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The angular velocity $\vec{\omega}$ lies along the axis of rotation. And the angular momentum $\vec{J}$ is the cross product of $\vec{r} \times \vec{p}$. Which according to me should also lie along the axis of rotation. But I read in a book that the direction of angular momentum vector and angular velocity vector are not the same. Why is it so?
Angular momentum is conserved when no external torque is applied, I've learned that a long time ago and know the derivation. Yet, I've now been wondering about the following case: Let's consider a (
Angular momentum at the center of mass is $$\mathbf{L}_{cm} = I_{cm} \mathbf{\omega}$$ Linear velocity of the center of mass is $$\mathbf{v}_{cm} = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{cm}$$ where $\mathbf{r}_{cm}$ is the location of the center of mass relative to A .
Objects moving in straight lines have angular momentum, and it is conserved (if the system is isolated). Angular momentum requires the specification of a reference point and its value depends on the reference point, so in the case of straight-line uniform motion the angular momentum is different depending on what is chosen for the reference point.
The direction of the magnetic moment is perpendicular to the plane of the loop. Seeing that the angular momentum is also perpendicular to that plane, and having shown that their magnitudes are proportional, is all it takes to show that two vectors are proportional.
Angular momentum is the "moment of momentum", meaning it gives us an idea of how far is the linear momentum vector applied at. Torques involve the moment arm of a force, and angular momentum involves the moment arm of momentum. Particle Mechanics. Take a single particle moving in a straight line (in the absence of external forces).
The angular momentum of the nucleus is the combined contribution of the spin-orbit angular momenta of the constituent particles. In order for an entity to have orbital angular momentum of its own it must some conceptual orbit: electrons in the atom, protons and neutrons in the nucleus, atoms in a molecule.
Torque is defined as the moment of force or turning effect of force about the given axis or point. It is measured as the cross product of position and force vector while as angular momentum is the rotational analogue of linear momentum. It is axial vector and is measured as the cross product of position and momentum vector.
The moment of (linear) momentum, $\vec r \times \vec p$, where $\vec p = m\vec v$ is typically called angular momentum. The moment of a force, $\vec r \times \vec F$, is typically called torque. The first moment of mass of a collection of particles, $\sum_i m_i \vec x_i$, is the total mass times the center of mass position.
$\vec L = \bf I \vec \omega$ is the angular momentum of the body in an inertial frame (the space axes) where $\bf I$ is the inertia tensor and $\vec \omega$ is the angular velocity of the body about the point of rotation. Let $\vec M$ be the total external torque in the inertial frame.