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.