A large bug of mass 5.0 g lands on the outside edge of a DVD*. The DVD has mass 9.0 g and radius 6.0 cm.
*DVDs are still a thing, right? Or, at least I expect that most of my 15-18 year old students know what a DVD is without further explanation. Or, I'm an old man.
I use this setup to introduce newton's second law for rotation, and the additive nature of rotational inertia... and then to discuss conservation of angular momentum.
(a) Does the bug's presence significantly affect the rotational inertia of the DVD?
By itself, the DVD is a disk, with rotational inertia (1/2)MR2. That gives 160 g*cm2 as the disk's inertia.
The bug adds its rotational inertia algebraically. The bug should be treated as a point object, whose rotational inertia is MR2. That gives 180 g*cm2 as the bug's inertia.
The rotational inertia of the bug-DVD system is then 340 g*cm2. The addition of the bug nearly doubles the DVD's rotational inertia; thus the presence of the bug is significant.
(b) Initially the bug and DVD are rotating at a constant angular speed. Then, the bug moves to a new position 3.0 cm from the DVD's center. Explain why and how the DVD's rotational speed changes.
No external torques about the center are exerted on the bug-DVD system ('cause no net force at all acts). Thus, angular momentum is conserved.
Angular momentum is Iω When the bug approaches the center of the disk, the bug's (and thus the system's) rotational inertia decreases because the R term in the inertia formula decreases. To keep angular momentum from changing, then, the ω term must increase. The DVD will speed up its angular velocity.
(c) Does the bug exert a torque about the DVD's center as it moves toward the new position?
Tricky. There's no EXTERNAL torque on the bug-DVD system. But angular momentum can still be conserved when internal torques act. The torque of the bug on the DVD would be internal to the bug-DVD system.
Consider the DVD by itself. It changes its angular speed. So by Newton's second law of rotation, it must experience a net torque.
What can possibly provide that net torque? The weight of the DVD and the normal force of the spindle on the DVD both act through the center of the DVD; they provide no lever arm, and thus no torque.
The only other possible provider of torque is the bug. But how, in terms of torque equaling force times lever arm, can the bug do that?
Since the bug rotates with the DVD, a static friction force must act between the DVD and the bug. That friction force acts tangent to the rotation of the disk, and thus has a lever arm with respect to the disk's center.