Question

A particle is going parallel to x-axis with constant speed V at a distance 'a' from the axis. Find its
angular velocity about an axis passing the origin O, at the instant when radial vector of the particle
makes angle with the x-axis​

Answer 1

Given:

A particle is going parallel to x-axis with constant speed V at a distance 'a' from the axis.

To find:

Angular velocity when radial vector makes an angle of $\theta$ with x axis.

Calculation:

Perpendicular distance of object from x axis be r ;

$\therefore \: \dfrac{r}{a} = \sin( \theta)$

$= > r = a \sin( \theta)$

Now , angular velocity is calculated by the ratio of tangential velocity to the perpendicular distance from axis ;

$\boxed{ \sf{angular \: velocity = \dfrac{tangential \: v}{perpendicular \: distance} }}$

$= > \sf{ \omega = \dfrac{v}{r} }$

$= > \sf{ \omega = \dfrac{v}{a \sin( \theta) } }$

So, final answer is:

$\boxed{ \red{\sf{ \omega = \dfrac{v}{a \sin( \theta) } }}}$