Angular variables like $\omega$, $\alpha$, etc are properties of the moving particle or the observer observing them?

Question

Angular displacement ($\theta$) of a moving particle is the angle through which its position vector rotates, with respect to a reference point.

Angular velocity ($\omega$) of the particle is the rate at which its position vector rotates. Its physical meaning is "how quickly an observer (i.e, the reference point) observing the particle turns his head" while observing the motion of the particle.

Similarly, angular acceleration ($\alpha$) is the rate at which angular velocity changes. Physical meaning : The rate at which observer's head slows down or speeds up as it turns, to keep observing the moving particle.

My question is, are these angular variables properties of the particle? Or the observer's?

Answer 1

...are these angular variables properties of the particle? Or the observer's?

Depends on the frame of refernce of the observer. If the observer is observing the motion through a translatory frame of reference (no matter whether inertial or not), all the angular parameters of the body will be the same as observed in the ground frame.

However, if the obeserver is observing through a rotatory frame of reference, then the angular parameters will come out to be different from the corresponding ones when measured from the ground frame.

But what does a "translatory" or "rotatory" frame of reference mean?

A translatory frame of reference is the one which does not have any angular velocity when observed in the ground frame. Note that here we are talking about the angular velocity of a frame, precisely, we are talking about the angular velocity of the co-ordinate axes which constitute that frame. Examples of translatory frame of reference:

• a frame moving with a constant velocity (and zero angular velocity)
• the frame of a block (connected to a spring) undergoing simple harmonic motion
• a frame moving in circles such that the direction of the co-ordinate axes is constant in time (imagine sitting on the London Eye)

A rotatory refernce frame is the one which has a non zero angular velocity with respect to the ground frame. Again, we are actuall seeing the angular velocity of the co-ordinate axes which make up that frame, and not of the frame itself. Examples:

• the frame of reference attached to a person sitting on a fan
• the frame of refence attached to a spinning ballerina
• the frame of refernce attached to a spinning top

Note: In all the above examples, there were actually two choices (about the type of frames) available to us. Either we could make our frame rotate with the ballerina (case 1) (in which case it would be a rotatory frame of reference), or we could just fix the directions of our axes and just fix the frame to the ballerina's position and not her rotation (case 2) (in which case it would be a translatory frame of reference). Since the above examples are of rotatory frame of reference, thus we are considering the case 1 of choosing the frame of reference.