Magnetic Potential Energy

In this lab, our purpose is finding an equation for magnetic potential energy.
As the picture shows, we set a cart on a air track so that there was not friction between cart and track.

Because the potential energy is caused by a force F for any system that has a non-constant PE, we made a relationship:

The r is the distance between cart and metal.

For making the relationship more clear, we need to apply different forces on cart and check each r. For measuring r, it is easy to do with ruler, but for the force applying on the cart, we need to use other approach. We use the gravity to apply. Because of the equation, F= mgsinx, we could find the angle and the mass of the cart, so that we can get the force. And then, we made a graph: 

From the graph, we can derive the equation:

and then, we could use integral to get the PE.