Answer
77.7k+ views
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.
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.
Recently Updated Pages
Name the scale on which the destructive energy of an class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write an article on the need and importance of sports class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Choose the one which best expresses the meaning of class 9 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)