Question

An electric motor in a crane while lifting a lord produces a tension of $4000\, N$ in the cable attached to the Load. If the motor winding the cable at the rate of $3\,m/s$, the power of the motor expressed in kilowatt units must be
A. 4
B. 3
C. 12
D. 6

Answer 1

Hint: We know that power is the work done per unit time. The power can also be calculated as the dot product of force and velocity.
$P = Fv\cos \theta $
Where $F$ is the force, v is the velocity, and $\theta $ is the angle made by the force with the direction of velocity.
In this case, the force applied by the motor is in an upward direction and the direction of motion of the load is also in the upward direction.

Complete step by step answer:
We know that the weight of the load is balanced by the tension in the cable used to lift the load.
Since the value of tension is given as $4000\,N$
The weight of the load is also $4000\,N$
It is given that cable is winded by the motor at a rate of $3\,m/s$.
This means the load is being pulled up at a speed of $3\,m/s$.
So, we can write
$v = 3\,m/s$
We know that power is defined as the work done per unit time.
$P = \dfrac{w}{t}$
Where $w$ is the work done and $t$ is the time.
Work is the product of force and displacement.
That is,
Where $F$ is the force and S is the displacement.
Here, we know that the ratio of displacement by time is the velocity.
$ \Rightarrow P = Fv\cos \theta $
Where $\theta $ is the angle between the force and velocity.
Here, the motor is applying force in the upward direction to lift the load.
And the motion of the load is also in the upward direction is same as that of the applied force.
Thus, the angle between the force and the direction of velocity is zero.
$\therefore \theta = {0^ \circ }$
On substituting the values, we get
$ \Rightarrow P = 4000 \times 3 \times \cos {0^ \circ }$
$\Rightarrow P = 12000\,W$
We got the power in watts.
We need to find the answer in kilowatts.
$ \Rightarrow P = \dfrac{{12000}}{{1000}}\,kW$
$ \Rightarrow P = 12\,kW$
This is the value of power in kilowatts.

$\therefore$ The power of the motor is 12 kW. Hence, the correct answer is option C.

Note:
While calculating the dot product take care of the angle between the components multiplied. In this question the force done by the motor is in the upward direction and since the load is being lifted upwards the direction of the velocity of the load is also in the upward direction. So, we got the angle between force and velocity as zero.
In case where the angle between the force and velocity is ${90^ \circ }$, then power will be zero in such case since $\cos {90^ \circ }$ is zero.