Mass is a property of physical objects which, roughly speaking, measure the amount of matter contained in an object. It is a central concept of classical mechanics and related subjects. In the SI system of measurement, mass is measured in kilograms.
Strictly speaking, mass refers to two quantities:
Inertial and gravitational mass have been experimentally proven to be equivalent, although they are conceptually quite distinct. Below, we will discuss the definitions and implications of each of these two quantities.
Inertial mass is determined using Newton's second and third laws of motion (see classical mechanics.) Given an object with a known inertial mass, we can obtain the inertial mass of any other object by making the two objects exert a force on each other. According to Newton's third law, the forces experienced by each object will have equal magnitude. This allows us to study how the two objects resist similar applied forces.
Suppose we have two objects, A and B, with inertial masses mA (which is known) and mB (which we wish to determine.) We will assume these masses to be constant. 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. According to Newton's second law,
where aA and aB are the accelerations of A and B respectively. To proceed, we must ensure that these accelerations are non-zero, i.e. that the forces between the two objects are non-zero. This may be done, for example, by having the two objects collide and performing our measurements during the collision.
Newton's third law states that the two forces are equal and opposite, i.e.
When substituted into the above equations, this yields the mass of B as
Thus, measuring aA and aB allows us to determine mA in terms of mB, as desired. Note that our above requirement, that aB be non-zero, allows this equation to be well-defined.
In the above discussion, we assumed that the masses of A and B are constant. This is a fundamental assumption, known as the conservation of mass, and is based on the expectation that matter can never be created or destroyed, only split up or recombined. (The implications of special relativity are discussed below.) 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.
Consider two objects A and B with gravitational masses MA and MB, at a distance of |rAB| apart. Newton's law of gravitation states that the magnitude of the gravitational force which each object exerts on the other is
where G is the universal gravitational constant. The above statement may be reformulated in the following way: given the acceleration g of a reference mass in a gravitational field (such as the gravitational field of the Earth), the gravitational force on an object with gravitational mass M has magnitude
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.
Experiments have found inertial and gravitational mass to be equal, to a high level of precision. These experiments are essentially tests of the well-known phenomenon, first observed by Galileo, that objects fall at a rate irrespective of their masses (in the absence of factors such as friction.) Suppose we have an object with inertial and gravitational masses m and M respectively. If gravity is the only force acting on the object, the combination of Newton's second law and gravitational law gives its acceleration a as
Therefore, all objects in the same gravitational field fall at the same rate if and only if the ratio of gravitational and inertial mass is always equal to some fixed constant. We may as well take this ratio to be 1, by definition.
In the special theory of relativity, "mass" refers to the inertial mass of an object as measured in the frame of reference in which it is at rest (which is known as its "rest frame[?]".) The above method for determining inertial masses remains valid, provided we ensure that the speed of the object is always much smaller than the speed of light, so that classical mechanics is valid.
This equation can be rearranged in the following way:
The classical limit corresponds to the situation in which the momentum p is much smaller than mc, in which case we can Taylor expand the square root, resulting in
The leading term, which is the largest, is the rest energy of the particle. Provided the mass is non-zero, a particle always has this minimum amount of energy regardless of its momentum. The rest energy is normally inaccessible, but it can be tapped by splitting or combining particles, as is done during nuclear fusion and fission. The second term is simply the classical kinetic energy, which can be demonstrated by using the classical definition of momentum
and substituting it into the above to give:
The relativistic energy-mass-momentum relation can also account for particles that are massless, which is an ill-defined concept in classical mechanics. When m = 0, the relation can be simplified to
where p is the relativistic momentum.