Question

you're standing on a small spherical asteroid that has a density of 2700 kg/m^3. you throw a baseball horizontally with a speed of 22 m/s and the ball orbits the asteroids in uniform circular motion. what is the asteroid's radius and what is the ball's period?

Answer 1

\[ \rho = 2700 kg/m^3\]
\[M=\rho V=\rho \frac{4}{3} \pi r^3\]
\[g_{*roid} = G\frac{M}{r^2}\]
\[ \sum F_{ball} = mg_{*roid}=ma_{centripetal}\]
\[g_{*roid}=a_{centripetal}\]
\[G\frac{M}{r^2}=\frac{v^2}{r}\]
\[G\frac{\rho \frac{4}{3} \pi r^3}{r^2}=\frac{v^2}{r}\]
solve for r (in terms of rho, G, pi, and v to make life simpler in the next part)

To find the period, use Kepler's third law for spherical orbits

\[ \frac{4\pi^2}{T^2} = \frac{GM}{r^3}\]


If you do it algebraically, keeping all of the constants and variables as letters, your solution will be prettier ^_^ (plus there won't be as many rounding approximations! ^^)

Answer 2

thank you so much