Question

The ratio of the moment of inertia of circular ring and circular disc having the same mass and radii about an axis passing the centre of mass and perpendicular to the plane is:
A. $1:1$
B. $2:1$
C. $1:2$
D. $4:1$

Answer 1

First of all, we will find the moment of inertia of the circular ring and circular disc. The mass and the radii of the ring and the disc are the same. We will divide both the expressions and manipulate accordingly to obtain the result.

Complete step by step solution:
In the given question, we are supplied with the following data:
There are two objects, one is a circular ring while the other is a circular disc.Both of them have the same mass and radius.Both the objects rotate about an axis which passes through the centre of mass and perpendicular to the plane. We are asked to find the ratio of their moment of inertia.

To begin with, we know that both of them will have a definite moment of inertia as they are rotating about the axis. We are given that the mass of the two objects are equal and radii are also the same.Let's proceed to solve the problem.Let the mass and the radii of the circular ring and the circular disc be $m$ and $r$ .

The formula which gives the moment of inertia of the circular ring is shown below:
${I_1} = m{r^2}$ …… (1)
Where,
${I_1}$ indicates the moment of inertia of the circular ring.
$m$ indicates the mass of the ring.
$r$ indicates the radius of the ring.

The formula which gives the moment of inertia of the circular disc is shown below:
${I_2} = \dfrac{1}{2} \times m{r^2}$ …… (2)
Where,
${I_2}$ indicates the moment of inertia of the circular disc.
$m$ indicates the mass of the disc.
$r$ indicates the radius of the disc.

Now we will divide the equations (1) and (2) and we get:
$\dfrac{{{I_1}}}{{{I_1}}} = \dfrac{{m{r^2}}}{{\dfrac{1}{2} \times m{r^2}}} \\\ \Rightarrow \dfrac{{{I_1}}}{{{I_2}}} = \dfrac{2}{1}$
$\therefore {I_1}:{I_2} = 2:1$

Hence, the ratio of their moment of inertia is $2:1$ .The correct option is B.
Note: While solving this problem, we should remember that the formulae of the moment of inertia of different bodies are different. It is focussed not only on the object's physical shape and its mass distribution, but also on the particular geometry of how the object rotates. So, in any case, the same object moving in various directions will have a different moment of inertia.