A cockroach of mass m lies on the rim of a uniform disk of mass 4.00 m that can

Question

A cockroach of mass m lies on the rim of a uniform disk of mass 4.00 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 0.390 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/K_O of the new kinetic energy of the system to its initial kinetic energy? (a) omega = _____________ (b) K/K_O = _____________

Explanation / Answer

a) let the radius of the disk be r

Initial moment of inertia

I = mr² + ½(4m)r²
I = 3mr²

moment of inertia after the cockroach moves toward the center

I = m(r/2)² + ½(4m)r²
I = mr²/4 + 2mr²
I = 2.25mr²

conservation of angular momentum

I = I
= I / I
= 3mr² / 2.25mr²
= 4 / 3 = 4*0.390/3 = 0.52 rad/s

b) K/K = ½I²/ ½I²
K/K = I²/ I²
K/K = (2.25mr²)(4 / 3 )²/ (3mr²)²
K/K = 2.25(4 / 3 )²/ 3²
K/K = 2.25(16² / 9 )/ 3²
K/K = 2.25(16 / 27)
K/K = 4/3 = 1.33

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