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 \omega²

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.