Question

The potential energy of gravitational interaction of a point mass m and a thin uniform rod of mass M and length L ,if they are located along the straight line at a distance 'a' from each other?

Answer 1

For this set up, we get the potential gravitational energy `dU` of the system, ie interaction between:

• small element of width `dx` and mass `dM`; and

• the mass m.

`dU = - (G \ m \ dM)/( a+ x)`

And `dM = M/L dx`

So `U = - (G m M)/L int_(x = 0)^L dx/( a+ x)`

`= -(G m M)/L ln (1+ L/a)`

With a point mass assumption and the same set up:

`U = - (GmM)/(a + L/2)`

They don't look that similar but if you play around with expansions or plot they do look alike.