Question

A crane pulls up a car of mass 500Kg to a vertical height of 4m. Find the work done by the crane.
A) $19.6{\text{J}}$
B) $19.6{\text{kJ}}$
C) $19600{\text{kJ}}$
D) ${\text{4900J}}$

Answer 1

As the crane pulls the car up from the ground, tension will build up in the arm of the crane. This tension will be balanced by the force due to gravity or the weight of the car. The work done by the crane will then be the dot product of the tension in its arm and the given vertical height to which the car is raised.

Formula used:
The work done by a force acting on a body is given by,
$W = F \cdot d$, where $F$ is the applied force and $d$ is the corresponding displacement of the body.

Complete step by step answer:
Step 1: Sketching a rough figure of the crane and the car and list the parameters given in the question.
The above figure depicts the forces in play as the car gets pulled up by the crane.
As seen from the figure, the weight is directed downwards and the tension in the arm of the crane is directed upwards.
The mass of the car is given to be $m = 500{\text{kg}}$.
The vertical height to which it is raised is given to be $d = 4{\text{m}}$.
Step 2: Expressing the force balance equation of the system under consideration to obtain the tension in the arm of the crane.
Here, the weight of the car $W = mg$ is balanced by the tension $T$ in the crane.
Thus the force balance equation gives $T = W = mg$ -------- (1)
Substituting for $m = 500{\text{kg}}$ and $g = 9.8{\text{m}}{{\text{s}}^{ - 2}}$ in equation (1) we get,
$\Rightarrow T = 500 \times 9.8 = 4900{\text{N}}$
Thus the tension in the arm of the crane is $T = 4900{\text{N}}$.
Step 3: Express the work done by the crane.
The work done by the crane can be expressed as $W = T \cdot d = Td$ ------- (2)
Substituting for $T = 4900{\text{N}}$ and $d = 4{\text{m}}$ in equation (2) we get,
$\Rightarrow W = 4900 \times 4 = 19600{\text{J}}$

Therefore, the work done by the crane is obtained as $W = 19600{\text{J}} = 19.6{\text{kJ}}$. So the correct option is B.

Note:
Here the force acting on the car is the tension in the arm of the crane and the displacement of the car is the vertical height to which the car is raised. Both are in the same direction or are parallel. So the angle between the force and the displacement of the car is zero i.e., $\theta = 0$. So $\cos \theta = 1$ and equation (2) is written as $W = T \cdot d = Td\cos \theta = Td$. The work done is thus positive. A negative work suggests that the displacement and the applied force are opposite in direction.