Question

Water is lifted from a well of depth $50m$. If the mass of the water is $20kg$ and the weight of the rope is $0.2kg$ per meter, the amount of work done is:
A.$15250J$
B.$12520J$
C.$12500J$
D.$125.2J$

Answer 1

We know that work is defined as the product of the force and displacement of an object. But in this question, we must observe that the direction of force and displacement is in a vertical direction rather than horizontal. When this type of force and displacement occurs where their direction is vertical then work done is equal to the potential energy possessed by the object. Here, in this question, two objects can possess potential energy: water and rope. Adding the two potential energies will give the desired result.

Complete answer:
The energy possessed by an object when it rises to a certain height against gravity is known as the gravitational potential energy. Let, the object of mass,$m$ get risen to a height,$h$ against gravity. So the force acting on the object due to gravity is given by:
$F = mg......(1)$
Where,
$F$ is the force experienced by the object.
$m$ is the mass of the object.
$g$ is the acceleration due to gravity.
Now, as the object gets lifted against gravity, some sort of work is done on the object. So, the work done is:
$W = F \times s......(2)$
Where,
$W$ is the work done.
$F$ is the force experienced by the body.
$s$ is the displacement.
Putting the values of force and displacement in equation (2) we get,
$W = mgh......(3)$
Now, from the question we have the height,$h$ to be $50m$ and mass of water to be $20kg$. The value of the acceleration due to gravity is taken to be $10m/{s^2}$ . So, the work done on the water is given as:
$W = mgh$
$\Rightarrow {W_{water}} = 20 \times 10 \times 50$
$\Rightarrow {W_{water}} = 10000J......(4)$
We have to find the total weight of the rope. From the question, we can deduce that the length of rope,$l$ is $50m$ and the weight of rope per meter is $0.2kg.$ Therefore the total weight of the rope is,
${m_{rope}} = 50 \times 0.2$
$\Rightarrow {m_{rope}} = 10kg$
As the rope is a long material, we assume it to be a concentrated load with the whole weight concentrated on the center of gravity of the rope to simplify the question. The center of gravity of the rope is concentrated at the half-length of the rope i.e. $25m$. Substituting the values of the mass of rope and the length at which the mass is concentrated we get,
$W = mgh$
$\Rightarrow {W_{rope}} = 10 \times 10 \times 25$
$\Rightarrow {W_{rope}} = 2500J......(5)$
Total work done is given by:
${W_{total}} = {W_{water}} + {W_{rope}}.......(6)$
Substituting the values of the work done on the water and the rope in equation (6), we get,
${W_{total}} = 10000J + 2500J$
$\Rightarrow {W_{total}} = 12500J$

So, option C is the correct answer.

Note:
Four types of potential energy can be possessed by an object. They are Gravitational potential energy, Nuclear potential energy, chemical potential energy, and Elastic potential energy. The energy stored in the nucleus of an atom is known as nuclear potential energy. Similarly, chemical energy is stored in between the links between the atoms of the compound. When an exothermic reaction takes place, this stored energy gets transformed into heat energy. The mechanical potential energy stored in the configuration of the material is known as elastic potential energy.