Question

A stationary wheel starts rotating about its own axis at a constant angular acceleration. If the wheel completes $50$ rotations in the first $2s$, then the number of rotations made by it in the next $2s$ is:
$\begin{aligned} & A.50 \\\ & B.100 \\\ & C.125 \\\ & D.150 \\\ \end{aligned}$

Answer 1

The problem needs to be solved by using the formula for angular displacement of a particle having uniform angular acceleration. So first, we need to find the total angular displacement in the said $2s$. Then we have to evaluate the number of rotations for that amount of angular displacement.
Formula used: $\theta ={{w}_{1}}t+\dfrac{1}{2}\alpha {{t}^{2}}$ , ${{\omega }_{2}}={{\omega }_{1}}+\alpha t$ ,$\theta =2\pi n$

Complete answer:
The angular displacement of a particle having initial angular velocity ${{\omega }_{1}}$ ,uniform angular acceleration $\alpha$in time t is given by
$\theta ={{w}_{1}}t+\dfrac{1}{2}\alpha {{t}^{2}}$ -----$(1)$ , also the final velocity after time t is given by ${{\omega }_{2}}={{\omega }_{1}}+\alpha t$----$(2)$
Now here, the wheel completes $50$ rotations in the first$2s$, that means the total angular displacement in radian is $\theta =2\pi \times 50rad=100\pi rad.$ Now putting the values of different quantities in$(1)$ we get
$\begin{aligned} & 100\pi =0\times 2+\dfrac{1}{2}\alpha \times {{2}^{2}} \\\ & or\alpha =50\pi rad{{s}^{-2}} \\\ \end{aligned}$
Now using the equation$(2)$ we get, the velocity acquired by the wheel in the first $2s$ ${{\omega }_{2}}=0+50\pi \times 2=100\pi rad{{s}^{-1}}$ . So to have the angular displacement in the next $2s$, we again use the equation $(1)$ and we get
$\begin{aligned} & {{\theta }_{1}}=100\pi \times 2+\dfrac{1}{2}50\pi \times {{2}^{2}}rad \\\ & or {{\theta }_{1}}=300\pi rad \\\ \end{aligned}$
So the number of complete rotations in next $2s$ is given by
$300\pi \div 2\pi =150$ .

So the correct option is D.

Note:
We must convert the total angular displacement in the number of rotations by dividing it by $2\pi$ .
Also for the next $2s$ the initial velocity must be calculated. The formulae can only be applied for particles having uniform angular acceleration, for non uniform accelerations the formulae will not work.