13 votes

How can a pulsar slow down?

Radiation can indeed take away angular momentum. Thinking about this just from a classical point of view the flux of energy carried by electromagnetic waves (in vacuum) is $$ \vec{S} = \frac{1}{\mu_0} ...
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  • 111k
10 votes

Precise definitions for higher spin operators

I am not really sure about the scope of this question and the type of answer OP is looking for but computing those higher spin matrices/representations successively is relatively straight forward: ...
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  • 2,571
8 votes
Accepted

Null conserved angular momentum

Yes, if the angular momentum of a particle is conserved and it is zero, the particle must move along a straight line. Indeed, by using the triple product identity, we have $$ {\bf r} \times ( {\bf r} \...
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  • 23.3k
7 votes

A squared quantum operator is the same as using the same operator twice? So can't we find $L^2$ operator by using the $\mathbf L$ operator twice?

The squared operator $L^2$ is taken from a dot product: \begin{align} L^2 & = \mathbf L \cdot \mathbf L = L_x^2 + L_y^2 + L_z^2 . \end{align} Within that dot product, each product of operators (...
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5 votes
Accepted

A squared quantum operator is the same as using the same operator twice? So can't we find $L^2$ operator by using the $\mathbf L$ operator twice?

Yes, of course. Consider $\hat r \cdot \mathbf L =0\require{cancel} $. Just do the operator dot product. Recall $$ \partial_\phi ~\hat \phi= -\cos\theta ~\hat\theta -\sin\theta ~\hat r, ~~~~ \...
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4 votes
Accepted

If a boy tries to walk on a circular disc , the disc rotates but the boy remains stationary. Why it violets the law of cons. of angular momentum?

Assuming the disk is on a friction-less axle at its center, when the boy starts to walk, he will exert a backward force on the disk, and it will exert a forward force on him. He will move and gain ...
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  • 11.5k
4 votes
Accepted

(Why) Is orbital angular momentum conserved for point masses?

Imagine you had a particle traveling at constant $\mathbf{v_0} = \dot{z_0}\mathbf{\hat{k}}$ upwards. At time $t = 0$, your particle is at $\mathbf{r} = z_0 \mathbf{\hat{k}}$ and you apply an ...
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  • 179
4 votes

Can conservation of angular momentum be proven?

I am answering the title Can conservation of angular momentum be proven? Conservation laws are called laws because they are axiomatic, experiments and observations of the last centuries have given ...
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  • 221k
4 votes
Accepted

Deriving an identity with rotation generators

Well, 3 ways, at least. Geometrically: two right angle rotation of $\hat y$ around the z-axis. Behold. Convince yourself this is a Lie algebra identity, so it should hold for all nontrivial ...
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4 votes
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What is the angular momentum of a particle rotating around an axis in 3D?

The definition $\vec{L}_i = m \vec{r}_i \times \vec{v}_i$ is the correct one. Note that this vector will not lie along $\hat{n}$, and that's as it should be! Your derivation above makes the mistake ...
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3 votes

What is the angular momentum of a particle rotating around an axis in 3D?

they all take each particle's angular momentum to be ri×(mivi) instead of Ri×(mivi). Which is correct and why? I can understand your confusion since both terms do not seem to be the same. I think the ...
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  • 377
3 votes
Accepted

"Precession" of a free electron

The talk about precession is misleading. The electron is in an eigenstate of the Hamiltonian, and is thus in a stationary state. The only time evolution is a global phase: $$ \exp{(-i\frac E {\hbar} t)...
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  • 23k
3 votes

How can a pulsar slow down?

If the pulsar slows down, its angular momentum decreases. This implies that there's some angular momentum radiated away. Rotational energy decreases too, of course. There could be several ...
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  • 6,696
3 votes
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Is It true that If the angular momentum is zero with respect to any point then the system is at rest?

Suppose we have two choices of origin point $\mathcal{O}$ and $\mathcal{O}'$, where $\vec{R}$ is the vector pointing from $\mathcal{O}$ to $\mathcal{O}'$. It is straightforward to show that the ...
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3 votes

Is It true that If the angular momentum is zero with respect to any point then the system is at rest?

I wrote this before I saw the other answers by Michael Seifert and John Alexiou had been posted so there is some overlap, but I'll post it anyway. The answer depends on what you mean by rest. Zero ...
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  • 7,267
3 votes
Accepted

Is angular momentum conservation Galilean invariant?

The answer is no. To begin with, a Galilean transformation in three dimensional Euclidean space(time) consists of space-time translation: $(t,\vec{x})\rightarrow(t+s,\vec{x}+\vec{a})$ spatial ...
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3 votes

Precise definitions for higher spin operators

Giving a completely precise definition of everything would take a large amount of time, and would likely not be helpful. So instead here I will spell out the general picture, and any terms which are ...
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  • 1,028
3 votes

Is there a limit to how close faraway objects can get due to the conservation of angular momentum?

The mass factor in moment of inertia varies with velocity as $\gamma m$ where $\gamma$ is the tangent-velocity dependent Lorentz factor. Angular momentum can thus take any value zero to infinity for ...
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  • 8,598
2 votes

How does gravity make us rotate about the rotation axis of earth?

Gravity always acts to the center of the Earth (if you assume the Earth as a sphere). The rotation (which has nothing to do with gravity) creates an additional centrifugal force acting outwards ...
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  • 1,529
2 votes
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The mathematics of different particle rotations

Consider a point in space, and the effect a rotation would have on that point. Because the point has no internal structure (in other words, it is characterized entirely by its spatial location), a ...
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  • 50.9k
2 votes
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Does an irregular rigid body can only rotate in three directions?

A general rigid body has 3 principal axis. Suppose $I_1<I_2<I_3$. If it rotates around $I_1$ or $I_3$ without external torques, the angular velocity doesn't change and is parallel to the angular ...
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2 votes

Confusion about the Wigner-Eckart theorem

In principle there exist tensor operators of arbitrary angular momentum/spin - if you consider that the angular momentum generators $L_i$ themselves are a vector operator, then e.g. $L_i$ in a spin-3/...
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  • 106k
2 votes
Accepted

Angular Momentum about a Point

If the angular momentum is calculated with respect to Q, both the position vector and the velocity vector must related to Q. So, the second expression is correct. If the momentum is calculated with ...
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1 vote

Is angular momentum conservation Galilean invariant?

But this doesn't make sense to me because the rotational invariance of a system doesn't seem to change when I change to a new inertial reference frame. It can change. Torque depends on origin. As an ...
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1 vote

Angular Momentum Conservation and Conservation of Energy

The sphere is initially slipping. Only when its speed drops to $5u/7$ does it start rolling. The sphere's speed drops because it is moving on a rough surface. The friction between the slipping sphere ...
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  • 2,014
1 vote
Accepted

How do the inertia tensor varies when a rigid body rotates in space?

This has been answered below, but consider the rotation matrix $\mathrm{R}$ whose columns represent the local $\hat{x}$, $\hat{y}$ and $\hat{z}$ axis in the world coordinates. $$ \mathrm{R} = \begin{...
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  • 2,601
1 vote

Does an irregular rigid body can only rotate in three directions?

You state "an irregular rigid body has only 3 axes such that $ \vec{L}_{cm} $ and $ \vec{\omega} $ are parallel." I presume you mean the principal axes? In terms of the Cartesian principal ...
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  • 5,878
1 vote

(Why) Is orbital angular momentum conserved for point masses?

Angular momentum(and torque) does not necessarily mean that there are rotations in place. you can define angular momentum for straight lines, however, that isn't very useful to solve any problem. If ...
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  • 354
1 vote

Is It true that If the angular momentum is zero with respect to any point then the system is at rest?

Angular momentum tells you where in space the axis of the momentum vector is. Consider a particle moving in a straight line. The magnitude of angular momentum $| \vec{L} |$ and the magnitude of ...
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1 vote
Accepted

Is change of angular momentum of two rotating disks colliding w.r.t. time $L(t) = \omega_0 I(1-e^{-t/C})$?

with $$ \underbrace{I\,\frac{d\omega}{dt}}_{\dot L}=\tau\quad \Rightarrow\\ I\,\int_{\omega_i}^{\omega_f}\,d\omega=I\,(\omega_f-\omega_i)=\int \tau\,dt$$ where $\omega_f~$ is the final angular ...
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