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# Gravitational Potential Energy

The gravitational force is conservative. We can see that it only depends on the end points by considering the work it takes to go between two points

 (1.16)

points in the radial direction, that is in the negative direction. The dot product . But is the component of in the radial direction. Call it . So

 (1.17)

In other words, the integral only depends on the radial coordinate , and is therefore independent of any meanderings at different angles, that is it's independent of the path.

From the definition of potential energy, we know that the potential energy is defined in relation to some reference point, say at radius . Let's set the potential at radius equal to zero so

 (1.18)

But point towards the center so it is

 (1.19)

Plugging this in we have
 (1.20)

To make life simple lets get rid of the first term by starting at infinity, so that . Then
 (1.21)

So after all this, the final form for the potential energy is pretty simple. The potential energy decreases as the two objects get closer together. It is inversely proportional to , unlike the force which is inversely propotional to . It is shown in figure 1.5.

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Next: escape velocity Up: Gravity Previous: solution
Joshua Deutsch 2003-03-05