A uniform disk with a mass of 1.0 kg and radius of 20 cm is rotating on friction

Question

A uniform disk with a mass of 1.0 kg and radius of 20 cm is rotating on frictionless bearings with a rotational speed of 10 rad/s when a clod of clay is dropped on a point 10 cm from the center of the disk, where it sticks. If the new angular velocity of the disk is 8.7 rad/s, what is the mass of the clay? Idisk=

A uniform disk with a mass of 1.0 kg and radius of 20 cm is rotating on frictionless bearings with a rotational speed of 10 rad/s when a clod of clay is dropped on a point 10 cm from the center of the disk, where it sticks. If the new angular velocity of the disk is 8.7 rad/s, what is the mass of the clay? Idisk= 1/2MR2

Explanation / Answer

Let:
I = moment of inertia of disk,
I1 = moment of inertia of disk and clay,
M = mass of disk,
m = mass of clay,
r = radius of disk,
r1 = distance of clay from centre,
f = initial frequency of rotation,
f1 = final frequency of rotation,
P = angular momentum.


Initially:
I = Mr^2 / 2
P = I * 2pi * f
= pi Mr^2 f.

Subsequently:
I1 = Mr^2 / 2 + m r1^2
P = I1 * 2pi * f1
= 2pi (Mr^2 / 2 + m r1^2) f1

Equating the two values of P:
pi Mr^2 f = 2pi (Mr^2 / 2 + m r1^2) f1
f1 = Mr^2 f / [ (Mr^2 + 2m r1^2) ]
= 0.850 * 0.17^2 * 18 / [ 0.850 * 0.17^2 + 2 * 0.130 * 0.0850^2 ]
= 16.7 Hz

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