Final answer:
The new angular velocity of the disk after the bug crawls to the center remains the same at 10.0 rad/s, due to the conservation of angular momentum and the negligible moment of inertia of the bug at the center.
Step-by-step explanation:
To determine the new angular velocity of the disk after the bug crawls to the center, we can use the principle of conservation of angular momentum. Since no external torques are acting on the system, the angular momentum before the bug moves is equal to the angular momentum after the bug moves. We can calculate angular momentum (L) using the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.
Before the bug moves, the angular momentum is just that of the rotating disk since the bug is at rest. We calculate this as:
Ldisk = Idisk ωinitial
Idisk = ½ M R2 = ½ (0.10 kg) (0.10 m)2
ωinitial = 10.0 rad/s
After the bug moves to the center, the new moment of inertia is the moment of inertia of the disk plus the moment of inertia of the bug at the center, which is negligible since it's at the rotation axis. Hence, the angular momentum remains the same, but since Idisk does not change, the angular velocity must increase to conserve angular momentum.
The new angular velocity (ωfinal) can be found using:
Linitial = Lfinal
(Idisk ωinitial) = (Idisk + Ibug at center) ωfinal
ωfinal = ωinitial (since Ibug at center is negligible)
So the new angular velocity remains the same, and the correct answer is: (a) 10.0 rad/s.