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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 4.10 Gravitational Potential Energy

Definition 4.10.1. Gravitational Potential Energy (General).

For a system with a small massive object (mass \(m\)) and a large massive object (mass \(M\)) 1 , the gravitational potential energy is
\begin{equation*} U_{grav} = -\frac{GmM}{r} \end{equation*}
where \(r\) is the distance between the centers of the two objects 2 .

Exercises Activities

1. Explanation: Zero.

When is the gravitational potential energy equal to zero?

2. Approximation: Near the Surface of the Earth.

When you are near the surface of the Earth, you can model the gravitational force as \(\vec{F}^g = mg (-\hat{y})\text{.}\) Use this to find the change in gravitational potential energy when an object goes from \(y_i = 0\) to \(y_f = h\text{.}\)
Tip.
Use the definition of potential energy as an integral. Remember to include the minus sign!
Answer.
Definition 4.10.2. Gravitational Potential Energy near the Surface of the Earth.
For a system with a small massive object (mass \(m\)) close to the surface of the Earth, the gravitational potential energy is \(U_{grav} = mgy\text{,}\) where \(y\) is the vertical distance from the location where \(U_{grav} = 0\text{.}\)
the mass of the large object must be distributed spherically
here \(r\) must be greater than the radius of the large massive object
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