Classical mechanics

Newton's Second Law

History of classical mechanics – Timeline of classical mechanics Branches Statics – Dynamics / Kinetics – Kinematics – Applied mechanics – Celestial mechanics – Continuum mechanics – Statistical mechanics Formulations Newtonian mechanics (Vectorial mechanics) Analytical mechanics: Lagrangian mechanics Hamiltonian mechanics Fundamental concepts Space – Time – Velocity – Speed – Mass – Acceleration – Gravity – Force – Impulse – Torque / Moment / Couple – Momentum – Angular momentum – Inertia – Moment of inertia – Reference frame – Energy – Kinetic energy – Potential energy – Mechanical work – Virtual work – D'Alembert's principle Core topics Rigid body – Rigid body dynamics – Euler's equations (rigid body dynamics) – Motion – Newton's laws of motion – Newton's law of universal gravitation – Equations of motion – Inertial frame of reference – Non-inertial reference frame – Rotating reference frame – Fictitious force – Linear motion – Mechanics of planar particle motion – Displacement (vector) – Relative velocity – Friction – Simple harmonic motion – Harmonic oscillator – Vibration – Damping – Damping ratio – Rotational motion – Circular motion – Uniform circular motion – Non-uniform circular motion – Centripetal force – Centrifugal force – Centrifugal force (rotating reference frame) – Reactive centrifugal force – Coriolis force – Pendulum – Rotational speed – Angular acceleration – Angular velocity – Angular frequency – Angular displacement Scientists Isaac Newton – Jeremiah Horrocks – Leonhard Euler – Jean le Rond d'Alembert – Alexis Clairaut – Joseph Louis Lagrange – Pierre-Simon Laplace – William Rowan Hamilton – Siméon-Denis Poisson

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In physics, mass (from Ancient Greek: î�á��î�î�) commonly refers to any of three properties of matter, which have been shown experimentally to be equivalent: Inertial mass, active gravitational mass and passive gravitational mass. In everyday usage, the Mass is an often taken to mean weight, but in scientific use, they refer to different properties.

The inertial mass of an object determines its acceleration in the presence of an applied force. According to Newton's second law of motion, if a body of mass m is subjected to a force F, its acceleration a is given by F/m.

A body's mass also determines the degree to which it generates or is affected by a gravitational field. If a first body of mass m₁ is placed at a distance r from a second body of mass m₂, each body experiences an attractive force F whose magnitude is

where G is the universal constant of gravitation, equal to 6.67×10^'11 kg^'1 m³ s^'2. This is sometimes referred to as gravitational mass (when a distinction is necessary, M is used to denote the active gravitational mass and m the passive gravitational mass). Repeated experiments since the seventeenth century have demonstrated that inertial and gravitational mass are equivalent; this is entailed in the equivalence principle of general relativity.

Special relativity provides a relationship between the mass of a body and its energy (E = mc²). Mass is a conserved quantity. From the viewpoint of any single observer, mass can neither be created or destroyed, and special relativity does not change this understanding. However, relativity adds the fact that all types of energy have an associated mass, and this mass is added to systems when energy is added, and the associated mass is subtracted from systems when the energy leaves. In nuclear reactions, for example, the system does not become less massive until the energy liberated by the reaction is allowed to leave whereby the "missing mass" is carried off with the energy, which itself has mass.

On the surface of the Earth, the weight W of an object is related to its mass m by

where g is the acceleration due to the Earth's gravity, equal to about 9.81 m s^'2. An object's weight depends on its environment, while its mass does not: an object with a mass of 50 kilograms weighs 491 newtons on the surface of the Earth; on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 newtons.

[edit] Units of mass

In the International System of Units (SI), mass is measured in kilograms (kg). The gram (g) is ¹–„₁₀₀₀ of a kilogram.

Other units are accepted for use in SI:

• The tonne (t) is equal to 1000 kg.
• The electronvolt (eV) is primarily a unit of energy, but because of the mass-energy equivalence it can also function as a unit of mass. In this context it is denoted eV/c², or simply as eV. The electronvolt is common in particle physics.
• The atomic mass unit (u) is defined so that a single carbon-12 atom has a mass of 12 u; 1 u is approximately 1.66×10^'27 kg.^{[note 1]} The atomic mass unit is convenient for expressing the masses of atoms and molecules.

Outside the SI system, a variety of different mass units are used, depending on context, such as the slug (sl), the pound (lb), the Planck mass (m_P), and the solar mass (M_�).

In normal situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same unit for both concepts. However, the distinction between mass and weight becomes important for measurements with a precision better than a few percent (because of slight differences in the strength of the Earth's gravitational field at different places), and for places far from the surface of the Earth, such as in space or on other planets.

A mass can sometimes be expressed in terms of length. The mass of a very small particle may be identified with its inverse Compton wavelength (1 cm^'1 ' 3.52×10^'41 kg). The mass of a very large star or black hole may be identified with its Schwarzschild radius (1 cm ' 6.73×10²⁴ kg).

[edit] Summary of mass concepts and formalisms

In classical mechanics, mass has a central role in determining the behavior of bodies. Newton's second law relates the force F exerted in a body of mass m to the body's acceleration a:

Additionally, mass relates a body's momentum p to its velocity v:

and the body's kinetic energy E_k to its velocity:

In special relativity, relativistic mass is a formalism which accounts for relativistic effects by having the mass increase with velocity.

Since energy is dependent on reference frame (upon the observer) it is convenient to formulate the equations of physics in a way such that mass values are invariant (do not change) between observers, and so the equations are independent of the observer. For a single particle, this quantity is the rest mass; for a system of bound or unbound particles, this quantity is the invariant mass. The invariant mass m of a body is related to its energy E and the magnitude of its momentum p by

where c is the speed of light.

[edit] Summary of mass related phenomena

The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties, however, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object.

• The Schwarzschild radius (r_s) represents the ability of mass to cause curvature in space and time.
• The standard gravitational parameter (î�) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
• Inertial mass (m) represents the Newtonian response of mass to forces.
• Rest energy (E₀) represents the ability of mass to be converted into other forms of energy.
• The Compton wavelength (î») represents the quantum response of mass to local geometry.

In physical science, one may distinguish conceptually between at least seven attributes of mass, or seven physical phenomena that can be explained using the concept of mass:^[1]

• The amount of matter in certain types of samples can be exactly determined through electrodeposition^{[disambiguation needed][clarification needed]} or other precise processes. The mass of an exact sample is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing mass," or mass deficit).

• Inertial mass is a measure of an object's resistance to changing its state of motion when a force is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater inertia.

• Active gravitational mass is a measure of the strength of an object's gravitational flux (gravitational flux is equal to the surface integral of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small 'test object' to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.

• Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two 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.

• Energy also has mass according to the principle of mass'energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.

• Curvature of spacetime is a relativistic manifestation of the existence of mass. Curvature is extremely weak and difficult to measure. For this reason, curvature wasn't discovered until after it was predicted by Einstein's theory of general relativity. Extremely precise atomic clocks on the surface of the earth, for example, are found to measure less time (run slower) than similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the Gravity Probe B satellite.

• Quantum mass manifests itself as a difference between an object's quantum frequency and its wave number. The quantum mass of an electron, the Compton wavelength, can be determined through various forms of spectroscopy and is closely related to the Rydberg constant, the Bohr radius, and the classical electron radius. The quantum mass of larger objects can be directly measured using a watt balance.

Inertial mass, gravitational mass, and the various other mass-related phenomena are conceptually distinct. However, every experiment to date has shown these values to be proportional, and this proportionality gives rise to the abstract concept of mass. If, in some future experiment, one of the mass-related phenomena is shown to not be proportional to the others, then that specific phenomena will no longer be considered a part of the abstract concept of mass.

[edit] Weight and amount

Anubis weighing the heart of Hunefer, 1285 BC

Weight, by definition, is a measure of the force which must be applied to support an object (i.e. hold it at rest) in a gravitational field. The Earth's gravitational field causes items near the Earth to have weight. Typically, gravitational fields change only slightly over short distances, and the Earth's field is nearly uniform at all locations on the Earth's surface; therefore, an object's weight changes only slightly when it is moved from one location to another, and these small changes went unnoticed through much of history. This may have given early humans the impression that weight is an unchanging, fundamental property of objects in the material world.

In the Egyptian religious illustration to the right, Anubis is using a balance scale to weigh the heart of Hunefer. A balance scale balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. The scale, by comparing weights, also compares masses. The balance scale is one of the oldest known devices for measuring mass.

The concept of amount is very old and predates recorded history, so any description of the early development of this concept is speculative in nature. However, one might reasonably assume^{[citation needed]} that humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection:

,

where w is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio:

, or equivalently .

Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the carob seed (carat or siliqua) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:

.

This example illustrates a fundamental principle of physical science: when values are related through simple fractions, there is a good possibility that the values stem from a common source.

Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).

The name atom comes from the Greek á�„�„î¿î�î¿ς/�¡tomos, î�-�„î�î�î�ω, which means uncuttable, something that cannot be divided further. The philosophical concept that matter might be composed of discrete units that cannot be further divided has been around for millennia. However, empirical proof and the universal acceptance of the existence of atoms didn't occur until the early 1900s.

Sodium and chlorine atoms in table salt (image obtained with an Atomic force microscope)

As the science of chemistry matured, experimental evidence for the existence of atoms came from the law of multiple proportions. When two or more elements combined to form a compound, their masses are always in a fixed and definite ratio. For example, the mass ratio of nitrogen to oxygen in nitric oxide is seven eights. Ammonia has a hydrogen to nitrogen mass ratio of three fourteenths. The fact that elemental masses combined in simple fractions implies that all elemental mass stems from a common source. In principle, the atomic mass situation is analogous to the above example of Roman mass units. The Roman pound and ounce were both defined in terms of different sized collections of carob seeds, and consequently, the two mass units were related to each other through a simple fraction. Comparatively, since all of the atomic masses are related to each other through simple fractions, then perhaps the atomic masses are just different sized collections of some common fundamental mass unit.

In 1805, the chemist John Dalton published his first table of relative atomic weights, listing six elements, hydrogen, oxygen, nitrogen, carbon, sulfur, and phosphorus, and assigning hydrogen an atomic weight of 1. And in 1815, the chemist William Prout concluded that the hydrogen atom was in fact the fundamental mass unit from which all other atomic masses were derived.

Carbon atoms in graphite (image obtained with a Scanning tunneling microscope)

If Prout's hypothesis had proven accurate, then the abstract concept of mass, as we now know it, might never have evolved, since mass could always be defined in terms of amounts of the hydrogen atomic mass. Prout's hypothesis; however, was found to be inaccurate in two major respects. First, further scientific advancements revealed the existence of smaller particles, such as electrons and quarks, whose masses are not related through simple fractions. And second, the elemental masses themselves were found to not be exact multiples of the hydrogen atom mass, but rather, they were near multiples. Einstein's theory of relativity explained that when protons and neutrons come together to form an atomic nucleus, some of the mass of the nucleus is released in the form of binding energy. The more tightly bound the nucleus, the more energy is lost during formation and this binding energy loss causes the elemental masses to not be related through simple fractions.

Hydrogen, for example, with a single proton, has an atomic weight of 1.007825 u. The most abundant isotope of iron has 26 protons and 30 neutrons, so one might expect its atomic weight to be 56 times that of the hydrogen atom, but in fact, its atomic weight is only 55.9383 u, which is clearly not an integer multiple of 1.007825. Prout's hypothesis was proven inaccurate in many respects, but the abstract concepts of atomic mass and amount continue to play an influential role in chemistry, and the atomic mass unit continues to be the unit of choice for very small mass measurements.

When the French invented the metric system in the late 1700s, they used an amount to define their mass unit. The kilogram was originally defined to be equal in mass to the amount of pure water contained in a one-liter container. This definition, however, was inadequate for the precision requirements of modern technology, and the metric kilogram was redefined in terms of a manmade platinum-iridium bar known as the international prototype kilogram.

[edit] Gravitational mass

Active gravitational mass is a property of the mass of an object that produces a gravitational field in the space surrounding the object, and these gravitational fields govern large-scale structures in the Universe. Gravitational fields hold the galaxies together. They cause clouds of gas and dust to coalesce into stars and planets. They provide the necessary pressure for nuclear fusion to occur within stars. And they determine the orbits of various objects within the Solar System. Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass. However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena.

[edit] Keplerian gravitational mass

Johannes Kepler 1610.

English Name		The Keplerian Planets
		Semi-major axis	Sidereal orbital period	Mass of Sun
Mercury		0.387 099 AU	0.240 842 sidereal year
Venus		0.723 332 AU	0.615 187 sidereal year
Earth		1.000 000 AU	1.000 000 sidereal year
Mars		1.523 662 AU	1.880 816 sidereal year
Jupiter		5.203 363 AU	11.861 776 sidereal year
Saturn		9.537 070 AU	29.456 626 sidereal year

Johannes Kepler was the first to give an accurate description of the orbits of the planets, and by doing so; he was the first to describe gravitational mass. In 1600 AD, Kepler sought employment with Tycho Brahe and consequently gained access to astronomical data of a higher precision than any previously available. Using Brahe's precise observations of the planet Mars, Kepler realized that the traditional astronomical methods were inaccurate in their predictions, and he spent the next five years developing his own method for characterizing planetary motion.

In Kepler's final planetary model, he successfully described planetary orbits as following elliptical paths with the Sun at a focal point of the ellipse. The concept of active gravitational mass is an immediate consequence of Kepler's third law of planetary motion. Kepler 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, or equivalently, that the ratio of these two values is constant for all planets in the Solar System. This constant ratio is a direct measure of the Sun's active gravitational mass, it has units of distance cubed per time squared, and is known as the standard gravitational parameter:

English Name		The Galilean moons
		Semi-major axis	Sidereal orbital period	Mass of Jupiter
Io		0.002 819 AU	0.004 843 sidereal year
Europa		0.004 486 AU	0.009 722 sidereal year
Ganymede		0.007 155 AU	0.019 589 sidereal year
Callisto		0.012 585 AU	0.045 694 sidereal year

In 1609, Johannes Kepler published his three rules known as Kepler's laws of planetary motion, explaining how the planets follow elliptical orbits under the influence of the Sun. On August 25 of that same year, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January of 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611 he had obtained remarkably accurate estimates for their periods. Later, the semi-major axis of each moon was also estimated, thus allowing the gravitational mass of Jupiter to be determined from the orbits of its moons. The gravitational mass of Jupiter was found to be approximately a thousandth of the gravitational mass of the Sun.

[edit] Galilean gravitational field

Galileo Galilei 1636.

The distance traveled by a freely falling ball is proportional to the square of the elapsed time.

Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects falling under the influence of Earth's gravity, and he was actively attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. Galileo used a number of scientific experiments to characterize free fall motion. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,^[2] but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass.^[3] In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two 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? The only convincing resolution to this question is that all bodies must fall at the same rate.^[4]

A later experiment was described in Galileo's Two New Sciences published in 1638. One of Galileo's fictional characters, Salviati, describes 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". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various angles to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows:

"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.".^[5]

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:

Galileo Galilei died in Arcetri, Italy (near Florence), on 8 January 1642. Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass. However, the relationship between Galileo's gravitational field and Kepler's gravitational mass wasn't comprehended during Galileo's life time.

[edit] Newtonian gravitational mass

Isaac Newton 1689.

Earth's Moon		Mass of Earth
Semi-major axis	Sidereal orbital period
0.002 569 AU	0.074 802 sidereal year	=
Earth's Gravity	Earth's Radius
0.00980665 km–„sec2	6 375 km

Robert Hooke published his concept of gravitational forces in 1674, stating that: 'all Coelestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" [and] "they do also attract all the other Coelestial Bodies that are within the sphere of their activity'. He further states that gravitational attraction increases 'by how much the nearer the body wrought upon is to their own center.'^[6] In a correspondence of 1679-1680 between Robert Hooke and Isaac Newton, Hooke conjectures that gravitational forces might decrease according to the square of the distance between the two bodies.^[7] Hooke urged Newton, who was a pioneer in the development of calculus, to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, Edmond Halley, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office.^[8] After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November of 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin: "On the motion of bodies in an orbit").^[9] Halley presented Newton's findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three book set, entitled Philosophiæ Naturalis Principia Mathematica (Latin: "Mathematical Principles of Natural Philosophy"). The first was received by the Royal Society on 28 April 1685-6, the second on 2 March 1686-7, and the third on 6 April 1686-7. The Royal Society published Newton's entire collection at their own expense in May of 1686-7.^[10]

Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, and proved the following relationship:

,

where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, î� is the gravitational mass (standard gravitational parameter) of the body causing gravitational fields, and r is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered.^[11]

[edit] Newton's cannonball

Newton's cannonball was a thought experiment used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book A Treatise of the System of the World. According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path. 'For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth.' ^[12]

Newton further reasons that if an object were 'projected in an horizontal direction from the top of an high mountain' with sufficient velocity, 'it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected.' Newton's thought experiment is illustrated in the image to the right. A cannon on top of a very high mountain shoots a cannon ball in a horizontal direction. If the speed is low, it simply falls back on Earth (paths A and B). However, if the speed is equal to or higher than some threshold (orbital velocity), but not high enough to leave Earth altogether (escape velocity), it will continue revolving around Earth along an elliptical orbit (C and D).

[edit] Universal gravitational mass and amount

Newton's cannonball illustrated the relationship between the Earth's gravitational mass and its gravitational field; however, a number of other ambiguities still remained. Robert Hooke had asserted in 1674 that: "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers", but Hooke had neither explained why this gravitating attraction was unique to celestial bodies, nor had he explained why the attraction was directed towards the center of a celestial body.

An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single powerful gravitational field directed towards the Earth's center.

To answer these questions, Newton introduced the entirely new concept that gravitational mass is 'universal': meaning that every object has gravitational mass, and therefore, every object generates a gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. With these assumptions in mind, Newton calculated what the overall gravitational field would be if a large collection of small objects were formed into a giant spherical body. Newton found that a giant spherical body (like the Earth or Sun, with roughly uniform density at each given radius), would have a gravitational field which was proportional to the total mass of the body,^[13] and inversely proportional to the square of the distance to the body's center.^[14]

Newton's concept of universal gravitational mass is illustrated in the image to the left. Every piece of the Earth has gravitational mass and every piece creates a gravitational field directed towards that piece. However, the overall effect of these many fields is equivalent to a single powerful field directed towards the center of the Earth. The apple behaves as if a single powerful gravitational field were accelerating it towards the Earth's center.

Newton's concept of universal gravitational mass puts gravitational mass on an equal footing with the traditional concepts of weight and amount. For example, the ancient Romans had used the carob seed as a weight standard. The Romans could place an object with an unknown weight on one side of a balance scale and place carob seeds on the other side of the scale, increasing the number of seeds until the scale was balanced. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound.

According to Newton's theory of universal gravitation, each carob seed produces gravitational fields. Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it should be 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. And since the Roman weight units were all defined in terms of carob seeds, then knowing the Earth's, or Sun's 'carob seed mass' would allow one to calculate the mass in Roman pounds, or Roman ounces, or any other Roman unit.

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. Figure 1 of Cavendish's paper.

This possibility extends beyond Roman units and the carob seed. The British avoirdupois pound, for example, was originally defined to be equal to 7,000 barley grains. Therefore, if one could determine the Earth's 'barley grain mass' (the number of barley grains required to produce a gravitational field similar to that of the Earth), then this would allow one to calculate the Earth's mass in avoirdupois pounds. Also, the original kilogram was defined to be equal in mass to a liter of pure water (the modern kilogram is defined by the man-made international prototype kilogram). Thus, the mass of the Earth in kilograms could theoretically be determined by ascertaining how many liters of pure water (or international prototype kilograms) would be required to produce gravitational fields similar to those of the Earth. In fact, it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. And if one were to collect an immense number of objects, the resulting sphere would probably be too large to construct on the surface of the Earth, and too expensive to construct in space. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, didn't occur until 1797, over a hundred years later. Cavendish found that the Earth's density was 5.448 – 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy,^[15] whereas its gravitational mass is known to over nine digits.^[16]

[edit] Inertial and gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational mass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory, gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational mass [...] remains as puzzling as ever".^[17]

[edit] Inertial mass

Inertial mass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial mass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of mass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "mass" in any essential way.

According to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion

where F is the force acting on the body and a is the acceleration of the body.^{[note 2]} For the moment, we will put aside the question of what "force acting on the body" actually means.

This equation illustrates how mass relates to the inertia of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of Newton's third law, which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects A and B, with constant inertial masses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote F_AB, and the force exerted on B by A, which we denote F_BA. Newton's second law states that

where a_A and a_B are the accelerations of A and B, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that

and thus

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial mass of an object. We choose a "reference" object and define its mass m_B as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

[edit] Newtonian Gravitational mass

The Newtonian concept of gravitational mass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance r_AB. The law of gravitation states that if A and B have gravitational masses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference mass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

This is the basis by which masses are determined by weighing. In simple spring scales, for example, the force F is proportional to the displacement of the spring beneath the weighing pan, as per Hooke's law, and the scales are calibrated to take g into account, allowing the mass M to be read off. A balance measures gravitational mass; only the spring scale measures weight.

[edit] Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational masses m and M respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

This says that 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 referred to as the 'universality of free-fall'. (In addition, the constant K can be taken to be 1 by defining our units appropriately.)

The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Lor�¡nd Eötvös, using the torsion balance pendulum, in 1889. As of 2008^[update], no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10^'12. More precise experimental efforts are still being carried out.

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially friction and air resistance, must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in free-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a vacuum, in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as David Scott did on the surface of the Moon during Apollo 15.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straght line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

[edit] Mass and energy in special relativity

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant mass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.

In as much as energy is conserved in closed systems in relativity, the mass of a system is also a quantity which is conserved: this means it does not change over time, even as some types of particles are converted to others. For any given observer, the mass of any system is separately conserved and cannot change over time, just as energy is separately conserved and cannot change over time. The incorrect popular idea that mass may be converted to (massless) energy in relativity is because some matter particles may in some cases be converted to types of energy which are not matter (such as light, kinetic energy, and the potential energy in magnetic, electric, and other fields). However, this confuses "matter" (a non-conserved and ill-defined thing) with mass (which is well-defined and is conserved). Even if not considered "matter," all types of energy still continue to exhibit mass in relativity. Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy appear without the other. "Matter" particles may not be conserved in reactions in relativity, but closed-system mass always is.

For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.^[18]

In bound systems, the binding energy must (often) be subtracted from the mass of the unbound system, simply because this energy has mass, and this mass is subtracted from the system when it is given off, at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. A familiar example is the binding energy of atomic nuclei, which appears as other types of energy (such as gamma rays) when the nuclei are formed, and (after being given off) results in nuclides which have less mass than the free particles (nucleons) of which they are composed.

The term relativistic mass is also used, and this is the total quantity of energy in a body or system (divided by c²). The relativistic mass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists.^[19] There is disagreement over whether the concept remains pedagogically useful.^[20][21][22]

For a discussion of mass in general relativity, see mass in general relativity.

[edit] Notes

[edit] References

• R.V. Eötvös et al., Ann. Phys. (Leipzig) 68 11 (1922)
• E.F. Taylor, J.A. Wheeler (1992). Spacetime Physics. New York: W.H. Freeman. ISBN 0-7167-2327-1. 

[edit] External links

• Stanford Encyclopedia of Philosophy: "The Equivalence of Mass and Energy" by Francisco Flores.
• "The Mysteries of Mass," Scientific American, July 2005.
• Usenet Physics FAQ by John Baez:
  • "Does mass change with velocity?"
  • "What is the mass of a photon?"
• "The Origin of Mass and the Feebleness of Gravity." Video of lecture by the Nobel Laureate Frank Wilczek.
• Okun, L. B., "Photons, Clocks, Gravity and the Concept of Mass." Slides for a talk.
• The Apollo 15 Hammer-Feather Drop.
• Online mass units conversion.