Question

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

• A. Parallel to position vector
• B. Perpendicular to position vector
• C. Directed towards origin
• D. Directed away from the origin

Answer 1

The correct option is B Perpendicular to position vector

The position vector of a particle,

$\overrightarrow{r}=asin\omega t\hat{i}+acos\omega t\hat{j}$

Velocity, $\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=a\omega cos\omega t\hat{i}-a\omega sin\omega t\hat{j}$

The dot product of velocity vector and position vector is

$\overrightarrow{v}.\overrightarrow{r}=(asin\omega t\hat{i}+acos\omega t\hat{j}).(a\omega cos\omega t\hat{i}-a\omega sin\omega t\hat{j})$

$\overrightarrow{v}.\overrightarrow{r}=-a^2\omega sin\omega tcos\omega t+a^2\omega sin\omega tcos\omega t$

$\overrightarrow{v}.\overrightarrow{r}=0$

$|\overrightarrow{v}||\overrightarrow{r}|cos\theta =0$

$cos\theta =0$

$\theta ={90}^0$

The velocity of the particle is perpendicular to the position vector.

The correct option is B.