# Mass

Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects.

Strictly speaking, there are three different quantities called mass:

• Inertial mass is a measure of an object's inertia: its resistance to changing its state of motion when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily.
• Passive gravitational mass is a measure of the strength of an object's interaction with the gravitational field. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass. (This force is called the weight of the object. In informal usage, the word "weight" is often used synonymously with "mass", because the strength of the gravitational field is roughly constant everywhere on the surface of the Earth. In physics, the two terms are distinct: an object will have a larger weight if it is placed in a stronger gravitational field, but its passive gravitational mass remains unchanged.)
• Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active 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. One of the consequences of the equivalence of inertial mass and passive gravitational mass is the fact, famously demonstrated by Galileo Galilei, that objects with different masses fall at the same rate, assuming factors like air resistance are negligible. The theory of general relativity, the most accurate theory of gravitation known to physicists to date, rests on the assumption that inertial and passive gravitational mass are completely equivalent. This is known as the weak equivalence principle. Classically, active and passive gravitational mass were equivalent as a consequence of Newton's third law, but a new axiom is required in the context of relativity's reformulation of gravity and mechanics. Thus, standard general relativity also assumes the equivalence of inertial mass and active gravitational mass; this equivalence is sometimes called the strong equivalence principle.

If one were to treat inertial mass mi, passive gravitational mass mp, and active gravitational mass ma distinctly, Newton's law of universal gravitation would take the form

[itex]m_ia=\frac{Gm_pm_a}{r^2}.[itex]
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## Units of mass

In the SI system of units, mass is measured in kilograms (kg). Many other units of mass are also employed, such as: grams (g), metric tons, pounds, ounces, long and short tons, quintals, slugs, atomic mass units, Planck masses, solar masses, and eV/c2.

The eV/c2 unit is based on the electron volt (eV), which is normally used as a unit of energy. However, because of the relativistic connection between (rest) mass and energy, E = mc2 (see below), it is possible to use any unit of energy as a unit of mass instead. Thus, in particle physics where mass and energy are often interchanged, it is common to use not only eV/c2 but even simply eV as a unit of mass (roughly 1.783 × 10-36 kg).

Because the gravitational acceleration is approximately constant on the surface of the Earth, a unit like the pound is often used to measure either mass or force (e.g. weight), although the pound is officially defined as a unit of mass.

For more information on the different units of mass, see Orders of magnitude (mass).

## Inertial mass

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

[itex] F = \frac{d}{dt} (mv) [itex]

where F is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means.

Now, suppose that the mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, the situation gets more complicated when we take special relativity into account. Another point to note is that, even in classical mechanics, it is sometimes useful to treat the mass of an object as changing with time. For example, the mass of a rocket decreases as the rocket fires. However, this is an approximation, based on ignoring pieces of matter which enter or leave the system. In the case of the rocket, these pieces correspond to the ejected propellent; if we were to measure the total mass of the rocket and its propellent, we would find that it is conserved.

When the mass of a body is constant, Newton's second law becomes

[itex] F = m \frac{dv}{dt} = m a [itex]

where a denotes the acceleration of the body.

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. As we have seen, Newton's second law states that

[itex]F_{AB} = m_A a_A \,[itex] and [itex]F_{BA} = m_B a_B \,[itex]

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

[itex]F_{AB} = - F_{BA} \,[itex]

Substituting this into the previous equations, we obtain

[itex]m_A = - \frac{a_B}{a_A} \, m_B[itex]

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 every other object in the universe by colliding it with the reference object and measuring the accelerations.

## Gravitational mass

The 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

[itex]|F| = {G M_A M_B \over |r_{AB}|^2}[itex]

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

[itex]F = Mg \,[itex]

This is the basis by which masses are determined by weighing. In simple bathroom scales, for example, the force F is proportionate to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the mass M to be read off.

## 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

[itex]K = \frac{M}{m} g[itex]

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 unlikely to be true; actually, he performed his experiments with balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Roland Eötvös, using the torsion balance pendulum, in 1889. To date, no deviation from universality, and thus from Galilean equivalence, has ever been found. More precise experimental efforts are still being carried out.

It should be noted that the universality of free fall only applies to systems in which gravity is the only force acting. 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, we all know that the feather will take much longer to reach the ground. This happens because the feather is not really in free fall: the force of air resistance on it is about as strong as the force of gravity. On the other hand, if the experiment is performed in a vacuum, where there is no air resistance, the hammer and the feather should fall at the same rate and reach the ground together. This demonstration was, in fact, carried out in 1971 during the Apollo 15 Moon walk, by Commander David Scott.

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 it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that inertial and gravitational masses are fundamentally the same thing. All of the predictions of general relativity, such as the curvature of spacetime, are ultimately derived from this principle.

## Relativistic relation among mass, energy and momentum

Special relativity is a necessary extension of classical physics. In particular, special relativity succeeds where classical mechanics fails badly in describing objects moving at speeds close to the speed of light.

In relativistic mechanics, the mass (m) of a free particle is related to its energy (E) and momentum (p) by the equation

[itex]\frac{E^2}{c^2} = m^2 c^2 + p^2[itex].

where c is the speed of light. This is sometimes referred to as the mass-energy-momentum relation.

The first thing to notice about this equation is that it can cope with massless objects (m = 0), for which it reduces to

[itex]E = pc \,[itex]

In classical mechanics, massless objects are an ill-defined concept, since applying any force to one would produce, via Newton's second law, an infinite acceleration - a nonsensical result. In relativistic mechanics, they are objects that are always travelling at the speed of light; an example being light itself, in the form of photons. The above equation says that the energy carried by a massless object is directly proportional to its momentum.

Let us now consider objects with non-zero mass. For these, the quantity m has a simple physical meaning: it is the inertial mass of the object as measured in its rest frame, the frame of reference in which its velocity is zero. (Note: massless objects do not possess a rest frame; they are moving at the speed of light in any frame of reference.) The way we would measure m is exactly the same as in classical mechanics, which we described above: bouncing it off a reference object and measuring the accelerations. As long as the velocity of each object remains much smaller than the speed of light during this procedure, relativistic corrections to classical mechanics will be utterly negligible.

In the rest frame, the velocity is zero, and thus so is the momentum p. The mass-energy-momentum relation thus reduces to

[itex]E = mc^2 \,[itex]

which states that the energy of an object as measured in its rest frame - its "rest energy" - is equal to its mass times the square of the speed of light.

Some books follow this up by stating that "mass and energy are equivalent", but this is somewhat misleading. The mass of an object, as we have defined it, is a quantity intrinsic to the object, and independent of our current frame of reference. The energy E, on the other hand, varies with the frame of reference; if the frame is moving at a high velocity relative to the object, E will be very large, simply because the object has a lot of kinetic energy in that frame. Thus, E = mc2 is not a "good" relativistic statement; it is true only in the rest frame of the object.

Some authors define a quantity known as the relativistic mass, which is basically the quantity E/c2. This makes the "equivalence" of "mass" and energy true by definition, though neither quantity is frame-independent! "Relativistic mass" was used in many early writings on relativity, and it is still used in books for laymen as well as introductory physics classes. However, the concept is downplayed or discouraged by many physicists nowadays, for reasons explained in the article on relativistic mass. Following the modern usage, whenever we refer to "mass" in this article we always mean the rest mass, unless otherwise identified.

Having defined the mass of an object, let us look at how it behaves when not at rest. We can arrange the mass-energy-momentum relation in the following way:

[itex]E = mc^2 \sqrt{1 + \left( {p \over mc} \right)^2}[itex]

When the momentum p is much smaller than mc, we can Taylor expand the square root, with the result

[itex]E = mc^2 + {p^2 \over 2m} + \cdots [itex]

The leading term, which is the largest, is of course the rest energy. The object always has this minimum amount of energy, regardless of its momentum. The second term is the classical expression for the kinetic energy of the particle, and the higher-order terms are basically relativistic corrections for the kinetic energy.

Under normal circumstances, the rest energy of an object is inaccessible, in the sense that it cannot be used to do mechanical work. When the object hits something, it can do work by transferring its momentum, and thus its kinetic energy, to whatever it hit. However, the rest energy depends only on the mass of the object, which does not change during collisions, so it cannot be transferred along with the kinetic energy.

On the other hand, it is possible to access the rest energy using processes that split or combine particles. The reason is that mass, as we have defined it, is not conserved during such processes. The simplest example is the process of electron-positron annihilation, in which an electron and a positron annihilate each other to produce a pair of photons: the electron and positron both have non-zero mass, but the photons are massless. Other examples include nuclear fusion and nuclear fission. Metabolism, fire and other exothermic chemical processes also convert mass to energy, however the mass change from these is negligible.

Energy, unlike mass, is always conserved in special relativity, so, roughly speaking, what is happening in these reactions is that the rest energy of the reactants is being transformed into the kinetic energy of the reaction products. The fact that rest energy can be liberated in this way is one of the most important predictions of special relativity.

## References

• R.V. Eötvös et al, Ann. Phys. (Leipzig) 68 11 (1922)

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