# Weak Equivalence Principle of Gravitation

by Ron Kurtus (updated 30 May 2023)

The Weak Equivalence Principle of Gravitation states that objects fall at the same rate, provided that are freely falling. There are several restrictions on the principle. The proof of this principle is pretty straightforward.

Questions you may have include:

• What is the Weak Equivalence Principle?
• What are its restrictions?
• What is the proof of that principle?

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

## Weak Equivalence Principle

The Weak Equivalence Principle (also called the Uniqueness of Free Fall Principle) states that gravitation causes objects to fall or move toward an attracting body at the same rate.

Objects fall at the same rate

In other words, objects dropped from the same height will travel at the same rate, irrespective of their masses.

## Restrictions

However, there are some restrictions on this principle.

### No outside forces

It is assumed that there are no outside forces such as air resistance acting on the falling objects. In other words, they are falling freely.

### Mass of each is much less than attracting body

A major restriction on the Weak Equivalence Principle is that the mass of each falling object must be much less than that of the attracting body.

The gravitational force causes both the falling object and the attracting body to move toward each other and their center of mass. Thus, the mass of the falling object much be so small with respect to the attracting body that the movement of the center of mass is negligible.

### Objects must be of similar size

Another restriction is that the objects must be similar in physical size, such that the center of mass for each is at approximately the same displacement from the attracting body. If the separations between the centers of mass are different, the objects would fall at slightly different rates.

In the illustration below, the distances to the large attracting body (such as the Earth), R1 and R2 are different enough to affect the rate of motion:

Exception is when objects are much different in size

This exception is seldom considered when studying the principle.

## Proof of principle

The proof of this principle is pretty straightforward.

Consider two objects that are approximately the same distance from a larger body. The Universal Gravitation Equation states:

F1 = Gm1M/R2

F2 = Gm2M/R2

where

• F1 and F2 are the forces on objects 1 and 2 respectively
• G is the Universal Gravitational Constant
• m1 and m2 are the masses of objects 1 and 2 respectively
• M is the mass of the attracting body
• R is the separation from the centers of the objects to the center of the attracting body

Since the force on an object is mass times acceleration, the equations reduce to:

F1 = m1a1 = Gm1M/R2

F2 = m2a2 = Gm2M/R2

thus:

m1a1 = Gm1M/R2

a1 = GM/R2

and

m2a2 = Gm2M/R2

a2 = GM/R2

(therefore)

a1 = a2

The acceleration for both objects is the same, and they will fall at the same rate.

## Summary

The Weak Equivalence Principle states that objects fall at the same rate, provided they are much smaller than the attracting body and are freely falling.

Physics is amazing

## Resources and references

Ron Kurtus' Credentials

### Websites

Equivalence Principle - Laboratory Tests of Gravitational Physics

Equivalence principle - Wikipedia

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

Gravitation Resources

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www.school-for-champions.com/science/
gravitation_equivalence_principle_weak.htm

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