# Angular Velocity Vector; Relationship to Angular Momentum

• Galois314
In summary, the conversation discusses the definition of angular velocity for point particles and how it relates to the particle's worldline and instantaneous quantities. The original definition states that angular velocity is parallel to the z-axis, but this contradicts the fact that it is an instantaneous quantity. A new definition is proposed, where angular velocity is defined as the rate at which the position vector "sweeps out" angle, and is always parallel to the angular momentum. This definition applies to any type of trajectory and is convenient as the constant of proportionality is a scalar rather than a rank-2 tensor. The conversation ends with two questions about why the original definition is used and why the new definition is not commonly used in physics. A link and citation are
Galois314
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 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)/r2.

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)?

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## What is the definition of angular velocity vector?

The angular velocity vector is a measure of how fast an object is rotating around a fixed axis. It is a vector quantity, meaning it has both magnitude and direction.

## How is angular velocity vector related to angular momentum?

Angular velocity vector and angular momentum are directly related. The angular momentum of an object is equal to the product of its moment of inertia and its angular velocity vector.

## How do you calculate the magnitude of angular velocity vector?

The magnitude of angular velocity vector can be calculated by dividing the angular displacement by the time it takes to make that rotation. It is typically measured in radians per second (rad/s).

## What is the direction of the angular velocity vector?

The direction of the angular velocity vector is perpendicular to the plane of rotation, following the right hand rule. This means that if you curl your fingers in the direction of rotation, your thumb will point in the direction of the angular velocity vector.

## How does the angular velocity vector change in an object's motion?

The angular velocity vector can change in both magnitude and direction as an object's motion changes. This can occur due to external torques acting on the object, changes in the object's moment of inertia, or changes in the object's angular momentum.

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