Question

A washing machine, starting from rest, accelerates within $3.14\,s$ to a point where it is revolving at a frequency of $2.00\,Hz$. Its angular acceleration is most nearly:
A. 0.1 \[rad/{s^2}\]
B. 0.637 \[rad/{s^2}\]
C. 2 \[rad/{s^2}\]
D. 4 \[rad/{s^2}\]
E. 6.28 \[rad/{s^2}\]

Answer 1

Hint: The number of times a repeated event occurs per unit of time is known as frequency. It's also known as temporal frequency, which highlights the difference between spatial and angular frequency. The unit of frequency is hertz (Hz), which equals one occurrence per second.

Complete step by step answer:
The time rate of change in angular velocity is referred to as angular acceleration in physics.There are two kinds of angular velocity: spin angular velocity and orbital angular velocity, so there are two types of angular acceleration: spin angular acceleration and orbital angular acceleration.

The angular acceleration of a rigid body around its centre of rotation is known as spin angular acceleration, while the angular acceleration of a point particle about a fixed origin is known as orbital angular acceleration.
It is given by the formula
\[\alpha = \dfrac{{\Delta \omega }}{{\Delta t}} \\
\Rightarrow \alpha = \dfrac{{{\omega _2} - {\omega _1}}}{{{t_2} - {t_1}}}\]
$\Rightarrow \alpha$=angular acceleration
$\Rightarrow {\Delta \omega}$=change in angular velocity
$\Rightarrow {\Delta t}$=change in time
$\Rightarrow \omega_{2}$=final angular velocity
$\Rightarrow \omega_{1}$=initial angular velocity
$\Rightarrow t_{2}$= final time
$\Rightarrow t_{1}$=initial timeFrequency of revolution of machine
\[\Rightarrow f = \dfrac{1}{T} \\
\Rightarrow f= \dfrac{\omega }{{2\pi }}\]
Given $f = 2$
So, \[\omega = 4\pi \]
This angular speed is attained in $3.14\,s$.
\[\alpha = \dfrac{{\Delta \omega }}{{\Delta t}}\]
\[\Rightarrow \alpha = \dfrac{{4\pi }}{{3.14}} \\
\therefore \alpha\approx 4.00\,{\rm{rad}}/{{\rm{s}}^2}\]

Hence option D is correct.

Note: A net external torque must induce angular acceleration in rigid bodies. For non-rigid bodies, though, this is not the case: A figure skater, for example, will increase her rotational speed (and therefore her angular acceleration) by contracting her arms and legs inwards, which requires no external torque.