Embibe Experts Solutions for Chapter: Rotational Mechanics, Exercise 4: Exercise - 4

The angular momentum of a particle relative to a certain point $O$ varies with time as $\overrightarrow{M}=\overrightarrow{a}+\overrightarrow{b}{t}^2$, where $\overrightarrow{a}$ and $\overrightarrow{b}$ are constant vectors, with $\overrightarrow{a}\perp \overrightarrow{b}$. Find the force moment $\overrightarrow{N}$ relative to the point $O$ acting on the particle when the angle between the vectors $\overrightarrow{N}$ and $\overrightarrow{\mathit{M}}$ equals $45°$.

The surface mass density (mass/area) of a circular disc of radius $R$ depends on the distance from the centre $x$ given as, $\sigma \left(x\right)=\alpha +\beta x$, where $\alpha$ and $\beta$ are positive constant then find its moment of inertia about the line perpendicular to the plane of the disc through its centre.

Calculate the moment of inertia of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to m and the radius of its base to $R$.

A force $\overrightarrow{F}=A\hat{\mathrm{i}}+B\hat{\mathrm{j}}$ is applied to a point whose radius vector, relative to the origin of coordinates $O$, is equal to $\overrightarrow{r}=a\hat{\mathrm{i}}+b\hat{\mathrm{j}}$, where $a$, $b$, $A$ and $B$ are constants, and $\hat{\mathrm{i}}$ and $\hat{\mathrm{j}}$ are the unit vectors along $x$ and $y$-axes. Find the moment $N$ and the arm $l$ of the force $F$, relative to the point $O$.

A uniform cylinder of radius $R$ and mass $M$ can rotate freely about a stationary horizontal axis $O$. A thin cord of length $l$ and mass $m$ is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length $x$ of the hanging part of the cord. The wound part of the cord is supposed to have its centre of gravity on the cylinder axis.

A vertically oriented uniform rod, of mass $M$ and length $l$, can rotate about its upper end. A horizontally flying bullet, of mass $m$, strikes the lower end of the rod and gets stuck on it; as a result, the rod swings through an angle $\alpha$. Assuming that $m\ll M,$ find
$\left(\mathrm{a}\right)$ the velocity of the flying bullet
$\left(\mathrm{b}\right)$ the momentum increment in the system "bullet-rod" during the impact; what causes the change of that momentum
$\left(\mathrm{c}\right)$ at what distance $x$, from the upper end of the rod, the bullet must strike, for the momentum of the system "bullet-rod" to remain constant during the impact.

A point $\mathrm{A}$is located on the rim of a wheel of radius $\mathrm{R} = 0.50 \mathrm{m}$ which rolls without slipping along a horizontal surface with velocity $\mathrm{v} = 1.00 \mathrm{m}/\mathrm{s}$ as shown in figure. Find:

(a) the modulus and the direction of the acceleration vector of the point $\mathrm{A}$ ;

(b) the total distance s traversed by the point $\mathrm{A}$ between the two successive moments at which it touches the surface.

Consider a bicycle in vertical position accelerating forward without slipping on a straight horizontal road. The combined mass of the bicycle and the rider is $M$and the magnitude of the accelerating torque applied on the rear wheel by the pedal and gear system is $\mathrm{\tau }$. The radius and the moment of inertia of each wheel is $R$ and $I$ (with respect to the axis) respectively. The acceleration due to gravity is $g$. 

(a) Draw the free diagram of the system (bicycle and rider ).

(b) Obtains the acceleration $a$ in terms of the above-mentioned quantities. $a=$

(c) For simplicity assume that the centre of mass of the system is at height $R$ from the ground and equidistant at $2R$ from the centre of each of the wheels. Let $\mathrm{\mu }$ be the coefficient of friction (both static and dynamic) between the wheels and the ground. Consider$M>>\frac{I}{R^2}$ and no slipping. Obtain the conditions for the maximum acceleration $a_{\mathit{m}}$ of the bike. $a_{\mathit{m}}=$

(d) For $\mathrm{\mu } = 1.0$ calculate $a_{\mathit{m}}$ . $a_{\mathit{m}}=$