D. C. Pandey Solutions for Chapter: Circular Motion, Exercise 1: Excercise 1

A particle moves such that its position vector $\overrightarrow{r}\left(t\right)=\cos \omega t\hat{\mathrm{i}}+\sin \omega t\hat{\mathrm{j}},$ where $\omega$ is a constant and $t$ is time. Then, which of the following statements is true for the velocity $\overrightarrow{v} \left(t\right)$ and acceleration $\overrightarrow{a} \left(t\right)$ of the particle?

A particle is moving along a circular path with a constant speed of $10 \mathrm{m} {\mathrm{s}}^{-1}$. What is the magnitude of the change in velocity of the particle, when it moves through an angle of $60°$ around the centre of the circle?

Two particles $A$ and $B$ are moving on two concentric circles of radii $R_1$ and $R_2$ with equal angular speed $\omega$. At $t=0$ their positions and direction of motion are shown in the figure. The relative velocity ${\overrightarrow{v}}_{\mathrm{A}}-{\overrightarrow{v}}_{\mathrm{B}}$ at $t=\frac{\pi }{2\omega }$ is given by,

If the radius of the circular path and frequency of revolution of a particle of mass $m$ are doubled, then the change in its kinetic energy will be ($E_{\mathrm{i}}$ and $E_{\mathrm{f}}$ are the initial and final kinetic energies of the particle, respectively),

A body of mass $m$ is performing a UCM in a circle of radius $r$ with speed $v$. The work done by the centripetal force in moving it through ${\left(\frac{2}{3}\right)}^{\mathrm{rd}}$ of the circular path is,

A ball uniformly moves one-fourth of a circle of radius $R$ in time $T$. Let $v_1$ and $v_2$ be the magnitudes of mean speed and mean velocity vector. The ratio $\frac{v_1}{v_2}$will be,