Question

The position vector of a particle is $\vec{r}=(a\cos \omega t)\hat{i}+(a\sin \omega t)\hat{j}$. The velocity of the particle is

• A. directed towards the origin
• B. directed away from the origin
• C. parallel to the position vector
• D. perpendicular to the position vector

Answer 1

Answer: (D)

Position vector of the particle
$(r)=(a\cos \omega t)\hat{i}+(a\sin \omega t)\hat{j}$
velocity vector
$\vec{v}=\frac{d\vec{r}}{dt}=(-a\omega \sin \omega t)\hat{i}+(a\omega \cos \omega t)\hat{j}$
$=\omega [(-a\sin \omega t)\hat{i}+(a\cos \omega t)\hat{j}]$
$\vec{v}\cdot \vec{r}=\omega [(-a\sin \omega t)\hat{i}+(a\cos \omega t)\hat{j})]\cdot [(a\cos \omega t)\hat{i}+(a\sin \omega t)\hat{j})]$
$=\omega \left[-a^2\sin \omega t\cos \omega t+a^2\cos \omega t\sin \omega t\right]=0$
Therefore velocity vector is perpendicular to the displacement vector.