Angular Velocity in terms of linear velocity

On roads we observe, many vehicles are travelling in a straight path.

When a car is travelling along a straight path, it changes its position from left to right or right to left in a period of time

The rate of change of the position of an object that is travelling along a straight path is called a linear velocity.

Linear velocity is the speed in a straight line (measured in m/s).

While angular velocity is the change in angle over time

Angular velocity only applies to objects that are moving along a circular path.

For instance, a racecar on a circular track, a giant wheel.

Angular velocity is 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.

In rotation, angular velocity is similar to linear velocity in translation.

For example, the planetary system of Jupiter and its moons (Ganymed, Europa, and IO).

The moons are in pure rotation while revolving around it.

Each moon is completing one revolution about Jupiter at a different time interval.

So, they have different angular velocities.

Let's study more about the angular velocity of any rotating body.

Consider one of the moon (Europa) of Jupiter. Let it be revolving at a distance $R$ from the Jupiter.

If in time $t$, it is displaced by an angle $\theta$. So, the linear displacement, $S$, for a small angle $\theta$ will be as,

Now, the linear displacement in terms of angular displacement will be as,

And on differentiating the displacements, we will get the velocities.

So, the linear velocity and angular velocity of an object revolving about an axis at a distance $R$ is as,

Now, let's see units of the different terms used till now.

The angular velocity is a vector quantity and thus have a direction.

In vector notation, the linear velocity, angular velocity and the radius of rotation are related as,

The direction of the angular velocity directs along the axis of rotation and given by the right-hand screw rule.

When we curl our fingers anticlockwise, the thumb points upwards direction and gives the direction of angular velocity $\omega$.

And while curling the fingers clockwise, the thumb points downward showing the direction of angular velocity.

For anticlockwise rotation, the angular velocity is taken as positive and for clockwise rotation, it is taken as negative.

Every point at the rigid body will have same angular velocity.

To understand this, let's consider a rod and locate three points O, P and Q, at the middle ends of the rod.

When the rod rotates about the point O with angular velocity $\omega$, the points P and Q will also rotate with it.

In time $t$, let the rod be displaced by an angle $\theta$.

So, the angular displacement of the points will also be the same and it will be $\theta$.

Since, the angular velocity is given by the time rate of change of angular displacement, the value of $\omega$ will be the same for each point.

Similarly, for every point of the rod, the angular velocity will be equal to $\omega$.

Revision

The angular displacement $\theta$ in radians in terms of linear displacement and radius is as,

The linear and angular velocity of any rotating body is as,

We apply the right-hand thumb rule to get the direction of angular velocity.

The End