Question

A cockroach of mass m lies on the rim of a uniform disk of mass 6.00 m that can rotate freely about its center like a merry-go-round. Initially the cockroach and disk rotate together with an angular velocity of 0.250 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?
(b) What is the ratio K/K0 of the new kinetic energy of the system to its initial kinetic energy?

Answer 1

a)

$$m_1\omega_f(0.5R)^2+0.5m_2\omega_f(R)^2\\=m_1\omega_i(R)^2+0.5m_2\omega_i(R)^2$$

$$m_1\omega_f(0.5)^2+0.5m_2\omega_f=m_1\omega_i+0.5m_2\omega_i\\m\omega_f(0.5)^2+0.5(6m)\omega_f=m\omega_i+0.5(6m)\omega_i\\\omega_f(0.5)^2+0.5(6)\omega_f=\omega_i+0.5(6)\omega_i\\\omega_f(0.5)^2+0.5(6)\omega_f=0.25+0.5(6)0.25$$

$$\omega_f=0.308\frac{rad}{s}$$

b)

$$\frac{K}{K_0}=\frac{\omega_f}{\omega_i}=\frac{0.308}{0.25}=1.23$$