# 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.

View More On Wikipedia.org
1. ### Angular momentum & Energy using Yukawa's potential

Hello there! I was doing my Gravitation problems and I found this problem that I'm unable to solve. Yukawa's theory for nuclear forces states that the potential energy corresponding to the attraction force produced by a proton and a neutron is: U(r) = \frac{k}{r}e^{-\alpha r},\ k<0,\ \alpha > 0...
2. ### Final angular velocity of a merry-go-round?

Homework Statement A child (mc = 36 kg) is playing on a merry-go-round (mm = 225 kg, R = 2.9m) that is initially at rest. The child then jumps off in a direction tangent to the edge of the merry-go-round. The child has a speed of 5.0 m/s just before she lands on the ground. What is the...
3. ### Time period of a pendulum made of two disks

Homework Statement Problem statement - [/B] Klepner and Kolenkow 6.15 : A pendulum is made of two disks each of mass M and radius R separated by a massless rod. One of the disks is pivoted through its center by a small pin. The disks hang in the same plane and their centres are a distance l...
4. ### Relative Body Angles of a Human Moving Through Space

Homework Statement Say a human is moving through space with constant acceleration due to gravity. There are no external forces/torques on the body other than the force of gravity. The person applies an internal torque at some joint, let's say the knees, so that they bend. Assume the rest of the...
5. ### Selection rules in electric dipole appoximation

Some literatures say that the selection rule in electric dipole approx. for angular momentum ##\Delta j = 0,-1,1## some other say ##\Delta l = -1,1##. I follow the notation used in my references, despite the difference I think since j and l are both angular momenta which fulfill angular momentum...
6. ### Angular Momentum Parking Gate

Homework Statement Homework Equations Torque = r X F The Attempt at a Solution r is 2.5m since that's the length of the red (minus 0.5), and F should be the weight, so ma? That gives something like (5 * 2.5/3) * 9.8 * 2.5, which doesn't give me an answer at all :(
7. ### Angular momentum of rigid body elements tensor

Homework Statement I was working through my text on deriving the tensor for Angular momentum of the sums of elements of a rigid body, I follow it all except for one step. Here is a great page which shows the derivation nicely - http://www.kwon3d.com/theory/moi/iten.html I follow clearly to the...
8. ### Center of rotation of a free rod

Suppose we have a free rod on a frictionless surface: if we hit it on a tip it will translate and rotate around its CM. What happens if we hit it at any other point between tip and CM? will it still rotate around CM?, if not, is it easy to find the center of rotation? If not, are the 3...
9. ### 2pi rotation of angular momentum eigenket

Homework Statement Prove that ## e^{2\pi i \mathbf{n\cdot J}/\hbar} |j,m\rangle = (-1)^{2j}|j,m\rangle ##. This equation is from Ballentine's QM book. The term in front of the ket state in the LHS is a rotation operator through ##2\pi## angle about an arbitrary direction ##\mathbf{n}##...
10. ### Conservation of Angular Momentum & Energy question

Question: A streetcar is freely coasting (no friction) around a large circular track. It is then switched to a small circular track. When coasting on the smaller circle its speed is: a) greater b) less c) unchanged Relevant Formulas: w = v/r KE = 1/2mv2 My teacher said the normal force from...
11. ### Synthesis Problem

A solid balsa cube of side length L = 16.0” and mass M = 8.60 kg is at rest on a horizontal table top. It is constrained to rotate about a fixed and frictionless axis, AB, along one edge of the cube. A bullet of mass m = 50.0 g is fired with speed v at the other side of the cube, at height a =...
12. ### How to calculate this angular velocity problem

A 2.3kg , 20-cm-diameter turntable rotates at 100 rpm on frictionless bearings. Two 500g blocks fall from above, hit the turntable simultaneously at opposite ends of a diagonal, and stick. What is the turntable's angular velocity, in rpm, just after this event? I first cacluted the angular...
13. ### Angular momentum to Power

As you can see, solid disk is rotating at steady angular speed, without any external force beeing applied. If i stop it gradualy within 60 second, by applying some linear load, how many kWh can i extract from this moving object?
14. ### Angular momentum commutator

Homework Statement Prove that ## [L_a,L_b L_b] =0 ## using Einstein summation convention. Homework Equations [/B] ## (1) [L_a,L_b] = i \hbar \epsilon_{abc} L_c ## ## (2) \epsilon_{abc} \epsilon_{auv} = \delta_{bu} \delta_{cv}- \delta_{bv} \delta_{cu}## ## (3) \epsilon_{abc} =...
15. ### Deriving commutator for angular momentum components

Homework Statement Prove that ## [L_a,L_b] = i \hbar \epsilon_{abc} L_c ## using Einstein summation convention. I think I have achieved the solution but I am not sure of my last steps, since this is one of my first excersises using this convention. Homework Equations [/B] ## (1)...
16. ### Impulse on a free rod

Homework Statement I have a rod of mass m and length l on a table without any kind of friction. I give it an impulse J in any point of distance d from the center of the rod, parallel to the table and perpendicular to the rod. Find the angular velocity ω and the velocity of the center of mass...
17. ### Commutatos and angular momentum

Homework Statement Let e and f be unit vectors. Le = eL is the definition of the component of angular momentum in direction e. Calculate the commutator [Le,Lf ] in terms of e, f and L Homework Equations [A,B]=(AB-BA) The Attempt at a Solution we know that L=r x p, in classical mechanics, and...
18. ### Angular momentum and Quantum Number l

The question: Consider two masses of 0.1 gm each, connected by a rigid rod of length 0.5 cm, rotating about their center of mass with an angular frequency of 800 rad/s. a.) What is the value of l corresponding to this situation? b.) What is the energy difference between adjacent l-values for the...
19. ### Angular Momentum: Puck Spinning

Homework Statement The puck in the figure shown below has a mass of 0.120 kg. Its original distance from the center of rotation is 40.0 cm, and it moves with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the...
20. ### Showing functions are eigenfunctions of angular momentum.

Homework Statement Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz. Homework Equations I know that Lz = -iћ(∂/∂φ) I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged...
21. ### Question about rotating reference frames

Hello, I have a few questions about rotation and relative motion. Suppose we transport the proverbial spinning ice skater used to demonstrate conservation of angular momentum to beginning physics students to a universe with only her and two planets. She is now spinning in deep space...
22. ### Selection Rules and Light with Orbital Angular Momentum

When we first learn of selection rules for atomic transitions, we learn that electrons have to change between states that differ in angular momentum by at most 1ħ, because photons have 1 unit of spin angular momentum. However, photons can have arbitrarily high integer quantities of orbital...
23. ### Possible partial waves for photoelectron of nitrogen

Homework Statement (a) The nitrogen atom has seven electrons. Write down the electronic configuration in the ground state, and the values of parity (Π), spin (S), orbital angular momentum (L), and total angular momentum (J) of the atom. (b) If an extra electron is attached to form the N–...
24. ### Just a confusion -- A mass falling onto a pivoting rod

Homework Statement A rod of mass ##M## and length ##l## is pivoted at the center(##O##) in horizontal position. An object of equal mass(having velocity ##v##) falls vertically on the rod at distance ##\frac{l}{4}## from ##O## and sticks to it. Find the angular velocity of the rod just after...
25. ### Angular momentum conservation

Homework Statement A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is IP. The person holds a spinning bicycle wheel with its axis horizontal. The wheel has a moment of inertia Iw and angular...
26. ### Example Problem in the book, why is tension ignored?

Homework Statement An Atwood machine consists of two masses, M and m, which are connected by an inelastic cord of negligible mass that passes over a pulley. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses M and m. Homework Equations...
27. ### System Analysis of Collisions and Angular Momentum

Consider a particle with velocity "v" has the collision with a rotating disc. How can I analyze the final angular velocity this system? If the mass of particle is very negligible related to the mass of rotating disc, definitely particle will turn back after collision. In this case, how can I...
28. ### Calculating the moment of inertia of a filled cylinder

Homework Statement For my IB higher level physics extended essay I will have to calculate the angular momentum of a cylinder rolling down a slope. The cylinder is made out of copper and will be filled with a known mass of car engine oil. I think i can obtain the angular velocity fairly easily...
29. ### Mouse Jumps Onto An Exercise Wheel

Homework Statement The vertical exercise wheel in a mouse cage is initially at rest, but can turn without friction around a horizontal axis through the center of the wheel. The wheel has a moment of inertia I=0.0004kg m2 and radius R = 0.06m An extremely smart pet mouse of mass m = 0.03 kg runs...
30. ### Angular momentum, block plus bullet

Homework Statement A wooden block of mass M resting on a frictionless, horizontal surface is attached to a rigid rod of length l and of negligible mass. The rod is pivoted at the other end. A bullet of mass m traveling parallel to the horizontal surface and perpendicular to the rod with speed v...
31. ### Solving for Angular momentum without mass

Homework Statement The problem is a Lagrangian problem that solves for a differential equation. I need to write a program to solve the Lagrangian numerically. My professor said you do not need mass for the program, but I'm not sure how. The problem is a vertical cone with a bead rolling around...
32. ### Maximum angular momentum for charged black hole?

Hi all I gather a normal black hole has maximum angular velocity at the point that the event horizon is moving at The speed of light. However what would be the maximum rotational velocity for a maximally charged black hole- for example one made purely of electrons? Thanks
33. ### Conceptual angular momentum question

The question is: A turtle is on a turntable. which is rotating on frictionless bearings at an angular velocity omega. The turtle walks towards the outside of the turntable (away from the center). Which of the following is true about the system's angular velocity omega and its angular momentum...
34. ### Linear and angular momentum

Homework Statement Two atoms of equal mass m, that move with the same speed but opposite direction, interact when they're in some region R of space, as in fig.1. After the interaction, one of the atoms moves with velocity ## \vec{V1} ## as in fig.2. a) Are the linear and angular momentum of...
35. ### Angular momentum Units . a very basic question

I have a very basic questions about units for angular momentum. The measure is in kg m^2/s Angular velocity is in radians/s and therefore radians do not appear in the units. Here is my question, can we leave this in degees/s? Sure its not used but is it wrong? If we are dealing with...
36. ### Gyroscope- Angular Momentum and Torque

I was reading on gyroscopes, and everything seemed to make sense: the spin angular momentum along the axis of the gyroscope changes due to the torque by gravity, causing precession. However, I can't understand why we are measuring angular momentum (the spin of the gyroscope) from the center of...
37. ### Clear misconception: Angular Momentum of Merry Go Round vs Ice Skater

Homework Statement Why is it that the horse closest to the axis of rotation on a merry go round spins slower than the horse on the outer edges of the merry go round? I saw this video on youtube and got confused at the first merry go round example: According to conservation of mass, isn't v...
38. ### Conserving Angular Momentum (Rigid Body Dynamics)

Lets say i have a rod (length = L) hinged at one end (A).It is initially at rest.Now if an impulse (J) acts on the other end (B),can i conserve the angular momentum about A(the hinge)? that is can i write: JL=Iw?(I=moment of inertia,w=angular velocity) this is what i saw in the book. My Doubt...