# Kinetic energy of Rotating Body (in 3D)

1. Dec 24, 2011

### Curl

Ok here is what I did:

$$T=\int \vec{\tau} \cdot d\vec{\theta } = \int\frac{\mathrm{d} \vec{L}}{\mathrm{d} t}\cdot d\vec{\theta } = \int \vec{L}\cdot d\vec{\omega} = \int I_{ij}\vec{\omega}\cdot d\vec{\omega}$$
Where I is the inertia matrix.

when I carry that out, I get:
T= 0.5*(I11wx2+I22wy2+I33wz2) + 2*(I12wxwy+I13wxwz+I23wzwy)

but the real answer has no factor of 2 in the second term. What's wrong?