What is Angular momentum: Definition and 1000 Discussions
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.
In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector; the latter is p = mv in Newtonian mechanics. Unlike momentum, angular momentum depends on where the origin is chosen, since the particle's position is measured from it.
Just as for angular velocity, there are two special types of angular momentum of an object: the spin angular momentum is the angular momentum about the object's centre of mass, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The total angular momentum is the sum of the spin and orbital angular momenta. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.
Angular momentum is an extensive quantity; i.e. the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body or a fluid the total angular momentum is the volume integral of angular momentum density (i.e. angular momentum per unit volume in the limit as volume shrinks to zero) over the entire body.
Torque can be defined as the rate of change of angular momentum, analogous to force. The net external torque on any system is always equal to the total torque on the system; in other words, the sum of all internal torques of any system is always 0 (this is the rotational analogue of Newton's Third Law). Therefore, for a closed system (where there is no net external torque), the total torque on the system must be 0, which means that the total angular momentum of the system is constant. The conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, and the precession of gyroscopes. In general, conservation limits the possible motion of a system but does not uniquely determine it.
In quantum mechanics, angular momentum (like other quantities) is expressed as an operator, and its one-dimensional projections have quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, implying that at any time, only one projection (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the axis of rotation of a quantum particle is undefined. Quantum particles do possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion.
Hello,
I've been doing some research on gravitational waves since their discovery, and I found that all of the places I looked were missing an important piece of information, that is: What is the mechanism by which angular momentum is being conserved.
All of places that I've searched will...
Two free electrons approach each other, so they start to emit photons due to bremsstrahlung. Where does the angular momentum carried away by the emitted photons come from?
Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators L^2 and L_z. We know that...
Two things I'd like to discuss:
1. The conservation of angular momentum. If you have two discs rotating on the same fixed rigid axis, will these nullify each other? I.e. Create no net angular momentum?
2. How / is it possible to convert angular momentum to linear momentum in the sense to be...
Hey, I am new to the forum, and would certainly appreciate help in understanding spin - although I realize that perhaps no one really understands spin.
If a beam of spin polarized electrons are absorbed in a target, then the target will start to rotate. This kind of makes sense on a...
Good morning all,
Recently in a modern physics course of mine, my professor was covering the topic of energy levels and ionization energies and it included a diagram very similar to this one:
While it is interesting to learn that these diagrams correspond to a very specific and strict set...
Let's say a train powered by electric third rail drives around without friction on a circular track, and light is shining out of the train windows, said light carries angular momentum, like light emitted from rotating or revolving things tend to do.
Where does that angular momentum come from...
Homework Statement
A block of mass m slides down a frictionless ramp from height h above the floor. At the base of the ramp it collides and sticks to the lower end of a uniform rod, length L, mass 2m, that is suspended about a pivot at point O, about which it is free to rotate. Express...
Homework Statement
Two objects with mass of m are connected with a rod with length 2l and with no mass. The center of the rod is pinned so that it can spin. Object with mass M comes with speed v and sticks to m. There is no friction.
1) What is the angular speed w after collision?
2) FInd the...
Homework Statement
Question - Models of global warming predict that large sections of the polar ice caps will melt. Explain what effect this will have on the rotation of the Earth, however slight.
Homework Equations
L = Iw
The Attempt at a Solution
Assuming polar ice caps protrude the earth...
If there is no net torque acting on a system total angular momentum of the system will be conserved as well as angular momentum of each body present in the system will be conserved.
And if there are two bodies /two charges present as a system and one of them (let's say body 1 )produces torque...
Homework Statement
I am trying to show that as the alpha particle is being scattered, the angular momentum magnitude at point M is m r^2 \frac{d \phi}{dt}.
Diagram:
Homework Equations
Linear Momentum: ##\vec L = \vec r \times \vec p ##
Repulsive Coulomb Force: ##\vec F = \frac{k 2Ze^2}{r^2}...
Putting spin on the ball improves it’s stability (the spin imparts angular momentum and it takes an outside torque to change angular momentum), but at a cost. As you know, for a given input of energy, spin consumes some of that energy and leaves less for translation.
Part A: So, How much more...
Homework Statement
There are two problems:
(A) Consider two identical billiard balls (spheres), each of mass M and radius R. One is stationary (ball 2) and the other rolls on a horizontal surface without slipping, with a horizontal speed v (ball 1).
Assume that all the frictional forces are...
Homework Statement : [/B]
A vector is perpendicular to B vector, and they stay still, relative to the body. No torque is applied on the asteroid, although he dissipates very little rotational kinetic energy, due to drag on dust clouds. It was also determined that the asteroid is a long body...
Homework Statement
Attached
Homework Equations
##J_+|j,m⟩ = ћ\sqrt{j(j+1)-m(m+1)} |j,m+1⟩##
##J_-|j,m⟩ = ћ\sqrt{j(j+1)-m(m-1)} |j,m-1⟩##
##J_z|j,m⟩ = mћ |j,m⟩##
##J^2|j,m⟩ = ћ^2j(j+1) |j,m⟩##
The Attempt at a Solution
Hi there,
For part a, the expression we're looking for is given, but...
Homework Statement
Is angular momentum of a projectile in projectile motion conserved ?[/B]Homework EquationsThe Attempt at a Solution
Gravitional force vector is acting along direction of displacement vector. Thus external torque is 0. Thus angular momentum is conserved throughout projectile...
Homework Statement
Is it possible for the respective angular momenta of each individual particle in a system to be zero, but the system's collective angular momentum be nonzero?
For example, a puck on a frictionless air table moves (without spinning) toward a point on a rod that is not the...
When a nucleus gamma decays, the gamma has its intrinsic spin of ##1\hbar##, but it can also carry away a significant amount of angular momentum in addition to that. Quadrupole radiation is very common, and in exceptional cases you can even get gammas with 5 or ##10\hbar##.
Now suppose I do the...
I've been working on my SR book http://www.lightandmatter.com/sr/ and its presentation of a suite of related mathematical tools such as the Levi-Civita tensor. This is aimed at the upper-division undergraduate level, although there are optional sections at the end of some of the chapters that go...
Homework Statement
https://scontent-sea1-1.xx.fbcdn.net/hphotos-xft1/v/t35.0-12/12414351_10206719685063143_386848762_o.jpg?oh=16c004481b7417fad921c37acc4942be&oe=56793416
Homework Equations
Angular momentum: H= Iw
Parallel axis theorem: Io = I + Md^2
Moment of Inertia of thin plate about it's...
Hi,
Two questions.
1) I'm having trouble understanding the stability of the stable Lagrangian points (L_4 and L_5); Wikipedia explains that if an object in the L_4 or L_5 of a planet is pushed closer towards the common center of gravity of the Sun and the planet, the increased speed that comes...
Homework Statement
A rod that is fixed to a vertical wall is attached through the middle of a solid disk. A piece of the disk crumbles off when the disk is spinning and rises vertically. The system is the chip and the spinning disk.
Assume no friction.
A. How does the direction of the angular...
Dear guys,
Recently, i am confused with a problem in my textbook of mechanics. The question is,
suppose there is a disk, placed horizontally, rotate about its center with angular velocity ω. A ball move with respect to the center of the disk in a trace of Archimedean spiral r=αθ. The angular...
Homework Statement
2. The attempt at a solution
I set the initial angular momentum of the disk = to the sum of : rod's angular momentum, angular momentum of disk, rod's center of gravity and disk's center of gravity. With the reference point being at B.
Why is the velocity of the rod's...
Homework Statement
A device consists of eight balls each of mass 0.6kg attached to the ends of low-mass spokes of length 2.1m, so that the radius of rotation of the balls is 1.05m. The device is mounted in the vertical plane. The axle is held up by supports that are not shown, and the wheel is...
I would like to prove that the angular momentum operators ##\vec{J} = \vec{x} \times \vec{p} = \vec{x} \times (-i\vec{\nabla})## can be used to obtain the commutation relations ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##.
Something's gone wrong with my proof below. Can you point out the mistake...
Homework Statement
[/B]Homework Equations
L = I ω
L= r x p
The Attempt at a Solution
For b)
My proffesor found the moment of inertia at the point at which the wheel touches the ground, and used the formula L = I ω
What I don't understand is why can't one use L= r x p to solve the...
Homework Statement
Indiana Jones standoff. Person A fires a 20g bullet at 500 m/s at Person B, who is holding a sword. The bullet sticks to the sword. The angular momentum of the sword is 2.225 kgm^2 / s. The moment of inertia about the center of mass of the sword is .7082 kgm^2. The...
Homework Statement
A metal bar of length r is attached to a pivot point at (0,0) in the XY plane. A steel ball is projected toward the free end of the rod and strikes it at the tip. Show how to calculate the angular acceleration of the bar after the collision and draw a vector diagram to scale...
I know that the angular momentum is conserved in the below example, but intuitively I am struggling. Anyway, here goes!
Homework Statement
A bullet that is traveling without spinning hits and sticks into a filled disk tilted on its side an arbitrary distance "d" away from the center of mass...
Homework Statement
A 2.0-m measuring stick of mass 0.175 kg is resting on a table. A mass of 0.500 kg is attached to the stick at a distance of 74.0 cm from the center. Both the stick and the table surface are frictionless. The stick rotates with an angular speed of 5.30 rad/s.
(a) If the...
Homework Statement
A particle of mass m moves under the influence of a central force
F(r)=-mk[(3/r^2)-2a/r^3]rhat
Show that if the particle is moving in a circular orbit of radius a, then its angular momentum is L=mh=m√(ka)
Homework Equations
L=mvr for circular orbit
The Attempt at a...
Starting with the orbital angular momentum of the ith element of mass, $ \vec{L}_I = \vec{r}_I \times \vec{p}_I = m_i \vec{r}_i \times \left( \omega \times \vec{r}_i\right) $, derive the inertia matrix such that
$\vec{L} =I\omega, |\vec{L} \rangle = I |\vec{\omega} \rangle $
I used a X b X c...
Homework Statement
Consider two forces F and -F that act at different points on an extended object. Show that the net force of this combination is zero and that the torque about any point P is independent of the location of P, and depends only on the separation of the two points at which the...
I've been a fan of this forum for a while, but never signed up for it, today I'm stuck on this problem and can't find anywhere other than Chegg but I don't have a premium account.
1. Homework Statement
A meter stick is at rest on frictionless surface. A hockey puck is going towards the 30cm...
Homework Statement
A thin ring of radius r is constrained to rotate with constant angular velocity ω as shown in attached picture. Let the linear mass density of the ring be ρ(θ)=ρ0(2+sin2θ) where ρ0 is constant.
a) Find the angular momentum L of the ring about O, at the instant the ring is in...
Homework Statement
What is the best way to evaluate the rotation of a angular momentum eigenket (## | l,m \rangle ##) around an Cartesian axis?
2. The attempt at a solution
I've tried to use the method in...
Angular momentum is conserved in a closed system. Is thermal isolation required too?
An example special case in mind is a closed cavity high speed rotation.
It contains high pressure gas and the thermal flow is driving convection currents creating turbulence and or a heat pumping loop...
Ok, so as far as I understand it, it is impossible to turn linear momentum (p) into rotational momentum (L), but I don't quite understand why. The main thought experiment I have in my head is this:
A ball in space is traveling with a momentum mbVb, and gravity and friction are assumed to be...
[Moderator's note: thread spun off from a previous one on a related but different topic.]
Well that's another question that has been puzzling me for quite some time and would love to know the answer to it...
Imagine 4 tubes of equal length, making a square, with 4 identical photons running...
Hi folks!First of all, English is not my native language so I hope there is not much misleading spelling mistakes, discrepancies and inaccuracy.
I’m currently working with a hobby-project of a flying drone/UAV with two adjustable angled rotors. I’m able to make it fly more or less in a way it...
Hey everyone,
I just made an account because I have a problem concerning angular momentum and precession.
In the picture below you see the vectors l1 and l2 that make up total orbital angular momentum L precess around L. I can get my head around why that is the case. The same for s1 and s2...
I know that the angular momentum of the moon with respect to the Earth can be calculated by L = Iw but if the L of the moon is k * Learth-sun the there will be way more eclipses, one of each every 28 days. So the momentum of the moon must change over time, how does it work?
Homework Statement
Show that the quantization of angular momentum implies that the kinetic energy of the electron is quantized as K=nhforb/2, where forb is the frequency of rotation. Assume circular orbit.
Homework Equations
Radial acceleration:
arad = v2/r = (4π2r/T = 2*π*v/Tr = nħ
KE =...
Homework Statement
A girl places a rock in a sling and begins spinning it in a circular path above her head. By exerting a force on the sling, she increases the rock's speed while it goes around her head a few times, before releasing the rock. Options:
1. She exerts a force on the rock that is...
Hi everyone,
I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function Y_{lm}(\theta,\phi) with
L_zY_{lm}=mħY_{lm} and L^2Y_{lm}=l(l+1)ħ^2Y_{lm}
where L_z=-iħ\frac{\partial}{\partial \phi}
and...