Question

A wheel is at rest. Its angular velocity increases uniformly and become $80\,rad{s^{ - 1}}$ after $5\sec $ The total angular displacement is:
A. $800\,rad$
B. $400\,rad$
C. $200\,rad$
D. $100\,rad$

Answer 1

Hint: In order to solve this question, we should know that when a body starts from rest its initial velocity is zero and here we will use the general equation of motion in terms on angular motion and them will determine the net total angular displacement covered by the wheel in given period of time.

Formula used:
If $\omega ,{\omega _0},\alpha ,t$ be the final angular velocity, initial angular velocity, angular acceleration, time taken by the body so we have equation of motion as $\omega = {\omega _0} + \alpha t$ and if $\theta $ is the angular displacement then we use $\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$

Complete step by step answer:
According to the question we have given, initial velocity of the wheel ${\omega _0} = 0$.
Final velocity after time of $t = 5s$ is $\omega = 80\,rad{s^{ - 1}}$ then using, $\omega = {\omega _0} + \alpha t$ we get,
$80 = 0 + \alpha 5$
$ \Rightarrow \alpha = 16rad{s^{ - 2}}$

Now, we have the angular acceleration, in order to know total displacement we use the formula, $\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$ and on putting the values of given parameters we get,
$\theta = 0(5) + \dfrac{1}{2}(16){(5)^2}$
on solving we get,
$\theta = 8 \times 25$
$\therefore \theta = 200\,rad$
So, the total displacement covered by the wheel in the time period of five seconds is $\theta = 200\,rad$.

Hence, the correct option is C.

Note: It should be remembered that, radian is a measurement of the angle covered by the moving body and here while solving such questions always notice the case when a body starts its motion from rest its initial velocity is zero and uniformly increasing velocity means the value of velocity is increasing with constant rate which is called angular acceleration of the body.