The third term in the expression for $\dot{I}{ij}\omega_j$(where $\dot{I}{ij}$ is the time derivative of the inertia tensor contracted with the angular velocity $\omega_j$) vanishes because the integral it contains is the first moment of mass, and the problem specifies that the rigid body is rotating about its center of mass.

Here is the breakdown of the vanishing term and the physical reason:

1. The Third Term

From Step 3 of the proof on the web page, the third term in the expanded expression for $\dot{I}_{ij}\omega_j$ is:

$$ \text{Third Term} = -\int_V \rho x_i \epsilon_{jpq} \omega_p x_q \omega_j dV $$

This term is manipulated to separate the angular velocity and position components:

$$ \text{Third Term} = -x_i \epsilon_{jpq} \omega_p \omega_j \int_V \rho x_q dV $$

2. The Vanishing Integral

The integral part of this term is:

$$ \int_V \rho x_q dV $$

• $\rho$ is the mass density.
• $dV$ is the infinitesimal volume element.
• $x_q$ is the $q$-th component of the position vector $\vec{x}$.

This integral, $\int_V \rho \vec{x} dV$, represents the first moment of mass (or static moment) of the body.

3. The Physical Reason

The integral $\int_V \rho x_q dV$ is related to the center of mass ($\vec{R}$) by the definition:

$$ M\vec{R} = \int_V \rho \vec{x} dV $$

where $M$ is the total mass of the rigid body.

The proof explicitly states that the rotation is about the center of mass (CM). If the origin of the coordinate system is chosen to be the center of mass, then the position vector of the center of mass is $\vec{R} = 0$.

Therefore, since the rotation is about the center of mass: