Problem 1

A satellite of mass

m

moves uniformly in a circular orbit of radius

r

measured from the center of an isolated, spherically symmetric planet of mass

M

. Neglect drag and the satellite's effect on the planet;

G

is the gravitational constant. (a) Identify the force and state the condition for a circular orbit. (b) Derive the orbital speed

v

and period

T

. (c) Check their dependence on

m

and their dimensions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{v=\sqrt{\frac{GM}{r}},\quad T=2\pi\sqrt{\frac{r^3}{GM}},\quad v,T\text{ independent of }m,\quad [v]=\mathrm{m\,s^{-1}},\ [T]=\mathrm{s}}

problems.proof.analysis

The boxed expressions give the orbital speed and period, respectively; neither depends on satellite mass $m$ . Their dimensions are velocity and time, as required.