Questions tagged [angular-momentum]
The conserved quantity arising from a rotational invariance. Combine with rotational-dynamics for the classical mechanics approach and quantum-mechanics for the QM interpretation
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Question regarding the angular velocity of a ball rolling off of a sphere
Here is the actual problem in its entirety:
A uniform ball with radius $r$ is rolling down from the top of a fixed sphere with the radius $R$. Its initial velocity is negligibly small. What will be ...
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Impulse on a rod hinged at a point
I have a situation that is as follows:
A rod of mass $M$ is hinged at one of its end A on a smooth horizontal surface and can rotate about A without friction. A particle of mass $m$ moving on the ...
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When can we use conservation of energy [closed]
In this question, we can use conservation of angular momentum about the centre of pulley and this gives the right answer.
My question is why can't we use conservation of energy by doing the following
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An ant is sitting at edge of a rotating disc, if ant reaches the other end, after moving along diameter, then angular velocity will?
an ant is sitting at the edge of a rotating disc, if the ant reaches the other end, after moving along the diameter, then the angular velocity will increase, decrease, or remain constant? will it ...
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Why the 'up' and 'down' quark have different amount of charge but same amount of spin? [closed]
Why the 'up' and 'down' quark have different amount of charge but same amount of spin nevertheless they are opposite as their charges?As I understood the magnetic moment is the ability of a particle ...
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Intuition behind torque, rotational inertia and angular momentum
I'm reading about conservation of linear momentum and angular momentum. I understand the idea that angular momentum should be thought of as the "rotational analogue" of linear momentum, just ...
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How to calculate the deflection angle of a pool ball based on distance, exit angle, and applied force? [closed]
I'm working on a pool game and would like to apply a somewhat realistic view, so i need to know how to calculate the angle that a pool ball returns after touching the edge of the table. the problem is ...
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Physical Pendulum: Sign of Torque
I was looking into some of the motion equations for a physical pendulum that oscillates in small angles (using the approximation $\sin(\theta) \approx \theta$). Specifically, I was interested in ways ...
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Forces on us on a rotating Earth
So the earth is constantly rotating but it doesn't need a force to rotate. It'll rotate indefinitely.(?) But we and other masses on earth need a force on us to continue rotating along with earth? And ...
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Newton's second law for rotating body with changing mass
Newton's second law for a body with changing mass given as $$F=ma + \frac{dm}{dt}v$$ I need the version for rotational motion. By inspection, it seems that it would be $$\tau = I\alpha + \frac{dI}{dt}\...
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Commutator of spin operators
Suppose we are given $\left[S_X, S_Y\right]$, $\left[S_Y, S_Z\right]$ and $\left[S_Z, S_X\right]$, that is the spin operator commutation relations, can we then determine the matrix representation of ...
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What happens when you replace an identity matrix with a matrix full of ones?
In physics, we often use resolutions of identity
$$\sum_n |n\rangle\langle n|=\mathbb{I}$$ to simplify expressions. Sometimes, the "full matrix" (for lack of a better term) $$\sum_{m,n}|m\...
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Composition of angular momentum (quantum): how do we know that finding common eigenspace of $J^2$ and $J_z$ is enough for degeneracy?
I have some basic question on composition of angular momentum (actually spin in my case), I forgot some basis.
The fundamental commutation relations between $J_x,J_y,J_z$ (the three components of the ...
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Reaction torque of angled nutrunner
I have a question regarding the behavior of an right-angled, air/electrical powered nutrunner. More specifically, the reaction torque that the operator is subject to during operation. When the ...
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Rotations of spherical harmonics and Wigner $D$-matrices
I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the ...
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Basics about angular momentum in quantum mechanics
As usual let $|l,m\rangle$ denote an eigenstate of $\vec{L}^2$ and $L_z$. I know that
\begin{align}
\vec{L}^2 |l,m\rangle &= \hbar^2 l(l+1) |l,m\rangle, \\
L_z |l,m\rangle &= \hbar m |l,m\...
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What is the conversion ratio of linear to angular momentum when a ball hits a rod in space?
If the ball hits the rod at 90 degrees then the rod will start spinning, while also following the original trajectory of the ball. On what factors does the ratio between the two types of momentum ...
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Spin without quantum mechanics?
In Emergence of spin from special relativity some answers discuss how spin can arise in non-relativistic quantum mechanics (let's not enter into those details here). However it is also argued that you ...
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Emergence of spin from special relativity
I have pulled up and read as many answered questions as I can find here on why spin emerges as a consequence of making quantum mechanics compatible with special relativity- and still have problems ...
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How to prove Nitrogen atom with total angular momentum $L=2$ and $L=1$ are not anti-symmetric?
I'm working on Problem 5.13(d) in "Griffiths 《Introduction of Quantum Mechanics》 2nd Edition".
It asked to determine the nitrogen electron configuration by Hund's rule. And here is the ...
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Why Goldstein's book is claiming that radius and angle doesn't contain time variable even there is $\dot{r}$ and $\dot{\theta}$?
$$L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2)-V(r)$$
$$p_\theta=\frac{\partial L}{\partial \theta}=mr^2\dot{\theta}$$
$$\dot{p}_\theta=\frac{d}{dt}(mr^2\dot{\theta})$$
Goldstein wrote that $\dot{P}_\...
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Pulling a mass moving in a circle inwards [closed]
In this question, I couldn't derive the right answer if I equate
$$\text{tension} = \text{radial acceleration} = \frac{v^2}{r}$$
Why shouldn't I equate that?
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Why is the rotation about COM? [duplicate]
Suppose a ring is given to us with no hinge as such. Now a bullet comes and strikes the ring and gets embedded in the ring.
The ring will now have linear momentum and some rotation going on. Okay, the ...
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General solution of a ball elastically colliding with a spinning rod
I am working on finding the general solution to a disc colliding with a thin spinning rod in two dimensions floating in free space. The collision is perfectly elastic. The width of the rod is ...
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Inequalities of orbital angular momentum eigenvalues (show that is bounded)
If we apply the rising/lowering operators for angular momentum to a state $|{l,m}\rangle$, we get:
\begin{align}
L_+|{l,m}\rangle = C_+(l,m)|{l,m+1}\rangle \\
L_-|{l,m}\rangle = C_-(l,m)|{l,m-1}\...
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Does Initial Angular Velocity Affect Loss of Angular Velocity in Collisions?
I derived the following formula for change in angular velocity after a collision using conservation of angular momentum.
$$I_1 ω_i+I_2 ω_2i=I_1 ω_f+I_2 ω_2f$$
$$I_1 ω_i-I_1 ω_f=I_2 ω_2f-I_2 ω_2i$$
$$...
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Why have we not found an interior Kerr solution?
The Schwarzschild interior solution was found not so long after the exterior solution was found. I understand that Kerr solution is significantly more complicated and there are more conditions at the ...
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Changing the RPM of a frictionless spinning wheel in a box
Imagine a spinning wheel built into a hand size vacuum box. There is no friction between the axe bearings of the wheel and the box. Let's say that the wheel rotates with 60 RPM. Am I right if I assume:...
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What is the physical significance of $\langle L_x\rangle=\langle L_y\rangle$ and $\langle L_x^2\rangle = \langle L_y^2\rangle$?
If we find the expectation value $$\langle L_x\rangle = \langle L_y\rangle = 0$$ and $$\langle L_x^2\rangle = \langle L_y^2\rangle,$$ what is the physical significance that their values are equal?
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Why is $J^P=0$ for all even-even nuclei?
I have the following question, for which I have answered fully, but I am questioning the logic behind finding the value of $J^P$ for ${}^{20}_{10}\mathrm{Ne}$:
N.B, when I wrote "Number per ...
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Gyroscope force
I read that angular momentum is just pointed perpendicular to the plane of rotation (disk) as a convention.
Force is the change in momentum over time.
Is there actually a force in the direction of the ...
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Calculate energy levels from term symbols
Term symbols offer an extremely compact way of describing the different energy levels of a system. It takes into account many corrections, including the central field, spin-orbit interaction, and ...
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On which quantum numbers the energy eigenvalues of a multielectron system depends?
We usually read the case for a hydrogen atom where we find that the potential energy depends only on the variable $r$ and the energy eigenvalues are dependent on principal quantum number only. What ...
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Potential Energy for Non-conservative Forces
I was studying for the GRE and came across the problem below.
A small puck of mass $M$ is attached to a massless string that drops through a hole in a platform, as shown in the diagram above. The ...
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Is $L=Iω$ always valid? [duplicate]
I've read that the formula for angular momentum $L=Iω$ is valid only for symmetrical bodies. The other way of calculating it would be to find the cross-product of radius and linear momentum. I was ...
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Spin Half in 4-Dimensions
The Stern Gerlach experiment in 3-Dimensions provides us with conditions on what properties Spin-Half vectors must satisfy, from which we can build our basis states in $x, y, \text{ and } z$. The ...
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Optimizing for "sudden" change in angular momentum
I'm trying to create an elaborate impossible-to-juggle juggling club, by triggering a sudden change in angular momentum via a mechanism inside of the juggling club while it is in mid-air. The ...
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Rotational and translational angular momenta of a rigid body when they are in different direction
can I find the angular momentum of a rigid body undergoing translation and rotation by simply adding the angular momentum of its center of mass as a point particle vectorally to the spin angular ...
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Will two bodies initially connected to and revolving around each other, start spinning when disconnected?
Two extended bodies are connected with a string and revolve around each other (that is, around the center of mass of this system). No gravity, no external forces.
The string is cut, and they start to ...
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How to determine which irrep the Hilbert space of states carry?
Is the following statement ([1]) correct?
[1] If the universe has a symmetry under a group $G$, does this mean the Hilbert space carries a unitary representation formed by taking the direct product of ...
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Coriolis acceleration and conservation of angular momentum
I am trying to do a derivation the results of angular momentum, but I am running into an issue.
To make things simple, lets start with the derivation of angular momentum:
$$L = mr^2\omega$$
If we ...
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Find force on the disc due to hinge at its centre when the particle reaches the highest point [closed]
A uniform circular disc of mass $M = 4\,\text{kg}$ and radius $R = 25\,\text{cm}$ is suspended in a vertical plane and hinged at its centre 'O'. It is free to rotate about a horizontal smooth and ...
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Question about angular momentum and its absorption by quantum particles
I was thinking a little about how the absorption of angular momentum occurs from the point of view of QM. For example, suppose we have an atom A and an electron $e^-$.
The electron $e^-$ is ejected ...
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Total number of states for a two electron system
Let's say, I have the following electronic configuration, $1s2s$, and I'm trying to find all the possible states.
By looking at the configuration, it is obvious, that in the $|l_1,m_{l_1},m_{s_1},l_2,...
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Magnetic moment and angular momentum of electron
I recently got to know about something really interesting.
These are as follows:
1: The magnetic moment of an electron is, $\cfrac{ev}{2πr}$, where $e$ is the charge of the electron, $v$ is its ...
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Triplet states and Fine Structure
If we have a system, with total spin angular momentum given by $S$, then we have spin multiplicity equal to, $2S+1$. This spin multiplicity basically tells us the different spin states, this system ...
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Atomic levels energy
While reading about $LS$ coupling, and the fine structure of atomic energy levels, of various electron-electron configurations, I came across two different representations.
For example, if we consider ...
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Physical meaning of the canonical conjugate momenta in spherical coordinates
In cartesian coordinates, a particle under an arbitrary potential $U(x,y,z)$ will have a Lagrangian
$$L=\frac{m}{2}\left(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}\right)-U(x,y,z)$$
Consequently, the ...
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Angular momentum in rotating non-inertial reference frame
I know in inertial reference frame rigid body we can write $L=I\omega$ where $L$ is the angular momentum of the rigid body, $I$ is the moment of inertia and $\omega$ is the angular velocity of the ...
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Difference between Kepler motion and uniform circular motion
I'm trying to verify a line of thought.
Consider, we have two cases. In the first case, we have a rotating disc about a point, and we have two points marked on the disc at distance $r_1$ and $r_2$ ...