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A motor requires $2\sec $ to go from a speed of$60rpm$ to$120rpm$ with a constant acceleration. Number of revolutions it takes in this time is
A. $1.50$
B. $4.5$
C. $3$
D. $6.0$

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Last updated date: 25th Jun 2024
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Answer
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Hint:-Recall the concept of angular velocity. It is the velocity at which the particle rotates around a canter or a point in the given time. It is also known as rotational velocity. It shows how fast the position of an object changes with time.

Complete step-by-step solution:
Step I:
Given that time $t = 2\sec $
$N1 = 60rpm$
1minute = 60seconds
$60rpm = \dfrac{{60}}{{60}} = 1 revolution per second$
Similarly $N2 = 120rpm$
$120rpm = \dfrac{{120}}{{60}} = 2 revolution per seconds$
Step II:
Formula for angular velocity is written as $\omega = 2n\pi $
Where $\omega $ is the angular velocity
${\omega _1} = 2{n_1}\pi $
${\omega _1} = 2 \times 1 \times \pi $
${\omega _1} = 2\pi rad/\sec $
Similarly, ${\omega _2} = 2 \times N2 \times \pi $
${\omega _2} = 2 \times 2 \times \pi $
${\omega _2} = 4\pi rad/\sec $
Step III:
Also the angular acceleration of the body is given by
$ \propto = \dfrac{{{\omega _2} - {\omega _1}}}{T}$
$ \propto = \dfrac{{4\pi - 2\pi }}{2}$
$ \propto = \dfrac{{2\pi }}{2}$
$ \propto = \pi rad/{\sec ^2}$
Step IV:
Angular displacement is the shortest angle between the initial and final positions for a given object having circular motion. It has both magnitude and direction. It is the angle of movement of a body in the circular path. So it is a vector quantity. It is known that if the angular acceleration, initial velocity and time are given, then angular displacement can be calculated using the formula
$\theta = \omega t + \dfrac{1}{2} \propto {t^2}$
Where $\theta $ is angular displacement
$\omega $ is the initial angular velocity
$t$ is the time taken
$ \propto $ is the angular acceleration
$\theta = 2\pi \times 2 + \dfrac{1}{2}\pi {(2)^2}$
$\theta = 4\pi + 2\pi $
$\theta = 6\pi $
Step V:
To measure an angle, a radian is used. There are $2\pi $ radians in one complete revolution. Hence,
Number of revolutions is given by $ = \dfrac{{6\pi }}{{2\pi }} = 3$.

Option C is the right answer.

Note:- It is to be noted that the terms angular acceleration and radial acceleration are different terms. Angular acceleration is the rate of change of angular velocity with time. An object with angular velocity will either rotate faster or slower. On the other hand, when an object undergoes circular motion then it shows radial acceleration.