A Texas cockroach of mass 0.157 kg runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has a radius 14.9 cm, rotational inertia 5.92 x 10-3 kg·m2, and frictionless bearings. The cockroach's speed (relative to the ground) is 2.92 m/s, and the lazy Susan turns clockwise with angular velocity ω0 = 3.89 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? (b) Is mechanical energy conserved as it stops?

-7.23 rad/s

Explanation:

Given that

Mass of the cockroach, m = 0.157 kg

Radius of the disk, r = 14.9 cm = 0.149 m

Rotational Inertia, I = 5.92*10^-3 kgm²

Speed of the cockroach, v = 2.92 m/s

Angular velocity of the rim, w = 3.89 rad/s

The initial angular momentum of rim is

Iw = 5.92*10^-3 * 3.89

Iw = 2.3*10^-2 kgm²/s

The initial angular momentum of cockroach about the axle of the disk is

L = - mvr

L = - 0.157 * 2.92 * 0.149

L = - 0.068 kgm²/s

This means that we can get the initial angular momentum of the system by summing both together

2.3*10^-2 + - 0.068

L' = - 0.045 kgm²/s

After the cockroach stops, the total inertia of the spinning disk is

I (f) = I + mr²

I (f) = 5.92*10^-3 + 0.157 * 0.149²

I (f) = 5.92*10^-3 + 3.49*10^-3

I (f) = 9.41*10^-3 kgm²

Final angular momentum of the disk is

L'' = I (f). w (f)

L'' = 9.41*10^-3w (f)

Using the conservation of total angular momentum, we have

-0.068 = 9.41*10^-3w (f) + 0

w (f) = - 0.068 / 9.41*10^-3

w (f) = - 7.23 rad/s

Therefore, the speed of the lazy Susan after the cockroach stops is - 7.23 and is directed in the opposite direction of the initial lazy Susan angular speed

b)

The mechanical energy of the cockroach is not converted as it stops