Solution Found!
Answer: CALC A child is pushing a merry-go-round. The
Chapter 9, Problem 5E(choose chapter or problem)
Problem 5E
CALC 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 ωz at t = 5.00 s and the average angular velocity ωav-z for the time interval t = 0 to t = 5.00 s. Show that ωav-z is not equal to the average of the instantaneous angular velocities at t = 0 and t = 5.00 s, and explain.
Questions & Answers
QUESTION:
Problem 5E
CALC 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 ωz at t = 5.00 s and the average angular velocity ωav-z for the time interval t = 0 to t = 5.00 s. Show that ωav-z is not equal to the average of the instantaneous angular velocities at t = 0 and t = 5.00 s, and explain.
ANSWER:
Solution 5E
Angular velocity is the rate of change of angular displacement.
Therefore, …..(1)
Given that, θ(t) =, where γ = 0.400 rad/s and β = 0.0120
(a) We can calculate the angular velocity as a function of time. This is done as follows.
Differentiating θ(t) = wrt time,
…..(2)
This is the required expression for angular velocity as a function of time.