Question

A cockroach of mass m lies on the rim of a uniform disk of mass 8.00m that can rotate freely about its center like a merry-go-round. Initially, the cockroach and disk rotate together with an angular velocity of 0.230 rad/s. Then the cockroach walks halfway 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

let cockroach mass = m

mass of merry go round = M = 8m

moment of inertia of merry go round Im = 1/2*Mr^2 = 1/2*8mr^2 = 4mr^2

moment of inertia of cockroach Ic = mr^2

initial moment of inertia of the system Ii = Im+Ic = 4mr^2+mr^2 = 5 mr^2

initial angular speed wi = 0.23 rad/s

now cocke=roach moves to the half the distance

then monent of inertia I'c = m(r/2)^2 = 1/4mr^2

final moment of inertia If = Im+I'c = 4mr^2+1/4mr^2 = 17mr^2/4

from the conservation of angular momentum

Li = Lf

Ii*wi = If*wf

wf = 5mr^2*0.23/17mr^2/4

wf = 20/17*0.23 = 0.27 rad/s

b) initial kinetic energy k0 = 1/2*Ii*wi^2

final kinetic energy = k = 1/2If*wf^2

k/ko = 1/2*17mr^2*0.27^2/1.2*5mr^2*0.23^2

k/ko = 4.25*0.27^2/5*0.23^2 = 1.17