Question

The wheel is making revolutions about its axis with uniform angular acceleration. Starting from rest, it reaches $100 rev/sec$ in $4 seconds$. Find the angular acceleration. Find the angle rotated during these four seconds.

Answer 1

Hint
We should know that the angular acceleration is the time rate of change of the angular velocity and is usually that is shown by alpha and is expressed in radians per second. Based on this concept we have to solve this question.

Complete step-by step answer
Let us consider the angular acceleration of the wheel to be $\alpha$.
Then initial angular velocity ($\omega _0$) = 0 and final angular velocity = $\omega$
$\omega = 100 \times 2\pi \;\dfrac{{rad}}{s} = 200\pi \;\dfrac{{rad}}{s}(As\;\omega = 100\;\dfrac{{rev}}{s})$
Now applying equations, we get:
$\omega = {\omega _0} + \alpha t$
$\Rightarrow 200\pi \;\dfrac{{rad}}{s} = 0 + \alpha (4)(t = 4s)$
$\Rightarrow \alpha = \dfrac{{200\pi }}{4} = \dfrac{{50\pi rad}}{{{s^2}}}$
The angle rotated in this time t is given by:
$\theta = {\omega _0}t + \dfrac{1}{2}a{t^2}$
$\theta = 0 + \dfrac{1}{2} \times 50 \times {4^2} = 25 \times 16 = 400\pi \;radians$
Hence, the answer is $400\pi \;radians$.

Note
The angular acceleration is also known as rotational acceleration and the expression that is formed is considered to be as a vector quantity. It is considered a vector quantity because it consists of a magnitude component and either of two defined directions or we can sense. At every point of a rigid body we find that there is the presence of rotational velocity and acceleration.