Question

MEDAL and FAN
"While you and your young cousin are sitting in the backyard looking up in the night sky, your cousin states, "If the moon were half as big it wouldn't have to move around the earth as fast to stay the same distance away."
True OR False? and why?
I just need help...

Answer 1

The question sounds tricky but it isn't! The moon is attracted to the Earth because of gravity. Assuming the moon is moving in a perfectly circular path, the force of gravity on the moon by the Earth is also the centripetal force acting on the moon. In other words:

\[F_c=F_G\]\[m \omega^2 r=\frac{GMm}{r^2}\]
In this equation, the small m is the mass of the moon, while the large M is the mass of the Earth. The angular speed of the moon is ω, and the distance between the moon and the Earth is r. If we rearrange this equation to solve for the angular speed, we can see how the angular speed will change as you manipulate other variables:

\[\omega=\sqrt{\frac{GM}{r^3}}\]
You can see that the mass of the moon reduces out of the equation completely! In other words, the angular speed of the moon DOES NOT depend on the mass of the moon - it only depends on the mass of the Earth and the distance between the moon and the Earth.

Does that make sense?

Answer 2

Yes it does!! thank you!!(: @matt101