Question

The spin dryer of a washing machine rotating at 15 r.p.s. slows down to 5 r.p.s. after making 50 revolutions. Find its angular acceleration.

Answer 1

Hint: One complete revolution is equal to an angle of 2$\pi$. Convert the angular velocities in the units of per second. Then use the equation $2\alpha \theta =\omega _{2}^{2}-\omega _{1}^{2}$ to find the angular acceleration of the spin dryer.

Formula used:
1 r.p.s = 2$\pi {{s}^{-1}}$
$2\alpha \theta =\omega _{2}^{2}-\omega _{1}^{2}$

Complete step by step answer:
The angular velocity of a rotating or a revolving body is the change in angle of rotation in one unit of time. One of the units of angular velocity is r.p.s.
Let us understand the meaning of r.p.s. The full form of r.p.s is revolution per second. It is a unit of revolution. It tells us how many rotations or revolutions the body completes in one second of time. 1 complete revolution is when the revolving body rotates by an angle of 2$\pi$. Therefore, 1 r.p.s = 2$\pi {{s}^{-1}}$.
Therefore,
15 r.p.s = $15\times 2\pi {{s}^{-1}}=30\pi {{s}^{-1}}$.
And
5 r.p.s = $5\times 2\pi {{s}^{-1}}=10\pi {{s}^{-1}}$.
The rate of change of angular velocity with time is called angular acceleration. The SI unit of angular acceleration is per second square (${{s}^{-2}}$).
Suppose a rotating body, changes its angular velocity from ${{\omega }_{1}}$ to ${{\omega }_{2}}$ and during this it rotates for an angle $\theta$. Then the relation between the angular velocities (${{\omega }_{1}}$ and ${{\omega }_{2}}$), angle of rotation ($\theta$) and its angular acceleration ($\alpha$) is given by,
$2\alpha \theta =\omega _{2}^{2}-\omega _{1}^{2}$ …… (i).
Let us consider the clockwise direction as positive direction and the anticlockwise as negative direction.
It is given that the angular velocity of the spin dyer changes from 15 r.p.s to 5 r.p.s.
This means ${{\omega }_{1}}=15rps=30\pi {{s}^{-1}}$ and ${{\omega }_{2}}=5rps=10\pi {{s}^{-1}}$.
It is said that the dyer completes 50 revolutions.
Hence, $\theta =50revolutions=50\times 2\pi =100\pi$.
Substitute all the known values in equation (i).
$\Rightarrow 2\alpha (100\pi )={{\left( 10\pi \right)}^{2}}-{{\left( 30\pi \right)}^{2}}$
$\Rightarrow 200\pi \alpha =100{{\pi }^{2}}-900{{\pi }^{2}}$
$\Rightarrow 200\pi \alpha =-800{{\pi }^{2}}$
$\Rightarrow \alpha =\dfrac{-800{{\pi }^{2}}}{200\pi }=-4\pi {{s}^{-2}}$.
The angular acceleration is negative because the dyer is slowing down.

Note: Rotation mechanics can be correlated with translation mechanics (motion in straight lines). We say that rotation motion is analogous to translation motion.
The quantity displacement is analogous to the angle of rotation.
Velocity is analogous to angular velocity.
Acceleration is analogous to angular acceleration.
Therefore, the kinematic equations for translation motion are also analogous to rotation motion.
$v=u+at$ is analogous to ${{\omega }_{2}}={{\omega }_{1}}+\alpha t$
$s=ut+\dfrac{1}{2}a{{t}^{2}}$ is analogous to $\theta ={{\omega }_{1}}t+\dfrac{1}{2}\alpha {{t}^{2}}$
$2as={{v}^{2}}-{{u}^{2}}$ is analogous to $2\alpha \theta =\omega _{2}^{2}-\omega _{1}^{2}$.
Remember that $\theta$, $\omega$, $\alpha$ are vectors. Hence, you must take care of the direction of the quantities.