Question

A wheel rotating with uniform angular acceleration covers $50$ revolutions in the first five seconds after the start. Find the angular acceleration and the angular velocity at the end of five seconds.

Answer 1

When a body moves along a straight line with constant acceleration the motion of the body can be studied using the equation of motion. In a rotational motion, the body rotates about a fixed axis. If the body is rotating with a constant angular acceleration, we can use the equation of motion for rotation also.

Complete step by step solution:
Let us first write the information given in the question.
Revolutions $= 50$, time taken $t = 5\sec$.
We have to calculate the angular acceleration and angular velocity at the end of $5\sec$.
Angular distance covered in $50$ seconds is given below.
$\theta = 2\pi \times 50 = 100\pi$ …………..(1)
Here, $\theta$ is the angular distance.
Let us use the following equation of motion for rotational motion.
$\theta = \omega t + \dfrac{1}{2}\alpha {t^2}$
Here, $\theta$ is the angular displacement,$\omega$ is the angular velocity,$\alpha$ is the angular acceleration, and $t$ is the time taken.
Let us substitute the values in this equation.
$\theta = (0)t + \dfrac{1}{2}\alpha {t^2} = \dfrac{{\alpha {t^2}}}{2}$ …………………(2)
Let us equate equations (1) and (2).
$\dfrac{{\alpha {t^2}}}{2} = 100\pi \Rightarrow \alpha = \dfrac{{200\pi }}{{25}} = 8\pi rad/{s^2}$
If we write this in terms of revolutions, we will divide by $2\pi$.
$\alpha = 4revolution/{s^2}$
Let us use another equation of motion to find the angular velocity.
$\omega ' = \omega + \alpha t$
Here, $\omega '$ is the final angular velocity, $\omega$ is the initial angular velocity, $\alpha$ is angular acceleration, and $t$ is the time.
Let us substitute the values. $\omega ' = 0 + 4 \times 5 = 20revolution/\operatorname{s}$
Hence, the angular acceleration and angular velocity are $4revolutions/{s^2}$ and $20revolution/s$.

Note:
Whenever the body starts from rest or comes to rest, we will assume the initial angular velocity and final angular velocity zero, respectively.
The distance covered in the rotation motion is measured by the angular displacement. In a complete revolution, the angular distance covered is $2\pi$.