Question

A particle at a distance of $1$m from 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.

Radius, $r=1$ $m$

Velocity is given by:

$\overrightarrow{V}=\dot{r}\hat{r}+r\dot{\theta }\hat{\theta }$

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

$⟹\frac{dr}{dt}\cdot \frac{dt}{d\theta }=r$

$⟹\dot{r}\cdot \frac{1}{\dot{\theta }}=r$

$⟹\dot{r}=r\dot{\theta }$

$\therefore \overrightarrow{V}=r\dot{\theta }\hat{r}+r\dot{\theta }\hat{\theta }$

$⟹\tan \theta =\frac{r\dot{\theta }}{r\dot{\theta }}=1$

$⟹\theta ={45}^o$