Question

A particle at a distance of $1m$ front the origin starts moving such that $dr/d\theta =r$. where $(r,\theta )$ are polar coordinates. Then the angle between resultant velocity and tangential velocity component is

• A. $30$ degrees
• B. $45$ degrees
• C. $60$ degrees
• D. dependent on where the particle is

Answer 1

The correct option is B $45$ degrees

Given, the particle is at $1m$ from thr origin
$\Rightarrow r=1$

$v_{tangential}=\frac{d\theta }{dt}$

$v_{radial}=\frac{dr}{dt}$

$\overrightarrow{v_{resultant}}=\overrightarrow{v_{radial}}+\overrightarrow{v_{tangential}}$

Let $\varphi$ be the angle between the resultant velocity and tangential velocity componment

Then $tan\varphi =\frac{v_{radial}}{v_{tangential}}$

$\Rightarrow tan\varphi =\frac{\left(\frac{dr}{dt}\right)}{\left(\frac{d\theta }{dt}\right)}$

$\Rightarrow tan\varphi =\frac{dr}{d\theta }$

Given

$\frac{dr}{d\theta }=r$

$\Rightarrow tan\varphi =r=1$

$\Rightarrow \varphi =ta{n}^{-1}\left(1\right)=45{}^{\circ }$

Hence, the correct answer is OPTION B.