Question

A wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest it reaches 100rev/sec in 4 seconds. Then the angle rotates during these 4 seconds.
$\begin{array}{rl}& \text{A}.100\pi rad\\ & \text{B}.200\pi rad\\ & \text{C}.300\pi rad\\ & \text{D}.400\pi rad\end{array}$

Answer 1

Hint: We are given that the initially wheel starts its rotational motion through rest means initially angular velocity is zero and it reaches 100rev/sec and time taken is 4 sec so we will used equation of angular motion we will find angular acceleration and then we will further proceed with third equation of angular motion to calculate angular displacement.
Formula used:
${\omega }^{\prime }={\omega }_{{}^{\circ }}+\alpha t$
$\theta =\omega t+{\displaystyle \frac{1}{2}}\alpha t^2$

Complete answer:
We are given with following details in the question:

T = time = 4s

Initial angular velocity
${\omega }_{{}^{\circ }}=0$

Final angular velocity
${\omega }^{\prime }=100rev/s$

$\alpha =$Angular acceleration.

We know the formula

$\begin{array}{rl}& {\omega }^{\prime }={\omega }_{{}^{\circ }}+\alpha t\\ & 100=0+\alpha \times 4\\ & \alpha ={\displaystyle \frac{100}{4}}\\ & \alpha =25rev{s}^{-2}\end{array}$

Here the angle rotated into three equations of kinematics

We know that
$\begin{array}{rl}& \theta =\omega t+{\displaystyle \frac{1}{2}}\alpha t^2\\ & =0+{\displaystyle \frac{1}{2}}\times 25\times 4^2\\ & =8\times 25\\ & =200\times 2\pi \\ & =400\pi \text{ rad}\end{array}$

Then the angle rotates during these 4 seconds is $400\pi \text{ rad}$.

So, the correct answer is “Option D”.

Additional Information:
There is some ambiguity in what is meant by kinematic expressions for rotational motion. We can also consider the angular displacement and angular velocity as functions of time, given constant angular acceleration, as the kinematic equations. We would consider these to be the result of the application of Newton’s second law for rotational motion about a fixed axis, in the case of constant torque and constant moment of inertia..

Note:
Angular displacement mathematically can be defined as angular velocity times time plus 1/2 angular acceleration times squared. This equation is analogous to the equation for linear displacement.
“Angular displacement can be also known by compounded of angular velocity and angular acceleration”.
The only major problem we face is dealing with torque. In every question of rotation the three things that we must know is how to deal with torque during rotational motion, rotational + translational motion and equilibrium condition