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.
Homework Statement
Show that ##|l, m\rangle## for ##l=1## vanishes for the commutator ##[l_i^2, l_j^2]##.
Homework Equations
##L^2 = l_1^2 + l_2^2 + l_3^2## and ##[l_i^2,L^2]=0##
The Attempt at a Solution
I managed to so far prove that ##[l_1^2, l_2^2] = [l_2^2, l_3^2] = [l_3^2, l_1^2]##. I...
1. Homework Statement
Determine the total magnitude of angular momentum Ho of the particle about point O. The velocity of the particle is 5.5 m/s.Homework Equations
Ho= r x mv
The Attempt at a Solution
The answer is 43.04. My question is, isn't the graph wrong? If you take the magnitude of the...
I am reading a proof of why
\left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z
Given a wavefunction \psi,
\hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
Hello
I need help to explain the affect of the cross product without the its current symbolism, but for angular momentum.
I can explain angular momentum in terms of the cross product of 3D space formulated like this:
|r| |v| * sin(angler.v) e-perp to r and v Eq.1
(I can explain this to...
The basic idea:
I am interested in the possibility of an azimuthally-directed Poynting vector component which drops with the inverse cube of the distance (or as 1/r^3), primarily because it suggests the possibility of emitting field angular momentum, allowing for a uni-directional torque to be...
Homework Statement
The red box only
Homework EquationsThe Attempt at a Solution
I suppose we have to show
L_3 (Π_1) | E,m> = λ (Π_1) | E,m>
and
H (Π_1) | E,m> = μ (Π_1) | E,m>
And I guess there is something to do with the formula given? But they are in x_1 direction so what did they have...
Hi.
I am revising my Mechanics: Dynamics by reading the Beer 10th edition textbook and Pytel 2nd edition
In Pytel pg 358 art. 17.3 the angular momentum about the mass center of a rigid body in general motion is being calculated...
A spiral phase plate can change the orbital angular momentum of a beam of light. Should I think of the beam of light carrying the orbital angular momentum or the photons that make up the beam light?
If the orbital angular momentum is carried by the individual photons what is being orbited, the...
Homework Statement
A 39.00 kg rod of length 2.8 m is hanging vertically by one of its ends that is free to swing in a complete circle about a frictionless axle/pivot. The rod has uniform mass density. Suddenly it is struck horizontally by a 5 kg putty that sticks to the center of...
If we define Si=(1/2)× (reduced Planck's const)×sigma
Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B})
Here (sigma)i is Pauli matrix.
Next one is, what will we get from simplifying
<Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...
Homework Statement
Wheels A and B as shown in the figure are connected by a massless belt that does not slip. The radius of A is R and the radius of B is r. What is the ratio of rotational inertias ##\frac{I_a}{I_b}## if the two wheels had the same angular momentum about their central axes...
Homework Statement [/B]
A thin uniform bar 2.00 m long and weighing 90.0 N is hanging from the ceiling by a frictionless pivot. It is suddenly struck 1.50 m below the pivot by a small 3.00 kg ball initially travellimg horizontally at 10.0 m/s. The ball rebounds and moves in oppossite direction...
Homework Statement
A child is initially sitting near the outer rim of a revolving merry-go-round. Suddenly, the child moves towards the center of the merry-go-round (while it is still revolving). For the merry-go-round+child system, let the symbols L and K refer to the magnitude of the angular...
Homework Statement
Consider an electron in a state described by angular wavefunction $$\psi(\theta,\phi)=\sqrt{\frac{3}{4 \pi}}\sin \theta \cos \phi$$ Here θ and φ are the polar and azimuthal angles, respectively, in the spherical coordinate system.
i. Calculate the probability that a...
1. A moving rough cylinder of radius a, and mass m collides with an identical cylinder, on a smooth horizontal surface. Its centre of mass moves with linear velocity v0, and its angular velocity is ω0. What is the motion of the cylinders after the collision?
I have be told that the answer to...
Homework Statement
A rotating flywheel slows down only because of friction in its bearings. At the initial time ti, the angular speed of the flywheel is 2.0 rad/s. The power due to friction at a later time tf is half of the power due to friction at time ti. The angular displacement of the...
I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953
They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with...
Hello.
The problem is this, what happens to angular momentum, tangential velocity and centripetal force when you change the center of rotation.
For example, if we have rotating hinged arm, weight at the end, with certain angular momentum and tangential speed etc. which then gets stopped at...
I have a couple of questions that i thougth this group could help me with.
1. A plane (SR71) takes off from the equator, with a lateral speed, relative to space of 1000mph. (earth rotational speed) say it takes an hour to get there so, its going 10,000mph or something. . Tt flys over the...
Homework Statement
The figure shows an overhead view of a 2.50-kg plastic rod of length 1.20 m on a table. One end of the rod is attached to the table, and the rod is free to pivot about this point without friction. A disk of mass 39.0 g slides toward the opposite end of the rod with an initial...
Homework Statement
Basically, I'm dealing with part d) in this document: https://s3.amazonaws.com/iedu-attachments-message/b663095a5021cb6aee55657de728a8d7_bfbe0ba9d2f10f8ac9ef9d049934c1da.jpg. I have found that the angular momentum only depends on spatial coordinate and it doesn't on time. Is...
Hi. I have come across the following statement - the eigenvalue equation for J+ is given by
J+ | j m > = ħ √{(j+1)-m(m+1)} | j , m+1>
My question is this - how can this be an eigenvalue equaton as the ket | j, m> has changed to | j , m+1> ? Surely the raising/lowering operators don't have...
Does a rotating magnetic field possesses angular momentum in the direction of rotation?
I suppose this comes down to a broad question about the physical nature of fields in general. I love the Einstein-de Haas effect, where an iron core spins in the opposite direction of the induced spin...
Hi.
To show that [ L2 , L+ ] uses the following commutators [ L2 , Lx ] = 0 and [ L2 , Ly ] = 0 . But if [ L2 , Lx ] = 0 this shows that L2 and Lx have simultaneous eigenstates ; but then should L2 and Ly not commute ?
Thanks
When analyzing the conservation of angular momentum of a particular system, should we use the same p.o.r. before and after or can we use different p.o.r.'s? As far as I know, we should always use the same reference, but sometimes I see several solutions that use different references in my...
I need websites or books that has quantum mechanical exercises in particular that finds the total angular momentum eigenvalues (for example two spin 1/2 systems). Do you know where I can train?
Homework Statement
If the Earth, with a radius of 6400 km, were collapsed into a sphere of the same mass, having a radius of 10 km, what would be its rotational period?
Homework Equations
L = Iw
The Attempt at a Solution
I can solve this if the moment of inertia is given but since it isn't I...
Hi there,
I've been trying to solve the following problem, which I found looks pretty basic, but actually got me really confused about the definition of angular momentum.
Problem
The trajectory of a point mass m is described by the following equations, in spherical coordinates:
r(t) = r_0 +...
Hello,
I'm playing around with simulating drones (quadcopters) in Gazebo (an open source robotics simulator).
The control system is made up of six PIDs (one for each degree of freedom) and I'm encountering trouble tuning the pids for pitch / roll control.
In this case, the linear x / y and...
Homework Statement
A door ( a rod of length ##L##, mass ##M##) rotates with angular velocity ##\omega## about a point ## H ##, and approaches a stop at ##S##. ##H## and ##S## are along the same line, and separated by a distance ## s ##. Show that the angular momentum of the door about the point...
Homework Statement
If the steel disk has mass of 200 kg and a radius of 2 meters you can make it spin by applying a force to the rim. This torque increases the angular momentum of the disk. Suppose the force is 20 Newtons. How long would you have to apply it to get the wheel spinning 5...
I have two balls spinning with v1, omega1 and v2, omega2. They collide elastically with no tangential slip, resulting in new values for v1, omega1 and v2, omega2. I have the two components v1 & v2 figured out in the plane of contact, where angular momentum does not come into play. But I am still...
Is the wavefront velocity if an OAM mode 1 light beam proportional to its wavelength?
I understand that the helical structure step length gives the wavelength of the beam. In this case, a small wavelength beam would travel much slower. The problem is, f=v/λ, but now v<c and if λ is shorter then...
It is known that particles with rest mass cannot travel at the speed of light.
Can we also say that particles that travel at subliminal velocity, like these OAM photons do, have mass?
It has been demonstrated [1] that these beams can be thought as made of photons that posses intrinsic OAM, and...
Homework Statement
“A bicycle wheel is mounted as in the lab and as shown to the right. This wheel has a mass of 6.55 kg, a radius of R = 38.0 cm, and is in the shape of a ring. A mass M = 1.85 kg is attached to the end of a string which is wrapped around an inner hub which has a radius r =...
Homework Statement
There is a man walking on a disk with mass 70 kg and speed 4 m/s. He walks on a circle with radius 1,5 m. How fast does the disk (mass 200 kg and radius 2 m) under him rotates (need to calculate angular velocity)[/B]
Homework Equations
angular momentum = J * w (J-moment...
Homework Statement
A 2.9-kg particle P is located at [(r)\vec] = 3.3 m [^(x)] + 1.8 m [^(y)] from the origin of the x-y coordinate system shown in the Figure. It moves with a velocity of [(v)\vec] = −4.1 m/s [^(x)] + 2.6 m/s [^(y)]. A force, [(F)\vec] = 2.7 N [^(x)] + 1.4 N [^(y)] acts on the...
Hi everyone
I need raising and lowering operators for l=3 (so it should be 7 dimensional ).
is it enough to use this formula:
(J±)|j, m > =sqrt(j(j + 1) - m(m ± 1))|j, m ± 1 >
The main problem is about calculating lx=2 for a given wave function , I know L^2 and Lz but I need L+ and L- to solve...
Homework Statement
A uniform solid sphere of radius R, rolling without sliding on a horizontal surface with an angular velocity ωo, meets a rough inclined plane of inclination θ=60°. The sphere starts pure rolling up the plane with an angular velocity ω. Find the value of ω.
Homework...
Hello.
Let's have two electrons with same orbital quantum number li and these electrons are in antiparallel; one electron has magnetic quantum number mi = a and and other electron has mi = -a (but we don't know which one has ml = a as we're in coupled representation to talk about total angular...
I can see how it would be conserved for the situation of a star turning into a white dwarf since the object is just contracting. Just like the classic ice skater example.
But what about a super nova? Say a star with spin up goes supernova and that the remaining black hole also has spin up but...
To radiate energy, the Poynting vector must not drop faster than with the inverse square of the distance. Under what circumstances can EM angular momentum be emitted to the vacuum of space (i.e. without being recovered via inductive coupling) and yet not lead to energy losses through radiation...
Homework Statement
A stationary, axisymmetric, spacetime has two Killing vector fields [ξt, ξφ] corresponding to translation along t or φ directions. A particle of unit mass moving in this spacetime has a four-velocity u = γ[ξt + Ωξφ].
(i) Explain why we can interpret this as a particle moving...
Homework Statement
Homework Equations
I= sum m r2
L= r p
or
L=I W
The Attempt at a Solution
I= m1 r12 + m2 r22
I= 5.20 (0.9)2+ 2.20(0.9)2= 5.994 kg.m2
Then I used the second equation of second momentum
L(Angular momentum) = I W
L= 5.994 x 4.60
In the solutions sheet, he used the first...
Homework Statement
(This is a problem I myself created, so it may sound a bit trivial/stupid.) A particle of mass m in the xy plane has velocity v and a radius vector r with respect to some origin. After some time Δt, the same particle has velocity v and a radius vector r' with respect to the...
I wonder what would the angular momentum vector look like for these gears.
As it rotates, there is no clear direction on where the angular momentum vector is pointing. This object is symmetric.
Here's the video about these.
Homework Statement
A 4 g bullet traveling at 500 m/s strikes a disk of mass 1 kg and
radius 10 cm that is free to rotate around an axis passing through its
center. The bullet’s incoming path is 5 cm above the rotation axis and
the bullet comes to rest in the position shown in the figure. At how...
So a light particle is orbiting a massive particle by gravity.
We take both particles as spot particles.
The light particle makes an eccentric orbit where maximum radius of the orbit equals 2 and minimum radius equals 1.
I suppose the mass of the massive particle such that the speed of the...
Homework Statement
Imagine that you are standing on the edge of a cliff looking out over the vista… a sudden gust of wind nudges you off balance and you start tilting out over the edge of the cliff…. Yikes! You start wind milling your arms to regain your balance. A) do you rotate your arms...