Question

Two rings of radius $R$ and $nR$ made of same material have the ratio of moment of inertia about on-axis passing through centre in $1{\text{ }}:{\text{ }}8$, The values of n is.
A) $2$
B) $\bar 2$
C) $4$
D) $\dfrac{1}{2}$

Answer 1

Hint : We can solve this problem by using the concept of moment of inertia.
- The moment of inertia is a physical quantity that describes how easily a body can be rotated about a given axis.
- That is, the moment of inertia $\left( I \right) = M{R^2}$where m represents the mass of the system and R is the radius.

Complete Step by Step Solution:
Moment of inertia $= I = M{R^2}$
$[M{\text{ }} = {\text{ }}Mass{\text{ }}of{\text{ }}ring$
$R{\text{ }} = {\text{ }}radius{\text{ }}of{\text{ }}rings]$

Now for the ring of radius R
Moment of inertia $\left( {{I_1}} \right) = M{R^2}$

${I_1} = \left( {\delta L} \right){R^2}$ [where $\delta =$linear density of wire
$L =$length of wire]
${I_1} = \delta \left( {2\pi R} \right).{R^2}$ [As the ring is circular and the total length of the wire $= 2\pi R$]
${I_1} = 2\pi {R^3}.\delta$
Now, for the ring of radius $\left( {nR} \right)$
Moment of inertia $\left( {{I_2}} \right) = M{\left( {nR} \right)^2}$
${I_2} = \left( {\delta L} \right){\left( {nR} \right)^2}$
[where $\delta =$ linear density of wire,
$L{\text{ }} = {\text{ }}length{\text{ }}of{\text{ }}wire]$
$\therefore \,\,{I_2} = \delta \left( {2\pi nR} \right){\left( {nR} \right)^2}$
LAs, the ring is circular and the total length of the wire $= 2\pi nR$
$\therefore \,\,{I_2} = 2\delta \pi {n^3}{R^3}$
Now, the ratio of moment of inertia is $1{\text{ }}:{\text{ }}8$
Therefore,
$\dfrac{{{I_1}}}{{{I_2}}} = \dfrac{{2\pi {R^3}\delta }}{{2\delta \pi {n^3}{R^3}}}$
$\Rightarrow \dfrac{1}{8} = \dfrac{1}{{{n^3}}}$
$\Rightarrow {n^3} = 8$
$\Rightarrow n = 2$
$\therefore \,\,\,n = 2$
Hence, the option $\left( A \right){\text{ }} = {\text{ }}2$ is correct.

Note:
- Linear density of mass represents a quantity of mass per unit length. Total mass is the product of length and linear density.
- The total length of a ring is equal to the perimeter of the circle that is $2\pi n$ where n is the radius of the circle.
- Moment of inertia is that property of matter which resists the change in its state of motion, such that a stationary object remains immovable and a moving object is moving at its current speed.