Question

A cockroach of mass m lies on the rim of a uniform disk of mass 7.00m that can rotate freely about its center like a merry-go-round. Initially the cockroach and disk rotate together with an angular velocity of 0.270 rad/s. Then the cockroach walks half way to the center of the disk.

(a) What then is the angular velocity of the cockroach-disk system? rad/s

(b) What is the ratio K/K0 of the new kinetic energy of the system to its initial kinetic energy?

Answer 1

here,

assuming mass disk = mass of cockroach = m

radius of disk, r = 7 m

angular velocity , w1 = 0.270 rad/s

initial moment of inertia, I1 = 0.5*mr^2 + mr^2 = 1.5*mr^2

Final moment of inertia, I2 = 0.5*mr^2 + m(r/2)^2 = 0.75mr^2

Part a:
From conservation of angular momentum :
I1 * w1 = I2 * w2

Final angular speed, w2 = (I1 * w1)/I2

Final angular speed, w2 = (1.5*mr^2 * 0.270)/(0.75*mr^2)

Final angular speed, w2 = 0.54 rad/s

Part B:
Final Ke/initial KE = 0.5*(I2 * w2^2)/0.5*(I1 * w1^2)

Final Ke/initial KE = 0.5*(0.75 * 0.54^2)/( 0.5*(1.5 * 0.27^2))

Final Ke/initial KE = 2

Final kinetic energy = 2 * initial kinetic energy