Question

The position vector Of a particle $\overrightarrow{R}$ as a function Of time is given by $\overrightarrow{R}$ = 4 sin(2$\mathrm{\pi }$t)$\hat{i}$ + 4 cos(2$\mathrm{\pi }$t)$\hat{j}$, where R is in meters, t is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along x-and y-directions, respectively. Which one of the following statements wrong for the motion of particle?

(1) Path of the particle is a circle of radius 4 m
(2) Acceleration vector is along —$\stackrel{\rightharpoonup }{R}$
(3) Magnitude of acceleration vector is $\frac{v^2}{R}$, where v is the velocity of particle
(4) Magnitude of the velocity of particle is 8 meter/second

Answer 1

$$\begin{gathered} R⃗=4\sin \left(2\pi t\right)iˆ+4\cos \left(2\pi t\right)jˆ \\ thisisthevectoreqnofthecircularmotioninclockwisedirection. \\ V=\frac{dR}{dt}=8\mathrm{\pi cos}\left(2\pi t\right)iˆ-8\pi \sin \left(2\pi t\right)j \\ a=\frac{dV}{dt}=-16{\mathrm{\pi }}^2\sin \left(2\mathrm{\pi t}\right)\mathrm{i}-16{\mathrm{\pi }}^2\cos \left(2\mathrm{\pi t}\right)\mathrm{j} \\ \mathrm{a}=-4{\mathrm{\pi }}^2\left(4\sin \left(2\mathrm{\pi t}\right)\mathrm{i}ˆ+4\cos \left(2\mathrm{\pi t}\right)\mathrm{j}ˆ\right) \\ \mathrm{a}=-4{\mathrm{\pi }}^2\left(\overline{\mathrm{R}}\right) \\ \mathrm{hence}\mathrm{accn}\mathrm{vector}\mathrm{is}\mathrm{along}-\overline{\mathrm{R}} \\ \mathrm{Regards} \end{gathered}$$