Gravitational potential energy – concepts & equations when reference varies from the planet’s surface to infinity

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Gravitational potential energy, Ep, is the energy of a mass due to its position within a gravitational field. Here on Earth, the Ep of an object at some point, x, above the ground is easily found as it is equal to the work done to move the object from the ground up to the point, x, as shown in figure 1.

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The gravitational potential energy equation derivation when reference point is the surface of the earth

Gravitational potential energy Ep at some point x above the ground
= work done to move to the point x with the vertical height h from the ground
= force required × distance moved (since work W = Fr)
Gravitational potential energy Ep = (mg) × h = mgh

Gravitational potential energy
fig 1: Gravitational potential energy

Hence, in this case, Ep = mgh. Here, We chose the ground as our starting point because this is our defined zero level; that is, the place where Ep = 0. Note that since work must be done on the object to lift it, it acquires energy.
Hence, at point x, Ep is greater than zero.

The gravitational potential energy on a larger, planetary scale when reference point is at infinity

On a larger, planetary-scale we need to rethink our approach.

Due to the inverse square relationship in the Law of Universal Gravitation, the force of attraction between a planet and an object will drop to zero only at an infinite distance from the planet.

For this reason, we will now choose infinity (or some point a very large distance away) as our level of zero potential energy.

There is a strange result of our choice of zero level.

Because gravitation is a force of attraction, work must be done on the object to move it from a point, x, to infinity; that is, against the field so that it gains energy, Ep. That means, at infinity the potential energy becomes the largest.

Therefore, Ep at infinity > Ep at point x
but Ep at infinity = 0
so that Ep at point x < 0
that is, Ep at point x has a negative value! (see figure 2)

figure 2: gravitational potential energy Ep for a planet
figure 2: gravitational potential energy Ep for a planet

Hence we can say, If we choose a planet’s surface as the zero level, Gravitational Potential Energy Ep at x has a positive value. If infinity is chosen as the zero level, Ep has a negative value.

Using the same approach as earlier, the gravitational potential energy, Ep, of an object at a point, x, in a gravitational field is equal to the work done to move the object from the zero energy level at infinity (or some point very far away) to point x. It can be shown mathematically that: Ep = – G m1m2 / r Where m1 is the mass of the planet, and m2 is the mass of the object. r is the distance between the planet and the object.

We have seen how gravitational potential energy Ep can be expressed as Ep = – G m1m2 / r

Take Away | Summary

1] If we choose a planet’s surface as the zero level or reference, Gravitational Potential Energy Ep at x has a positive value.
If infinity is chosen as the zero level, Ep has a negative value.

2] Gravitational potential energy Ep = (mg) × h = mgh (if we choose a planet’s surface as the zero level or reference)

3] Gravitational potential energy Ep can be expressed as Ep = – G m1m2 / r (if infinity is chosen as the zero level)

Gravitational potential energy – concepts & equations when reference varies from the planet’s surface to infinity

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