(A) The moment of inertia of a ring of mass $M$ and radius $R$ about an axis passing through its center and perpendicular to its plane is $I = MR^2$.G

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(A) The moment of inertia of a ring of mass $M$ and radius $R$ about an axis passing through its center and perpendicular to its plane is $I = MR^2$.Given mass per unit length is $\lambda$. The mass of the first ring is $M_1 = \lambda(2\pi R)$ and its radius is $R_1 = R$.So,$I_1 = M_1 R_1^2 = (2\pi R \lambda) R^2 = 2\pi \lambda R^3$.The mass of the second ring is $M_2 = \lambda(2\pi nR)$ and its radius is $R_2 = nR$.So,$I_2 = M_2 R_2^2 = (2\pi nR \lambda) (nR)^2 = 2\pi \lambda n^3 R^3$.The ratio is given as $\frac{I_1}{I_2} = \frac{1}{8}$.Substituting the expressions: $\frac{2\pi \lambda R^3}{2\pi \lambda n^3 R^3} = \frac{1}{n^3} = \frac{1}{8}$.Therefore,$n^3 = 8$,which gives $n = 2$.

Class 11 Physics System of Particles and Rotational Motion Moment of Inertia and Radius of gyration

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

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$2$
B
$4$
C
$1$
D
$3$

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System of Particles and Rotational Motion

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Moment of Inertia and Radius of gyration

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List-$I$ List-$II$

$(a)$ $MI$ of the rod (length $L$,mass $M$,about an axis $\perp$ to the rod passing through the midpoint) $(i) \frac{8ML^2}{3}$

$(b)$ $MI$ of the rod (length $L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends) $(ii) \frac{ML^2}{3}$

$(c)$ $MI$ of the rod (length $2L$,mass $M$,about an axis $\perp$ to the rod passing through its midpoint) $(iii) \frac{ML^2}{12}$

$(d)$ $MI$ of the rod (length $2L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends) $(iv) \frac{2ML^2}{3}$

Choose the correct answer from the options given below:

List-$I$	List-$II$
$(a)$ $MI$ of the rod (length $L$,mass $M$,about an axis $\perp$ to the rod passing through the midpoint)	$(i) \frac{8ML^2}{3}$
$(b)$ $MI$ of the rod (length $L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends)	$(ii) \frac{ML^2}{3}$
$(c)$ $MI$ of the rod (length $2L$,mass $M$,about an axis $\perp$ to the rod passing through its midpoint)	$(iii) \frac{ML^2}{12}$
$(d)$ $MI$ of the rod (length $2L$,mass $2M$,about an axis $\perp$ to the rod passing through one of its ends)	$(iv) \frac{2ML^2}{3}$

MediumJEE MAIN 2021

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

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