Angular momentum Definition and 188 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.

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  1. E

    Angular Momentum

    I need some clarification on a homework problem related to angular momentum. I understand how to calculate the angular momentum by using L= IW but when calculating the moment of intertia for the particle i don't understand why to use .5m as the radius instead of .4m due to being placed at the...
  2. G

    Came up with a problem that I can't solve

    Imagine a hoop with mass M and radius R that will only roll without slipping on the floor. Place a point object with mass m on top of the hoop and then the system starts from at rest. Question: where does m leave M? If one fixes the hoop or let the hoop slide, solutions can be found using high...
  3. RoboNerd

    Implementing angular momentum approach in problem

    Homework Statement An object is in uniform circular horizontal motion at the end of a chord of length L. Its tangential speed is v. The chord is pulled into length 0.5L in such a way that the tension in the chord remains constant. As a result, the tangential speed: a) remains constant b)...
  4. RoboNerd

    Angular momentum of a satellite

    Homework Statement A satellite is in a circular orbit of radius R from the planet's center of mass around a planet of mass M. The angular momentum of the satellite in its orbit is: I. directly proportional to R. II. directly proportional to the square root of R III. directly proportional to...
  5. M

    Total angular momentum state using two ways

    Consider addition of two angular momenta J = J1 + J2 , with j1=j2=1. Find the eigenstates of the total angular momentum I jm > in terms of the product states I j1 m1 j2 m2 > in two ways (a) Make use of the tables of the Clebech _Gordan coefficients (b) The state with m1 = m2 = 1 must be a...
  6. xXhumans0monstersXx

    Spinning Ice Skater Question

    So basically, I was doing my AP Physics 1 homework and came across the spinning ice skater question yet again. The question states, "An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body, in which of the following ways are the...
  7. F

    [Rotational dynamics] cube sliding on a dish

    1. Homework Statement A small cube is sliding on a round dish (see attached figure) . The cube is always in contact with the (vertical) edge of the dish (which prevents the cube from falling outside the dish itself). There is friction between the cube and the dish. The dish can rotate around...
  8. N

    Find distance from COM using torque

    Homework Statement Word for word, from the problem: "A person’s center of mass is easily found by having the person lie on a reaction board. A horizontal 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman...
  9. S

    Angular momentum conservation and constant velocity as expla

    I'm confused about situations involving rotating frames in which the angular momentum is conserved and the initial velocity does not change. I'll make an example. Take a rotating carousel (constant angular velocity) with no friction on it and a ball. At the initial time instant the ball has the...
  10. i_hate_math

    Conservation of Angular momentum problem

    Homework Statement In the figure, a small 0.235 kg block slides down a frictionless surface through height h = 0.471 m and then sticks to a uniform vertical rod of mass M = 0.470 kg and length d = 2.36 m. The rod pivots about point Othrough angle θ before momentarily stopping. Find θ. Homework...
  11. S

    Central force and acceleration in the polar direction

    Consider a central force. The central force is radial by definition, so ##\vec{F}=f(r) \hat{r}##. Therefore, by definition, the acceleration caused by the force, in the direction of ##\hat{\theta}## must be zero, ##\vec{a_{\theta}}=0##. In presence of central force angular momentum is...
  12. S

    Coriolis force and conservation of angular momentum

    I'm trying to understand the relations between the existence of Coriolis force and the conservation of angular momentum. I found this explanation on Morin. I do not understand the two highlighted parts. In particular it seems that Coriolis force is there to change the angular momentum of the...
  13. S

    Angular acceleration in rigid body rotation due to a torque

    For the rotation of a rigid body about a fixed axis z the following holds. $$\vec{\tau_z}=\frac{d\vec{L_z}}{dt}= I_z \vec{\alpha} \tag{1}$$ Where \vec{\tau_z} is the component parallel to the axis z of a torque \vec{\tau} exerted in the body; \vec{L_z} is the component parallel to the rotation...
  14. S

    Forces that cause acceleration due to conservation laws

    I find difficulties in identify the forces acting behind the acceleration of objects that are considered consequence of conservation principles (for istance of KE and angular momentum). I'll make an example to explain. The same string-mass system is linked to a rod. In case (a) a force pull the...
  15. S

    Acceleration only due to conservation of angular momentum

    I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force. Consider for istance planetary motion. The angular momentum \vec{L} of the planets is conserved and that means \mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta} is conserved too...
  16. S

    Component of angular momentum perpendicular to rotation axis

    Consider the rotation of a rigid body about a fixed axis z, not passing through a principal axis of inertia of the body. The angular momentum \vec{L} has a parallel component to the z axis (called \vec{L_z}) and a component perpendicular to it (called \vec{L_n}). I have some doubts on...
  17. S

    Torque on barbell when angular momentum is not constant

    Homework Statement [/B] Consider a barbell with two equal masses m that rotates around a vertical axis z not passing through its center with angular velocity \vec{\omega}. The barbell is forced to stay in this position by an appropriate support. Identify the forces exerting torques on the...
  18. J

    Angular momentum of hydrogen atom with Schrodinger Equation

    If we were to assume that the electron moves around the proton with radius a, the Schrodinger equation becomes: ##\frac{1}{a^2}\frac{d^2\psi}{d\phi^2} + \frac{2m}{\hbar^2}|E|\psi = 0## The question in my textbook asks me to solve the above equation to obtain values of energy and angular...
  19. S

    Torque on rigid body when angular momentum is not constant

    I 'd like to clarify some doubts about the rotational motion around a fixed axis of a rigid body, in the case the angular momentum vector \vec {L} is not parallel to the angular velocity \vec {\omega} . In particular, consider a horizontal barbell with two equal masses m , forced to rotate...
  20. RoboNerd

    Quick Question on Kepler & angular momentum conservation

    Homework Statement Homework Equations I guess kepler's law but most importantly conservation of angular momentum are key here. The Attempt at a Solution [/B] I put down E as the answer, but the solutions have D as the correct answer. Why is this the case? Thanks in advance for the help!
  21. hackhard

    B Can body start spinning without ext cause

    suppose a platform "P" is rotating about the z-axis wrt x-axis . another platform "Q" rotating about z-axis below "P" with same angular velocity wrt x-axis standing on Q , P is at rest wrt Q, After some time rotational inertia of P about z-axis starts changing with time standing on Q , P will...
  22. T

    Question about cause of gravitational waves

    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...
  23. L

    Eigenstates of Orbital Angular Momentum

    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...
  24. J

    Revolving light source and creation of angular momentum

    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...
  25. K

    Conservation of Angular Momentum; angle of rotation

    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...
  26. A

    Collision of rolling billiard balls

    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...
  27. J

    [Orbital mechanics] Asteroid angular momentum

    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...
  28. J

    Conservation of Angular Momentum for a Satellite

    Homework Statement Homework Equations Angular momentum: H= Iw Parallel axis theorem: Io = I + Md^2 Moment of Inertia of thin plate about it's...
  29. M

    Finding Angular Momentum

    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...
  30. WhiteG7

    Conservation of Angular momentum and linear momentum

    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...
  31. Leef

    CFD Thermodynamics flow in Angular momentum system

    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...
  32. S

    Changing Momentum p -> L -> -p?

    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...
  33. jaketsi

    Thinking about 3D rotating

    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...
  34. J

    Precession of angular momentum in vector model

    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...
  35. Universh13

    Moon's angular momentum

    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?
  36. H

    Moment of Inertia and Bike stability

    So is a Bike driver stable when the bike is running because the bike wheels has a certain moment of inertia about the horizontal axis ,which might alter(mi gets lesser) if the direction of the axis changes ? Thanks in advance
  37. Chrono G. Xay

    Calculate the 'Feel' of a Drumhead?

    As another of my personal music projects, I have wondered if it would not be possible to calculate the 'feel' of a drumhead (i.e. the amount of 'give' expressed as transverse displacement 'z' that an equally pre-tensioned circular membrane of radius 'r' experiences when struck on its plane at a...
  38. M

    Conservation of angular momentum invariance

    Homework Statement Given a reference frame O' moving at a constant speed $\vec{V}$ in relation to another reference frame O, I want to prove that ##\vec{r_{1B}} \times m_1\vec{v_{1B}} + \vec{r_{2B}} \times m_2\vec{v_{2B}} = \vec{r_{1F}} \times m_1\vec{v_{1F}} + \vec{r_{2F}} \times...
  39. B

    Possible values for L_x

    Homework Statement I've a physical system with ##l=1## and I have to calculate the values I can obtain if I measure ##L_x## and their probability. Homework Equations I know that: - the values I can obtain are ##\ m=0, \pm 1## - ##\displaystyle L_x=\frac{L_+ + L_-}{2}## - ##L_x|1, m>_x=\hbar...
  40. Thom_Silva

    A Rotating motor compressing a spring

    Hello, I've recently came across this video (), where the authors use a motor to compress springs and therefore achieve locomotion. I've been thinking why is there a resulting downward net force. But i can't really figure it out. Thank you for time :) See the video from 1.16 minutes and...
  41. D

    Cylinder with point mass angular momentum

    Homework Statement A uniform cylinder of mass M and radius R can be rotated about a perpendicular axle through its centre. A particle of mass m is attached to the cylinder's rim. The system is rotated with angular velocity w about the axle, which is held in a fixed direction during the motion...
  42. vinicius0197

    Deducing Kepler's second law from Newton's laws?

    I've searched a little bit and found that I can derive kepler's third law from Newton's law of gravitation. That's okay. But I want to deduce kepler's second law too: "An imaginary line joining a planet and the sun sweeps out an equal area of space in equal amounts of time". I know it's possible...
  43. H

    How does the velocity of a mass spun around a pencil change?

    Suppose a mass ##m## is attached to the end of a string whose other end is attached to a cylindrical pencil. The mass is then spun around the pencil in a circle (whose centre coincides with the centre of the pencil) such that the string wraps around the outer surface of the pencil, thereby...
  44. G

    Canonical definition of Angular Momentum,

    Let's start with an arbitrary solid body rotating around a fixed axis of rotation with angular velocity ##\vec \omega## in the ## \hat z## direction. For simplicity, let's say the origin O is on the axis of rotation. Take a look at the picture I sketched in the next post. Tried my best to be...
  45. O

    Physics Experiment Help (Torque, Angular Momentum, etc)

    < Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown > Hey, So I am not sure if this is in the right section but feel free to move it. We are conducting an experiment at school at the moment and are having difficulty understanding all the theory...
  46. M

    A Question Regarding Black Holes

    Hello people, I have a question regarding black holes. The way i understand it, black holes form in supernovas, and they occur because the gravitational pull of the stellar remnant is so great that nothing can stop it, and it basically collapses down to a singe point, virtually nothing... Now...
  47. S

    Net Angular Momentum of Satellite with Reaction Wheel

    I am modelling the attitude dynamics of a satellite. The satellite has a reaction wheel in 1 plane to help control the attitude. There is significant debate about the equation for the net angular momentum of the satellite and what inertia tensors should be used regarding parallel axis theorems...
  48. kzf

    Kinematics derivation of conservation of angular momentum

    Homework Statement A welding robot consists of an arm (thin rod) that can rotate about the origin point O, and a welding tip, which can freely move along the rod, from the outermost point of the arm A all the way to the center point O. The design invokes two electric motors, one to turn the...
  49. K

    Calculating angular speed of a ball after collision

    Hi, I've been wondering is there anyway of calculating the angular speed of a ball after there is a collision of it and another mass. For example a baseball bat hitting the ball. I have not looked up on angular momentum, but is angular momentum involved in this? Based on common sense, I think...
  50. omega5

    Rings on a rod: angular momentum conceptual question

    Homework Statement A uniform rod of mass 3.15×10−2kg and length 0.380m rotates in a horizontal plane about a fixed axis through its center and perpendicular to the rod. Two small rings, each with mass 0.250kg , are mounted so that they can slide along the rod. They are initially held by catches...