Period of falling through asteroid vs orbit

1. The problem statement, all variables and given/known data

2. Relevant equations

See below

3. 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?

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