OPERATIONAL DEFINITION OF FORCE

An operational definition is a demonstration of a process – such as a variable, term, or object – in terms of the specific process or set of validation tests used to determine its presence and quantity. The term was coined by Percy Williams Bridgman (see Operationalization). Properties described in this manner must be sufficiently accessible, so that persons other than the definer may independently measure or test for them at will.[citation needed] An operational definition is generally designed to model a conceptual definition. The most operational definition is a process for identification of an object by distinguishing it from its background of empirical experience. The binary version produces either the result that the object exists, or that it doesn't, in the experiential field to which it is applied. The classifier version results in discrimination between what is part of the object and what is not part of it. This is also discussed in terms of semantics, pattern recognition, and operational techniques, such as regression.

For example, the weight of an object may be operationally defined in terms of the specific steps of putting an object on a weighing scale. The weight is whatever results from following the measurement procedure, which can in principle be repeated by anyone. It is intentionally not defined in terms of some intrinsic or private essence. The operational definition of weight is just the result of what happens when the defined procedure is followed. In other words, what's being defined is how to measure weight for any arbitrary object, and only incidentally the weight of a given object.

Operationalize means to put into operation. Operational definitions are also used to define system states in terms of a specific, publicly accessible process of preparation or

validation testing, which is repeatable at will. For example, 100 degrees Celsius may be crudely defined by describing the process of heating water until it is observed to boil. An item like a brick, or even a photograph of a brick, may be defined in terms of how it can be made. Likewise, iron may be defined in terms of the results of testing or measuring it in particular ways.

One simple, every day illustration of an operational definition is defining a cake in terms of how it is prepared and baked (i.e., its recipe is an operational definition). Similarly, the saying, if it walks like a duck and quacks like a duck, it must be some kind of duck, may be regarded as involving a sort of measurement process or set of tests

ACCELERATION

"Accelerate" redirects here. For other uses, see Accelerate (disambiguation).

For the waltz composed by Johann Strauss, see Accelerationen.

Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0.

Components of acceleration for a planar curved motion. The tangential component at is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector. The centripetal component ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

In physics, and more specifically kinematics, acceleration is the change in velocity over time.[1] Because velocity is a vector, it can change in two ways: a change in magnitude and/or a change in direction. In one dimension, i.e. a line, acceleration is the rate at which something speeds up or slows down. However, as a vector quantity, acceleration is also the rate at which direction changes.[2][3] Acceleration has the dimensions L T−2. In SI units, acceleration is measured in metres per second squared (m/s2).

In common speech, the term acceleration commonly is used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, a change in the direction of velocity also is an acceleration: for rotary motion, the change in direction of velocity results in centripetal (toward the center) acceleration; where as the rate of change of speed is a tangential acceleration.

In classical mechanics, for a body with constant mass, the acceleration of the body is proportional to the resultant (total) force acting on it (Newton's second law):

where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration.

[edit] Average and instantaneous acceleration

Average acceleration is the change in velocity (Δv) divided by the change in time (Δt). Instantaneous acceleration is the acceleration at a specific point in time.

[edit] Tangential and centripetal acceleration

See also: Local coordinates

The velocity of a particle moving on a curved path as a function of time can be written as:

with v(t) equal to the speed of travel along the path, and

a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t) and the changing direction of ut, the acceleration of a particle moving on a curved path on a planar surface can be written using the chain rule of differentiation as:

where un is the unit (outward) normal vector to the particle's trajectory, and R is its instantaneous radius of curvature based upon the osculating circle at time t. These components are called the tangential acceleration and the radial

acceleration, respectively. The negative of the radial acceleration is the centripetal acceleration, which points inward, toward the center of curvature.

Extension of this approach to three-dimensional space curves that cannot be contained on a planar surface leads to the Frenet-Serret formulas.[4][5]

[edit] Relation to relativity

After completing his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant proper acceleration are actually feeling themselves being accelerated, so that, for example, a car's acceleration forwards would result in the driver feeling a slight push backwards. In the case of gravity, which Einstein concluded is not actually a force, this is not the case; acceleration due to gravity is not felt by an object in free-fall. This was the basis for his development of general relativity, a relativistic theory of gravity.

UNITS OF MASS AND ACCELERATION

For other uses, see Mass (disambiguation).

In physics, mass (from Ancient Greek: μᾶζα) 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, mass is often taken to mean weight, but care should be taken to distinguish between the two terms in scientific use, as they actually refer to different properties.

The inertial mass of an object determines its acceleration in the presence of an applied force. According to Isaac 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, the first body experiences an attractive force F given by

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 = mc 2 ). As a consequence of this relationship, the total mass of a collection of particles may be greater or less than the sum of the masses of the individual particles.

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 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.

An object with mass is said to be massive.[1]

Contents [hide]

1 Units of mass

2 Summary of mass concepts and formalisms

3 Summary of mass related phenomena

4 Weight and amount

5 Gravitational Mass

o 5.1 Keplerian gravitational mass

o 5.2 Galilean gravitational field

o 5.3 Newtonian gravitational mass

5.3. 1 Universal gravitational mass and amount

6 Inertial and

gravitational mass

o 6.1 Inertial mass

o 6.2 Newtonian Gravitational mass

o 6.3 Equivalence of inertial and gravitational masses

7 Mass and energy in relativity

8 Notes

9 References

10 External links

[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 (mP), 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 Ek

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 above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.

The Schwarzschild radius (rs) represents the ability of mass to cause curvature in space and time.

The standard gravitational parameter (μ) 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 (λ) 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:[2]

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.

The amount of matter in certain types of samples can be exactly determined through electrodeposition 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).

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. (Note: Quantum mass can be represented in many different mathematical formalisms. The Schrödinger equation is a classical formalism in which kinetic energy is equal to the classical momentum squared divided by the classical mass. The Klein–Gordon equation is a relativistic formalism in which the invariant mass is equal to the difference of the energy squared and the momentum squared.)

Inertial mass, passive and active 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

Main article: weight

Anubis weighing the heart of Hunefer, 1285 BC

The concepts of passive gravitational mass and atomic mass grew out of the much older concepts of weight and amount. Weight, by definition, is a measure of the force that must be applied to support an object (i.e. hold it at rest) in a gravitational field. Gravitational fields typically change only slightly over short distances, and the gravitational field of the earth 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 changes went unnoticed through much of history. This may have given early humans the impression that weight was unchanging, and a fundamental property of all objects.

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. When two objects are near each other they experience almost identical gravitational fields. Hence, if they have similar masses then their weights will also be similar. The two sides of a balance scale are close enough that the scale, by comparing weights, also compares masses. The balance scale measures mass and is one of the oldest known devices used for this purpose.

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 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 found to be 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 found to be 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 ἄτομος/átomos, α-τεμνω, 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 universally acceptance of the existence of atoms didn’t occur until the early 1900’s.

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 have never 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, and the atomic mass unit continues to be the unit of choice for very small mass measurements. (Although, the definition of the atomic mass unit is no longer tied to the hydrogen atom).

When the French invented the metric system in the late 1700s, they used an amount to define their mass unit. The gram was originally defined to be equal in mass to the amount of pure water contained in a one milliliter container. This definition, however, was inadequate for the precision requirements of modern technology, and the metric kilogram was redefined in terms of a manmade physical object.

[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

Main article: Kepler's laws of planetary motion

Johannes Kepler 1610.

EnglishName

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

E 1 1

arth .000 000 AU

.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 proved 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. Kepler discovered that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit; this is his third law of planetary motion. The concept of active gravitational mass is an immediate consequence of this law, since the square of the orbital period is proportional to the cube of the semi-major axis, then the ratio of these two

values must be 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:

EnglishName

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

Main article: Galilean moons

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 objects were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial objects 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.

Galileo’s use of scientific instrumentation went beyond astronomical observations and the telescope. Sometime prior to 1638, Galileo had 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 [3], 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.[4] 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.[5]

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.".[6]

This dialogue provides compelling and realistic solutions to the problems that had plagued earlier measurements. Previous experimenters were unable to measure the short time periods involved in free fall motion. Aristotle, a Greek philosopher, appears to have performed experiments with objects falling in water, but the frictional forces of the water had confused his experimental results. Galileo, on the other hand, by using a nearly frictionless inclined plane, was able to slow the acceleration without introducing additional frictional forces, and thus he was able to measure the fall times with his ingenious “water clock”. Galileo found that the distance

traveled by an object in free fall is always proportional to the square of the elapsed time:

[edit] Newtonian gravitational mass

Isaac Newton 1689.

Earth's Moon

Mass of EarthSe

mi-major axis

Sidereal orbital period

0.002 569 AU

0.074 802 sidereal year

=

Earth's Gravity

Earth's Radius

0.00980665 km⁄sec

2

6 375 km

Galileo Galilei died in Arcetri, Italy (near Florence), on 8 January 1642. Isaac Newton was born almost a year later on 4 January 1643, in Woolsthorpe-by-Colsterworth, England. Prior to Newton’s birth, Johannes Kepler had shown that the planets follow elliptical paths under the influence of the Sun’s gravitational

mass, and Galileo had shown that objects in free fall under the influence of the Earth’s gravitational field have a constant acceleration. However, the relationship between Kepler’s gravitational mass and Galileo’s gravitational field hadn’t been fully comprehended.

In 1600 AD, Johannes Kepler had speculated that forces from Earth might govern the Moon’s motion. He composed an essay in which he wrote: "In Terra inest virtus, quae Lunam ciet" ("There is a force in the Earth which causes the Moon to move").[7] Robert Hooke, a contemporary of Isaac Newton, published his own ideas about gravitational forces in 1674 AD, 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”. Hooke further states that a celestial body’s gravitational attraction increases “by how much the nearer the body wrought upon is to their own center.”[1] In a correspondence of 1679-1680 between Robert Hooke and Isaac Newton, Hooke describes his concept of gravitation and conjectures that the strength of the gravitational force might decrease according to the square of the distance between the two bodies.[8] 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 inverse square hypothesis was correct. Newton’s own investigations of gravity verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. 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,

μ is the gravitational mass (standard gravitational parameter) of the body causing gravitational fields,

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. [2]

Isaac Newton kept quiet about his discoveries until 1684, at which time he revealed to a friend, Edmond Halley, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office [3]. 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")[4]. Halley presented Newton’s findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton eventually presented 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 resolved to publish Newton’s books at their own expense, and published the entire collection in May of 1686-7 [5].

Main article: 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 gravitational fields, a dropped object 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.” [6]

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 Coelestial 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 an 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), composed of “concentric spherical” surfaces (the density at a given radius being roughly uniform), would have a gravitational field which was proportional to the total mass of the body [7], and inversely proportional to the square of the distance to the body’s center [8].

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 the apple towards the Earth’s center.

Newton’s concept of universal gravitational mass puts gravitational mass on an equal footing with weight. Humans, since ancient times, have known that the weight of a collection of similar objects is proportional to the number of objects in the collection; and consequently, weights are recorded in terms of amounts. For example, the weight of a horse might be 1200 lbs. Meaning that if the horse were placed on one side of a balance scale and a collection of 1 lb weights were placed on the other side, then it would take 1200 weights to balance the horse. Newton’s concept of universal gravitational mass suggests

that the gravitational mass of a collection of similar objects is also proportional to the number of objects in the collection:

,

where:

μn is the gravitational mass of a collection of similar objects and,

n is the number of objects in the collection.

Gravitational mass and weight are proportional. Consequently, gravitational mass and weight can be defined in terms of amounts of a common standard. For example, if one were to measure the standard gravitational parameter of a 1200 lb horse and found it to be 36 millimeters cubed per second squared, then one could quickly deduce that the standard gravitational parameter of a 600 lb pig would be 18 millimeters cubed per second squared, and the standard gravitational parameter of a 100 lb child would be 3 millimeters cubed per second squared. Measuring the standard gravitational parameter and weight of one object allows one to estimate the standard gravitational parameter of every other object whose weight is known.

[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".[9]

[edit] Inertial mass

This section uses mathematical equations involving differential calculus.

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 mA and mB. 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 FAB, and the force exerted on B by A, which we denote FBA. Newton's second law states that

where aA and aB 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 aA 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 mB 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 rAB. The law of gravitation states that if A and B have gravitational masses MA and MB 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 Loránd Eötvös, using the torsion balance pendulum, in 1889. As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the accuracy 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 relativity

Main article: Mass 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 relativistic definition(s) of mass are quantities which are conserved: this means they do not change over time, even as some types of particles are converted to others. The incorrect popular idea that mass may be converted to (massless) energy in relativity is due to the fact that 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, even if not "matter" all these types of energy still continue to exibit mass in relativity when they a created from matter—whether they are considered "matter" or not. Whether these types of "pure" energy are created from matter, or matter is created from them, system mass does not change in the process. Matter particles may not be conserved in reactions in relativity, but closed-system mass always is[citation needed].

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.[10] There is disagreement over whether the concept remains pedagogically useful.[11][12]

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

[edit] Notes

1. ̂ Since the Avogadro constant NA is defined as the number of atoms in 12 g of carbon-12, it follows that 1 u is exactly 1/(103 NA) kg.

2. ̂ Newton's second law is valid only for bodies of constant mass.