Exercise - 4

A particle moves along a closed trajectory in a central field of force where the particle's potential energy $U=k{r}^2$ ($k$ is a positive constant, $r$ is the distance of the particle from the centre $O$ of the field). Find the mass of the particle if its minimum distance from the point $O$ equals $r_1$ and its velocity at the point farthest from $O$ equals $v_2$.

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 $45°$.

A plank of mass $m_1$, with a uniform sphere of mass $m_2$ placed on it, rests on a smooth horizontal plane. A constant horizontal force $F$ is applied to the plank. With what accelerations will the plank and the centre of the sphere move, provide there is no sliding between the plank and the sphere?

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 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.

A uniform sphere of mass $\mathrm{m}$ and radius $\mathrm{r}$ rolls without sliding over a horizontal plane, rotating about a horizontal axle $\mathrm{OA}$. In the process, the centre of the sphere moves with velocity along a circle of radius $\mathrm{R}$ Find the kinetic energy of the sphere.

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 $a_{\mathit{m}}$ . $a_{\mathit{m}}=$