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

T=\frac{2\pi r}{v}=2\pi\sqrt{\frac{r^3}{GM}}

problems.proof.analysis

One revolution covers $2\pi r$ , so $T=2\pi r/v$ . Substitute the speed just derived and simplify to obtain the period.