Kevin S. Huang

2009: Problems 21-22

Topic: Gravity
Concepts: Gravitational potential energy

Solution:

Recall the gravitational potential energy between two masses $M$ and $m$ separated by distance $r$ is

U=-\frac{GMm}{r}

In our case, we have

U=-\frac{3GM^2}{d}

so the answer is D.

Topic: Gravity
Concepts: Center of mass, Circular orbits

Solution:

The gravitational force $F$ on $3M$ is given by

F=\frac{G(3M)M}{d^2}

This provides the centripetal acceleration so

F=(3M)a_c=3M\omega^2r

Since the stars are orbiting their center of mass, $r=d/4$ for $3M$ . Using $\omega=2\pi/T$ , we have

\frac{3GM^2}{d^2}=3M\left(\frac{2\pi}{T}\right)^2\frac{d}{4}

\frac{4GM}{d^3}=\frac{4\pi^2}{T^2}

T=\pi\sqrt{\frac{d^3}{GM}}

so the answer is A.