From a circular ring of mass $M$ and radius $R\text{,}$ an arc corresponding to a $90°$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $k$ times $M{R}^2\text{.}$ Then, the value of $k$ is

Three solid sphere each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of the side of length $d$. The ratio $\frac{I_0}{I_{\mathrm{A}}}$ of moment of inertia $I_0$ of the system about an axis passing the centroid and about the centre of the spheres $I_{\mathrm{A}}$ and perpendicular to the plane of the triangle is

A particle of mass $m$ is fixed to one end of a light spring having force constant $k$ and unstretched length $l$. The other end is fixed. The system is given an angular speed $\omega$ about the fixed end of the spring such that it rotates in a circle in gravity-free space. Then, the stretch in the spring is

As shown in the figure, a bob of mass $m$ is tied by a massless string whose other end portion is wound on a flywheel (disc) of radius $r$ and mass $m$. When released from rest the bob starts falling vertically. When it has covered a distance of $h$ the angular speed of the wheel will be

A rigid massless rod of length $3 l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be

A circular disc $D_1$ of mass $M$ and radius $R$ has two identical discs $D_2$ and $D_3$ of the same mass $M$ and radius $R$ attached rigidly at its opposite ends (see figure). The moment of inertia of the system about the axis $OO\text{'}$ passing through the centre of $D_1$, as shown in the figure will be