A disk with moment of inertia I1 = 10 kg-m2 rotates with angular speed ?o = 10 r

Question

A disk with moment of inertia I1 = 10 kg-m2 rotates with angular speed ?o = 10 rev/sec about a frictionless, vertical axle. A second disk with half the moment of inertia (I2 = 0.5I1), initially not rotating, drops onto the first disk, as shown. The two disks eventually reach the same final angular speed ?

a. Determine ?, in units of rev/s.

b. Calculate how much energy, in units of Joules, is lost to deformation by computing the change ?K in the kinetic energy of the system. What percentage of the initial energy of the system is this loss ?

Explanation / Answer

a) let w is the final angular speed of the two disks.

Apply conservation of angular momentum

I1*wo = (I1+I2)*w

w = I1*wo/(I1+I2)

= 10*10/(10 + 0.5*10)

= 6.67 rev/s

b) wo = 10 rev/s = 10*2*pi rad/s = 62.8 rad/s

w = 6.67 rev/s = 6.67*2*pi rad/s = 41.9 rad/s

Ki = 0.5*I1*wo^2 = 0.5*10*62.8^2 = 19719.2 J

Kf = 0.5*(I1+I2)*w^2 = 0.5*(10 + 5)*41.9^2 = 13167 J

loss of kinetic energy = ki - kf

= 19719.2 - 13167

= 6552.2 J

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