A particle moves such that its position vector $$\overrightarrow r \left( t \right) = \cos \omega t\widehat i + \sin \omega t\widehat j$$ where $\omega$ is a constant and t is time. Then which of the following statements is true for the velocity $\overrightarrow v \left( t \right)$ and acceleration $\overrightarrow a \left( t \right)$ of the particle :
Solution
$$\overrightarrow r \left( t \right) = \cos \omega t\widehat i + \sin \omega t\widehat j$$
<br><br>$\overrightarrow v = {{d\overrightarrow r } \over {dt}}$ = $- \omega \sin \omega t\,\widehat i + \omega \cos \omega t\widehat j$
<br><br>$\overrightarrow a = {{d\overrightarrow v } \over {dt}}$ = $- {\omega ^2}\cos \omega t\,\widehat i - {\omega ^2}\sin \omega t\widehat j$
<br><br>= $$ - {\omega ^2}\left( {\cos \omega t\,\widehat i + \sin \omega t\widehat j} \right)$$
<br><br>= $- {\omega ^2}\overrightarrow r$
<br><br>$\therefore$ $\overrightarrow a$
is antiparallel to $\overrightarrow r$ and it's direction towards the origin.
<br><br>$\overrightarrow v .\overrightarrow r =$ $$\omega \left( { - \sin \omega t\cos \omega t + \cos \omega t\sin \omega t} \right)$$ = 0
<br><br>So $\overrightarrow v \bot \overrightarrow r$.
About this question
Subject: Physics · Chapter: Kinematics · Topic: Motion in a Straight Line
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