A Texas cockroach of mass 0.17 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius 15 cm, rotational inertia 5.5 10-3 kg·m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.0 m/s, and the lazy Susan turns clockwise with angular velocity ω0 = 3.0 rad/s. The cockroach finds a bread crumb on the rim and, of course, stops.
(a) What is the angular speed of the lazy Susan after the cockroach stops?
3.8
To find the angular speed of the lazy Susan after the cockroach stops, we can use the principle of conservation of angular momentum.
The formula for angular momentum is given by:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
Initially, the total angular momentum of the system (cockroach + lazy Susan) is given by:
L_initial = (moment of inertia of cockroach * angular velocity of cockroach) + (moment of inertia of lazy Susan * angular velocity of lazy Susan)
Since the cockroach stops, the final angular velocity of the cockroach is 0 rad/s. Therefore, the total angular momentum of the system after the cockroach stops is given by:
L_final = (moment of inertia of lazy Susan * angular velocity of lazy Susan)
Since angular momentum is conserved, we have:
L_initial = L_final
Substituting the given values:
(meaning moment of inertia of cockroach * angular velocity of cockroach = moment of inertia of lazy Susan * angular velocity of lazy Susan )
(moment of inertia of cockroach * 0) + (moment of inertia of lazy Susan * 0) = (moment of inertia of lazy Susan * angular velocity of lazy Susan)
The moment of inertia of the cockroach is negligible compared to the moment of inertia of the lazy Susan, so we can ignore the first term.
0 + (moment of inertia of lazy Susan * 0) = (moment of inertia of lazy Susan * angular velocity of lazy Susan)
0 = (moment of inertia of lazy Susan * angular velocity of lazy Susan)
Solving for the angular velocity of the lazy Susan:
angular velocity of lazy Susan = 0 / moment of inertia of lazy Susan
Since anything divided by zero is undefined, the angular velocity of the lazy Susan after the cockroach stops is undefined.