Newton defined the mass of an object as the quantity of matter it possessed. A small rock, for example, has a mass—a fixed, unchanging quantity of matter. If you were to take that rock along with you on a trip to the moon, it would have the same quantity of matter (the same mass) that it had on Earth. Its weight, however, would be less on the moon. The rock's weight on earth was the pull that the earth's gravity exerted on it. On the moon, its weight, as measured with a spring scale, will be less because the moon does not pull on it as strongly as the earth.
Defining the mass of an object as the quantity of matter it possesses is not a very good scientific definition. A better one can be found in Newton's second law of motion. If a constant force is applied to an object on a frictionless, horizontal surface, the object accelerates—its velocity increases uniformly with time. If a force twice as large is applied to the same object, its acceleration doubles as well. The object's acceleration is proportional to the force applied to it. We might write: Fα a where F is the force applied to the object and a is the acceleration of the object while the force acts. The symbol α means that the two quantities, force and acceleration, are proportional; that is, if the force doubles the acceleration doubles.
Additional experiments show that force and acceleration are always proportional for any object; however, the same force applied to a baseball and a bowling ball will provide the bowling ball with a much smaller acceleration than the baseball. To convert the proportionality Fα a to an equation requires a proportionality constant so that we may write proportionality constant x a = F.
If the proportionality constant is to reflect the difference between the baseball and the bowling ball, we might write proportionality constant = f(F, a), or m = f(F, a).
Here, m is defined as the inertial mass of the object. It shows that a bowling ball requires a much bigger force than a baseball to produce the same acceleration.
Mass then can be defined as a ratio of force to acceleration. We define one kilogram to be an inertial mass that accelerates at one meter per second per second when a force of one newton is applied to it. If the same force (one newton) is applied to a two kilogram mass, its acceleration is only 0.5 meter per second per second.
If two objects acquire the same acceleration when the same force is applied to them, they have the same inertial mass. It makes no difference whether one is made of lead and the other of aluminum, their inertial masses are identical.
It is a common practice to measure mass on an equal arm balance. Two masses that balance are said to have the same gravitational mass because the gravitational pull on each of them is the same. Measuring inertial and gravitational masses are very different procedures. Inertial masses can be measured anywhere and are totally independent of gravity. Gravitational masses can be determined only in a gravitational field and there is no acceleration. Are the two kinds of masses related? Experiments have shown to within one part in ten billion, that two objects with the same gravitational mass have the same inertial mass.
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Rogers, Eric M. Physics for the Inquiring Mind. Princeton: Princeton University Press, 1960. pp. 105-134.
Sears, Zemansky, and Young. College Physics. 6th ed. Reading, MA: Addison-Wesley, 1985. pp. 59-64.
White, Harvey. Modern College Physics. Princeton, NJ: D. Van Nostrand, 1956. pp. 460-463.