Problem — True or false about moment of inertia

Problem

State whether the following statements are true or false, justifying in one line: (a) the moment of inertia of a body depends on the chosen axis of rotation; (b) in a rigid body in pure rotation about a fixed axis, all points have the same velocity; (c) the angular momentum of an isolated system is conserved even if internal forces change the moment of inertia; (d) rolling without slipping implies $v_\text{CM} = \omega R$.

Solution

(a) True. $I = \sum m_i r_i^2$ depends on the distances $r_i$ from the axis: changing the axis changes the distances and hence $I$.

(b) False. All points have the same angular velocity $\omega$, but different linear velocities: $v = \omega r$ increases with distance from the axis. The edge moves faster than the centre.

(c) True. If there are no external torques, $L$ is conserved. Internal forces can change $I$ and $\omega$ (as with the skater), but the product $L = I\omega$ stays fixed.

(d) True. This is the definition of the rolling constraint: the contact point does not slip, so the velocity of the centre is linked to the angular velocity by $v_\text{CM} = \omega R$.

$\ev{\text{(a) T}\quad\text{(b) F}\quad\text{(c) T}\quad\text{(d) T}}$

Links

Topics: Rotational dynamics Concepts: Moment of inertia · Angular velocity · Conservation of angular momentum · Rolling