Physics GRE - #1

It is possible that the Newtonian theory of gravitation may need to be modified at short range. Suppose that the potential energy between two masses $m$ and $m'$ is given by $$V(r)=-\frac{Gmm'}{r}(1-ae^{-\frac{r}{\lambda}})$$
For short distances $r\ll \lambda$ calculate the force between $m$ and $m'$.

• $F=-\frac{Gmm'}{r^2}$
• $F=-\frac{Gmm'}{r^2}(1-a)$
• $F=-\frac{Gmm'}{r^2}(1+a)$
• $F=-\frac{Gmm'}{\lambda r}$

To get the force, we take the negative derivative of the potential function, $-V^\prime(r)$. $$\begin{eqnarray*} -V^\prime(r) & = & -\frac{Gmm'}{r^2}(1-ae^{-r/\lambda})-\frac{Gmm'}{r}(ae^{-r/\lambda}) \\                      & = & -\frac{Gmm'}{r^2}\left(1-ae^{-r/\lambda}\left(1+\frac{r}{\lambda}\right)\right) \end{eqnarray*}$$
When $r\ll \lambda$, we have $$F(r)\approx -\frac{Gmm'}{r^2}(1-a).$$