Question

A stationary wheel starts rotating about its own axis at uniform angular acceleration 8 rad/$s^2$. The time taken by it to complete 77 rotations is
A. 5.5 sec
B. 7 sec
C. 11 sec
D. 14 sec

Answer 1

Hint: From the kinematical equations of rotational mechanics, we know that, $\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$
Where, initial angular velocity = ${\omega _0}$, angular acceleration = $\alpha $and angular displacement in time t sec is $\theta $.

Complete step by step answer:
As wheel starts rotating from rest or stationary state, initial angular velocity ${\omega _0}$= 0 rad/sec
According to the question, angular acceleration, $\alpha $= 8 rad/$s^2$
Let us assume, at time t sec. The wheel has completed 77 rotations.
1 rotation = $2\pi $rad
Therefore, angular displacement, $\theta = 77 \times 2\pi = 154\pi $rad
From the equation of rotational mechanics, $\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$
Putting the values in the equation,
$154\pi = 0 \times t + \dfrac{1}{2} \times 8 \times {t^2}$
$\implies 4{t^2} = 154 \times \dfrac{{22}}{7}$
$\implies {t^2} = \dfrac{{22 \times 22}}{4}$
$\therefore t = \dfrac{{22}}{2} = 11$sec
So, in $11$ seconds the wheel will complete 77 rotations.

So, the correct answer is “Option C”.

Note:
Wheel is rotating so keep in mind that angular displacement should be expressed in radians not in number of rotations. If the wheel is not in rest or stationary state initially then there will be a certain value of ${\omega _0}$ which we have to consider in the problem.