A particular horizontal turntable can be modeled as a uniform disk with a mass o

Question

A particular horizontal turntable can be modeled as a uniform disk with a mass of 230 g and a radius of 22.0 cm that rotates without friction about a vertical axis passing through its center. The angular speed of the turntable is 2.00 rad/s. A ball of clay, with a mass of 50.0 g, is dropped from a height of 40.0 cm above the turntable. It hits the turntable at a distance of 15.0 cm from the middle, and sticks where it hits. Assuming the turntable is firmly supported by its axle so it remains horizontal at all times, find the final angular speed of the turntable-clay system.

Explanation / Answer

Angular momentum is conserved , so

Li = Lf

I = moment of inertia of the turnable along the vertical axis passing through its center

w is the initial angular speed of the turnable

Iw + r'mv = I'w'

I' = moment of inertia of the turnable and clay system along the vertical axis passing through its center

M = mass of turnable = 230 g

m = mass of the clay = 50.0 g

r = radius of turnable = 22cm

I' = Mr^2/2 + m(0.15)^2

= 0.0067

I = 0.011

0.0011*2 + 0.15*0.05*2.82 = 0.0067 * w'

0.02335 = 0.0067 * w'

w' = 3.48

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