Embibe Experts Solutions for Chapter: Rotational Mechanics, Exercise 3: Exercise - 3

A uniform circular disc of mass $50 \mathrm{kg}$ and radius $0.4 \mathrm{m}$ is rotating with an angular velocity of $10 \mathrm{rad} {\mathrm{s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $6.25 \mathrm{kg}$  and radius $0.2 \mathrm{m}\text{,}$ are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in $\mathrm{rad} {\mathrm{s}}^{-1}$) of the system is

A uniform wooden stick of mass $1.6 \mathrm{kg}$ of length $l$ rests in an inclined manner on a smooth, vertical wall of height $h\left(<l\right)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of $30°$ with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $\frac{h}{l}$ and the frictional force $f$ at the bottom of the stick are $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

The position vector$r$ of a particle of mass $m$ is given by,
the following equation, $\overrightarrow{r}\left(t\right)=\alpha t^3\hat{\mathrm{i}}+\beta t^2\hat{\mathrm{j}} \mathrm{m}$, where $\alpha =\frac{10}{3} \mathrm{m} {\mathrm{s}}^{-3}, \beta =5 \mathrm{m} {\mathrm{s}}^{-2}$ and $m=0.1 \mathrm{kg}.$ At, $t=1 \mathrm{s},$ which of the following statement (s) is are true about the particle?
 

Angular momentum of the particle rotating with a central force is constant due to:

A mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming pulley to be a perfect uniform circular disc, the acceleration of the mass $m$, if the string does not slip on the pulley, is

A hoop of radius $r$ and mass $m$ rotating with angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

A bob of mass $m$ attached to an inextensible string of length $l$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega \mathrm{rad} {\mathrm{s}}^{-1}$ about the vertical. About the point of suspension

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