Question

A grinding wheel attained a velocity of \[20rad/sec\] in \[5sec\] starting from rest. Find the number of revolutions made by the wheel.
A. $\dfrac{\pi }{{25}}$ revolutions per second
B. $\dfrac{1}{\pi }$ revolutions per second
C. $\dfrac{{25}}{\pi }$ revolutions
D. None

Answer 1

Hint: We know about linear motion, when some force is applied to an object it starts moving. Similarly, we have rotational motion. When we fix one end of the object then it cannot move from its place but can only rotate. The force applied here is called torque.

Complete step by step answer:
Let us first write the information given in the question.
Initial velocity ${\omega _0} = 0$, velocity attained $\omega = 20rad/\sec $, time = \[5sec\].
We have to calculate the number of revolutions made by the wheel.
We have the following equation of motion.
${\omega _0} - \omega = \alpha t$
Where, ${\omega _0}$ is the initial angular velocity, $\omega $is the final angular velocity, $\alpha $ is the angular acceleration and $t$ is the time.
Let us first calculate the angular acceleration using the above equation of motion.
$\alpha = \dfrac{{0 - 20}}{5} = - 4rad/{s^2}$
A negative sign shows the deceleration, i.e., acceleration is working opposite to the direction of motion.
Let us find the angular displacement using the following equation of motion.
$\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$
Here $\theta $is the displacement and other terms have their usual meaning.
Let us substitute the values.
$\theta = 0 \times 5 + \dfrac{1}{2} \times 4 \times {\left( 5 \right)^2} \Rightarrow \theta = 50rad$
Now, we know one revolution will be done when $2\pi $ radian is completed. So, to calculate the number of revolutions we will divide the angular displacement with $2\pi $radian.

Number of revolutions = $\dfrac{{50}}{{2\pi }} = \dfrac{{25}}{\pi }$

So, the correct answer is “Option C”.

Note:
All the laws of motion which are applicable in linear motion are also applicable in rotational motion.
Rotational analogs of velocity are angular velocity, inertia is a moment of inertia, acceleration is angular acceleration and displacement is angular displacement.