- For alternative meanings see mass
(disambiguation).
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 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.)
Inertial and gravitational mass have been experimentally shown to be
equivalent, as accurately as we can measure, although they are conceptually quite distinct. This equivalence between inertial and
gravitational mass is at the heart of the theory of general
relativity. Below, we will discuss the definitions and implications of each of these two quantities.
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 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.
Gravitational Mass
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.
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.
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
-
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.
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.
See also
External links
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