(D) Let the mass per unit length of the wire be $\lambda$.The circumference of loop $P$ is $2\pi r$,so its mass is $M_P = \lambda(2\pi r)$.The moment

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(D) Let the mass per unit length of the wire be $\lambda$.The circumference of loop $P$ is $2\pi r$,so its mass is $M_P = \lambda(2\pi r)$.The moment of inertia of loop $P$ about an axis passing through its center and perpendicular to its plane is $I_P = M_P r^2 = \lambda(2\pi r)r^2 = 2\pi\lambda r^3$.The circumference of loop $Q$ is $2\pi(nr)$,so its mass is $M_Q = \lambda(2\pi nr) = n M_P$.The moment of inertia of loop $Q$ about a similar axis is $I_Q = M_Q (nr)^2 = (n M_P)(n^2 r^2) = n^3 M_P r^2 = n^3 I_P$.Given that $I_Q = 8 I_P$,we have $n^3 I_P = 8 I_P$.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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From a uniform wire,two circular loops are made: $(i)\ P$ of radius $r$ and $(ii)\ Q$ of radius $nr$. If the moment of inertia of $Q$ about an axis passing through its center and perpendicular to its plane is $8$ times that of $P$ about a similar axis,the value of $n$ is (diameter of the wire is very much smaller than $r$ or $nr$).

A
$8$
B
$6$
C
$4$
D
$2$

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System of Particles and Rotational Motion

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Moment of Inertia and Radius of gyration

Three particles of mass $m$ each are placed at the vertices of an equilateral triangle $ABC$ of side length $\ell$. What is the moment of inertia of the system about the axis $AX$ passing through vertex $A$ and perpendicular to side $AB$ in the plane of the triangle?

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$A$ long slender rod is welded to a thin circular disc of diameter $0.5 \ m$ at a point on its circumference. The rod is in the same plane as that of the disc and forms a tangent to the disc. The radius of gyration of the disc about the rod (in $m$) is

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$A$ circular disc is to be made by using iron and aluminium,so that it acquires a large moment of inertia about its geometrical axis. This is possible with:

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An annular ring has mass $10 \ kg$ and inner and outer radii are $5 \ m$ and $10 \ m$ respectively. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is

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Moment of inertia of a thin uniform rod of length $L$ and mass $M$ rotating about the perpendicular axis passing through its centre is $I$. If the same rod is bent in the form of a ring,its moment of inertia about the diameter is $I_1$. If $I_1 = xI$,then the value of $x$ is:

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Three particles of mass $m$ each are placed at the vertices of an equilateral triangle $ABC$ of side length $\ell$. What is the moment of inertia of the system about the axis $AX$ passing through vertex $A$ and perpendicular to side $AB$ in the plane of the triangle?

$A$ long slender rod is welded to a thin circular disc of diameter $0.5 \ m$ at a point on its circumference. The rod is in the same plane as that of the disc and forms a tangent to the disc. The radius of gyration of the disc about the rod (in $m$) is

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