Gravitation Near Earth's Surface

[slayer16]

Homework Statement

The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator just barely proved the centripetal force needed for the rotation. a) show that the corresponding shortest period of rotation is

T=(√3π)/(√Gp)

where p is the uniform density of the spherical planet. b) Calculate the rotation period assuming a density of 3 g/cm^3.

Homework Equations

Newton's law of gravitation.
Equation for centripetal acceleration and force.
Equation for period.

The Attempt at a Solution

I do not understand the first sentence really well. If someone could elucidate the first question, so I can understand, then I might be able to tackle the problem.

 

[collinsmark]

The I think the statement is saying, if a planet spins fast enough it will start to fling off material into space. This is because the centripetal force needed to keep material attached to the planet (at its equator) exceeds the gravitational force (i.e. there isn't enough gravity to hold onto the material at the equator, so it is flung out into space). So in order for the planet to keep its material, the centripetal acceleration at a point on the planet's equator must be less than or equal to the gravitational acceleration. The problem is asking you to solve the for case where they are equal. The solution is:

$$T = \sqrt{\frac{3 \pi}{G \rho}}$$

(It's asking you to derive this formula.) I believe the formula is correct for a perfectly spherical planet.

[Edit: Or another way to phrase the problem, how fast would a planet have to spin such that your would feel weightlessness merely by being on the equator?]