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.
We are taught that all fermions have spin ##\frac{1}{2}##, short for spin angular momentum ##\frac{\hbar}{2}##, which can be added to the orbital angular momentum. Considering spin is a kind of angular momentum, it must be dependent on the mass (or moment of inertia) of the particle. However...
We all know we need to apply conservation of angular momentum here. This necessarily leads to a difference in mechanical energy. Since initial rotational inertial is less than final rotational inertia, there is a loss of mechanical energy. However, I have not been able to convince myself what's...
So my book states torques perpendicular to the fixed axis of rotation tend to tilt the axis , however we assume sufficient restraints exist so these torques are simply ignored.
It follows that angular momentum perpendicular to axis remians constant.
(See image )
My question is that if a rod is...
First off, I do know how to solve this problem. We use the principle of conservation of angular momentum about the centre of mass of the system which comprises of the loop and the bullet to obtain option B. My doubt is, why do we just not use the principle about the centre of the loop? Where is...
Hello,
As far I know, in a closed system both, linear and angular monentums, are conserved.
İmagine such a scenario: everything is motionless, both momentums zero initially, then from a disk are fired (compressed spring push) two equal mass balls at same speed but opposite direction. Now balls...
We know from Kepler's Second Law, that a line from a planet to the Sun sweeps an equal area in equal time. The planet's velocity increases when it orbits closer to the Sun. The area swept is a triangle 1/2 r v sin (theta) = constant, asserted by Kepler, based on observations by Tycho Brahe...
I am struggling with the latter, and think that I somehow need to assume ##f## is real-valued to proceed?
My work:
The position distributions are equal since
$$P_{-m}(\mathbf{x}) = |\Psi_{-m}(\mathbf{x})|^2 = |f(r)Y_l^{-m}(\theta,\phi)|^2 = |f(r)(-1)^m(Y_l^{m})^*|^2 =...
This is a spin-off of a similar problem posted here in which the cylinder gathers snow as it rolls down an incline. I think one has to understand the snow-gathering process before attempting the more complicated case. A horizontal surface makes that easy but because it is a different problem, I...
First, I can say that the velocity of the mass leaving the system is equal to ##w_o## (k) x ##L/2## (i) which tells me that its velocity will be w_o L/2 (j)
Now, since the net external force is equal to 0, (linear) momentum is conserved, so:
At first the velocity of the center of mass was 0 and...
I just calculated the Lagrangian of a particle of mass ##m## in a radially symmetric potential ##V(r)##. I thought it would be a good idea to switch to spherical coordinates for that matter. What I get is
$$
L = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\varphi}^2...
Hi.
Question as in the summary.
Spin has no obvious classical interpretation but it is often a conserved quantity and considered as some sort of angular momentum. What do you need to establish that spin is a conserved quantity? I'm finding references to situations where spin is not a...
How should I calculate the angular momentum carried by a current carrying circular wire? Is it correct to consider the angular momentum of the electrons moving with drift velocity? Like
##L = n m_e v_{drift} r## where ##r## is radius of the loop, and ##n## is total number of electrons moving in...
I've already solved the orbital speed by equating the kinetic and potential energy in the circle orbit case.
$$\frac{1}{2}mv^2 = \frac{1}{2}ka^2.$$And so $$v^2 = \frac{k}{m}a^2$$Now when the impulse is added, the particle will obviously change course. If we set our reference point in time just...
Here is the problem statement along with the figure.
Here, I take the right-ward and anti-clockwise directions to be positive.
After the ball collides with the wall, its angular velocity remains the same and its velocity changes direction while remaining the same in magnitude.
Using the...
Dear People,
I have a question. I have a rotating tube like a line that has two end and one of them is the center of rotation (like a watch arrow just tube), and inside the tube a mass that is moving towards the center of rotation. So the masses moving along the line aka along the length of the...
I understand how a massive, electrically charged spinning ball would have both angular momentum and a magnetic dipole, and i can see how the
Stern–Gerlach experiment shows that the magnetic dipole of an electron is quantized.
What kind of experiment demonstrates
a connection between electron...
Hi, I have a question.
Let us say we have the wave function as with eigen value and base eigen value of:
##!\psi >\:=\:\frac{1}{6}\left(4!1,0,0>\:+\:3!2,1,1>\:-1!2,1,0\:+\:\sqrt{10}!2,1,-1>\right)##
I need to find <Ly^2>
the solution of the problem according to answers, is demanding that...
Cohen tannoudji. Vol 1.pg 702"Now, let us consider an infinitesimal rotation ##\mathscr{R}_{\mathbf{e}_z}(\mathrm{~d} \alpha)## about the ##O z## axis. Since the group law is conserved for infinitesimal rotations, the operator ##R_{\mathbf{e}_z}(\mathrm{~d} \alpha)## is necessarily of the form...
I read that quantum spin is the measure of the angular momentum of a quantum object. Suppose you have a rotating Thing 1. Quantum objects bounce off of it then collide with Thing 2. Will this transfer angular momentum from Thing 1 to 2, causing it to rotate?
For this problem,
Why for part (a) the solution is,
Is the bit circled in red zero because since the putty is released at a very small distance above the rod it velocity is negligible?
Also for part (d) the solution is
I did a computation of the initial and finial kinetic energies of the...
I have typed up the main problem in latex (see photo below)
It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.
The section Kepler’s Second Law here describes the above equation.
In this problem,
##\text{r = D, m = M and v = V}##
What is the way to go about finding out ##\theta## as shown in Figure 13.21?
So i was able to solve the angular velocity part but i don't know how to find the velocity of centre of mass . For the first part i simply conserved momentum about COM because if i consider the particles as a part of the same system as rod the collision are internal forces . I am mainly...
Specifically given a purely magnetic hamiltonian with some associated vector potential :
$$ H = \dfrac{1}{2m} (\vec{p} - q\vec{A}) $$
How can I deduce if $$ \vec{L} = \vec{r} \times \vec{p}$$ is conserved? ( $$\vec{p} = \dfrac{\partial L}{\partial x'}$$, i.e. the momentum is canonical)
i,j,k arevector
I know L=P*r=m*v*r=m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k=mabωk.
but why m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k.I need some detail.
please help me.
I think that the quantum numbers are l=1 and ml=0, so I write the spherical harmonic Y=Squareroot(3/4pi)*cos(theta).
I would like to know how to compute the wave function at t=0, then I know it evolves with the time-evolution operator U(t), to answer the first request.
I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ...
On page 36 we read the following:
In the above text...
I have tried this same approach three times and I got the same answer. I can't figure out what's wrong. Btw answer is 12mu/(3+cos2α)
And yes, sorry for my shitty handwriting. If you can't understand the reasoning behind any step then please let me know.
My attempt/questions:
I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get:
##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##.
I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
Many texts state that in an elliptic orbit you can find angular momentum magnitude as
$$ L = r m v = m r^2 \frac {d \theta} {dt} $$
I wonder if
$$ v = r \frac {d \theta} {dt} $$
is valid at every point. I understand this approximation in a circumference or radius r but what about an arc...
Hello everyone!
I've been watching the following Walter Lewin lecture, the part that illustrates my question is part 17:19 of the video
Most things have made sense during this lecture, but one persistent question I have is the following: why does the bicycle tilt toward the inside of the...
I am using the following formula to solve this problem.
$$ L_a= L_c + \text { (angular momentum of a particle at C of mass M)}$$
Because the point C is at rest relative to point A, so the second term in RHS of above equation is zero. Hence, the angular momentum about A is same as angular...
I think the the time given doesn't matter since no torque is acting on the system, but not sure. Therefore, all we need is to determine the angular momentum about the axis passing through O and perpendicular to the plane of disk. This will involve finding the moment of inertia of smaller disk...
Summary:: I'd like to check my understanding of standard problems where a billiard ball resting on a plane is hit horizontally at some height above its center
So the situation is that a ball of mass ##m## and radius ##r## is at rest on a horizontal surface. There is friction between the ball...
Hello everyone, I have a doubt regarding the conservation of angular momentum.
When dealing with collisions between two objects, if the net external force is zero we know that the linear momentum is conserved; even when the system is not isolated, for instance because of gravity acting on the...
I thought the answer is B because the angular momentum in conserved in all 3 pictures.
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Suppose we have a rotating body like a bicycle wheel in space away from gravity. This body stops after a while due to friction between the wheel and wheel axles. Is not the conservation of angular momentum violated?
One of the component of angular momentum operator is ##\hat{L}_{x}=\hat{y} \hat{P}_{z}-\hat{z} \hat{P}_{y}##
I want it's position representation.
My attempt :
I'll find the representation of the first term ##\hat{y} \hat{P}_{z}##. The total representation is the sum of two terms.
The...
I'm studying orbital angular momentum in the quantum domain, and I've come up with the Robertson uncertainty relation for the components of orbital angular momentum. Therefore, I read that it is necessary to pay attention to the triviality problem, because in the case where the commutator is...
I am trying to build a simulation of a car engine and wheels for a game project.
My model is currently this:
Engine outputs a torque -> this spins up a flywheel over time (the physics step of 1/60s) -> the flywheel is coupled with the clutch and thus transmission -> the transmission multiplies...
I have a problem in understanding angular momentum equation (mrv), especially the part where radius is involved.
imagine an elastic collision occurred between sphere of mass (M) attached to a string forming a circle of radius (R) and moving with velocity (V) and another stationary sphere having...
In other words, is there a rotational orientation of each atom in a monatomic gas (and corresponding rotational speed conserving angular momentum) that affects collisions, or does a substance need to have at least 2 atom particles to have the orientation/rotational ability to have particle...
The classic way to go about this problem would be to use Kepler's laws and thus find the new time period of earth.
However I encountered this question in a test on rotational motion which deals with conservation of angular momentum.
The equation used here would be I1ω1= I2ω2
Replacing I with MR2...
Considering an atom within a rigid body, does the angular momentum of an electron within the atom vary when the body is put in motion?
My intuition is that, whether considered in a classical sense or quantum sense, the speed of a given electron in its motion within an atom will be constant and...
If you put a hydrogen atom in a box (##\psi=0## on the walls of the box), spherical symmetry will be broken so ##n##,##l##,##m_l## are no longer guaranteed to be good quantum numbers. In general, the new solutions will be a linear combination of all the ##|n,l,m_l\rangle## states. I know that...
Here is my depiction of the initial state:
Note that the presence of ##f_k## means the ball is initially slipping. We also know that the linear and angular speeds of the ball are increasing in time. At some point, the ball should stop slipping.
The condition for no slipping is that the speed...