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.