Inertial and Gravitational Mass

Definition of Inertial Mass

Knowing that inertia is the property that makes objects resist motion, we can put together our understanding of inertia and bodies with mass to arrive at the following definition.

The definition of inertial mass should sound familiar to you — this is just an application of Newton’s first law!

The Inertial Mass Equation

The inertial mass equation relates to the acceleration of a moving object and is therefore just a rearrangement of Newton’s second law:

$m=\frac{F_{\mathrm{net}}}{a}$

Returning to our example of a cup resting on a table, try to think about what forces are in play, and how this relates to inertial mass.

A cup resting on a flat surface will resist motion until an external force, such as the push of a hand, causes a change in motion, Pixabay CC0 1.0

The forces acting on the cup are:

• The gravitational force, the weight of the cup, is pulling it down.

• The normal force, the contact force between the cup and the table, is perpendicular to the table’s surface.

• Static friction, the force resisting a change in motion between the two materials in contact, opposes the direction of movement when we push.

When these forces are balanced, there can be no movement of the cup. Until we apply a force greater than the force of friction, the cup will remain at rest. Now, imagine giving the cup a light, forward push with a finger. We’ve added an applied force that causes a forward acceleration — the cup now has increasing momentum, meaning its mass is in motion. The force needed to move a small cup isn’t much; the cup has a small inertial mass and doesn’t resist a change in motion very well.

What about pushing a much larger object, like the table itself, or a couch? You’ll need to apply a much greater force than you did to shift the cup forward. With a greater inertial mass, bigger objects like furniture resist a change in motion much more, and if you’re pushing heavy furniture across carpeted flooring, you’ll find that the frictional force makes the moving job even harder!

Let’s do an example using the inertial mass equation with a moving object, given the net force and acceleration.

Finding the inertial mass isn’t complicated when you’re already familiar with Newton’s first and second laws!

What is Gravitational Mass?

The force of gravity is the force causing the acceleration of all objects on Earth downwards at$9.8\mathrm{m}/{\mathrm{s}}^2$. If we measure the mass of an object using only gravity, we’re calculating gravitational mass.

The definition of gravitational mass might sound similar to measuring weight, but these are not the same. Gravitational mass is measured using a true balance, which compares a known mass to a mass we want to measure. Scales, such as a bathroom or clinic scale, measure your weight due to gravity — in other words, the downward force of your body’s mass from Earth’s gravitational field. A balance is designed to measure gravitational mass itself, since a comparison of two masses isn’t changed by the force of gravity. When the instrument is in equilibrium, the masses on both sides are equivalent, no matter where you measure it.

Gravitational mass is measured using a balance, where a known mass is placed on one side. Smaller or unknown masses are added until the scale is balanced, Pixabay CC0 1.0

The Gravitational Mass Equation

The gravitational mass equation is once again an application of Newton’s second law:

$m=\frac{F_g}{a_g}$

The difference here is that we’re using the local acceleration due to gravity, and the gravitational force instead of the net force. Physics problems solving for gravitational mass will look just like the example we went through for inertial mass. In fact, you likely have already solved for gravitational mass already when given the weight of some object on Earth to start — you just might not have realized it then!

The Relationship between Gravitational Mass and Inertial Mass

Have we determined a difference between gravitational mass and inertial mass measurements? We learned that calculating inertial mass requires us to know the net force, yet gravitational mass only requires the gravitational force. These calculations aren’t quite the same, so there should be a difference between gravitational and inertial mass, right? The answer to this question is that inertial and gravitational mass are equivalent: experimentally, no difference has been found.

The Equivalence of Inertial and Gravitational Mass

The equivalence of inertial and gravitational mass means a fundamental relationship exists. If the inertial mass and gravitational mass of an object are measured at rest, they will be identical. If the same object is moving, even at very high velocities, both measurements will still be the same. The mass of an object will not change as it accelerates, or if the laboratory we are in is accelerating!

This relationship formed the basis of Einstein’s equivalence principle, a statement of the equivalence of inertial and gravitational masses. The equivalence principle was a key concept that helped develop our modern understanding of gravity.