Question

Imagine a universe in which the law of gravitation has the inverse - first - power form

$\overrightarrow{F}=\frac{GMm}{r}$

(a) Show that the speed of a planet in a circular orbit is independent of the orbit radius. (b) Show that in such a universe Kelper's third law would take the form
$\frac{T_1}{T_2}=\frac{r_1}{r_2}.$

Answer 1

Let $M$ is a star and $m$ is a planet

(a) Now, $F=\frac{GMm}{r}=\frac{m{v}^2}{r}$

$\therefore v=\sqrt{GM}$

So, $v$ is independent of $r$

(b) Now, $T=\frac{2\pi r}{v}=\frac{2\pi r}{\sqrt{GM}}$

So, $T\propto r$

$\therefore \frac{T_1}{T_2}=\frac{r_1}{r_2}$