# What are the necessary conditions for relative angular velocity to be defined? [closed]

I have certain doubts related to the definition of relative angular velocity .My textbook defines it in the manner given below:-

$$\omega_{AB}=\frac{v_{AB}}{r_{AB}}$$

$$\omega_{AB}$$ is the relative angular velocity of $$A$$ w.r.t $$B$$ and $$v_{AB}$$ is the component of relative velocity of $$A$$ w.r.t $$B$$ perpendicular to the line joining them.

So I want to make sure that is it necessary that the particle relative to which angular velocity is defined should execute circular motion relative to a fixed axis or not?

I have another doubt that whether the motion of both the particles should be in the same plane or not?

So,please provide the necessary conditions for relative angular velocity to be defined as I have searched a lot in my textbooks and websites but I am not able to find the good content on this topic.

## closed as off-topic by Aaron Stevens, Jon Custer, mpv, ZeroTheHero, tpg2114♦Jun 30 at 11:10

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• Your answer can easily be found online – Aaron Stevens Jun 18 at 18:14
• Like everything ... – InfiniteLooper Jun 18 at 18:29
• @Aaron Stevens but I want the clear concept on relative angular velocity but not angular velocity. – Unique Jun 18 at 18:31
• @InfiniteLooper there is a minimum amount of content on relative angular velocity on internet. – Unique Jun 18 at 18:32
• You can use your definition at any point in time since $V_{AB}$ and $r_{AB}$ are defined at all points in time. It is as if you are picking point $B$ to be your instantaneous reference point and calculating the angular velocity of $A$ about this point. I don't understand the issue here. – Aaron Stevens Jun 18 at 18:37

## 2 Answers

For two particles $$A$$ and $$B$$ in 3D the right formula for the angular velocity of $$B$$ w.r.t $$A$$ is

$$\overrightarrow \omega = \frac{\overrightarrow{r} \wedge \overrightarrow{v}}{r^2}$$ Where :

• $$\overrightarrow r$$ is the vector $$\overrightarrow{AB}$$ with length $$r$$, the distance between $$A$$ and $$B$$
• $$\overrightarrow{v}$$ is the velocity of $$B$$

This is a vector in your space with dimension the inverse of a time. This is exactly what you are after if you want to define an angluar speed (in $$rad.s^{-1}$$ for ex)

Furthermore, the vector part here is necessary to capture the fact that the particle may turn around the other in many ways. In other terms, the abolute value of the angular speed is not enough to characterize your movement, and you need a vectorial data.

Note that you can define this number for any trajectory of $$B$$ such that $$r$$ never vanishes, that is $$B$$ never meets $$A$$

Now, for a circular motion of $$B$$ around $$A$$ you angular speed will be independent of time. For example you can compute the angular speed of $$B$$ going straight (in the referential of $$A$$) it is a good exercise. This will depend on time and will vanishes when times flows to infinity.

For two points in 2D you does not have a vectorial product as in 3D but you just need a scalar value to characterize the angular speed and you wrote down the formula.

If two particles are in 3D and moves amon a plane, with normal $$\overrightarrow{k}$$ then $$\overrightarrow \omega = \omega k$$ where $$\omega$$ is given by you.

• Your formula is for angular velocity and not for angular momentum. – nasu Jun 18 at 20:46

If you're interested in concepts, then there are some point that could be of help for you.

Please note that I never heard of "relative angular velocity" before. There is good reason for this: it makes no sense to speak of angular velocity wrt a point.

To define angular velocity (actually velocity too) a reference frame, not a point, must be specified. This is because once fixed B a reference frame could rotate in any way keeping B steady, and of course velocities and angular velocities would change accordingly.

To answer your question, however, I'd need some context. I dare say that in point mechanics angular velocity is of very doubtful use. An exception may be circular motion, but the definition you're reporting seems to be thought for more general motions, otherwise the "perpendicular component" clause would be superfluous.

AFAIK the domain where angular velocity is really important is motion of rigid bodies. But I wouldn't get into that chapter unless you ensure me it's of real present interest for you. I only say that in case of rigid motion, once fixed the frame, there is but one angular velocity for the body - and it's just that that makes it a very useful concept.

I'm afraid you'll think that I haven't answered your questions, but I did my best to show you, so to say, an environment, without which those questions are ill-posed.