In the figure the lower disk, of mass 440 g and radius 3.9 cm , is rotating at 1

Question

In the figure the lower disk, of mass 440 g and radius 3.9 cm , is rotating at 180 rpm on a frictionless shaft of negligible radius. The upper disk, of mass 280 g and radius 2.0 cm , is initially not rotating. It drops freely down onto the lower disk, and frictional forces bring the two disks to a common rotational speed.

Find the final common frequency in rpm.   

Express your answer using two significant figures.

Find the percentage of the initial kinetic energy lost to friction.

Express your answer using two significant figures

Explanation / Answer

Let b the common angular frequency.

Angular momentum of the system is conserved since net external torque is zero.

I11 = (I1 + I2)

=> (m1R12/2)1 = (m1R12/2 + m2R22/2)

=> = [(m1R12) / (m1R12 + m2R22)]1

=> = [(440 * 3.92) / (440 * 3.92 + 280 * 22)] * 180

=> = 154.2 rpm

Initial kinetic energy, K1 = I112/2 = [440 * 3.92 / 2] * 1802 / 2 = 5.421 * 107 g-cm2

Final kinetic energy, K2 = (I1 + I2)2/2 = [(440 * 3.92 + 280 * 22) / 2] * 154.22 / 2 = 4.644 * 107 g-cm2

Percentage energy lost to friction = [(K1 - K2)/K1] * 100 = {[(5.421 - 4.644) * 107] / (5.421 * 107)} * 100 = 14.33 %

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