Question

A washing machine is switched off when it is rotating at an angular speed of \[50rev/s\]. It slows down under uniform acceleration and stops after completing 150 revolutions. What was the value of deceleration and how much time it took to come to rest?

Answer 1

Hint It is given that a washing machine was switched off which was running at \[50rev/s\], and it exactly completed 150 revolutions before stopping. Find time taken by assuming \[\theta \]as 150 revs. Later find angular retardation using angular speed and time.

Complete Step By Step Solution
It is given that the washing machine runs at \[50rev/s\]. Therefore, this is the angular velocity of our washing machine. In the question , it is mentioned that the machine was allowed to run for sometime and was switched off. From the time of start till the switch off time, it was able to complete 150 revolutions. Now , we can find the time period using,
\[\theta = (\dfrac{{\omega + {\omega _0}}}{2})t\], where \[\theta \]is total number of revolutions produced before coming to rest and \[{\omega _0}\]is initial angular velocity and \[\omega \]is final angular velocity.
\[ \Rightarrow 150 = (\dfrac{{50 + 0}}{2})t\]
Taking known terms on one side we get
\[ \Rightarrow \dfrac{{300}}{{50}} = t\]
\[ \Rightarrow t = 6s\]
Now, we know that the machine was switched on for a total time of 6 seconds. In seconds, the machine accelerated for \[50rev/s\]and was able to complete 150 revolutions, before coming to halt. Now, the deceleration period from where the machine slows down is given as,
\[\omega = {\omega _0} + \alpha t\], where \[{\omega _0}\]is the initial deceleration velocity, \[\omega \] is final deceleration velocity and \[\alpha \]is angular acceleration.
Substituting the values we get,
\[ \Rightarrow 0 = 50 - \alpha (6)\](here \[\alpha \]is minus which indicates deceleration)
\[ \Rightarrow \alpha = \dfrac{{50}}{6}rev/{s^2}\]
Further simplifying we get ,
\[ \Rightarrow \alpha = \dfrac{{25}}{3}rev/{s^2}\]
Therefore the machine runs for a period of 6 seconds and undergo angular deceleration of \[\dfrac{{25}}{3}rev/{s^2}\]when the power is switched off.

Note Angular acceleration is defined as the ratio of rate of change of angular velocity to the change in time period , in accordance to the angular velocity. Like every other kinematic quantity, angular acceleration is also a vector quantity.