Moment of inertia

A thin rod of mass $M$ and length $a$ is free to rotate in horizontal plane about a fixed vertical axis passing through point O. A thin circular disc of mass $M$ and of radius $a/4$ is pivoted on this rod with its center at a distance $a/4$ from the free end so that it can rotate freely about its vertical axis, as shown in the figure. Assume that both the rod and the disc have uniform density and they remain horizontal during the motion. An outside stationary observer finds the rod rotating with an angular velocity $\Omega$ and the disc rotating about its vertical axis with angular velocity $4\Omega$. The total angular momentum of the system about the point O is $\left(\dfrac{Ma^2\Omega}{48}\right)n$.

The value of $n$ is ____.

Moment of inertia (M.I.) of four bodies, having same mass and radius, are reported as ;

I$_1$ = M.I. of thin circular ring about its diameter,
I$_2$ = M.I. of circular disc about an axis perpendicular to disc and going through the centre,
I$_3$ = M.I. of solid cylinder about its axis and
I$_4$ = M.I. of solid sphere about its diameter.

Then :

A rod of mass $m$ and length $L$, pivoted at one of its ends, is hanging vertically. A bullet of the same mass moving at speed $v$ strikes the rod horizontally at a distance $x$ from its pivoted end and gets embedded in it. The combined system now rotates with angular speed $\omega$ about the pivot. The maximum angular speed $\omega_M$ is achieved for $x = x_M$. Then

A ring and a disc are initially at rest, side by side, at the top of an inclined plane which makes an angle 60° with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $(2 - \sqrt3)/\sqrt{10}$ s, then the height of the top of the inclined plane, in metres, is ______. Take $g = 10\ \mathrm{m\,s^{-2}}$.

The moment of inertia of a uniform cylinder of length $l$ and radius R about its perpendicular bisector is I. What is the ratio $l/R$ such that the moment of inertia is minimum ?

A uniform disc of radius R and mass M is free to rotate only about its axis. A string is wrapped over its rim and a body of mass m is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is :

Moment of inertia of an equilateral triangular lamina ABC, about the axis passing through its centre O and perpendicular to its plane is $I_o$ as shown in the figure. A cavity DEF is cut out from the lamina, where D, E, F are the mid points of the sides. Moment of inertia of the remaining part of lamina about the same axis is :

A circular hole of radius $\dfrac R4$ is made in a thin uniform disc having mass M and radius R, as shown in figure. The moment of inertia of the remaining portion of the disc about an axis passing through the point O and perpendicular to the plane of the disc is :

A cubical block of side 30 cm is moving with velocity 2 ms$^{-1}$ on a smooth horizontal surface. The surface has a bump at a point O as shown in figure. The angular velocity (in rad/s) of the block immediately after it hits the bump, is :

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as $\rho_A(r)=k\left(\dfrac rR\right)$ and $\rho_B(r)=k\left(\dfrac rR\right)^5$, respectively, where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_A$ and $I_B$, respectively. If $\dfrac{I_B}{I_A}=\dfrac n{10}$, the value of $n$ is

From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is :

Consider a thin uniform square sheet made of a rigid material. If its side is 'a', mass m and moment of inertia I about one of its diagonals, then :

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $F=0.5$ N are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in $\mathrm{rad\,s^{-1}}$ is

A mass 'm' is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release ?

A cylinder of mass $M_c$ and sphere of mass $M_s$ are placed at points A and B of two inclines, respectively. (See Figure). If they roll on the incline without slipping such that their accelerations are the same, then the ratio $\dfrac{\sin\theta_c}{\sin\theta_s}$ is :

A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$, as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_O$ and $I_P$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\dfrac{I_P}{I_O}$ to the nearest integer is

Two solid cylinders $P$ and $Q$ of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder $P$ has most of its mass concentrated near its surface, while $Q$ has most of its mass concentrated near the axis. Which statement(s) is(are) correct?

A metal rod of length 'L' and mass 'm' is pivoted at one end. A thin disk of mass 'M' and radius 'R' (< L) is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case A) The disc is not free to rotate about its center and (case B) the disc is free to rotate about its center. The rod-disc system performs SHM in vertical plane after being released from the same displaced position. Which of the following statement(s) is (are) true?

Four solid spheres each of diameter $\sqrt5$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The moment of inertia of the system about the diagonal of the square is $N\times10^{-4}\ \mathrm{kg\,m^2}$, then $N$ is

It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to $\dfrac4\pi\times10^{11}$ Hz. Then the moment of inertia of CO molecule about its center of mass is close to (Take $h=2\pi\times10^{-34}$ J s)