Question

If the angular momentum of a rotating body about a fixed axis is increased by 10%. Its kinetic energy will be increased by:
A. 10%
B. 20%
C. 21%
D. 5%

Answer 1

Hint: In this question, we first know about the basics of kinetic energy. We will use the relation between the rotational kinetic energy and the angular momentum to find the increase in the kinetic energy. Further, we will discuss the basics of angular momentum and energy.

Formula used:
$K = \dfrac{{{L^2}}}{{2I}}$

Complete step by step solution:
As we know that the kinetic energy of an object is the energy that it possesses due to motion of the object. It can be defined as the work needed to accelerate a body or object of given mass from rest to its velocity. When an object gains this energy during its acceleration, the object maintains this kinetic energy unless its speed or velocity changes.
Here we are given that the momentum of a rotating body about a fixed axis is increased by 10%, which means:
$\dfrac{{\Delta L}}{L} = 10\% $
We know that, the expression for the rotational kinetic energy in terms of angular momentum is given as:
$K = \dfrac{{{L^2}}}{{2I}}...............(1)$
Here, L is the angular momentum, I is the moment of inertia and K is the rotational kinetic energy.
From equation (1), we can write:
$\eqalign{& \dfrac{{\Delta K}}{K} = 2\dfrac{{\Delta L}}{L} \cr
  & \therefore \dfrac{{\Delta K}}{K} = 2 \times 10\% = 20\% \cr} $
Therefore, the correct option is B) i.e., the kinetic energy will be increased by 20%.

So, the correct answer is “Option B”.

Additional Information: Angular momentum is the rotational equivalent to linear momentum, given by:
$L = mvr$
Here, L is linear momentum, m is mass, v is velocity and r is the radius of an object.
It is a conserved quantity. The total angular momentum of a closed system remains constant. Angular momentum is also the product of inertia and angular velocity. If angular momentum remains constant than the angular velocity of the object must increase
Further, as we know that energy is defined as the ability to do work. Energy can be found in many things and can take different forms, like-kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure.
We also know that, according to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another. Here, in this question electrical energy is transferred to heat energy.

Note: Here, the initial and final linear momentum is equal when there is no torque i.e., conserved angular moment. Also, total energy is given by the sum of potential energy and kinetic energy. The unit of energy here is electron volt which is $1.602 \times {10^{ - 19}}Joule$. The S.I unit of energy is Joule. At the maximum height the potential energy is maximum and kinetic energy is zero