Gravitational potential energy of planet

[ACLerok]

Find U, the gravitational potential energy of the object at a distance R from the center of the planet, with the gravitational potential energy taken to be zero when the object is infinitely far away from the planet. Express your answer in terms of R, m, M, and G, the universal gravitational constant.

 

[FZ+]

Pretty straightforward.

Write out the equation for gravitational force.

Integrate wrt distance, putting in correct limits.

 

[M98Ranger]

The gravitational potential energy of an object at a distance R from the center of a planet can be calculated using the formula U = -(GmM)/R, where G is the universal gravitational constant, m is the mass of the object, M is the mass of the planet, and R is the distance between the object and the center of the planet.

In this formula, the negative sign indicates that the gravitational potential energy is a negative value, meaning that it decreases as the distance between the object and the planet increases. This makes sense intuitively, as the farther away an object is from a planet, the less gravitational potential energy it possesses.

Using this formula, we can see that when the object is infinitely far away from the planet (R = ∞), the gravitational potential energy becomes zero. This is because as R approaches infinity, the fraction (GmM)/R approaches zero, resulting in a gravitational potential energy of zero.

Therefore, the expression for the gravitational potential energy of an object at a distance R from the center of a planet, with the gravitational potential energy taken to be zero when the object is infinitely far away, is U = -(GmM)/R.