# Is angular velocity about any point in a rigid body always the same?

Suppose I have a disc that is in translational as well as rotational motion.

Question 1:

If we talk about the angular velocity of the disc, about which point is this angular velocity measured? (I mean to ask that is there any convention that if not mentioned angular velocity of a rigid body is the differential of angle measured with respect to a fixed point, say, the centre of mass of the body?)

Question 2:

Now suppose I have value of angular velocity about the centre of mass. Then will angular velocity of the body about any other point say P at a distance d from the centre of mass have the same value? If so then why?

• All points at rest with the rigid body give rise to the same angular velocity. Commented Jul 12 at 18:12
• The answer to point 2 is yes. There are a lot of similar questions with answers. For example: physics.stackexchange.com/a/680029/195949 Commented Jul 12 at 19:42
• Commented Jul 12 at 21:40
• To obtain the angular velocity, you need the rotation matrix of the rigid body, the rotation matrix is independent of the point that you choose, thus the angular velocity of any point is the same
– Eli
Commented Jul 13 at 7:18

Doubt 1

It is measured about the axis of rotation.

Doubt 2

Yes.

For a rigid body all points will complete one revolution, $$2\pi$$ radians, about any axis of rotation in exactly the same amount of time. Therefore they all have the same angular velocity in radians per second.

What does vary is the speed of the particles. The farther from the axis of rotation the greater the speed of the particle since the distance the particle has to travel per rotation is greater.

Hope this helps.

Each part of the rotating disk will have the same angular velocity. However, they will not all have the same tangential velocity (which will increase as you move further away from the axis of rotation). Tangental and angular velocity are related by $$\omega = \frac{v}{r}$$ when you move away from the axis of rotation, r increases and keeps $$\omega$$ constant. This answers both of your doubts.

If the body is rigid, then no point on or in it is allowed to move with respect to any other point. If one point moves, all other points must move. Rotation of the object insists that there is an axis piercing the object about which all these points are rotating around. If we let the rigid body be free, this axis will always pierce its center of mass. This is because rotation about any other point will result in the center of mass moving translationally (in a sense, orbiting this arbitrary point). Linear momentum is conserved, so this can't be the case.

In other words, the axis is on the center of mass because this is the only way the object can orbit itself while conserving angular and linear momentum.