Angular Speed About Instantaneous Axis Through Mass Center

To determine the correct statement regarding the angular speed about an axis passing through the centre of mass, we utilize fundamental principles of rigid body kinematics:

1. Kinematics of a Rigid Body: For any two points $P$ and $Q$ on a rigid body, the velocity of point $P$ relative to point $Q$ is expressed as: $\vec{v}_P = \vec{v}_Q + \vec{\omega} \times \vec{r}_{P/Q}$ where $\vec{\omega}$ represents the angular velocity vector of the rigid body and $\vec{r}_{P/Q}$ is the position vector of point $P$ relative to point $Q$.

2. Uniqueness and Invariance of Angular Velocity: The angular velocity vector $\vec{\omega}$ is an intrinsic, state-defining vector property of the entire rigid body at a given instant. It is independent of the choice of reference point or coordinate axis chosen to describe the motion (whether it is calculated about the Instantaneous Axis of Rotation, the Center of Mass, or any other point).

3. Application to the Problem: Since $\vec{\omega}$ is unique for a rigid body at any instant regardless of the axis chosen, the magnitude of the angular velocity about an axis passing through the centre of mass is equal to the magnitude of the body's angular speed, which is $\omega$, in both cases.

Thus, the angular speed about the instantaneous axis passing through the centre of mass is $\omega$ for both cases.

Correct Option: D