### Exercise 2.1: Rotational kinetic energy

**Show that the rotational kinetic energy can also be written as:**

**where $I$ is the moment of inertia about the line through the center of mass with direction $\hat{\omega}$ and $\omega$ is the instantaneous rate of rotation.**

$$\require{cancel}$$

Let $\hat{\omega}$ be one of the basis vectors of the coordinate system, $\hat{e_0}$, and with its origin at the center of mass of the body. For a constituent particle in the body, $(\xi_\alpha^1)^2 + (\xi_\alpha^2)^2 = \xi_\alpha^\perp$ will equal to the distance from the particle to the line $\hat{e_0}$ which is the same as the line $\hat{\omega}$. Therefore, the inertia component, $I_{00}$ is:

where $I$ is the moment of inertia about the line $\hat{\omega}$. Also, by definition, the components of the angular velocity vector, $\vec{\omega}$, are:

since we defined $\vec{\omega}$ as being in the direction $\hat{\omega} = \hat{e}_0$. Therefore the kinetic energy is:

All the other terms in the expression for $T$ cancel out as the components $\omega^1$ and $\omega^2$ are equal to zero. Therefore,