Question

A stone of mass m kg is whirled in a vertical circle of radius 20 cm. The difference in the kinetic energies at the lowest and the topmost positions is
A. 4 mg joules
B. 0.4 mg joules
C. 40 mg joules
D. Sufficient data not available

Answer 1

To solve this problem, there are two concepts to be applied: Law of conservation of energy and Work-energy theorem.
The Law of conservation of energy states that energy can neither be created nor destroyed but can be changed from one form to another.
The work-energy theorem gives us the idea that work and energy are interconvertible.

Complete step by step answer:
The energy is related to the work being done. There is a theorem that relates these quantities known as the Work-Energy theorem, which states that the net work done on an object is equal to the change in kinetic energy of the system.
Thus, we see that when work is done on an object, it gets converted to an increase in the kinetic energy of the system and this does not manifest from nowhere. The potential energy inside the body starts decreasing which leads to the increase in the kinetic energy, thus obeying the law of conservation of energy which states that energy can neither be created nor destroyed.
This energy that is acquired by the objects when work is done on them, is called mechanical energy.
The energy possessed by an object by virtue of its position is potential energy and the same energy gets converted into kinetic energy when set into motion or when work is done on it. The net energy at any point i.e. the sum of the kinetic and potential energy and is equal to the mechanical energy.
Mechanical Energy = Kinetic Energy + Potential Energy
So, if the total energy of the system were to remain the same, then the kinetic energy lost by the body will be equal to the gain in potential energy.
$\Rightarrow - KE = PE$
When the stone is whirled, the bottom point has zero kinetic energy and the top has the highest kinetic energy. This change in kinetic energy will result in a gain in the potential energy of the system.
If h is the distance between the bottom and topmost points of the circle,
Kinetic energy lost = Potential Energy gained = $mgh$
Since the stone is rotated in the circle, the topmost and bottom points will be equal to the diameter of the circle.
$\Rightarrow h = D = 2R$
Given radius, $R = 20cm = 0 \cdot 2m$
Substituting, we get,
Kinetic energy lost, $KE = mg(2R) = mg(2 \times 0 \cdot 2) = 0 \cdot 4mg$ joules.

Hence, the correct option is Option (B).

Note:
The purpose of having mechanical energy in a body is to do work. The work is done by a body at the expense of mechanical energy.
For example, when we eat food, the chemical energy in the food gets converted to thermal energy and thus, mechanical energy. The mechanical energy is spent by us when we move because we are doing work. When we do the work, the mechanical energy, which can be thought of as our bank or credit balance, is being spent to perform the work, which can be thought of as a bank transaction.
Mechanical energy is highly effective and convertible easily to other forms of energy viz. thermal energy, electric energy, etc. making the study of the mechanical energy classified into a vast discipline of engineering known as Mechanical Engineering. This branch of Engineering mainly deals with the conversion of energy from mechanical to other forms through objects known as machines.