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.
There's enough angular momentum in electron spin to get a 1cm radius ring of silver atoms to turn with a period of order days after relaxing from spin-up into randomness. (assuming you could get all of it to show up externally, and not end up in microscopic rotations or l quantum numbers.)
I...
Homework Statement
A 10 g bullet traveling at 400 m/s strikes a 10 kg , 1.2-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door immediately after impact?
Homework Equations
p[/B]= mv
L = Iω...
Homework Statement
The figure shows a uniform thin rigid plank of length 2b which can roll
without slipping on top of a rough circular log of radius a. The plank is initially
in equilibrium, resting symmetrically on top of the log, when it is slightly
disturbed. Find the period of small...
Hello,
Suppose that the angular momentum of a system can take the values 0, 1, 2. One carries out a measurement of ##J_z## on this system.
What can be said about the state of the system after the measurement? To what extent can it be perfectly certain if ##J_y## and ##J_x## do not commutate...
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I have been brushing up my Rigid Body Dynamics.
I tried computing the angular speed with respect the Center of Mass (CM) using the usual split of kinetic energy and also the split of Angular momentum using the CM.
First, a simple case: Two particles of mass M each separated by a distance...
Hi, me and a friend were discussing calendars and how they go wrong. Apparently one orbit around the sun happens during, on average, 365.242189 rotations around Earth's axis. The persian calendar almost nails it, with a 1 sec per year error, because it is based on star observations rather than...
Homework Statement
A uniform thin rod of length ##2l## and mass ##m## lies on a horizontal table. A horizontal impulse ##J## is given to the rod at one end. There is no friction. The total kinetic energy of the rod after impulse will be ?
Homework Equations
##Jl=I\omega##
##J=mv_{cm}##...
Why the tangential velocity of a particle increase if there are no external torque acting on it and its angular momentum is conserved?
I know that L = I.ω (angular momentum equals moment of inertia times angular velocity)
and v = ω.r (tangential velocity equals angular velocity times the...
Lets say I have a massive rod laying on a table with little froction, screwed into the table on one side to become our pivot point, and I lay next to it with my feet pointing towards the rod.
First scenario: I position myself very close to the pivot point and push, the rod rotates and I move...
Considering the angular momentum of a collapsing star preventing it from resulting in a black hole by degeneracy pressure, are there ekpyrotic universe models that include angular momentum and degeneracy pressure as key factors of cosmic inflation?
Hello!
Excuse my ignorance. The forum is full of difficult questions so I even feel a bit ashamed of posting this, But that is the only way I can learn.
I do not understand some concepts stated below in the images, and I am not able to grasp what is being said. For instance, I do not know why...
Homework Statement
I have the following problem to solve:
A 1.8m board is placed in a truck with one end resting against a block secured to the floor and the other one leaning against a vertical partition. The angle the Determine the maximum allowable acceleration of die truck if the board...
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
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
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 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...
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 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...
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 =...
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...
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...
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...
Homework Statement
Two ice skaters of mass ##m = 50\,\mathrm{kg}## each are moving towards each other frictionless on parallel paths with a distance of ##3\,\mathrm{m}##. They both have a velocity of ##v_o=10\,\frac{\mathrm m}{\mathrm s}##.
Skater 1 is holding a massless rod of length...
Homework Statement
Obtain the matrix representation of the ladder operators ##J_{\pm}##.
Homework Equations
Remark that ##J_{\pm} | jm \rangle = N_{\pm}| jm \pm 1 \rangle##
The Attempt at a Solution
[/B]
The textbook states ##|N_{\pm}|^2=\langle jm | J_{\pm}^\dagger J_{\pm} | jm \rangle##...
Homework Statement
In classical physics, a system's magnetic moment can be written like so: \mu = g\frac{Q}{2M}L, where ##Q## is the total charge, ##M## is the total mass of the system and ##L## the angular momentum.
a) Show, that for a cylinder (##I = \frac{1}{2}MR^2##) spinning around its...
Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly)
1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be...
Homework Statement
Prove that ##[L_i,x_j]=i\hbar \epsilon_{ijk}x_k \quad (i, j, k = 1, 2, 3)## where ##L_1=L_x##, ##L_2=L_y## and ##L_3=L_z## and ##x_1=x##, ##x_2=y## and ##x_3=z##.
Homework Equations
There aren't any given except those in the problem, however I assume we use...
Homework Statement
a spool of radius R1 and R2 (R2>R1) is kept on hortizontal surface. A force f= 2t N (where t is time ) acts on the inner radius tagentially find the angular momentum of the system about the bottomost point of the spool.
Homework Equations
v=u+at
W=Wi+alpha(t)
L=IW+mvr...
I am currently trying to model collisions of rigid balls. I have successfully been able to calculate collisions that only deal with linear momentum, but have run into trouble when I want to calculate angular momentum (e.g. when ballA glances the top of ballB, both balls should start spinning a...
Hi guys,
This is basically a quick question to hopefully find some pointers on a topic I've been browsing the internet on to not much success. Basically the topic I'm trying to find more information on is the angular momentum problem with the "nebular hypothesis" of the formation of our solar...
Homework Statement
Suppose we have a wavefunction with n=4. If we measure the orbital angular momentum along the z-direction(no spin in this problem) and get 2*hbar then what are the possible values of the total angular momentum and what is the most general wavefunction after the measurement...
Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.0×104 m/s when at a distance of 2.6×1011 m from the center of the sun, what is its speed when at a distance of 5.2×1010 m .
Express your answer using two significant figures
I applied...
Hello everyone, currently working on a physics project.
I was very curios about waterwheels and actually have an overshot waterwheel setup. I was testing how the efficiency of a waterwheel would be effected by the drop height of water onto the waterwheel. Do any of you have any idea how I can...
This (photo) is a very typical example of conservation of angular momentum, but my trouble arrises from trying to prove that the difference of energy will have to correspond to work, by calculating the work done by you to alter the moment of inertia. I have spent a lot of time in this, but I...
Hi there,
A friend of mine is creating a physics engine in java (mostly as a challenge I believe).
Today he asked me a question about the results of collisions between objects. For example, imagine that these 2 objects collide:
The results of such a collusion would be something like this...
Hello!
I'm trying to derive the linear velocity vector from the position vector and the angular momentum vector. I've seen on the internet that V = W x R (V,W and R are all vectors and x is the cross product) but I cannot for the life of me derive it! I've tried doing it by writing out the...
I know that magnetic fields can align objects with spin. In that case, if we suspend an object and turn on a magnetic field such that a significant number of electrons become aligned with the field, could we observe a macroscopic change in angular momentum?
This is rewording of a question on a test I've just done.
A vinyl record on a turntable has radius R=0.15 m, mass M=1.5 kg. The angular speed is reduced from 33.3 rev per minute to zero as a result of an applied torque, in 7 seconds.
Moment of inertia given as I=1/2 M R^2
Calculate a) angular...
Hello all,
I can understand the mathematics of this phenomena
First, one can solve the Euler equations of motion numerically, using Runge-Kutta and plot the motion.
Also, the path of the angular velocity vector will like on the kinetic energy ellipsoid and the angular momentum vector...
For an equations such as this what goes into the θ?
θ = sinθ or θ = θ?
Let's say if the angle of displacement = 45° do I just plug 45° as θ into the equation below or should it be sin(45°)?
Or is it θ = S/R ?
ωf2 = ωi2 + 2 α (θf - θi)
I am having trouble visualizing which two tires of a car will be pushed down based on the angular momentum and torque of that car. Let's say if its angular momentum is point OUT while its torque is pointing UP in relations to the picture below.
My guess is it's the two right wheels of the...
Homework Statement
The period of a comet is 75.8 years. The perihelion distance is 0.596 AU (1 AU = 1.5 ⋅ 1011 m).
The velocity at perihelion is vp = 5.45 ⋅104 m/s.
a) Find the length of the major semi-axis of the elliptical orbit.
b) Find the aphelion distance and the velocity at aphelion...