Gravitational Force Between Two Objects

by Ron Kurtus (updated 30 May 2023)

You can find the gravitational force between two objects by applying the Universal Gravitation Equation, provided you know the mass of each object and their separation.

With this equation, you can make calculations to determine such things as the force between the Earth and the Moon, as well as between two large masses.

Questions you may have include:

What is the Universal Gravitational Equation?
What is the force of attraction between the Earth and the Moon?
What is the force of attraction between two other objects?

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

Universal Gravitation Equation

The Universal Gravitation Equation is:

F = GMm/R²

where

F is the force of attraction between two objects in newtons (N)
G is the Universal Gravitational Constant = 6.674*10⁻¹¹ N-m²/kg²
M and m are the masses of the two objects in kilograms (kg)
R is the separation in meters (m) between the objects, as measured from their centers of mass

Force attracting Earth and Moon

To calculate the gravitational force pulling the Earth and Moon together, you need to know their separation and the mass of each object.

Distance

The Earth and Moon are approximately an average of 3.844*10⁵ kilometers apart, center to center.

(Note that the orbit of the Moon around the Earth is not a true circle, so an average separation is used. This also means that the force of attraction varies.)

Since the units of G are in N-m²/kg², you need to convert the units of R to meters.

R = 3.844*10⁸ m

Mass of each object

Let M be the mass of the Earth and m the mass of the Moon.

M = 5.974*10²⁴ kg

m = 7.349*10²² kg

Force of attraction

Thus, the force of attraction between the Earth and Moon is:

F = GMm/R²

F = (6.674*10⁻¹¹ N-m²/kg²)(5.974*10²⁴ kg)(7.349*10²²kg)/(3.844*10⁸ m)²

F = (2.930*10³⁷N-m²)/(1.478*10¹⁷ m²)

F = 1.982*10²⁰ N

Note: Notice how all the units, except N, canceled out.

Attraction between Earth and Moon

Result of force

This considerable force is what holds the Moon in orbit around the Earth and prevents it from flying off into space. Inward force of gravitation equals the outward centrifugal force from the motion of the Moon.

(See Circular Planetary Orbits for more information.)

Also, the gravitational force from the Moon pulls the oceans toward it, causing the rising and falling tides, according to the Moon's position.

(See Gravitation Causes Tides on Earth for more information.)

Force attracting large objects

In the same manner, you can calculate the gravitational force attracting two large objects.

Suppose you had an object with mass of 100 kg, another with a mass of 200 kg, and the separation of their centers was 2 meters.

F = GMm/R²

F = (6.674*10⁻¹¹ N-m²/kg²)(100 kg)(200kg)/(2 m)²

F = 33370*10⁻¹¹N

Simplify:

F = 3.3*10⁻⁷N

That is a very small force attracting the objects together. However, even a smaller force was measured in the Cavendish Experiment to Measure Gravitational Constant.

Summary

You can apply the Universal Gravitation Equation to show the force of attraction between two objects, such as the force between the Earth and the Moon, as well as between two large masses.

Think clearly and logically