How do I find the period of a planet's rotation on its axis given ?

[UniBoy5]

Homework Statement

Consider a planet with uniform mass density p. If the planet rotates too fast, it will fly apart. Show that the minimum period of rotation is given by

T^2 = 3(pie)/Gp

Homework Equations

F = ma = Gm1m2/R^2 (Equation 1)

a = v^2/R (Equation 2)

v= 2(pie)R/T (Equation 3)

m= (4/3(pie)R^3)p

The Attempt at a Solution

I tried putting equation 2 into equation 1. I only included the mass of the planet (m). I don't know if this is right. After finding v, I solved for T^2. My answer was not correct. Please help, thanks!

 

[mgb_phys]

Slightly the wrong equation.
The planet will fly apart when the centrifugal force at the equator is equal to the gravity.
what's the equation for the outward force on a spinning object?

 

[UniBoy5]

No idea... that's why I'm asking for help!

 

[UniBoy5]

I don't see the need for using centrifugal forces since the situation is assumed to being viewed in an inertial frame.

 

[mgb_phys]

ignoring all the hair splitting stuff about centrifugal force vs centripetal accelration

There is an equation in your textbook or on google involving centrifugal force and rotation rate,
this is what is going to cause the planet to fly apart.
At the point it does this - this force is equal to gravity.
so set the two equations equal and solve exactly as you did above,

You are almost correct, it's just your definition of 'a' in f=ma that's wrong.
Actually since the equation must involve only rotation rate and radius and have the units of 'a' - you could guess it.