Answer to Question #288883 in Physics for Jihyo

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 = 0.406 rad/s and β = 1.30×10⁻² 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

$\text{ a)}\space \omega(t)=\theta'(t)=\gamma+3\beta t^2.\\ \text{ b)}\space \omega(0)=\gamma=0.406 \text{ rad/s}.\\ \text{ c)}\space \omega_\text z(5)=0.406 +3·0.012·5^2=1.38\text{ rad/s}.\\ \space\space\space\space\space\space\omega_\text{av}=[\omega_\text z(5)-\omega(0)]/t,\\ \space \space\space\space\space\space\omega_\text{av}=[1.38-0.406]/5=0.195\text{ rad/s}.$

The average speed is different because we use a different formula to calculate it.