Question

A particle moves so that its position vector is given by $\vec{r}=\cos \omega t\hat{x}+\sin \omega t\hat{y}$ . Where $\omega$ is a constant. Which of the following is true ?

• A. Velocity and acceleration both are parallel to $\vec{r}$
• B. Velocity is perpendicular to $\vec{r}$ and acceleration is directed towards the origin.
• C. Velocity is perpendicular to $\vec{r}$ and acceleration is directed away from the origin.
• D. Velocity and acceleration both are perpendicular to $\vec{r}$

Answer 1

Answer: (B)

$\vec{r}=\cos \omega t\hat{x}+\sin \omega t\hat{y}$
$\vec{v}=-\omega \sin \omega t\hat{x}+\omega \cos \omega t\hat{y}$
$\vec{a}=-{\omega }^2\cos \omega t\hat{x}+\omega \sin \omega t\hat{y}=-{\omega }^2\vec{r}$
$\vec{r}.\vec{v}=0$ hence $\vec{r}\perp \vec{v}$
$\vec{a}$ is directed towards the origin.