Question

A wheel is rotating at $900rpm$, about its axis. When the power is cut off, it comes to rest in $1\text{ minute}$. The angular retardation in $rad/{{\sec }^{2}}$ is:
$A)\text{ }\pi \text{/2}$
$B)\text{ }\pi \text{/4}$
$C)\text{ }\pi \text{/6}$
$D)\text{ }\pi \text{/8}$

Answer 1

This problem can be solved by using the direct formula for the angular acceleration of the body by using the equation of rotational motion for constant angular acceleration in terms of the angular velocities and the time taken. The angular retardation is the magnitude of the negative angular acceleration.
Formula used:
$\alpha =\dfrac{{{\omega }_{f}}-{{\omega }_{i}}}{t}$

Complete answer:
We will use the direct formula for the angular acceleration of the body for motion under constant angular acceleration.
The angular acceleration $\alpha$ for a body when its angular acceleration changes from ${{\omega }_{i}}$ to ${{\omega }_{f}}$ in a time $t$ is given by
$\alpha =\dfrac{{{\omega }_{f}}-{{\omega }_{i}}}{t}$ --(1)
Now, let us analyze the question.
The initial angular velocity of the wheel is ${{\omega }_{i}}=900rpm=900\times \dfrac{2\pi }{60}rad/\sec =30\pi rad/\sec$ $\left( \because 1rpm=\dfrac{2\pi }{60}rad/\sec \right)$
Since, the wheel finally comes to rest, the final angular velocity is ${{\omega }_{f}}=0$.
The time taken for this change is $t=\text{1 minute = 60 seconds}$. $\left( \because 1\text{ minute = 60 seconds} \right)$
Let the angular acceleration of the wheel be $\alpha$.
Therefore, using (1), we get the angular acceleration $\alpha$ as
$\alpha =\dfrac{0-30\pi }{60}=\dfrac{-30\pi }{60}=\dfrac{-\pi }{2}rad/{{\sec }^{2}}$
Now, the angular retardation will be nothing but the magnitude of the negative acceleration.
Therefore, the angular retardation will be $\left| -\dfrac{\pi }{2} \right|=\dfrac{\pi }{2}rad/{{\sec }^{2}}$.

Therefore, the correct answer is $A)\text{ }\pi \text{/2}$.

Note:
Students often make the mistake of not assigning the correct signs while writing the angular acceleration and hence, do not imply whether the acceleration is actually retardation or not. In fact, a negative acceleration means that the body is under retardation. This confusion can be avoided by strictly adhering to the formula (1) and not always subtracting the initial acceleration or initial angular acceleration from the final angular acceleration or final angular acceleration. This will always result in a negative value of the acceleration or angular acceleration if the body is retarding.