Relative Angular Velocity

What is angular velocity?

Angular velocity can be defined as the rate of change of angular position in time. It is a vector quantity. Its direction is perpendicular to the position vector and the direction of its velocity.

From vector analysis, we know that the rate of change of a vector quantity A in time when it is rotated about a point with a constant angular velocity ω is given by,

dA/dt = ω X A

Therefore the rate of change of position vector r with respect to a point is given by

 dr/dt = ω X r

⇒ v = ω X r

V here is the velocity with respect to that point. This is the standard relation between angular velocity and velocity vector. If the direction of the position vector and angular velocity is mutually perpendicular, we can write the above equation as v = ωr. Here, in this case, the reference point is at rest.

Angular velocity can also be defined as the rate of change of angle in time.

ω = dΦ/dt

Relative velocity:

The relative velocity of a body can be defined as the velocity of that body in the frame of reference of another body that is also moving. The relative velocity of a body A with respect to a body B travelling at an angle Φ with B can be written as,

vAB = ( va2 + vb2 – 2vavbcosΦ)½

Relative angular velocity in rotation of a rigid body.

The angular velocity of a point on a rigid body about its axis of rotation will always be equal to the angular velocity of the rigid body. But if we take the point of reference away from the axis of rotation, then the relative angular velocity of a point on that rigid body will change depending on the point we choose.

For example, let us take a look at the rotation of the disc shown in the figure below,

 Here we have considered two points on the disc. Let the two points be called p and q. Here the disc rotates at an angular velocity ω. So the angular velocity of points p and q will be given by

vp = ωrp  and vq = ωrq ( rp and rq are the distance of p and q from the centre, respectively.)

Here we need to find the relative angular velocity of point q with respect to p. Therefore, let the angular velocity of q with respect to p be written as ωqp. So ωqp will be given by,

   ωqp =( perpendicular component of vqp ) / (rq – rp) , (vqp = relative velocity between p and q)

Therefore, ωqp = (vq – vp)/(rq – rp)

⇒ ωqp = (ωrq – ωrp)/(rq – rp)

⇒ ωqp = ω

This is the value of the relative velocity between the two points. Since the value is independent of the distance between the two points, we can say that ωqp = ω will be the relative angular velocity between any two points on that disc.

Further, we can use this analysis to assert that the value of relative angular velocity between two points on a rigid body will be equal to the angular velocity of the rotation of that body.

We can also say that the relative velocity is the rate of change of position vector with respect to a desired frame of reference. 

Note: We got the relative velocity here to be in a simple form, but for bigger calculations, the expression for relative angular velocity will become more complicated. 

Relative angular velocity between two bodies moving at an angle with another.

Let us consider two bodies x and y moving at speed vx and vy, respectively, as shown in the figure below.

Here we need to find the perpendicular component of the relative velocity between x and y.

On resolving the velocity vector perpendicular to the line joining the two bodies and parallel to the line joining the two bodies, we get the perpendicular component of the relative velocity to be

vxy丄 = vysinΦ1 + vxsinΦ2  

Therefore the relative angular velocity between the two bodies will be,

ωxy = vxy丄 / rxy

⇒ ωxy =( vysinΦ1 + vxsinΦ2) / rxy

Note: There is a difference between the rate of change of angle between two points and relative angular velocity. Even though it may look the same, the rate of change of angle between two points is simply the difference in their angular velocity and relative angular velocity that governs the rate of rotation of one point with respect to another.

Conclusion:

The relative angular velocity between two points is the rate of rotation of one point with respect to the other. This value may not always be equal to the difference between their respective angular velocities. The relative angular velocity between two points in a rigid body will be equal to the angular velocity of the rigid body. The relative velocity between two random bodies moving at a random angle will depend on their separation and their relative orientation. We always find the perpendicular component of the relative velocity to find the relative angular velocity.