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
Two pucks are lying on ice where they can slide and rotate with almost no friction. A string is tied to both pucks but it is tied to the middle of the first puck and wrapped around the second puck. You pull on both strings with the same force, F. The first puck moves without...
I have a 5.0 m tractrix and am trying to work out angular momentum and total angular momentum for two hitchpoint speeds 60 & 70 km/h.
My result shows a higher total angular momentum for the lower speed.
This is not what I expected.
Here are my equations
Positions:
Derivatives
Angular velocity...
I'm trying to deduce the angular momentum ( for a rigid body ) on my own, and here is the problem I face.
By introducing the angular momentum of a tiny piece in rigid body (" i ") as :
Li = ri × pi
Li = ri × mi vi --------------------------------- [ Line 1 ]
Li = ri × mi ri ωi
To find the...
I can't find on any good source (such as a textbook) a precise specification about the cases when Hund's rules (especially Hund's third rule) for an electronic configuration of atom are valid (the rules help to select the lowest energy state of a configuration).
As far as I understood:
Hund’s...
I've started on "Noether's Theorem" by Neuenschwander. This is page 35 of the 2011 edition.
We have the Lagrangian for a central force:
##L = \frac12 m(\dot{r}^2 + r^2 \dot{\theta}^2 + r \dot{\phi}^2 \sin^2 \theta) - U(r)##
Which gives the canonical momenta:
##p_{\theta} = mr^2...
Hello,
I encountered the following statement in my lecture notes and there is a couple of things I don't understand:"Let's consider two particles with spins ##s_1 = \frac{1}{2}## and ## s_2 = 1## with a spherically symmetric interaction potential. Assume these two particles are in a two...
Homework Statement
Not a HW problem, but a "me re-thinking things" problem. Please tell me where my thinking is flawed:
You have an ice skater with no net external torques acting on him/her. (We are analyzing the time after they have to get an external torque on them by pushing off of the...
Homework Statement
A uniform thin rod AB is equipped at both ends with the hooks as shown in the figure and is supported by a frictionless horizontal table. Initially the rod is hooked at A to a fixed pin C about which it rotates with a constant angular velocity w1 . Suddenly end B of the rod...
Homework Statement
A particle (5 kg) moves with constant velocity 2 m/s along the straight line 2y=3x+4, the angular momentum of the particle about origin is?
Homework Equations
L=r x p
The Attempt at a Solution
For a 2d problem we take the component of velocity perpendicular to the point...
I have a disc that is rotating due to air being blown at its outer radius. The incoming relative velocity of the air is high, therefore the effect of friction supersedes the effect of conservation of angular momentum. The tangential portion of this velocity decreases due to the friction as it...
Homework Statement
A disk of radius ##r## and mass ##m## rolls down an inclined plan. It reaches the end of the plane with velocity ##v_{f}## and collides with a vertical rod of length ##L## and mass ##M## sticking with it. See figure.
What is the angular momentum magnitude and direction...
Homework Statement
A ball of mass ##m## is attached to a massless string of length ##L##. The ball is released from rest as shown in the figure and as it reaches the bottom of the circle, the string wraps around a nail which is a distance ##d## below the center of the circle. What is the...
1. The problem statement
I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian.
The Attempt at a Solution
I get...
Homework Statement
A 2.4 kg particle-like object moves in a plane with velocity components vx = 25 m/s and vy = 80 m/s as it passes through the point with (x, y)coordinates of (3.0, −4.0) m. (Express your answers in vector form.)
(a) What is its angular momentum relative to the origin at this...
Homework Statement
A rod of length D sits at rest on a friction less table. A ball of mass M strikes the end of the rod with a speed V and rebounds with a speed 3v/4 causing the rod to rotate counterclockwise around a fixed axis at one end. The rotational inertia of the rod is I
Homework...
According to the book "transport phenomena" by Lightfoot, Byron and Stewart if you take the cross product of the equation of motion (for very small element of fluid) and the position vector ##r## you get the equation of change of angular momentum. After some manipulation of vectors and tensors...
Homework Statement
I'd like to show, if possible, that rotational invariance about some axis implies that angular momentum about that axis is conserved without using the Lagrangian formalism or Noether's theorem. The only proofs I have been able to find use a Lagrangian approach and I'm...
Hello! I got a bit confused about the fact that the whole the description of spin (and angular momentum) is done in the z direction. So, if we are told that a system of 2 particles is in a singlet state i.e. $$\frac{\uparrow \downarrow -\downarrow \uparrow }{2}$$ does this mean that measuring...
My intuition is if an object is orbiting a centre, it is accelerating as the direction of its vector constantly changes, i.e a ball orbiting a stick because they are tied by a string. I don't understand why Earth's spin does not slow down, if we think of Earth as lots of individual atoms, those...
Homework Statement
Consider two pairs of operators Xα, Pα, with α=1,2, that satisfy the commutation relationships [Xα,Pβ]=ihδαβ,[Xα,Xβ]=0,[Pα,Pβ]=0. These are two copies of the canonical algebra of the phase space.
a) Define the operators $$a_\alpha =...
Homework Statement
A certain odd-parity shell-model state can hold up to a maximum of 4 nucleons. What are its values of J and L? What about an odd-parity shell-model state with a maximum of 6 nucleons?
Homework Equations
Parity = (-1)L
J = L+S
Total angular momentum, J, is equal to orbital...
Hello. I'm currently entering into a Physics II class at the start of my third semester at UCONN (my first semester was introductory modern physics - kinetic theory, hard-sphere atoms, electricity and magnetism, scattering, special relativity, Bohr model, etc), and finished Physics I off with...
Homework Statement
A rod (mass M, length L) is placed vertically on a smooth horizontal surface. Rod is released and after some time velocity of COM is v downwards and at this moment rod makes 60 degrees with horizontal. Find angular momentum of rod about Instantaneous center of rotation...
Homework Statement
A circular ring (2m, R) with a small insect of mass m on its periphery, is placed upon smooth horizontal surface (axis of rotation passing through center and perpendicular to the ground i.e disk is lying horizontally)
. The insect starts moving with velocity v w.r.t ground...
Homework Statement
suppose you're sitting on a rotating stool holding a 2kg mass in each outstretched hand, if you suddenly drop the masses, will your angular velocity increase, decrease or remain the same?
Homework Equations
dL/dt=net torque
when net torque is 0, L=constant=Iw
therefore...
Homework Statement
Assertion- If linear momentum of particle is constant, then its angular momentum about any axis will also remain constant
Reason-Linear momentum remains constant when net force is 0, angular momentum remains constant when net torque is zero
which of these statements is/are...
Homework Statement
A uniform rod (M, L) is rotated about a point L/3 from its left end. Angular momentum about O
Homework Equations
1) L=I(cm)w for purely rotating body
2) L(orbital)= M*v(cm)*perpendicular distance(r)
3) L(spin)= I*w
The Attempt at a Solution
I got the correct answer in two...
Homework Statement
A disk is undergoing pure rolling motion with speed v. The radius of the disk being R and mass M. Then the angular momentum of the disk about the
1)bottom most and
2)top most point
Homework Equations
1) L(orbital) = m*v*r where v is the velocity of cm which is...
Homework Statement
A 7.3-cm-diameter baseball has mass of 150 g and is spinning at 230 rad/s .
Treating the baseball as a uniform solid sphere, what is its angular momentum?
I'm about to pull my hair out because I feel like I understand everything about this problem perfectly and yet I'm still...
Homework Statement
Four particles of mass 1 Kg each, are moving on a plane with the velocities given in the figure.
Homework EquationsThe Attempt at a Solution
First I calculated the position of the CoM:
Xcm=7/4(i + j)
Then I calculated the velocity of the CoM:
Vcm= ½i + ¼j
For the internal...
Hello,
in classical physics orbital angular momentum is defined as the cross product of the position vector 'r' and the momentum 'p'. A friend told me that all moving objects must have orbital angular momentum (even if it is moving along a straight line). That statement confuses me a lot...
From a wiki's vis-viva equation page, it is given that the specific angular momentum h is also equal to the following:
h = wr^2 = ab * n
How can ab * n be derived to be equal to the angular momentum using elliptical orbit energy/momentum/other equations without having to use calculus or...
Homework Statement
Four equal discs of mass ocuppy the vertices of a square frame made by four rigid bars of length and negligible mass. The frame is at rest on a horizontal table, and it can move with negligible friction. An instantaneous impulse is transmitted to one of the masses, in the...
Homework Statement
4 persons each with mass m stand out on the edge of the carousel that rotates with angular velocity W0. carousel has mass 4m, radius r and inertia I = 2mr^2. The 4 persons then go all the way to the center of the carousel.
Show that the final angular velocity W1 = 3W0
See...
Homework Statement
A point charge q sits at the origin. A magnetic field ##\mathbf{B} (\mathbf{r})=B(x,y)\mathbf{\hat{z}}## fills all of space. The problem asks us to write down an expression for the total electromagnetic field angular momentum ##\bf{L_{EM}}##, in terms of q and the magnetic...
Homework Statement
A long board is free to slide on a sheet of frictionless ice. A skater skates to the board (laid horizontally relative to the skater's motion) and hops onto one end, causing the board to slide an rotate. In this situation, are angular and linear momentum conserved...
In the Dirac equation, the wave-function is broken into four wave-functions in four entries in a column of a matrix. Since there are four separate versions of the wave-function, does each version have the spin angular momentum of h-bar/2? This seems overly simplistic. How does spin angular...
Hi, I found some back of envelop calculations which show that Jupiter accounts for over 60% of the solar system's angular momentum.
http://www.zipcon.net/~swhite/docs/astronomy/Angular_Momentum.html
Is that correct?
A previous thread here on the subject ( now locked for some reason ) claimed...
1. At the instant of the figure, a 6.70 kg particle P has a position vector of magnitude 4.30 m and angle θ1 = 43.0° and a velocity vector of magnitude 3.40 m/s and angle θ2 = 32.0°. Force , of magnitude 7.40 N and angle θ3 = 32.0° acts on P. All three vectors lie in the xy plane. About the...
Homework Statement
I am working on the derivation of Kepler's Second Law based on torque and angular momentum. I understand that the vector "L" is equal to the mass (m) times the cross product of the vector "r" and the vector "v." The source I am following then states that
L = mrvtheta. I do...
Homework Statement
[/B]
Parts (c) and (f) are the ones I'm having trouble with;
Homework EquationsThe Attempt at a Solution
[/B]
For (c), I assume the problem is meant to involve using the result from part (b), which was H = g(J2 - L2 - S2)/2 .
I was trying just to do it by first showing...
Homework Statement
Homework Equations
Li = Lf
L = I*omega
K = (1/2)*(I)*(omega)^2
The Attempt at a Solution
[/B]
Given that there are no non-conservative forces in action, I am assuming that the two kinetic energies should be equal. However, as shown by my work above, the two values...
Homework Statement
A cylinder with radius ##R## and height ##h## which has a distributed charge on its surface with density ##\sigma## spins over its axis with angular velocity ##\omega##.
If the cylinder has a mass density ##\rho##, find the relationship between magnetic momentum and angular...
Homework Statement
I've spent at least 1.5 hours on this problem trying to figure out what i did wrong and I can't find anything. With an exam in two days plus another chapter to go through.
Regardless, here are the problem(6) and answer, as well as my work. Hope you can read it, and the...
Homework Statement
I have a basic problem where a child jumps tangentially onto the outer edge of a stationary merry-go-round, and you have to use conservation of momentum to find the final angular speed of the merry-go-round.
But the next part of the question asks "how much mechanical energy...
I've learned that in a crystal, the crystal momentum is conserved. When one considers the electrons as Bloch waves, they have a momentum that doesn't commute with the Hamiltonian and they have well definite energies, hence they cannot have a well definite momentum, because there is no basis in...
Homework Statement
Homework Equations
F=ma
τ = Iα = rF
v=rω, a=rα
L = Iω
Center of Mass/Moment of intertia equations
The Attempt at a Solution [/B]
So right now I've tried to model the force acting on the ring as it goes around the peg, but I think centripetal force is involved and I'm not...
Homework Statement
If we have a hollow ball completely filled with water which is rolling without slipping on a horizontal ground. If the water freezes which of the parameter will remain unchanged-
angular speed, angular momentum, linear momentum, kinetic energy, total energy
Homework...
Homework Statement
Homework EquationsThe Attempt at a Solution
Hi All,
My problem is that when I calculate this integral or use software to do it for me I get (3*i*pi)/16, when I've been told that the answer is 1/2i giving a probability of 1/4. Would someone be able to point out where my...
I explained this thinking to a meteorologist once and she couldn't give me an answer. Any physicists want to give it a shot?
I find the typical explanation of tornadoes perplexing (that's a polite way of saying I don't believe it). The explanations I've seen claim that tornadoes start out...