# 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?

Apr 22, 2017

For this set up, we get the potential gravitational energy $\mathrm{dU}$ of the system, ie interaction between:

• small element of width $\mathrm{dx}$ and mass $\mathrm{dM}$; and

• the mass m.

$\mathrm{dU} = - \frac{G \setminus m \setminus \mathrm{dM}}{a + x}$

And $\mathrm{dM} = \frac{M}{L} \mathrm{dx}$

So $U = - \frac{G m M}{L} {\int}_{x = 0}^{L} \frac{\mathrm{dx}}{a + x}$

$= - \frac{G m M}{L} \ln \left(1 + \frac{L}{a}\right)$

With a point mass assumption and the same set up:

$U = - \frac{G m M}{a + \frac{L}{2}}$

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