Question

A playground merry-go-round is a large disk of mass 10.0 kg and radius 2.00 m. The merry-go-round is initially at rest. A physics teacher pushes on the rim for 24.0 s to accelerate it with constant angular acceleration until the merry-go-round has completed a total of 9.00 full revolutions.

a. What is the final angular velocity after 24.0 s?

b. How much total work did the teacher do on the merry-go-round? (You may assume the axle for the merry-go-round is frictionless.)

Answer 1

here,

the mass of merry-go-round, m = 10 kg

the radius of merry-go-round , r = 2 m

a)

time taken , t = 24 s

the angle covered , theta = 9 rev = 9 * 2pi rad

theta = 56.52 rad

let the angular acceleration be alpha

using second equation of motion

theta = w0 * t + 0.5 * alpha * t^2

56.52 = 0 + 0.5 * alpha * 24^2

solving for alpha

alpha = 0.196 rad/s^2

using first equation of motion

the final velocity , w = w0 + alpha * t

w = 0 + 0.196 * 24 rad/s

w = 4.71 rad/s

the final angular velocity is 4.71 rad/s

b)

using Work energy theorm

the work done by teacher , W = change in kinetic energy

W = 0.5 * I * w^2 = 0.5 * (0.5 * m * r^2) * w^2

W = 0.5 * ( 0.5 * 10 * 2^2) * 4.71^2 J

W = 221.8 J