Question

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)= γt+ βt³, where γ = 0.400 rad/s and β = 0.0120 rad/s³.

a) Calculate the angular velocity of the merry-go-round as a function of time.

b) What is the initial value of the angular velocity?

c) Calculate the instantaneous value of the angular velocity ω_z at t=0 to t= 5.00 s. Show that the w_av-z is not equal to the average of the instantaneous angular velocities at t=0 and t=5.00 s, and explain why it is not.

Answer 1

$\text{ a)}\space \omega(t)=\theta'(t)=\gamma+3\beta t^2.\\ \text{ b)}\space \omega(0)=\gamma=0.4\text{ rad/s}.\\ \text{ c)}\space \omega(5)=0.4+3·0.012·5^2=1.3\text{ rad/s}.\\ \space\space\space\space\space\space\omega_\text{av}=[\omega(5)-\omega(0)]/t=[1.3-0.4]/5=0.18\text{ rad/s}.$