Question

A particle is moving with constant speed v along the x-axis in a positive direction. Find the angular velocity of the particle about the point \[\left( {0,b} \right).\] When position of the particle is \[\left( {a,0} \right).\]

Answer 1

Hint: Angular velocity defined as the rate of velocity at which an object or a particle is rotating around a centre or a specific point in a given time period. The symbol of angular velocity is omega \[(\omega ).\] It is also known as rotational velocity. Here, to find the angular velocity, we will first draw the figure. Then by applying the formula of angular velocity, we will get our required answer.

Complete step by step answer:
We have been given that a particle is moving with constant speed v along the x-axis in a positive direction. We need to find the angular velocity of the particle about the point
\[\left( {0,8} \right),\] when position of the particle is \[\left( {a,0} \right).\]
Let us construct a figure using the given information by representing the motion of particle along x-axis, we get

So, from the above figure, we get \[{r^2} = {a^2} + {b^2}\] by using Pythagoras Theorem.
Now, we know that, Angular velocity, \[\omega = \dfrac{{v_{perp}}}{{r}}\]
where, ${v_{perp}}$ = perpendicular velocity
r = distance between the position of the particle and the point about which the angular velocity is to be calculated.
On putting the values in the above formula, we get
\[
{\omega = \dfrac{{vsin\theta }}{r}........(\because {v_{perp}} = \sin \theta )} \\
{\omega = \left( {\dfrac{v}{r}} \right)\left( {\dfrac{b}{r}} \right).......(\because \sin \theta = \dfrac{b}{r})} \\
{\omega = \dfrac{{vb}}{{{r^2}}}} \\
\]
Now on using \[eq.\left( 1 \right),\] we get
\[\omega = \dfrac{{vb}}{{{a^2} + {b^2}}}\]

Thus, the angular velocity of the particle about the point \[\left( {0,b} \right)\] is $\dfrac{{vb}}{{{a^2} + {b^2}}}.$

Note: Students should note that angular velocity is always equal to the perpendicular velocity divided by radius which actually means that velocity and radius should be perpendicular to each other.