Physics

Problem

A rigid body rotates about a fixed axis in such a way that its instantaneous angular velocity varies with time according to

\omega = \omega_{0} + bt^{2},

where

\omega_{0}

and

b

are positive constants and

t

is time. Answer the following. a. (i) Obtain an expression for the angular displacement

\theta(t)

measured from

t = 0

. b. (ii) Derive an expression for the average angular velocity between

t = 0

and an arbitrary time

t = T

. c. (iii) Show that the ratio of the instantaneous angular velocity at

t = T

to the average angular velocity over the same interval is

\dfrac{\omega_{0} + bT^{2}}{\omega_{0} + \tfrac{1}{3}bT^{2}}

. d. (iv) For

\omega_{0} = 2\,\text{rad\,s}^{-1}

b = 0.5\,\text{rad\,s}^{-3}

and

T = 2\,\text{s}

, calculate the numerical value of the ratio obtained in part (iii).