Question

A quarter horsepower motor runs at a speed of $ 600rpm $ . Assuming $ 40\% $ efficiency of the motor, the work done by the motor in one rotation is
(A) $ 7.46J $
(B) $ 7400J $
(C) $ 7.46erg $
(D) $ 74.6J $

Answer 1

Hint : To solve this question, we need to calculate the energy produced by the motor in one rotation by using the value of the power given in the question. Then we have to use the value of the efficiency given to calculate the final value of the power.

Formula used: The formula used to solve this question is given by
 $\Rightarrow E = Pt $ , here $ E $ is the energy produced due to a constant power of $ P $ in time $ t $ .

Complete step by step answer
The angular speed of the motor is given to be equal to $ 600rpm $ .
This means that the motor takes one minute to complete $ 600 $ rotations. We know that there are sixty seconds in a minute. So the time required for $ 600 $ rotations is equal to $ 60s $ . So we get the time required to complete one rotation as
 $\Rightarrow t = \dfrac{{60}}{{600}}s $
 $ \Rightarrow t = 0.1s $ ………………………...(1)
So it takes $ 0.1s $ to complete one rotation.
Now, we know that the energy is related to the power as
 $\Rightarrow E = Pt $ ………………………...(2)
According to the question, the power of the motor is equal to one quarter of a horsepower. So this means that we have
 $\Rightarrow P = \dfrac{1}{4}hp $
We know that one horsepower is equal to $ 746 $ watts. So the power is
 $\Rightarrow P = \dfrac{1}{4} \times 746{\text{ W}} $ ………………………...(3)
Substituting (1) and (3) in (2) we get the energy produced by the motor in one rotation as
 $\Rightarrow E = \dfrac{1}{4} \times 746 \times 0.1J $
On solving we get
 $\Rightarrow E = \dfrac{{74.6}}{4}J $ ………………………...(4)
But the motor is only $ 40\% $ efficient. So only $ 40\% $ of the energy produced by the motor is converted into work. This gives us the work done by the motor as
 $\Rightarrow W = \dfrac{{40}}{{100}}E $
From (4)
 $\Rightarrow W = \dfrac{{40}}{{100}} \times \dfrac{{74.6}}{4}J $
On solving we get
 $\Rightarrow W = 7.46J $
Thus the work done by the motor in one rotation is equal to $ 7.46J $ .
Hence, the correct answer is option A.

Note
We should not forget to calculate the time in seconds. The angular speed in this question is given in revolutions per minute. But the SI unit of time is second. So the time has to be converted into seconds.