Mass
- For alternative meanings, see mass (disambiguation).
Strictly speaking, there are two different quantities called mass:
- Inertial mass is a measure of an object's inertia, which is 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.
- Gravitational mass is a measure of the strength of an object's interaction with the gravitational force. Within the same gravitational field, an object with a smaller gravitational mass experiences a smaller force than an object with a larger gravitational mass. (This quantity is sometimes confused with weight.)
Table of contents |
2 Gravitational Mass 3 General Relativity: Equivalence of Inertial and Gravitational Masses 4 Consequences of Special Relativity 5 See also |
Inertial Mass
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 m_{A} (which is known) and m_{B} (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 F_{AB}, and the force exerted on B by A, which we denote F_{BA}. According to Newton's second law,
- .
Newton's third law states that the two forces are equal and opposite, i.e.
- .
- .
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.
Gravitational Mass
Consider two objects A and B with gravitational masses M_{A} and M_{B}, at a distance of |r_{AB}| apart. Newton's law of gravitation states that the magnitude of the gravitational force which each object exerts on the other is
- .
General Relativity: Equivalence of Inertial and Gravitational Masses
At the heart of the general theory of relativity is the Principle of Equivalence, which states that it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the inertial and gravitational masses are rigorously equivalent (and, with a proper choice of units, numerically equal). From this equivalence, the many other predictions of general relativity (such as the curvature of spacetime, etcetera) are derived. A weaker form of this principle, merely stating that inertial and gravitational mass are equal, was considered by Galileo and Newton. The first high precision experimental verification of the principle was carried out by Roland Eotvos in 1889.
This theory was a response to the experimental observation that the inertial and gravitational masses are 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
Consequences of Special Relativity
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.
For more details, see the relativistic mass article.
In relativistic mechanics, the mass of a free particle is related to its energy and momentum by the following equation:
- .
This equation governs the mechanics of massless particles such as photons, the particles of light.
See also
External links
- Usenet Physics FAQ
- Mass & energy
- Photons, Clocks, Gravity and the Concept of Mass by L.B.Okun
- Conversions of english and american mass units to metric units
- Mass conversion, milligrams or micrograms to kilograms, weight of water, volume units, capacity measures and liter - Prefixes