A child is pushing a merry-go-round. The angle through which the merry-go-round

Question

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to (t)=t+t3, where = 0.356 rad/s and = 1.40×102rad/s3.

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

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

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

c) Calculate the instantaneous value of the angular velocity z at t= 5.45 s .

d) Calculate the average angular velocity avz for the time interval t=0 to t= 5.45 s .

Explanation / Answer

PART A)

angular velocity = w(t)

to find angular velocity, differentiate theta(t) given in the question.  

w(t) = (gamma)+(3*beta*t^2) rad/s

PART B)

w0 occur when t=0

from equation in PART (A) omega(t) = (gamma)+3(beta)*t^2

substitute gamma = 0.356, beta = 0.014 and t=0

w0 = 0.356 + (3*0.0140*0^2) = 0.356 rad/s

PART C)

the working is as in PART B

w(5) = (gamma)+3(beta)*t^2 = 0.356 + (3*0.014*(5.45)^2) = 1.6035 rad/s

PART D)

the formula of average angular velocity for the time interval t = 0 to t = 5.45 seconds is

(theta(5.45) - theta(0)) / ( 5.45 - 0 )

(0.356*(5.45) + (0.014*(5.45^3)) - 0) / (5.45 - 0) = 0.7718 rad/s

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