rotational kinematics - Fundamental definition of angular velocity for particle and rigid body

Several textbooks I've read so far seem to give different definitions of angular velocity vector of a particle moving with a velocity $\vec{v}$ in 3 d space.

One tells me to take reference line through the origin and simply find the rate of change of angle made by position vector with that line. I feel like this is valid only for 2d motion but I am not sure. Besides, how would you assign a direction to this? Another definition tells me to find the rate of change of angle about an axis of rotation but I don't understand the physical significance in this case as it's a complicated way of finding an angle.

EDIT:(to clarify first definition)

In the first case, we are taking a reference vector and finding the rate of change of the angle made by the position vector with it. This is not same as considering an axis. For example, let's say the angle is 90 degrees and.the particle moves in a circle, this means that the reference vector is the "axis". In this case, the angle made with reference vector is constant so angular velocity is 0. This is not possible when considering axis.

Are they different kinds of angular velocities or something? In general, when I say angular velocity of a particle in 3d about so and so, what am I referring to and what is the so and so?

Also, is the definition for **angular velocity ** of rigid bodies nothing but an extension of a particle or is it entirely new?

I gave the background thinking a blunt question would be closed because of how silly it sounds. Everything is too messy now so,

Condensed version of my question

When talking about angular velocity of a particle, do we always say "about an axis" ? It seems like there are infinitely many axes in case of a single particle at a given time, so is it useful to talk about angular velocity of a particle? Why can't I talk of angular velocity about a point? I understand that angular velocity itself is not a fundamental vector like velocity as it is up to us to define that angle. So what is the insight behind defining angular velocity as rate of change of angle about the axis which is the line around which all other points perform circular motion (I am assuming that I the definition of axis of rotation)?

Answer

You are trying to define angular velocity from linear velocity. It some way this is a backward way of thinking. Linear velocity is different at different points in a rotating frame. The intrinsic quantity is the rotation, and the measured quantity is the linear velocity. This is similar to how a force is an intrinsic quantity and the torque of the force is measured at different points. Also similar to momentum, and how angular momentum is measured at different points.

Consider the following framework:

1. 
   

   Linear velocity is the manifestation of rotation at a distance.

   
   

   For a particle on a rigid body, or a particle riding on a rotating frame the velocity vector $\mathbf{v}$ is a function of position $$\mathbf{v} = \mathbf{r} \times {\boldsymbol \omega}$$ _{Here $\boldsymbol \omega$ is the angular velocity vector, and $\mathbf{r}$ is the position of the axis of rotation relative to the particle}

   
   


2. 
   

   Pure translation is a special case of the above.

   
   

   When ${\boldsymbol \omega} \rightarrow 0$ and $\mathbf{r} \rightarrow \infty$ because the axis of rotation is located at infinity then all points move with the same linear velocity $\mathbf{v}$.

   
   



3. 
   

   Angular momentum is the manifestation of momentum at a distance.

   
   

   For a particle on a rigid body, or a particle riding on a rotating frame the angular momentum $\mathbf{L}$ about the rotation axis is a function of position $$\mathbf{L}= \mathbf{r} \times \mathbf{p} $$ _{Here $\mathbf{p}$ is the momentum vector, and $\mathbf{r}$ is the position of the axis of momentum (motion of center of mass) relative to where angular momentum is measured.}

   
   


4. 
   

   Pure angular momentum is a special case of the above When two bodies of equal and opposite momentum vectors combine (like in a collision) the resulting momentum is zero, but the angular momentum is finite. This is equivalent to non moving particle at infinity with $\mathbf{p} \rightarrow 0$ and $\mathbf{r} \rightarrow \infty$.

   
   


5. 
   

   Torque is the manifestation of force at a distance

   
   

   A force $\mathbf{F}$ applied at a distant location $\mathbf{r}$ has an equipollent torque of $$ {\boldsymbol \tau} = \mathbf{r} \times \mathbf{F} $$ _{Here $\mathbf{r}$ is the position of the force relative to the measuring point.}

   
   
   


6. 
   

   A force couple is a special case of the above

   
   

   When two equal and opposite forces act on a body, it is equivalent to a single zero force at infinity with $\mathbf{F} \rightarrow 0$ and $\mathbf{r} \rightarrow \infty$.