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# Inertial and Gravitational Mass

Mass, and mass in motion, is what makes our macroscopic lives and experiences possible — but is there a difference between an object in free fall and a car accelerating with an engine? Lets go over the definitions of inertial and gravitational mass and find out if there is a difference.

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## What is Inertial Mass?

Inertia is the physical property that keeps objects at rest and unchanged. Without some external force acting on a cup resting on a table, the cup will remain still forever; a cup cant just get up and walk away, or push itself to fall off the edge of the table. The concept of inertial mass is based on the idea that all bodies with mass have this inherent resistance to a change in motion.

### 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.

Inertial mass is a property of mass that determines an objects resistance to changes in motion from an external force. A body at rest will move if enough force is applied. A body moving with a constant velocity can experience a change in velocity or direction from an external force.

The definition of inertial mass should sound familiar to you — this is just an application of Newtons 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 Newtons second law:

$m=\frac{{F}_{\mathrm{net}}}{a}$

To calculate the inertial mass of a moving object, youll need the net force, so free-body diagrams can be a useful tool for solving these problems if you arent given the net force right away.

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 tables 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. Weve 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 isnt much; the cup has a small inertial mass and doesnt resist a change in motion very well.

What about pushing a much larger object, like the table itself, or a couch? Youll 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 youre pushing heavy furniture across carpeted flooring, youll find that the frictional force makes the moving job even harder!

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

A box is pushed across a floor with a net force of$25.5\mathrm{N}$, causing the box to accelerate at$6.25\mathrm{m}/{\mathrm{s}}^{2}$. What is the inertial mass of the box?

To solve for the inertial mass, well want to use Newtons second law:

$m=\frac{{F}_{\mathrm{net}}}{a}$

Since we already have the net force, all we need to do is use the equation for inertial mass, Newtons second law.

$m=\frac{25.5\mathrm{kg}\mathrm{m}/{\mathrm{s}}^{2}}{6.25\mathrm{m}/{\mathrm{s}}^{2}}=4.08\mathrm{kg}$.

Finding the inertial mass isnt complicated when youre already familiar with Newtons 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, were calculating gravitational mass.

Gravitational mass is the mass of an object calculated using the objects response to the force of gravity alone.

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 bodys mass from Earths gravitational field. A balance is designed to measure gravitational mass itself, since a comparison of two masses isnt 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 Newtons second law:

$m=\frac{{F}_{g}}{{a}_{g}}$

The difference here is that were 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 arent 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 Einsteins 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.

## Inertial and Gravitational Mass - Key takeaways

• Inertial mass is a measurement of an objects resistance to changes in motion until enough external force is applied.
• Objects with greater inertial mass are more resistant to changes in motion, requiring a greater applied force in order to be moved.
• Inertial mass can be calculated if we know the net force and acceleration with Newtons second law.
• Gravitational mass is the mass of an object measured using the force and acceleration due to gravity alone.
• Gravitational mass can be measured using a true balance, or calculated using the gravitational force and acceleration with Newtons second law.
• Gravitational mass and inertial mass have experimentally been found to be equivalent, meaning the two values will remain identical regardless of whether or not an object is at rest or traveling at high velocities.

#### Flashcards in Inertial and Gravitational Mass 21

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What is gravitational mass and inertial mass?

Gravitational mass is a measurement of mass due to the force and acceleration of gravity alone. Inertial mass is a measurement of the resistance of a mass to changes in motion.

Are inertial and gravitational mass the same?

Inertial and gravitational mass measurements are the same. This equivalence has been experimentally verified. If we measure an object’s inertial and gravitational mass at rest, the measurements will be equal and remain identical, even in motion at high velocities.

What are some examples of inertial and gravitational mass?

Some examples of inertial mass include measuring the mass of an object being pushed given the net force and forward acceleration. A couch pushed across a carpeted floor has a greater inertial mass and therefore resistance to motion than a smooth cup pushed across a polished table. An example of gravitational mass is comparing a known mass to an unknown mass on a balance.

What is the formula to calculate gravitational mass?

The formula to calculate gravitational mass is m=F/a, where F is the gravitational force and a is the acceleration due to gravity.

## Test your knowledge with multiple choice flashcards

What is gravitational mass?

What is inertial mass?

How does the force of friction relate to the measurement of inertial mass?

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