B M Sharma Solutions for Chapter: Rigid Body Dynamics I, Exercise 2: DPP 2.2

A ring of mass $M$ and radius $R$ lies in the $x-y$ plane with its centre at the origin, as shown. The mass distribution of ring is non-uniform such that, at any point $\mathrm{P}$ on the ring, the mass per unit length is given by $\lambda ={\lambda }_0{\cos }^2\left(\theta \right)$ (where ${\text{λ}}_0$ is a positive constant). Then, the moment of inertia of the ring about $z$ axis is:

Seven identical discs are arranged in a hexagonal, planar pattern so as to touch each neighbour, as shown in the figure. Each disc has mass $m$ and radius $r$. What is the moment of inertia of the system of seven discs about an axis passing through the centre of the disc and normal to plane of all discs?

A uniform disc of radius $R$ lies in the $x-y$ plane, with its centre at the origin. Its moment of inertia about $z$-axis is equal to its moment of inertia about the line$y=x+c.$ The value of $c$will be

The moment of inertia of a hollow cubical box of mass $M$ and side $a$, about an axis passing through the centres of two opposite faces, is equal to

The moment of inertia of a uniform thin rod of mass $m$ and length $l$ about axes $PQ$ and $RS$, passing through the centre of the rod $C$ and in the plane of the rod, are $I_{\mathrm{PQ}}$ and $I_{\mathrm{RS}}$, respectively. Then, $I_{\mathrm{PQ}}+I_{\mathrm{RS}}$ is equal to

A square plate of edge $\frac{a}{2}$ is cut out from a uniform square plate of edge $a$, as shown in the figure. The mass of the remaining portion is $M.$The moment of inertia of the shaded portion about an axis passing through $O$ (centre of the square of side $a$) and perpendicular to plane of the plate is

With reference to the figure of a cube of edge $a$ and mass $m,$ state which of the following statements are correct. ($\mathrm{O}$ is the centre of the cube)

A square plate of mass $M$ and side length $l$ may either rotate in $x‐z$ plane about a point $A$ or may rotate by falling towards $x‐y$ plane. If $I_1$and $I_2$ are moments of inertia in the two cases, respectively, find the correct option.