- #1

Galois314

- 18

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I read in several places that if, for example, a point particle exhibits uniform circular motion about the z-axis within an osculating plane not equal to the x,y plane, then the angular velocity still points along the z-axis, even though the angular momentum does not (it precesses about the z-axis). However, this contradicts the other fact that I learned, which is that angular velocity is an instantaneous quantity and does not depend on the entire worldline of the particle.

An instantaneous quantity only depends on an infinitesimal segment of the worldline of the particle, and not on the entire worldline. For example, if we take instantaneous velocity at time t, then it only depends on the infinitesimal displacement in an infinitesimal time neighborhood about t, and not on the rest of the particle's worldline outside the infinitesimal neighborhood. However, if we take angular velocity as defined in the example above, we know that at any point in time t, the particle instantaneously travels in a straight line tangent to the circle of rotation. If we are only given what the particle is doing in an infinitesimal neighborhood about time t, then we cannot figure out what the circle of rotation is, and thus, we cannot determine the axis of rotation. Hence, the angular velocity as defined in the example above depends on the entire worldline of the particle, in contradiction to the fact that it is an instantaneous quantity.

That being said, I realized that if angular velocity is defined the following way instead, then it

Let the angular velocity of a particle be the instantaneous rate at which the position vector from the origin "sweeps out" angle. Consider a time t and an infinitesimal time neighborhood about t. Then, in the infinitesimal time dt, the direction of the radius vector will change by an infinitesimal angle dθ. The magnitude of the angular velocity shall be dθ/dt, and the direction of the angular velocity shall be normal to the plane containing the initial and final radius vectors. This plane turns out to also be the plane spanned by the instantaneous position and velocity vectors of the particle at time t. It is obvious at this stage that this definition of angular velocity is instantaneous and does not depend on what the particle is doing outside the infinitesimal neighborhood. It turns out that this angular velocity

There are two main advantages of this new definition of angular velocity for point particles: This definition applies to any type of trajectory, regardless of whether the particle is actually rotating about an axis or not (just like the angular momentum). This reflects the fact that this angular velocity is

This brings me to my two main questions:

Why probably do people say the angular velocity is an instantaneous quantity and is always along the axis of rotation, even though these conditions contradict each other, as I reasoned above? Why do physicists not use this new definition of angular velocity when dealing with point particles (I have yet to see a physics university website or textbook define angular velocity this way)?

Thank you for answering.

An instantaneous quantity only depends on an infinitesimal segment of the worldline of the particle, and not on the entire worldline. For example, if we take instantaneous velocity at time t, then it only depends on the infinitesimal displacement in an infinitesimal time neighborhood about t, and not on the rest of the particle's worldline outside the infinitesimal neighborhood. However, if we take angular velocity as defined in the example above, we know that at any point in time t, the particle instantaneously travels in a straight line tangent to the circle of rotation. If we are only given what the particle is doing in an infinitesimal neighborhood about time t, then we cannot figure out what the circle of rotation is, and thus, we cannot determine the axis of rotation. Hence, the angular velocity as defined in the example above depends on the entire worldline of the particle, in contradiction to the fact that it is an instantaneous quantity.

That being said, I realized that if angular velocity is defined the following way instead, then it

*is*instantaneous:Let the angular velocity of a particle be the instantaneous rate at which the position vector from the origin "sweeps out" angle. Consider a time t and an infinitesimal time neighborhood about t. Then, in the infinitesimal time dt, the direction of the radius vector will change by an infinitesimal angle dθ. The magnitude of the angular velocity shall be dθ/dt, and the direction of the angular velocity shall be normal to the plane containing the initial and final radius vectors. This plane turns out to also be the plane spanned by the instantaneous position and velocity vectors of the particle at time t. It is obvious at this stage that this definition of angular velocity is instantaneous and does not depend on what the particle is doing outside the infinitesimal neighborhood. It turns out that this angular velocity

*is*parallel to the angular momentum and is given by the vector (r × v)/r^{2}.There are two main advantages of this new definition of angular velocity for point particles: This definition applies to any type of trajectory, regardless of whether the particle is actually rotating about an axis or not (just like the angular momentum). This reflects the fact that this angular velocity is

*instantaneous*. The second advantage is that since the angular velocity is always parallel to the angular momentum, the constant of proportionality is a scalar rather than a rank-2 tensor, which is convenient. If we had used the original definition of angular velocity, then since the angular velocity is not parallel to the angular momentum, the moment of inertia had to be a rank-2 tensor.This brings me to my two main questions:

Why probably do people say the angular velocity is an instantaneous quantity and is always along the axis of rotation, even though these conditions contradict each other, as I reasoned above? Why do physicists not use this new definition of angular velocity when dealing with point particles (I have yet to see a physics university website or textbook define angular velocity this way)?

Thank you for answering.

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