A bug of mass m b =30g sits at the dge of a level, solid disk rotating of mass m

Question

A bug of mass mb=30g sits at the dge of a level, solid disk rotating of mass md=120g and radius r=10cm rotating counter clockwise about its center axis at a rate of 331/3 (33.3333333) revolutions per second. Then the bug crawls directly towars the center of the disk until it is 2.0 cm from the center axis of the disk in 15 seconds.

When the bug is 2 cm from the center of the disk,

(1) What is the angular velocity of the disk?

(2) What is the angular momentum of the disk?

(3)What is the kinetic energy of the disk?

(4) What is the speed of the bug?

(5) What is the avergae angular acceleration of the disk as the bug crawls from th rim of the disk to 2 cm from the center axis?

(6) How much work does the bug do during its crawl?

Explanation / Answer

Initial Moment of inertia of disk + bug = 1/2 M*r^2 + m*r^2
I1 = 1/2 * 0.120 * 0.1^2 + 0.03 * 0.1^2
I1 =  0.0009 Kg m^2

Initial Angular Velocity,
w1 = 33.33 * 2* = 209.4 rad/s


Final Moment of inertia of disk + bug = 1/2 M*r^2 + m*r^2
I2 = 1/2 * 0.120 * 0.1^2 + 0.03 * 0.02^2
I2 = 0.000612 kg m^2

Final Angular Velocity, w2 = ?

Using Momentum Conservation,
Initial Angular Momentum = Final Angular Momentum
I1*w1 = I2 * w2
0.0009 * 209.4 = 0.000612 * w2
w2 = 307.9 rad/s
w2 = 49 rev/s

Angular Momentum of disk =  1/2 * 0.120 * 0.1^2 * 307.9
Angular Momentum of disk = 0.185 Kg m^2

Kinetic Energy of the disk, K.E = 1/2 * I * w2^2
Kinetic Energy of the disk, K.E = 1/2* 0.120 * 0.1^2 * 307.9^2
Kinetic Energy of the disk, K.E = 56.9 J

Speed of bug, v = r*w2
v = 0.02 * 307.9
v = 6.16 m/s

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