Usually when we speak of an object's mass we do not distinguish whether we are referring to its inertial mass or its gravitational mass. This is because the quantity of matter present in an object, i.e., its mass, does not depend on the method by which it is measured.
Gravitational mass
is measured with the use of a double-pan or triple-beam balance. It is a static measurement - that is, a measurement that can only be accurately recorded when the system is in a state of rest. This method involves placing an unknown mass on the pan and using countermasses to return the balance to equilibrium.
This type of measurement only works in the presence of gravity and is actually based on the torque produced by the product of the weights and their lever-arms' distance from the axis of rotation. Since torques produce rotation, when the clockwise torque caused by the countermasses equals the counterclockwise torque caused by the unknown mass, we say that the balance is in equilibrium. Since the balance will then be in a state of rest, we can read the correct value for the unknown's gravitational mass from the balance's scale.
Inertial mass
is measured with the use of an inertial balance, or spring-loaded pan. It is a dynamic measurement - that is, a measurement that can only be accurately recorded while the system is in a state of motion. This method capitalizes on an object's inertia, or its tendency to continue in its current state of motion, as a means of quantifying the amount of matter present.
The pan is first calibrated by counting the number of vibrations in a specified amount of time produced by two objects whose masses are known. From this information, the period (represented with the variable **T**) of each object's mass is calculated by dividing the total amount of time by the total number of vibrations. Period is usually measured in terms of seconds per vibration. These two periods are then plotted on a graph of T^{2} vs Mass.
Subsequent knowledge of the vibrational period of any unknown mass will allow its inertial mass to be interpolated from this calibration graph. This type of balance will measure an object's inertial mass even in the absence of gravity.
A dramatic use of these two definitions of mass can be illustrated when we state that all freely falling bodies experience the same acceleration. When you use net F = ma for a projectile in freefall, **net F** equals the force of gravitational attraction between the object and the Earth; that is, the object's weight. Weight is calculated as the product of the object's gravitational mass and the Earth's gravitational field strength, **g**.
wt = mg When we look at the other side of the equation, ma, then we are talking about the object's inertial mass - its resistance to a change in its state of motion, that is, its resistance to being accelerated. This mass is a measure of how much inertia must be accelerated.
net F = ma -m_{gravitational}g = m_{inertial}a
Since we can experimentally determine that all freely-falling bodies experience the same acceleration, that is, a = -g, we have proof that
m_{gravitational} = m_{inertial}
and there is no need to distinguish between the two definitions. The value of an object's mass is unique, independent of its method of measurement. |