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# Equivalence of Inertial Mass and Gravitational Mass

by Ron Kurtus (revised 17 March 2018)

One of the Gravitational Equivalence Principles is the * equivalence of inertial mass and gravitational mass*. This principle states that mass determined by inertia is the same as mass determined by gravitation. You can see this by examining the forces from both inertial mass and gravitational mass.

Questions you may have include:

- How is mass determined by inertia?
- How is mass determined by gravitation?
- What is the equivalence of inertial and gravitational mass?

This lesson will answer those questions. Useful tool: Units Conversion

## Mass determined by inertia

The mass of an object can be determined by its inertia or resistance to acceleration.

F_{I}= m_{I}a

where

**F**is the force needed to overcome inertia_{I}**m**is the inertial mass_{I}**a**is the acceleration on the object

Thus, the inertial mass is:

m_{I}= F_{I}/a

This comes from *Newton's Second Law of Motion*, which states: "The acceleration of an object of constant mass is proportional to the force acting upon it." In other words, the inertial mass is related to the force and its acceleration.

(See Newton's Laws of Motion for more information.)

## Gravitational mass

The mass of an object can be determined from the gravitation force on it from another mass. The gravitational force between two objects is:

F_{G}= Gm_{G}M/R^{2}

and the gravitational mass of one object is:

m_{G}= F_{G}R^{2}/GM

where

**F**is the gravitational force on the object_{G}**G**is the Universal Gravitational Constant**m**is the gravitational mass of the object_{G}**M**is the mass of the larger attracting object**R**is the separation between the objects, as measured from their centers of mass

Note that **GM/R ^{2}** is the acceleration of

**m**toward

_{G}**M**. Thus, when

**R**is the radius of the Earth:

GM/R^{2}= g

where **g** is the acceleration due to gravity.

## Equivalence

Since the time of Newton, scientists have wondered if the inertial mass (**m _{i}**) was the same as the gravitational mass (

**m**). Many experiments verify the equivalence.

_{g}Albert Einstein stated that a gravitational force, as experienced locally while on a massive body such as the Earth, is actually the same as the pseudo-force experienced by an observer in an accelerated frame of reference.

Einstein used the equivalence of inertial and gravitational mass as a basic framework for the *General Theory of Relativity*.

Since **F _{I} = m_{I}a** and

**F**, experiments show that

_{G}= Gm_{G}M/R^{2}**F**, and thus

_{i}= F_{g}**m**.

_{I}= m_{G}## Summary

The *equivalence of inertial and gravitational mass* states that mass determined by inertia is the same as mass determined by gravitation. This has been proven by experiments.

Physics is amazing

## Resources and references

### Websites

**Equivalence Principle** - Laboratory Tests of Gravitational Physics

**Equivalence principle** - Wikipedia

**Elevator, rocket, and gravity: the equivalence principle** - Einstein online

### Books

**Top-rated books on Gravitation**

## Questions and comments

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## Equivalence of Inertial Mass and Gravitational Mass