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

a_c=\frac{v^2}{r},\quad F_g=\frac{GMm}{r^2}

problems.proof.analysis

Uniform circular motion requires inward acceleration $v^2/r$ . Gravity $GMm/r^2$ points inward and is the only centripetal force in this model.

problems.proof.pitfall. Do not add a separate centripetal force; gravity provides it.