# Moment of inertia question

Homework Statement

Consider a rigid body with an inertia tensor I =[30, 0, 0; 0, 40, 0; 0, 0, 20] N m s^2 and angular velocity w=10j+10k rad/s. Determine the moment of inertia about an axis parallel to w and find the rotational kinetic energy.

The attempt at a solution

I'm not sure if the fact that this is the principal moment of inertia matters. Also, I thought this had something to do with the parallel axis theorem, but neither mass nor any distances are given. Any help would be appreciated.

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marcusl
Gold Member
The fact that you have I along the principal axes will simplify the calculations but is not otherwise important.

Hint: look at how I transforms under coordinate transformation (i.e., rotation).

You will want to solve for the scalar form of I in the direction of w. So you will need to convert w to a unit vector and matrix multiply it on both sides of I to get a scalar value:

$$I = \hat{\omega}^T \bar{I} \hat{\omega}$$

Knowing the scalar form for I and the magnitude of angular velocity you can simply get the rotational kinetic energy.