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## Homework Statement

## Homework Equations

See below

## The Attempt at a Solution

My book derives the period of oscillation through the tunnel:

T = 2pi/w = 2pi*sqrt(m/k) = 2pi*sqrt(3/(4pi*G*p)) = sqrt(3pi/(G*p))

Where p is the density of the asteroid, and G is the Newton's gravitational constant.

I know that the orbit velocity is found by equating the gravity force to the necessary centripetal force:

mg = mv^2/r

v = sqrt(r*g)

So the period is 2*pi*r/v = 2pi*sqrt(r/g)

I know these two periods are equal. Can anyone help me with putting them and similar terms and proving that they are?