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= 0.400rad/s and β= 1.25×10⁻² rad/s^3.   

Part A

Calculate the angular velocity of the merry-go-round as a function of time.Express your answer in terms of the variables β, γ, and t. Part B.   What is the initial value of the angular velocity?

Express your answer in terms of the variables β, γ, and t.

Part C

Calculate the instantaneous value of the angular velocityωz at t= 4.90s

Part D

Calculate the average angular velocity

Part D

Calculate the average angular velocity ωav−z for the time interval t=0

Answer 1

Part A

Angular velocity is $\omega=\frac{d\theta}{dt}=\frac{d(\gamma t+\beta t^3)}{dt}=\gamma+3\beta t^2$

Part B

Initial value of the angular velocity

$\omega(t=0))=\gamma+3\beta \times0^2=\gamma=0.4rad/s$

Part C

Instantaneous value of the angular velocity is

$\omega(t=4.9))=\gamma+2\beta \times4.9^2=0.4+3\times 1.25\times10^{-2}\times4.9^2=1.3rad/s$

Part D

Average angular velocity is

$\omega_{av}\\=\frac{1}{4.9-0}\int_0^{4.9} \omega(t)dt\\ =\frac{1}{4.9}\int_0^{4.9} \gamma+3\beta t^2 dt\\ =\frac{1}{4.9}\left[\gamma t+\beta t^3 \right ]_0^{4.9}\\ =\frac{1}{4.9}\left[0.4 t+1.25\times10^{-2} t^3 \right ]_0^{4.9}\\ =0.7rad/s$

Pat E

At t=0, $\omega_{av}=\gamma=0.4rad/s$