Questions in rotational-motion

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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 centre 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)
One quarter sector is cut from a uniform circular disc of radius R. This sector has mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is
Two discs of same thickness but of different radii are made of two different materials such that their masses are same. The densities of the materials are in the ratio $1 : 3$. The moments of inertia of these discs about the respective axes passing through their centres and perpendicular to their planes will be in the ratio
A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis $XX'$ is
If solid sphere and solid cylinder of same radius and density rotate about their own axis, the moment of inertia will be greater for (L = R)
Two point masses of 0.3 kg and 0.7 kg are fixed at the ends of a rod of length 1.4 m and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum is located at a distance of
A circular disc A of radius r is made from an iron plate of thickness t and another circular disc B of radius 4r is made from an iron plate of thickness t/4. The relation between the moments of inertia ${{I}_{A}}$ and ${{I}_{B}}$ is
A thin wire of length $l$ and mass $M$ is bent in the form of a semi-circle. What is its moment of inertia about an axis passing through the ends of the wire
If $I_1$ is the moment of inertia of a thin rod about an axis perpendicular to its length and passing through its centre of mass, and $I_2$ is the moment of inertia of the ring formed by bending the rod, then
Four solids are shown in cross section. The sections have equal heights and equal maximum widths. They have the same mass. The one which has the largest rotational inertia about a perpendicular through the centre of mass is

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