Gravitational Potential

Gravitational Potential

Recall the idea of a potential energy E = mgh, of a body of mass m and at height h above the ground. This is the work done to bring the body from ground to the height. It is useful because we can use it to calculate the kinetic energy if it falls to the ground. Since this is just equal to the potential energy, it is easier than using the acceleration equations to find the velocity first, and then using mv².

Another feature of the potential energy mgh, is that a body tends to move from higher to lower potential energy, or that the weight points from higher to lower potential energy. These are just abstract ways to say that a body that is higher up falls down.

The above features of potential energy seem rather obvious. Consider now what happens if this height h is not just a few metres, but a few times the radius of the Earth. The weight no longer remains the same. What is the potential energy then? Is the idea useful for this more difficult case?

To answer these questions, lets see what the potential energy should be at a point P, at distance r from a sphere of mass M.

If we think of the sphere as the Earth, we may try and use the idea for mgh. The the potential energy could be the work done to bring a body of mass m from the surface of the sphere, to P. What if another sphere of the same mass has a different radius? We expect the same gravitational field strength at P, but the potential energy would be different.

The usual way to define the potential energy at P, is the work done to bring the body from very far away, to P. "Very far away" here means so far that the gravitational field strength from M is practically zero. That seems like a really strange definition. It does make things simpler if the sphere has a different radius but the same mass - then the potential energy at P is the same.

"Very far away" is often called infinity. We shall see on another page that such a strange definition does not really make calculations any more difficult than if we start from the surface of the sphere, once we get used to a few rules.

Recall that gravitational field strength is a convenient quantity, because it gives the force per unit mass. It depends on the field rather than the body in the field, unlike the force. So it gives a direct measure of the strength of the field.

Likewise, it is convenient to think of the potential energy per unit mass. This also does not depend on the mass of a body in the field. So it tells us something about the field itself. This quantity is called the "potential," without the "energy."

Summarising, the potential at a point P, is the work done to bring unit mass from infinity to P.

Likewise, the potential energy at a point P, is the work done to bring unit mass from infinity to P.

The Physics Notes