Question

On the application of a constant torque, a wheel is turned from rest through $400\,radians$ in $10\,s$.
(i) Find angular acceleration.
(ii) If the same torque continues to act, what will be the angular velocity of the wheel after $20\,s$ from start?

(A) (i) $8\,rad{s^{ - 2}}$, (ii) $160\,rad{s^{ - 1}}$
(B) (i) $16\,rad{s^{ - 2}}$, (ii) $160\,rad{s^{ - 1}}$
(C) (i) $8\,rad{s^{ - 2}}$, (ii) $80\,rad{s^{ - 1}}$
(D) (i) $4\,rad{s^{ - 2}}$, (ii) $80\,rad{s^{ - 1}}$

Answer 1

In order to calculate the angular acceleration use the angular displacement formula below and substitute the known values. From the angular acceleration obtained, use the angular velocity formula to know the value of the angular velocity.
Useful formula:
(1) The angular displacement is given by,
$\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$
Where $\theta$ is the angular displacement, ${\omega _0}$ is the initial angular velocity, $\alpha$ is the angular acceleration and $t$ is the time taken.
(2) The angular velocity is given by
$\omega = \alpha t$
Where $\omega$ is the angular velocity.

Complete step by step solution:
It is given that the
Wheel is turned through the distance in angular displacement, $\theta = 400\,rad$
Time taken for the turning, $t = 10\,s$
The angular acceleration is obtained by using the formula (1),
$\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$
The angular velocity at the initial is considered as $0$.
$400 = \left( 0 \right)\left( {10} \right) + \dfrac{1}{2}\alpha {\left( {10} \right)^2}$
By simplifying the above equation, we get
$400 = \dfrac{1}{2}\left( {100\alpha } \right)$
$\alpha = \dfrac{{800}}{{100}}$
By performing the division in the above step,
$\alpha = 8\,rad{s^{ - 2}}$
Hence the angular acceleration obtained is $8\,rad{s^{ - 2}}$
(ii) Using the formula(2) for calculating the angular velocity at $20\,s$ time.
$\omega = \alpha t$
Substituting the values,
$\omega = 8 \times 20$
$\omega = 160\,rad{s^{ - 1}}$
Hence the angular velocity at $20\,s$ is $160\,rad{s^{ - 1}}$.

Thus the option (A) is correct.

Note: The formula (1) of the angular displacement is applicable only at the time of the constant torque. In case if the torque is non constant and if it varies, then the angular acceleration also varies with the time. This is because the torque is the ability of the body to cause angular rotation.