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**Hint**It is given that the motor runs at a given speed. Now, at a reduced efficiency, we need to calculate work done. Find out the angular velocity using RPM and find out the equation for torque. Use a work done formula with respect to torque and angle of rotation.

**Complete step by step answer**

It is given that a quarter horse power motor runs at a specified speed. We know that horse power is one of the standard units of power and it is equal to 746 watts. Now the overall power is a quarter of one horse power.

\[ \Rightarrow P = \dfrac{{1Hp}}{4} = \dfrac{{746}}{4} = 186.5Watt\]

Now, it is given that the motor runs at a speed of 600 rotations per minute. We can use this to calculate angular velocity of the motor, using the formula,

\[ \Rightarrow \omega = \dfrac{{2\pi N}}{{60}}\] where N is the speed of motor in RPM

Substituting the values we get,

\[ \Rightarrow \omega = \dfrac{{2\pi \times 600}}{{60}} = 20\pi rad/s\]

Now, work done for a rotary motion can be defined as the product of the torque of the motor and the angle the motor rotates per revolution. Mathematically, we can give it as

\[ \Rightarrow W = \tau \times \theta \]

Torque of a motor is given as the ratio between the power of the motor and the angular velocity at which the motor rotates per second. We can represent this as,

\[\tau = \dfrac{{Power}}{\omega }\]

Substituting this in the work done equation, we get

\[ \Rightarrow W = \dfrac{{Power}}{\omega } \times \theta \]

From the given data and the identified values, substitute for power, angular velocity and \[\theta \]. Since, it is a complete rotation, \[\theta \] will be equal to \[2\pi \]radians. Now,

\[ \Rightarrow W = \dfrac{{186.5}}{{20\pi }} \times 2\pi \]

Cancelling the common terms, we get

\[ \Rightarrow W = \dfrac{{186.5}}{{10}}\]

\[ \Rightarrow W = 18.65J\]

This is work done at \[100\% \] motor efficiency. It is given that the motor rotates at \[40\% \]efficiency. Thus, work done at the given efficiency is

\[ \Rightarrow W = 18.65J \times 0.4\]

\[ \Rightarrow W = 7.46J\]

**Thus, option (a) is the right answer.**

**Note**Torque is defined as the moment of force that carries a required tendency of force that can rotate the body at a specified direction when applied. The direction of force is perpendicular to that of the central rotation axis.

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