at each point of application is zero, constraint forces can’t perform work.
Examples:
— In a pulley system such as the Atwood machine, the internal forces on the rope and at the supporting pulley don’t do any work on the system. Therefore, work should only be evaluated for the gravitational forces acting on the bodies.
— The centripetal force exerted inwards by a string attached to a ball in uniform circular motion constrains the ball sideways to circular motion, which restricts its movement away from the centre of the circle. This means the force does zero work because it is perpendicular to the velocity of the ball.
— Consider a book on a table, if external forces are applied to the book, it slides on the table. That means the force exerted by the table constrains the book from moving downwards. The force exerted by the table supports the book upwards and is perpendicular to its movement, implying that this constraint force does not perform work.
— The magnetic force on a charged particle can be evaluated as F = q*v*B, where q = Charge, v = velocity of the particle, B = Magnetic field. The result of a cross product is always perpendicular to both of the original vectors, so F ⊥ v. The dot product of 2 perpendicular vectors is always zero, so the work W = F*v = 0, and the magnetic force doesn’t zero work. Although the direction of motion can be changed but the speed can never change.
How is work mathematically calculated?
For moving objects, we integrate the quantity of work/time (power) along the trajectory of the point of application of the force. Hence, at any instant, the rate of the work done by a force (measured in Joules / second, or Watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. This is known as instantaneous power. Resnick and Halliday (1966) implied that, according to the fundamental theorem of calculus, the total work along a path is the time-integral of instantaneous power applied along the trajectory of the point of application.
In general, work is the result of a force on a point that follows a curve X, with a velocity (v), at each instant. The small amount of work (δW) occurring over an instant of time (dt) is evaluated as:
δW = F*ds = F*v dt
— F*v: Power over the instant dt
Summing these small amounts of work over the trajectory of the point yields the work:
— C = The trajectory from x(t1) to x(t2).
This integral is computed along the trajectory of the particle, and is therefore said to be path dependent.
If the force is always directed along this line, and magnitude of the force is F, then this integral simplifies to:
W = ∫C F ds
— s = Displacement along the line.
If F is constant, in addition to being directed along the line, then the integral simplifies further to:
W = ∫C F ds = F* ∫C ds = F*s
We can generalise this formula for a constant force force that is not directed along the line, followed by the particle. This implies that the dot product:
F*ds = F*cos(θ) ds
— θ: The angle between the force vector and the direction of movement
W = ∫C F*ds = F*s*cos(θ).
If a force is applied to a body at an angle of 90° from the velocity vector, no work is done at all, since the cosine of 90 degrees is zero. Hence, no work can be performed by gravity on a planet with a circular orbit. Moreover, no work can be done on a body in circular motion at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge.
— Work done by a variable force
If force is changing, or if the body is moving along a curved path (e.g. rotating and not necessarily rigid), then solely the path of the application point of the force is relevant for the work done. Furthermore, only the component of the force parallel to the application point velocity is doing work (positive in the same direction, and negative in the opposite direction of the velocity). This component of force is represented by the scalar quantity “scalar tangential component” or F*cos(θ), where θ is the angle between the force and the velocity. This leads to the general definition of work:
“The work of a force is the line integral of its scalar tangential component along the path of its application point.”
In the case of a varying force, the work done can be evaluated using calculus. If the force is given by F(x) (a function of x) then the work done by the force along the x-axis from a to b is:
A force couple results from equal and opposite forces that act on 2 different points of a rigid body. The sum of these forces may cancel out, but their effect on the body is the couple or torque (T). The work of the torque is expressed as:
— T*ω = The power over the instant δt.
The sum of these small amounts of work over the trajectory of the rigid body yields the work:
This integral is calculated along the trajectory of the rigid body with an angular velocity ω that varies with time, meaning it is ‘path dependent’.
If the angular velocity vector maintains a constant direction, then we can express it as:
ω—>= *φ*S
— φ: The angle of rotation about the constant unit vector (S). So, the work of the torque becomes:
— C: The trajectory from φ(t1) and φ(t2)
This integral depends on the rotational trajectory φ(t), and is therefore ‘path-dependent’.
If the torque (T) is aligned with the angular velocity vector, that means T = τ*S. Moreover, if both the torque and angular velocity are constant, then the work is computed as:
To understand this integral, we have to consider the torque as arising from a force of constant magnitude F, being applied perpendicularly to a lever arm at a distance r. This force acts through the distance along the circular arc, s = r*φ, so the work done is:
W = F*s = F*r*φ
When we include the torque τ = F*r, we obtain:
W = F*r*φ = τ*φ
Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.
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This diagram shows a force of constant magnitude and perpendicular to the lever arm.
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The relationship between work and potential energy
The scalar product of a force (F) and the velocity (v) of its point of application defines the power input to a system at an instant of time. If we integrate this power over the trajectory of the point of application, C = x(t), then it defines the work input to the system by the force.
— Path dependence
Therefore, the work done by a force (F) on an object that travels along a curve (C) is represented by the line integral:
- dx(t): This defines the trajectory C
- v: Velocity along this trajectory
This integral requires the path along which the velocity is defined, so the work evaluated shows it is to be ‘path dependent’.
dW/dt = P(t) = F*v
This time derivative of the integral for work yields the instantaneous power.
— Path independence
If the work for an applied force is independent of the path, then the work done by the force defines a potential function. According to the gradient theorem, this function evaluates the work done by the force at the beginning and end of the trajectory of the point of application. This leads to a potential function, U(x), that can be computed at the points x(t1) and x(t2) to calculate the work over any trajectory between these 2 points. This function is defined with a negative sign so that positive work is a reduction in the potential:
— The function, U(x), is called the potential energy associated with the applied force. The force derived from such a potential function is conservative. Since both gravity and spring forces have potential energies, that means the gradient of work becomes:
- F = Force that can be derived from a potential.
- U = Potential that defines a force F at every point x in space. This means the set of forces is called a ‘force field’.
The gradient of the work, or potential, in the direction of the velocity V of the body can yield the power applied to a body by a force field.
P(t) = -∇U*v = F*v
— Work by gravity (in space)
In the absence of other forces, gravity induces a constant downward acceleration of every freely moving object. The gravitational acceleration near Earth’s surface is g = 9.8m/s2 and the gravitational force on an object of mass m is Fg = mg. For simplicity’s sake, consider the gravitational force is focused at the centre of mass of the object.
If an object is displaced upwards or downwards a vertical distance (y2 - y1), the work W done on the object by its weight mg is:
W = Fg*(y2 - y1) = Fg*Δy = -m*g*Δy
- Fg = Weight (Newtons, or pounds)
- Δy = Change in height y
Notice that the work done by gravity depends only on the vertical movement of the object. If friction is not present in this case, it doesn’t affect the work done on the object by its weight.
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This schematic shows gravity F = mg does work W = mgh along any descending path.
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The force of gravity exerted by a mass ‘M’ on another mass ‘m’ is expressed as:
F = -(G*M*m)*r/r3
- r = Position vector from M to m.
If a mass (m) moves at the velocity (v), then the work of gravity on this mass as it moves from position r(t1) to r(t2) can be evaluated as:
Notice that the position and velocity of the mass (m) are expressed as:
er & et = Radial and tangential unit vectors directed relative to the vector from M to m.
We can use this to simplify the formula for work of gravity.
Since the solution uses the fact that
(d/dt)*r-1 = r-2r* = -r*/r2
The function becomes:
U = -(G*M*m)/r
This is the gravitational potential function, also known as gravitational potential energy. The negative sign indicates work being gained from a loss of potential energy.
— Work by a spring
Imagine a spring that exerts a horizontal force F = (-k*x,0,0) proportional to its deflection in the x direction independent of the body’s motion. We use the body’s velocity, v = (vx, vy, vz), to yield the work of this spring on a body moving along the space with the curve X(t) = (x(t), y(t), z(t)).
For a simple case, imagine the contact with the spring occurs at t = 0. This leads to the integral of the product of the distance x and the x-velocity, x*vx, as (1/2)*x2. Notice velocity disappears in this equation. This means the work is the product of the distance and the spring force, which depends on distance, leading to the x2 result.
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This diagram illustrates the forces in springs assembled in parallel.
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- Work by a gas
- P = Pressure
- V = Volume
- a & b = Initial and final volumes
What is the work-energy principle?
This principle states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. Young & Freedman (2008) explained the work W done by the resultant force on a particle equals the change in the particle’s kinetic energy (Ek).
W = ΔEk = 0.5*m*v22 - 0.5*m*v12
— v1 = Speed of the particle before the work is done
— v2 = Speed of the particle after the work is done
— m = Particle’s mass
The work-energy principle was derived from the Newton’s 2nd law and the resultant force on a particle. Paul (1979) evaluated the scalar product of the forces with the velocity of the particle computes the instantaneous power added to the system.
Ensuring no velocity component in the direction of the constraint force is the key for constraints defining the direction of the particle’s movement. This meant the constraint forces don’t contribute to the instantaneous power. The time integral of this scalar equation computes work from the instantaneous power, and kinetic energy from the scalar product of velocity and acceleration. Whittaker (1904) asserted the fact that the work–energy principle eliminates the constraint forces underlying Lagrangian mechanics.
In general systems, work can change the potential energy of a mechanical device, the thermal energy in a thermal system, or the electrical energy in an electrical device. In such cases, work transfers energy from one place to another or one form to another.
— Derivation for a particle moving along a straight line
When the resultant force (F) is constant in magnitude and direction, and parallel to the particle’s velocity, the particle is moving with constant acceleration (a) along a straight line. Recall that Newton’s 2nd law links the net force and the acceleration, and the particle displacement (s) is given as:
s = (v22 - v12) / (2*a)
The work of the net force can be evaluated from the product of its magnitude and the particle displacement. If we substitute the above equations, we get:
Other derivations are shown below:
Below is the vertical displacement derivation:
W = F*S = m*g*h
For cases of rectilinear motion, the net force F is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle. This integrates the work along the path of the particle.
— General derivation of the work-energy theorem for a particle
For any net force acting on a particle moving along any curvilinear path, its work equals the change in the kinetic energy of the particle. Below is the equation of the work-energy theorem”.
The identities, a*v = 0.5*(dv2/dt), v2 = v*v, and a = dv/dt can be combined to yield:
dv2/dt = d(v*v)/dt = (dv/dt)*v + v*(dv/dt) = 2*(dv/dt)*v = 2a*v
— Derivation for a particle in constrained movement
In particle dynamics, the first integral of Newton’s 2nd law of motion yields a formula that equates work applied to a system to its change in kinetic energy. We can separate the resultant force used in Newton's laws into forces that are applied to the particle and forces imposed by constraints on the particle’s movement. Since the work of a constraint force is zero, only the work of the applied forces would be considered for the work-energy principle.
Imagine a particle (P) that follows the trajectory X(t) with a force F acting on it. Let’s isolate the particle from its environment, so it is exposed to constraint forces R, then Newton's Law is expressed as:
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— m = Particle’s mass
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i. Vector formulation
Note that n number of dots denotes its nth time derivative. So the scalar product of each side of Newton's law with the velocity vector yields:
— This is due to the constraint forces being perpendicular to the particle velocity.
If we integrate their equation along its trajectory from the point X(t1) to the point X(t2), this leads to:
The left side of this equation describes the work of the applied force as it acts on the particle along the trajectory from time t1 to time t2. It can be rewritten as:
This integral is computed along the trajectory X(t) of the particle and is therefore ‘path dependent’.
On the other hand, the right side of the 1st integral of Newton's equations can be simplified to become:
Then we integrate it explicitly to yield the change in kinetic energy:
— The particle’s kinetic energy is defined by the scalar quantity:
ii. Tangential and normal components
We can resolve the velocity and acceleration vectors into tangential and normal components along the trajectory X(t):
Given:
Then, the scalar product of velocity with acceleration in Newton’s 2nd law becomes:
Given the kinetic energy of the particle is defined by the scalar quantity:
This results in the work-energy principle for particle dynamics as, W = ΔK, which is generalised to arbitrary rigid body systems.
— Moving in a straight line (skidding to stop)
Imagine a vehicle moving along a straight horizontal trajectory under the action of a driving force and gravity that sum to F. The constraint forces between the vehicle and road define R as:
If we let the trajectory be along the X-axis, then X = (d, 0) and the velocity is V = (v, 0), then R*V = 0, and F*V = Fx*v, where Fx is the component of F along the X-axis, then:
Fx*v = m*v.*v
If we integrate both sides of the above equation, we get:
If Fx is constant along the trajectory, then the integral of velocity is distance, meaning:
Fx*(d(t2) - d(t1)) = 0.5*m*v2 *(t2) - 0.5*m*v2 *(t1)
For example, imagine a car skids to a stop, given k as the coefficient of friction and W as the weight of the car. Then the force along the trajectory is Fx = −k*W. The velocity (v) of the car can be calculated from the length (s) of the skid using the work–energy principle,
k*W*s = (W/2g)*v2 , or v = (2*k*s*g)0.5
This formula incorporates the mass of the vehicle as m = W/g.
— Coasting down a mountain road (gravity racing)
Imagine a vehicle initially at rest before rolling down a mountain road. We can use the work-energy principle to evaluate the minimum distance traversed by the vehicle to reach a certain velocity, e.g. 60 kph. Obviously, rolling resistance (friction) and air drag slows the vehicle down, which increases the actual distance required.
Let the trajectory of the vehicle following the road be X(t), a curve in 3D space. If the force acting on the vehicle that pushes it down the road is the constant force of gravity F = (0, 0, W), and the force of the road on the vehicle is the constraint force R, then Newton’s 2nd law gives:
The scalar product of this equation with the velocity, V = (vx, vy, vz) gives the equation:
W*vz = m*(*V)*V
— V = Magnitude of V
Since R*V = 0, the constraint forces between the vehicle and the road cancel from this equation, meaning they don’t do any work. If we integrate both sides of the above equation, this leads to:
Since the weight force (W) is constant along the trajectory and the integral of the vertical velocity is the vertical distance, that means:
W*(Δz) = 0.5*m*V2
Because V(t1) = 0, this result is independent on the shape of the road followed by the vehicle.
For example, if the downwards gradient of a slope is 2%. That means the altitude decreases 2 ft (0.61m) for every 100 ft (30.48m) traveled. Because the angles are small, the sin and tan functions are approximately equal. Therefore, the distance (s) in feet down a 2% grade to reach the velocity (V) is at least:
s = Δz / 0.02 = 25*(V2/g)
This formula incorporates the weight of the vehicle as W = m*g.
How does one calculate the work of forces acting on a rigid body?
The work forces acting at various points on a single rigid body can be evaluated from the work of a resultant force and torque. Consider the forces F1, F2 ... Fn act on the points X1, X2 ... Xn in a rigid body.
The movement of the rigid body define the trajectories o f Xi, i = 1, ..., n, which is denoted by the set of rotations [A(t)] and the trajectory d(t) of a reference point in the body. Let the coordinates xi i = 1, ..., n define these points in the moving rigid body’s frame of reference (M). This leads to the trajectories traced in the fixed frame (F) given as:
Xi(t) = [A(t)]*xi + d(t), i = 1,…,n.
The velocity of the points Xi along their trajectories are:
— ω = Angular velocity vector obtained from the skew symmetric matrix [Ω] = Å*AT, known as the angular velocity matrix.
The small amount of work by the forces over the small displacements (δri) are evaluated through the approximation of the displacement by δr = vδt.
δW = F1*V1*δt + F2*V2*δt + … + Fn*Vn*δt
or
This formula can be rewritten as:
— F & T = Resultant force and torque applied at the reference point d of the moving frame M in the rigid body.
How is potential energy computed?
Given a force field (F(x)), the work integral can be computed using the gradient theorem to find the scalar function associated with potential energy. Evaluation of the work integral requires the introduction of a parameterised curve γ(t)=r(t) from γ(a) = A to γ(b) = B, and computation of:
For the force field (F), let v = dr/dt, the gradient theorem would help obtain:
The power applied to a body by a force field can be calculated from the gradient of the work, or potential, in the direction of the velocity (v) of the point of application.
P(t) = -∇U*v = F*v
Burton Paul (1979) listed examples of work computed from potential functions include gravity and spring forces.
Describe potential energy for near Earth gravity
For small changes in height, gravitational potential energy is evaluated using the formula:
Ug = m*g*h
— m = Mass in kg
— g = Local gravitational field (9.8 metres per second squared on earth)
— h = Height above a reference level in metres
— U = Energy in Joules
In classical physics, gravity exerts a constant downward force F = (0, 0, Fz) on the centre of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r(t) = (x(t), y(t), z(t)), such as the track of a roller coaster can be obtained using its velocity, v=(vx, vy, vz):
— The integral of the vertical component of velocity is the vertical distance.
— The work of gravity depends only on the vertical movement of the curve r(t).
Describe potential energy for a linear spring
A horizontal spring exerts a force (F = (−k*x, 0, 0)) that is proportional to its deformation in the axial or x direction. The work of a spring on a body moving along the space curve s(t) = (x(t), y(t), z(t)) can be obtained using its velocity, v = (vx, vy, vz):
Let the contact with the spring occur at t = 0, the integral of the product of the distance x and the x-velocity, x*vx, is x2/2. The following function is the potential energy of a linear spring:
U(x) = 0.5*k*x2
The elastic potential energy is the potential energy of an elastic object deforming under tension or compression. This occurs when the elastic force acts to restore the object to its original shape, which is often the electromagnetic force between the atoms and molecules that constitute the object. If the stretch is released, the energy is transformed into kinetic energy.
Describe potential energy for gravitational forces between 2 bodies
The gravitational potential function, also known as gravitational potential energy, is:
U = -G*M*m/r
— The negative sign indicates work is gained from a loss of potential energy.
The gravitational force between 2 bodies of mass M and m separated by a distance (r) is given by Newton’s law:
F = -(G*M*m/r2)*r^
— r^ = Vector of length 1 pointing from M to m
— G = Gravitational constant
Let the mass (m) move at the velocity (v), the work of gravity on this mass as it moves from position r(t1) to r(t2) can be evaluated by:
The position and velocity of the mass (m) are:
r = r*er
v = r**er + r*θ**et
— er = Radial unit vectors directed relative to the vector from M to m
— et = Tangential unit vectors directed relative to the vector from M to m
We can use these equations to simplify the formula that can yield work of gravity.
This calculation uses the following differential equation:
d(r-1)/dt = -(r-2)*r* = -r*/r2
What is gravitational potential energy (GPE)?
Gravitational potential energy is associated with work done to elevate objects against Earth's gravitational force. If an object falls from a point to another inside a gravitational field, the force of gravity does positive work on the object, hence the GPE subsequently decreases by the same amount.
Imagine a tennis ball is placed on the ground next to the table. If I elevate the tennis ball from the floor to the table, my mechanical force works against the gravitational force. If the ball falls off the table down to the floor, the “falling” energy the ball receives is provided by the gravitational force. Therefore, this potential energy is used to accelerate the mass of the ball and is converted into kinetic energy. When the ball contacts the ground, the kinetic energy it exerts is converted into heat, deformation and sound on impact.
The factors contributing to an object’s GPE include its height relative to a specified reference point, its mass, and the strength of the gravitational field it is situated in. Hence, a tennis ball positioned on a table has less gravitational energy than a bowling ball lying on the same table. Nevertheless, if we place the tennis ball on top of the same table on the Moon’s surface, it has less GPE than an identical tennis ball on top of an identical table placed on the Earth’s surface because the Moon’s gravity is weaker than that of the Earth.
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A trebuchet uses the gravitational potential energy of the counterweight to throw projectiles over 200 m.
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When the distance changes slightly relative to the distances from the centre of the gravitational field source, any variation in field strength is negligible. Although the strength of a gravitational field varies with location, physicists assume the force of gravity on a particular object is constant. The gravitational acceleration near the surface of the Earth is assumed to be a constant g = 9.8 m/s2 (“standard gravity”). This means the expression for gravitational potential energy can be derived using the W = Fd equation for work.
WF = -ΔUF
The amount of GPE carried by an elevated object is equal to the work done against gravity in lifting it. The work done equals the force required to move the object upward multiplied with the vertical distance it is moved. Since the upward force required to move an object at a constant velocity is equal to the weight (m*g) of an object, the work done in raising it through a height h is the product m*g*h. Feynman (2011) accounted only the mass, gravity, and altitude is his equation:
U = m*g*h
— U = Potential energy of the object relative to its being on the Earth's surface (J)
— m = Mass of the object (kg)
— g = Acceleration due to gravity (m/s2)
— h = Altitude of the object (m)
Thus, the potential difference is:
ΔU = m*g*Δh
What is the general formula?
If the distance varies significantly, it is invalid to approximate g as constant. This requires calculus and the general mathematical definition of work to determine GPE. Computation of the GPE requires integration of the gravitational force, which is evaluated by Newton’s law of gravitation , with respect to the distance r between the 2 bodies. Therefore, the GPE of a system of masses m1 and M2 at a distance r using gravitational constant G is:
U = -G*m1*M2/r + K
— K = An arbitrary constant dependent on the choice of datum from which potential is measured
If we let K = 0 (i.e. in relation to a point at infinity), this simplifies calculations, but U would always be negative. Given this formula for U, for all n*(n - 1)/2 pairs of 2 bodies, the total potential energy of a system of n bodies can be calculated by summing the potential energy of the system of those 2 bodies.
If we consider the system of bodies as a combined set of small particles the bodies are made out of, then applying the previous result on the particle level would yield the negative GPE. This GPE is significantly more negative than the total potential energy of the system of bodies because it includes the negative gravitational binding energy of each body. The potential energy of this system of bodies is the negative of the energy required to divide the bodies from each other to infinity. Simultaneously, the gravitational binding energy required to split all particles from each other to infinity.
U = -m*[G*(M1/r1) + G*(M2/r2)]
Hence,
U = -m*ΣG*M/r
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This graph illustrates Gravitational potential summation.
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Why is gravitational energy always negative?
Like all potential energies, differences in gravitational potential energy matter for most physical purposes, and zero point is considered arbitrary. Because there is no criterion for preferring 1 particular finite r over another, there is only 2 reasonable choices for the distance at which U becomes zero: r = 0 and r = ∞. Choosing U = 0 would seem counterintuitive because it would lead to negative gravitational energy, however it allows the GPE values to be finite, albeit negative.
The singularity at r = 0 in the formula for GPE suggests that the only other apparently reasonable alternative choice of convention, with U = 0 for r = 0, leads to positive potential energy. However it would be infinitely large for all non-zero values of r, which makes calculations of potential energies beyond the limits of the real number system.
What are the applications of gravitational potential energy?
GPE has a number of practical applications, including:
— The generation of pumped-storage hydroelectricity
— Powering clocks using falling weights
— Counterweights lifting up an elevator, crane, or sash window.
— Roller coasters
— Descent in transportation such as vehicles e.g. an automobile, truck, railroad train, bicycle, airplane, and fluid in a pipeline.
I’ll discuss other types of potential energy in other blog posts.
What is power in physics?
In physics, power is the amount of energy transferred or converted per unit time.
In other words, power is the rate with respect to time at which work is done. This means it is the time derivative of work.
P = dW/dt
— P = Power
— W = Work
— t = Time
If a constant force (F) is applied throughout a distance (x), the work done is defined as W = F*x. In this case, power can be expressed as:
P = dW/dt = d(F*x)/dt = F*(dx/dt) = F*v
If the force varies over a 3D curve C, then the work is expressed in terms of the line integral:
What are the units of power?
The dimension of power is energy over time. In the International System of Units (SI), the unit of power is the watt (W), which is equal to 1 J/s (Joule/second). Other common and traditional measures of power include:
- Horsepower (hp), which equals about 745.7 Watts
- Ergs per second (erg/s)
- Foot-pounds per minute
- dBm, a logarithmic measure relative to a reference of 1 milliwatt
- Calories per hour
- BTU per hour (BTU/h)
- Tons of refrigeration
What is average power?
When 1 kg of coal is burned, it releases more energy than 1kg of TNT being detonated. Although a TNT detonation releases energy more quickly, it delivers more power than burning coal. So, if ΔW is the amount of work performed during a period of time of duration Δt, the average power (Pavg) over that period is:
Pavg = ΔW/Δt
— Pavg is the average amount of work done or energy converted per unit of time.
The instantaneous power is then the limiting value of the average power as the time interval Δt approaches zero.
If power (P) is constant, then the amount of work performed during a period of duration t is:
W = P*t
What is mechanical power?
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One metric horsepower is required to lift 75 kg by 1 metre in 1 second.
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Power in mechanical systems combines forces and movement. In addition, power is the product of a force on an object and the object’s velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is also the time derivative of work, which is the work done by a force F on an object travelling along a curve (C).
- x = Defines the path C
- v = Velocity along this path
If the force (F) can be derived from a potential (conservative), then we can apply the gradient theorem to obtain:
WC = U(A) - U(B)
— A & B = Beginning and end of the path along which the work was done, respectively.
The power of any point along the curve (C) is the time derivative:
P(t) = dW/dt = F*v = -dU/dt
In 1 dimension, this can be simplified to:
P(t) = F*v
In rotational systems, power is the product of the torque (τ) and angular velocity (ω):
P(t) = τ*ω
— ω = Radians per second
— * = Scalar product
In fluid power systems such as hydraulic actuators, power is:
P(t) = p*Q
— p = Pressure in pascals (N/m2)
— Q = Volumetric flow rate (m3/s)
If a mechanical system doesn’t experience any power losses, that means the input power equals the output power.
Consider the input power to a device as a force (FA) acting on a point moving with velocity (vA) and the output power as a force (FB) acting on a point moving with velocity (vB). If the system doesn’t have any power losses, that means:
P = FB*vB = FA*vA
and the mechanical advantage of the system (output force per input force) is:
MA = FB/FA = vA/vB
For a rotating system:
P = TA*ωΑ = TΒ*ωΒ
— ΤΑ = Torque of the input
— ωΑ = Angular velocity of the input
— ΤB = Torque of the output
— ωB = Angular velocity of the output
This yields the mechanical advantage:
MA = TB/TA = ωA/ωB
These equations define the maximum performance of a device in terms of velocity ratios determined by its physical dimensions.
I’ll delve into the other types of power such as electric power, peak power, and radiant power in detail in another post.
What is kinetic energy?
https://en.wikipedia.org/wiki/Kinetic_energy
In physics, the kinetic energy of an object provides it the drive to move. It is defined as the work required to accelerate a body of mass from rest to a non-zero velocity. If a body moves at constant velocity, its kinetic energy level is maintained unless its velocity changes.
The word “kinetic” comes from the Greek word κίνησις kinesis, meaning “motion". Brenner (2008) traced back the dichotomy between kinetic energy and potential energy to Aristotle’s concepts of actuality and potentiality.
In classical mechanics, the principle of kinetic energy, E = m*v2, was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva. Willem’s Gravesande proved this principle by releasing weights from different heights into a block of clay. The weights’ penetration depth was found to be proportional to the square of their impact speed. Émilie du Châtelet published his explanation on the findings of Willem’s experiment.
In their present scientific definitions, the first use of the terms “kinetic energy” and “work” date back to the mid-19th century. In 1829, Gaspard-Gustave Coriolis published his Du Calcul de l'Effet des Machines, which outlined the mathematics of kinetic energy. Between 1849 and 1851, William Thomson, later Lord Kelvin, coined the term "kinetic energy”.
Describe Newtonian kinetic energy
— Rigid bodies
In classical mechanics, the kinetic energy of a point object or a non-rotating rigid body depends on the mass and speed of the body.
Ek = 0.5*m*v2
- m = Mass (kg)
- v = Speed (or velocity) of the body (m/s)
- EK = Kinetic energy (Joules)
e.g. If a 70kg man runs at 3 m/s, its kinetic energy is 0.5*70*32 = 0.5*70*9 = 315 J.
When a pitcher throws a baseball, they perform work on it to increase the baseball’s speed as it leaves the hand. The moving baseball would contact an object such as bat and contact it, doing work on it. The kinetic energy of a moving object is equal to the work needed to accelerate it from rest to a specific velocity or the work the object can do while decelerating to rest: Net force x displacement = Kinetic energy.
F*s = 0.5*m*v2
Since the kinetic energy increases with the square of the speed, an object that doubles its speed gains 4 times as much kinetic energy.
The kinetic energy of an object is also related to its momentum.
Ek = p2/2m
- p = Momentum
- m = Mass of the body
Translational kinetic energy is related to rectilinear motion of a rigid body with constant mass (m), with its centre of mass moving in a straight line with speed (v).
Et = 0.5*m*v2
- m = Mass of the body
- v = Speed of the body’s centre of the mass
The kinetic energy of systems depends on the frame of reference selected. The one giving the minimum value of that energy is the centre of momentum frame, i.e. the reference frame in which the total momentum of the system is zero.
The work done to accelerate a particle with mass (m) during the infinitesimal time interval (dt) can be evaluated by the dot product of force (F) and the infinitesimal displacement (dx)
F*dx = F*v dt = (dp/dt)* v dt = v*dp = v*d(m*v)
- The assumptions include the relationship p = m*v and the validity of Newton’s 2nd law.
If we apply the product rule:
d(v*v) = (dv)*v + v*(dv) = 2*(v*dv)
Therefore, under the assumption of constant mass so that dm = 0), we yield:
v*d(m*v) = 0.5*m*d*(v*v) = 0.5*m*d*v2 = d*(0.5*m*v2)
Since this is a total differential, it can be integrated to obtain kinetic energy. If we assume the object was at rest at time 0, then the integral can be integrated from time 0 to time t. Since the work done by the force to accelerate the object from rest to velocity (v) is equal to the work necessary to perform the reverse:
The kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal change of the body’s momentum (p). It assumes the body begins with zero kinetic energy at rest (motionless).
— Rotating bodies
If a rigid body (Q) is rotating about any line through the centre of mass, then it’s evident it has rotational kinetic energy (Er), which is the sum of the kinetic energies of its moving parts.
- ω = Body’s angular velocity
- r = Distance of any mass dm from that line
- I = Body’s moment of inertia
— Fluid dynamics
In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the ‘dynamic pressure’ at that point.
Ek = 0.5*m*v2
If w divide this equation by V, the unit of volume is:
Ek/V = 0.5*(m/V)*v2
q = 0.5*ρ*v2
- q = Dynamic pressure
- ρ = Density of the incompressible fluid
— Frame of reference
The speed, and hence the kinetic energy of a single object is frame-dependent, meaning it can take any non-negative value relative a suitable inertial frame of reference. Sears (1968) stated that a bullet passing an observer has kinetic energy relative to the observer (i.e. the reference frame). If the observer is moving at the same velocity as the bullet, the bullet is considered stationary, meaning it has zero kinetic energy. Conversely, the total kinetic energy of a system of objects can’t be minimised to zero by a suitable choice of the inertial reference frame. In other cases, the total kinetic energy has a non-zero minimum, since no inertial reference frame is selected in which all the objects are stationary. Hence, this minimum kinetic energy supplies the system’s invariant mass, independent of the reference frame.
The total kinetic energy of a system depends on the inertial frame of reference. In other words, it is the sum of the total kinetic energy in a centre of momentum frame and the kinetic energy the total mass would have as long as it’s concentrated in the centre of mass.
If V is the relative velocity of the centre of mass frame i in the frame k, then if:
v2 = (vi + V)2 = (vi + V)*(vi + V) = vi*vi + 2*vi*V + V*V = vi2 + 2*vi*V + V2 ,
That means,
However, if we let ∫(vi2/2) dm = Ei, the kinetic energy in the centre of mass frame ( ∫vi dm) would be the total momentum, which is zero. Then we let the total mass ∫dm = M and substitute it into the above equation to obtain:
Ek = Ei + M*V2/2
This means the kinetic energy of a system is minimal relative to the centre of momentum reference frames, i.e. frames of reference in which the centre of mass is stationary. In other frames of reference, there was supplementary kinetic energy corresponding to the total mass moving at the speed of the centre of mass.
— Rotation in systems
In some cases, the total kinetic energy of a body is the sum of the body's centre-of-mass translational kinetic energy and the energy of rotation around the centre of mass (rotational energy):
Ek = Et + Eτ
- Ek = Total kinetic energy
- Et = Translational kinetic energy
- Eτ = Rotational energy or angular kinetic energy in the rest frame
Describe relativistic kinetic energy of rigid bodies
If a body’s speed is a fraction of the speed of light, relativistic mechanics is used to calculate its kinetic energy. The expression for linear momentum is further modified in special relativity theory.
If m is an object’s rest mass, v is its velocity and v is its speed, c is the speed of light in vacuum, the formula for linear momentum:
p = m*γ*v
— γ = 1/(1 - v2/c2)0.5
Integrating by parts leads to:
Ek = ∫v*dp = ∫v*d(m*γ*v) = m*γ*v * v - ∫m*γ*v*dv = m*γ*v2 - 0.5*m* ∫γ*d*(v2)
Since γ = (1 - v2/c2)-0.5
Ek = m*γ*v2 - (-0.5*m*c2) * ∫γ*d*(1 - v2/c2)
= m*γ*v2 + m*c2*(1 - v2/c2)0.5 - E0
— E0 = Constant of integration for the indefinite integral
If we simplify the equation, it leads to:
Ek = m*γ*[v2 + c2*(1 - v2/c2)] - E0
= m*γ*(v2 + c2 - v2) - E0
= m*γ*c2 - E0
E0 is evaluated through observation when v = 0, γ = 1 and Ek = 0, yielding:
E0 = m*c2
This leads to the formula:
Ek = m*γ*c2 - m*c2 = m*c2 / (1 - v2/c2) - m*c2
This formula demonstrates the work done to accelerate an object from rest approaches infinity as the velocity approaches the speed of light. Therefore, it is impossible to accelerate an object across this boundary.
The mathematical by-product of this calculation is the mass-energy equivalence formula, which shows the body at rest must have energy content.
Erest = E0 = m*c2
At a low speed (v << c), the relativistic kinetic energy roughly equals the classical kinetic energy, which is evaluated using binomial approximation or selecting the first 2 terms of the Taylor expansion for the reciprocal square root:
Ek ~ m*c2*(1 + 0.5*v2/c2) - m*c2 = 0.5*m*v2
Therefore, the total energy (Ek) can be separated into the rest mass energy plus the Newtonian kinetic energy at low speeds.
When objects move much slower than light, the first 2 terms of the series becomes prevalent. The next term in the Taylor series approximation is small for low speeds:
Ek ~ m*c2*(1 + 0.5*v2/c2 + 0.375*v4/c4) - m*c2 = 0.5*m*v2 + 0.375*m*v4/c2
The relativistic relation between kinetic energy and momentum is:
Ek = (p2*c2 + m2*c4)0.5 - m*c2
This can be expanded as a Taylor series, in which the first term is the simple expression from Newtonian mechanics:
Ek ~ 0.5*p2/m - 0.125*p4/(m3*c2)
This suggests that the formulae for energy and momentum are not special and axiomatic, but rather they emerge from the equivalence of mass and energy and the principles of relativity.
General Relativity
If we use the formula:
gαβ*uα*uβ = -c2
— Four-velocity of a particle: uα = dxα/dr
— r = Proper time of the particle
If the particle has momentum:
pβ = m*gβα*uα
as it passes by an observer with four-velocity (uobs), then the expression for total energy of the particle as observed (measured in a local inertial frame) is:
E = -pβ*uobsβ
The kinetic energy can be expressed as the total energy minus the rest energy:
Ek = -pβ*uobsβ - m*c2
If a metric is diagonal and spatially isotropic (gtt, gss, gss, gss). Since
uα = (dxα/dt) * (dt/dr) = vα*ut
— uα = Ordinary velocity measured with respect to the coordinate system.
This leads to:
-c2 = gαβ*uα*uβ = gtt*(ut)2 + gss*v2*(ut)2
Solving for ut yields:
ut = c*[-1/(gtt + gss*v2)]
Therefore for a stationary observer (v = 0)
uobst = c*(-1/gtt)
This leads to kinetic energy having the expression:
Ek = -m*gtt*ut*uobst - m*c2 = m*c2*[gtt/(gtt + gss*v2)] - m*c2
If we factor out the rest energy, this obtains:
Ek = m*c2*[gtt/(gtt + gss*v2) - 1]
This equation can be simplified to the special relativistic case for the flat-space metric where:
gtt = -c2
gss = 1
In the Newtonian approximation to general relativity:
gtt = -(c2 + 2φ)
gss = 1 - 2φ/c2
— φ = Newtonian gravitational potential
This means clocks run slower and measuring rods are shorter near massive bodies
Describe kinetic energy in quantum mechanics
In quantum mechanics, observables such as kinetic energy are represented as operators. For a particle of mass (m), the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator (p^). The kinetic energy operator in the non-relativistic case is:
T^ = 0.5*p^2/m
If p^ substitutes p in the classical expression for kinetic energy in terms of momentum, then:
Ek = 0.5*p2/m
In the Schrödinger picture, p^ becomes -i*ћ*∇ where the derivative is evaluated with respect to position coordinates. Thus:
The expectation value of the electron kinetic energy, <T^>, for a system of N electrons described by the wave function |ψ> is a sum of 1-electron operator expectation values:
- me = Mass of the electron
- ∇i2 = Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons.
The density functional formalism of quantum mechanics requires knowledge of the electron density only. Given an electron density ρ(r), the exact N-electron kinetic energy functional is unknown. Nevertheless, for the specific case of a 1-electron system, the kinetic energy is expressed as:
- T[ρ] = von Weizsäcker kinetic energy functional
Describe classical mechanics beyond Newton’s laws
Classical mechanics also describes the more complex motions of extended non-point-like objects, such as Euler’s laws extending Newton’s laws. The concepts of angular momentum rely on the same calculus used to describe 1-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object with changing mass. Alternative formulations of classical mechanics were expounded by Lagrange and Hamilton. These, along with other modern formulations, bypass the concept of “force”, and instead refer to other physical quantities, such as energy, speed and momentum, to describe mechanical systems in generalised coordinates. In electromagnetism, Newton’s 2nd law for current-carrying wires doesn’t work unless it includes the electromagnetic field contribution to the system’s total momentum, expressed by the Poynting vector divided by c2, c = Speed of light in free space. I’ll delve into these topics later in another post.
What are the limits of its validity?
Many branches of classical mechanics are simplified or approximated versions of more accurate forms. e.g. General relativity and relativistic statistical mechanics. Geometric optics is an approximated version of the quantum theory of light, which doesn’t have a superior “classical” form.
Both quantum mechanics and classical mechanics break down at the quantum level with many freedoms of freedom. Instead, quantum field theory (QFT) is used to handle tiny distances and enormous speeds with many degrees of freedom, as well as any change in the number of particles throughout the interaction.
Statistical mechanics is used to deal with large degrees of freedom at the macroscopic level. They describe the behaviour of large (but countable) numbers of particles and their interactions as a whole at the macroscopic level. They are mainly used in thermodynamics for systems lying outside the bounds of the assumptions of classical thermodynamics.
Special relativity handles cases of high velocity objects that approach the speed of light.
Parameterised post-Newtonian formalism and general relativity handle cases of objects with significant mass (i.e. their Schwarzschild radius is not negligibly small for a given application), in which deviations from Newtonian mechanics becomes apparent. However, there is no theory of quantum gravity that unifies general relativity and quantum field theory to describe objects that are minuscule and significantly dense.
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This diagram illustrates the domain of validity for Classical Mechanics.
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I’ll discuss quantum mechanics, quantum field theory and special relativity in another post.
