The mass of an object (for example a particle, a rigid body, or some amount of a gas or fluid) is a quantity assigned to it that specifies, roughly speaking, how much matter the object contains. The concept of mass fulfills two roles. First, it indicates how much force is needed to accelerate the object. Second, the gravitational force between two objects depends on the masses of the objects. These two usages of `mass' are conceptually distinct, but are very closely related. Mass is often denoted by the letters m {\displaystyle m} or M {\displaystyle M} . The SI unit for mass is the kilogram (kg). One kilogram is defined as the mass of the international prototype of the kilogram , which is made from an alloy of platinum and iridium and is kept in the Bureau international des poids et mesures in Paris. In classical mechanics, there are two kinds of mass. The first one, called inertial mass , appears in the best-known version of Newton's second law, F = m a {\displaystyle F=ma} . This law says that the force F {\displaystyle F} needed to give an object an acceleration a {\displaystyle a} is proportional to a {\displaystyle a} , the proportionality constant being defined as the inertial mass of the object. The other kind of mass, gravitational mass , is the mass occurring in Newton's law of gravitation, F = G M 1 M 2 / r 2 {\displaystyle F=GM_{1}M_{2}/r^{2}} . (We use the symbol M {\displaystyle M} to indicate the difference with the inertial mass, m {\displaystyle m} .) This equation states that to any object we can associate a quantity M {\displaystyle M} , the gravitational mass, such that the gravitational force between two objects with gravitational masses M 1 {\displaystyle M_{1}} and M 2 {\displaystyle M_{2}} at a distance r {\displaystyle r} is proportional to M 1 M 2 / r 2 {\displaystyle M_{1}M_{2}/r^{2}} . Both Galilei and Newton realised it is not a priori clear that the two masses assigned to an object should be equal. It is an empirical fact (tested in many experiments from Galilei's time up to now) that the gravitational mass is always proportional to the inertial mass, and a good choice of units then makes the numerical values of the inertial and gravitational masses always equal to each other. In general relativity, Einstein's strong equivalence principle is assumed, which says that all experiments must give the same results in every inertial (free-falling) reference system. This implies, among other things, that the inertial and gravitational masses of objects must be equal. Strictly speaking, one could split the concept of gravitational mass into two distinct concepts, namely those of passive gravitational mass and active gravitational mass . The passive gravitational mass of an object then measures its acceleration due to a gravitational field, while the active gravitational mass measures the strength of The Gravitational Field produced by the object. However, in classical mechanics these two masses are equal because of the symmetry of the law of gravitation together with Newton's third law. In general relativity, where Newton's laws are no longer valid, the strong equivalence principle implies that the inertial mass, the passive gravitational mass, and the active gravitational mass all coincide.
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