Question 585: Centre of Mass and Rotational Motion

Question

Two rings of radius 'R' and 'nR' made of same material have the ratio of moment of inertia about an axis passing through its centre and perpendicular to the plane is $$1: 8$$. The value of '$$n$$' is (mass per unit length $$=\lambda$$)

Accepted Answer

To find the value of $n$ given the two rings of radius 'R' and 'nR' and knowing that the ratio of their moments of inertia is $1:8$, we first establish the formula for the moment of inertia of a ring about an axis passing through its centre and perpendicular to the plane of the ring.
The moment of inertia ($I$) of a ring about an axis passing through its centre and perpendicular to the plane of the ring is given by the formula:
$I = mR^2$
where $m$ is the mass of the ring and $R$ is its radius.
Given the mass per unit length $\lambda$, the mass of the first ring with radius $R$ can be expressed as:
$m_1 = \lambda \cdot 2\pi R$
The moment of inertia of the first ring, therefore, is:
$I_1 = (\lambda \cdot 2\pi R) \cdot R^2 = 2\pi\lambda R^3$
The second ring has a radius of $nR$. Therefore, its mass $m_2$ is:
$m_2 = \lambda \cdot 2\pi(nR)$
And the moment of inertia of the second ring is:
$I_2 = (\lambda \cdot 2\pi(nR)) \cdot (nR)^2 = 2\pi\lambda n^3 R^3$
It's given that the ratio of their moments of inertia is $1:8$, so:
$\frac{I_1}{I_2} = \frac{1}{8}$
Substituting the expressions for $I_1$ and $I_2$, we get:
$\frac{2\pi\lambda R^3}{2\pi\lambda n^3 R^3} = \frac{1}{8}$
Simplifying this expression, we see that $\pi$, $\lambda$, and $R^3$ cancel out, leaving us with:
$\frac{1}{n^3} = \frac{1}{8}$
Solving for $n$, we find:
$n^3 = 8$
Taking the cube root of both sides gives us:
$n = 2$
Therefore, the value of $n$ is $2$, which corresponds to
Option A
.

Answer

2

Answer

4

Answer

1

Answer

3

Centre of Mass and Rotational Motion

Two rings of radius 'R' and 'nR' made of same material have the ratio of moment of inertia about an axis passing through its centre and perpendicular to the plane is $1: 8. The value of 'n' is (mass per unit length =\lambda$)

Choose the correct answer: