Question

A cockroach of mass m lies on the rim of a uniform disk of mass 6.00m that can rotate freely about its center like a merry-go-round. Initially the cockroach and disk rotate together with an angular velocity of 0.210 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?

______________

(c) What accounts for the change in the kinetic energy? (Select all that apply.)

friction

centripetal force

gravity

cockroach does negative work on the disc

centrifugal force

cockroach does positive work on the disc

Answer 1

Using conservation of angular momentum,

Moment of inertia of disk = I = mr²/2 = 3mr²

Initial moment of inerta of cockroach =I₁= mr²

Final moment of inerta of cockroach = I₂ =m(r/2)² =mr²/4

Initial angular velocity of the system = ω₁ = 0.21 rad/s

Let ω₂ be the final angular velocity of the system

(a) Using conservation of angular momentum,

(I+I₁)ω₁ = (I+I₂)ω₂

=> (4mr²)(0.21) = (13/4)(mr²)ω₂

=> ω₂ = 0.26 rad/s

(b) Initial kinetic energy = K₀ = (1/2)(I+I₁)ω₁² = 0.088mr²

Final kinetic energy = K = (1/2)(I+I₂)ω₂² = 0.109mr²

K/K₀ = 1.24

(c) The cockroach does positive work on the disc