# Relative angular velocity of point with respect to another point

What is relative angular velocity of one point, say A, with respect to another point, say B? Both the points lie on the same rigid body which is rotating with constant angular velocity ω about a fixed axis.

Edit:

Here is the figure

The above body is rigid. For simplicity consider the rod joining A and B to be massless. So is the relative angular velocity of A with respect to B be zero? And if this is the case then how my question is different from Relative angular velocity

I think i'm missing something.

• I will return with a proper answer. Aug 30 '20 at 9:41
• They are at rest wrt each other thus answer is $0$ Aug 30 '20 at 14:26
• This question should be helpful. Aug 30 '20 at 14:29
• @Eli's answer below gives a mathematical proof that the relative angular velocity is zero. From an intuitive point of view, imagine sitting at point B and looking at point A. As the rigid body rotates, A remains fixed your field of view, so no angular velocity with respect to you. Aug 30 '20 at 14:30
• @dark_prince Even I feel the same as others and the answer should be zero. Aug 30 '20 at 14:37

I tried above to give an intuitive explanation of why the answer was zero. I will try to do the same for the new answer, ω (I'm flexible !). So again, imagine you are sitting at point B and looking at point A. As the rigid body rotates, A remains fixed your field of view, which led me before to say the relative angular velocity was zero. But as the object rotates, A faces different directions in the environment of the object so it appears to rotate once for each rotation of the object (as our moon rotates once a month despite always showing the same face to us). Make sense?

• Yes, perfectly! So that's what happening even if B's head remain fixed A will appear to move. Aug 31 '20 at 23:25
• Another thing to note: relative velocity should not be zero to have a non zero relative angular velocity. Imagine B to be situated below A at the same distance from axis; here relative angular velocity will be zero because relative velocity is zero. Your analogy fits here if we look from B, A will face the same direction i.e. upwards. B will always look upwards wrt to the environment Sep 1 '20 at 0:00
• I'm not following what you mean by below A, but it would seem no matter where B is an observer on B would notice that a spot on A faced in succession 360 degrees around the environment of the object thus again appearing to rotate once for each rotation of the object. Sep 1 '20 at 0:49
• Assume a cylinder, A is on top face and B is on bottom face, both at the boundary(distance from the axis is same). Now B will look at A in upward direction and wrt to the environment B's view will not change. So B will notice the angular velocity of A is zero. (here relative velocity of A wrt B is zero and so is relative angular velocity) Sep 1 '20 at 1:07
• The axis of rotation of the cylinder is through the center of the faces? If A had a mark on it, it seems to me that B would still see that mark in turn face 360 degrees around the environment of the object for each rotation of the object. So A would still have a relative angular velocity of ω. But @Eli's proof assumes they lie in a plane perpendicular to the axis of rotation ?? Sep 1 '20 at 2:35

The relative angular velocity$$~\vec{\omega}_{r}~$$ can obtain from this equation:

$$\vec{\omega}_{r}=\frac{\vec{R}_{AB}\times \vec{V}_{AB} }{\vec{R}_{AB}\cdot \vec{R}_{AB}}\tag 1$$

with :

$$\vec{R}_{AB}=\vec{R}_{B}-\vec{R}_{A}$$ $$\vec{V}_{AB}=\vec{V}_{B}-\vec{V}_{A}$$

equation (1)

$$\vec{\omega}_{r}=\frac{\left(\vec{R}_{B}-\vec{R}_{A}\right)\times \left(\vec{V}_{B}-\vec{V}_{A}\right) }{\vec{R}_{AB}\cdot \vec{R}_{AB}}\tag 2$$

with $$~\vec{V}_A=\vec{\omega}\times \vec{R}_A~$$ and $$~\vec{V}_B=\vec{\omega}\times \vec{R}_B~$$

equation (2)

$$\vec{\omega}_{r}=\frac{\vec{R}_{AB}\times (\vec{\omega}\times \vec{R}_{AB})}{\vec{R}_{AB}\cdot \vec{R}_{AB}}=\frac{(\vec{R}_{AB}\cdot \vec{R}_{AB})\vec{\omega} - ( \vec{R}_{AB}\cdot \vec{\omega})\vec{R}_{AB}}{\vec{R}_{AB}\cdot \vec{R}_{AB}}\tag 3$$

Now if A and B lie in the plane perpendicular to ω then $$\vec{R}_{AB}\cdot \vec{\omega} = \vec{0}$$

equation (3) becomes:

$$\vec{\omega}_{r} = \frac{(\vec{R}_{AB}\cdot \vec{R}_{AB})\vec{\omega}}{\vec{R}_{AB}\cdot \vec{R}_{AB}} = \vec{\omega}$$

thus the relative angular velocity is ω.

• Please see edited question. Aug 31 '20 at 12:11
• @dark_prince $\overrightarrow{a}\times \overrightarrow{b}\times \overrightarrow{a}=-\overrightarrow{b}\times \overrightarrow{a}\times \overrightarrow{a}=\overrightarrow{0}$
– Eli
Aug 31 '20 at 13:28
• Aug 31 '20 at 13:39
• You can't do that. Search vector triple product Aug 31 '20 at 13:40
• O.k I agree with you . I will correct my results
– Eli
Aug 31 '20 at 14:07