Two rings of radii R and nR made up of same material have the ratio of moment of inertia about an axis passing through centre is 1: 8. The value of n is

Question

Two rings of radii R and nR made up of same material have the ratio of moment of inertia about an axis passing through centre is 1: 8. The value of n is (A) 2 (B) 2 √2 (C) 4 (D) (1/2)

Tardigrade Admin · Accepted Answer

The moment of inertia of circular ring whose axis of rotation is passing through its centre is

I = m R^{2}

∴ I_{1} = m_{1} R^{2}

and

I_{2} = m_{2} (n R)^{2}

Since, both have same density

∴ \frac{m ^{2}}{2 π ( n R ) \times A} = \frac{m _{1}}{2 π R \times A}

where

A

is cross-section area of ring.

∴ m_{2} = n m_{1}

∵ \frac{I _{1}}{I _{2}} = \frac{m _{1} R ^{2}}{m _{2} ( n R ) ^{2}} = \frac{m _{1} R ^{2}}{m _{1} n ( n R ) ^{2}} = \frac{1}{n ^{3}}

∵ \frac{I _{1}}{I _{2}} = \frac{1}{8}

(Given)

∴ \frac{1}{8} = \frac{1}{n ^{3}}

or

n = 2

Q. Two rings of radii $R$ and $n R$ made up of same material have the ratio of moment of inertia about an axis passing through centre is 1 : 8. The value of $n$ is

2

$22$

4

$\frac{1}{2}$

Q. Two rings of radii R and nR made up of same material have the ratio of moment of inertia about an axis passing through centre is 1 : 8. The value of n is

2

22​

4

21​

Q. Two rings of radii $R$ and $n R$ made up of same material have the ratio of moment of inertia about an axis passing through centre is 1 : 8. The value of $n$ is

$22$

$\frac{1}{2}$