Energy of a solid rotating sphere

[wduff]

Homework Statement

A system of particles, forming a sphere of uniform mass density $$\rho$$ and radius R, rotates around the axis of the sphere with angular velocity $$omega$$(t) calculate the energy of the system.

Homework Equations

we were told to solve the problem with this integral:

E = $$\int$$ $$\rho$$ v² (r) d³r

The Attempt at a Solution

I took $$\rho$$ = M/( 4/3 $$\pi$$ R³)

v³ = r² $$\omega$$²

and d³ r = r^2 sin $$\theta$$ dr d$$\phi$$ d$$\theta$$

integrating from 0 to R for r, from 0 to pi for theta, and from 0 to 2 pi for phi, I got

3/5 M R^2 $$\omega$$^2

I think the answer ought to be 1/5 MR^2 w^2, right? Thanks for the help.

 

[tiny-tim]

Welcome to PF!

Hi wduff! Welcome to PF!

(have a pi: π and an omega: ω and a theta: θ and a phi: φ and a rho: ρ and try using the X² tag just above the Reply box )

You've used v = ωr, but it's not the same r, is it?

(and it's 2/5 … see http://en.wikipedia.org/wiki/List_of_moments_of_inertia)

 

[wduff]

Aha! Thanks a ton tiny tim! And yeah sorry about the terrible formatting I'm sure I'll get the hang of it.