# Gravitational potential energy and attraction

Gravitational potential energy usually arises due to the force of attraction experienced by a body. When a small body is placed between to massive bodies (not of equal masses) it is attracted by both. So when we consider the small body’s gravitational potential energy should we take the sum of the potential energies due to the two massive bodies or their difference? I get this question because the forces of attraction on the small body are in the opposite directions

## Answers and Replies

Doc Al
Mentor
Potential energy is a scalar, so you just add them.

Hang on, P.E is the work done against the gravitational field to take your test particle there against the field, once you choose your coordinates the other mass is working against the Gravitational field, hence you subtract them not add them.

In a sense you are right - you do "add them", however only once you've carefully defined your coordinate system such that the P.E components from each will be of different signs.

The sea on the earth is a perfect example of this. when the moon gets close to it its gravitational P.E decreases, hence it rises (less effective pull from the earth).

Please someone correct me if I'm wrong.

Doc Al
Mentor
Hang on, P.E is the work done against the gravitational field to take your test particle there against the field, once you choose your coordinates the other mass is working against the Gravitational field, hence you subtract them not add them.

In a sense you are right - you do "add them", however only once you've carefully defined your coordinate system such that the P.E components from each will be of different signs.
The gravitational PE between two masses is given by:

$${PE} = - \frac{Gm_1 m_2}{r}$$

Where r is the distance between them. (Note that when they are infinitely far apart the PE is taken to be 0.)

If you have two massive bodies, M1 & M2, the change in PE of the system when you bring a small mass close to them is:

$${PE} = (-\frac{GM_1 m}{r_1}) + (-\frac{GM_2 m}{r_2})$$

The sea on the earth is a perfect example of this. when the moon gets close to it its gravitational P.E decreases, hence it rises (less effective pull from the earth).
The reason for the rising of the sea is the difference in the moon's gravitational field strength acting on the sea compared to the earth.

I've thought about it and you're right, energy is a scalar and position does not come into it, the test particle being between the two massive objects is just a special case of the more general. Thanks for the clarification