Question

The position vector of the particle is $r\left(t\right)=acos\omega t\hat{i}+asin\omega t\hat{j}$ , where $a$ and $\omega$ are real constants of suitable dimensions. The acceleration is

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

Answer 1

Answer: (A)

Given that, $r(t)=a\cos \omega t\hat{i}+a\sin \omega t\hat{j}$
$\because \vec{v}=\frac{dr(t)}{dt}=-a\omega \sin \omega t\hat{i}+a\omega \cos \omega t\hat{i}j$
$\vec{a}=\frac{d\vec{v}}{dt}=-a{\omega }^2\cos \omega t\hat{i}-a{\omega }^2\cos \omega t\hat{j}$
$\vec{a}\cdot \vec{v}=a^2{\omega }^3\sin \omega t\cos \omega t-a^2{\omega }^3\sin \omega t\cos \omega t$
$\Rightarrow \vec{a}\cdot \vec{v}=0$
Above result implies that acceleration is perpendicular to velocity.