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.
I have typed up the main problem in latex (see photo below)
It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.
I have tried this same approach three times and I got the same answer. I can't figure out what's wrong. Btw answer is 12mu/(3+cos2α)
And yes, sorry for my shitty handwriting. If you can't understand the reasoning behind any step then please let me know.
I am using the following formula to solve this problem.
$$ L_a= L_c + \text { (angular momentum of a particle at C of mass M)}$$
Because the point C is at rest relative to point A, so the second term in RHS of above equation is zero. Hence, the angular momentum about A is same as angular...
I think the the time given doesn't matter since no torque is acting on the system, but not sure. Therefore, all we need is to determine the angular momentum about the axis passing through O and perpendicular to the plane of disk. This will involve finding the moment of inertia of smaller disk...
Summary:: I'd like to check my understanding of standard problems where a billiard ball resting on a plane is hit horizontally at some height above its center
So the situation is that a ball of mass ##m## and radius ##r## is at rest on a horizontal surface. There is friction between the ball...
Hello everyone, I have a doubt regarding the conservation of angular momentum.
When dealing with collisions between two objects, if the net external force is zero we know that the linear momentum is conserved; even when the system is not isolated, for instance because of gravity acting on the...
I thought the answer is B because the angular momentum in conserved in all 3 pictures.
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Considering an atom within a rigid body, does the angular momentum of an electron within the atom vary when the body is put in motion?
My intuition is that, whether considered in a classical sense or quantum sense, the speed of a given electron in its motion within an atom will be constant and...
Here is my depiction of the initial state:
Note that the presence of ##f_k## means the ball is initially slipping. We also know that the linear and angular speeds of the ball are increasing in time. At some point, the ball should stop slipping.
The condition for no slipping is that the speed...
https://www.physicsforums.com/threa...f-a-translating-and-rotating-pancake.1005990/
So,I think I posted this in the wrong place. So, I will move it to here.
Here, in post #6, it is stated that ##\int R dm = M R##. As far as I know, R change from time to time and it is not constant. Hence, isn't...
I am currently reading David Morin book and found this statement :
##\,\,\,\,\,\,\,\,## "It is important to remember that you are free to choose your origin from the legal possibilities of fixed points or the CM"
Is it really alright to choose the center of a...
I have read Classical Mechanics book by David Morin, and there are some parts that I do not understand from its derivation.
Note :
## V## and ##v## is respectively the velocity of CM and a particle of the body relative to the fixed origin , while ##v'## is velocity of the particle relative to...
While reading in the book of Introduction to Quantum Mechanics by David Griffith in the section of Fine structure of Hydrogen: spin- orbit coupling, he said that the average value of S operator is considered to be the projection of S onto J. I could not understand why he assumed that. please...
What we know:
The ball is dropped at the tip A with some speed ##v_0## and rebounds with speed ##v##. This collision produces an angular impulse, changing the angular momentum of the bar with the flywheels.
Solution inspired by an answer provided by @TSny in the similar question.
Angular...
Consider the system of the mass and uniform disc.
Since no external forces act on the system, the angular momentum will be conserved. For elastic collision, the kinetic energy of the system stays constant.
Measuring angular momentum from the hinge:
##\vec L_i = Rmv_0 \space\hat i + I \omega_0...
Hi, I have the following problem:
A homogeneous disc with M = 1.78 kg and R = 0.547 m is lying down at rest on a perfectly polished surface. The disc is kept in place by an axis O although it can turn freely around it.
A particle with m = 0.311 kg and v = 103 m/s, normal to the disc's surface at...
Ok, I know there are a lot of strange things in our solar system. Can anyone explain why the small planets spin so slowly? and why does Jupiter spin so quickly? It seems like a ball of debris, getting smaller and smaller, would increase its speed like an ice-skater pulling their arms in...
I was first wondering wether we can solve this question by applying conservation or energy or not but after googling it I found that we can't apply conservation of energy since there will be some energy lost in this case. I don't know how this energy is getting lost.
My second doubt was if we...
Summary:: Would energy method give us a different answer from conservation of angular momentum?
Hello,
I do not know how to type equations here. So, I typed my question in Word and attached it here. Please see photos.
Note: This question is not a homework. I did not find it in textbooks or...
Hi, I have just joined the forum. Thank you all for being a part of such places so that people like me can get answers to the questions on their minds!
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I have been trying to understand how a quadcopter yaws. Referring to the figure below which is bird's eye view of...
In quantum mechanics one sees what J^2 can offer but why do we even consider looking at the eigenstates and eigenvalues of J^2 and a component of J, say J_z? Why don't we just use J?
Lets do it for the left (the right will be similar): ##r_{left}=[(L-a\sin\theta)\sin\phi,(L+a\cos\theta)\cos\phi]## so ##v_{left}=[-a\dot{\theta}\cos\theta\sin\phi+(L-a\sin\theta)\dot{\phi}\cos\phi,-a\dot{\theta}\sin\theta\cos\phi-(L+a\cos\theta)\dot{\phi}\sin\phi]##. Is this right?
Hello all! Hope everyone's been doing well!
My question relates to the nebular theory of solar system formation. It is generally accepted that via the nebular hypothesis, matter in a nebula contracts on its own gravity and begins to spin, but I'm having trouble understanding why it must begin...
I have the moment of inertia for the core(initial) and full body(final) but my answer for the moment of inertia for the arms(initial) was incorrect.
Arms(initial) moment of inertia:(1/12)(6)(1.7^2)=1.445 this is incorrect for some reason
Core(initial) moment of inertia: .9558
Full...
Summary: Consider a train carriage rolling along a curve that forms a left turn on the track. The carriage speed is directed along the y-axis (into the plane of the paper) in the figure. The trolley will have a tendency to curl in the curve in the specified direction. A flywheel is inserted...
Relevant Equations:
Angular momentum density stored in an electromagnetic field: $$\vec{l}_{em} = \epsilon_0[\vec{r} \times (\vec{E} \times \vec{B})]$$
Electric field of an electric charge: $$\frac{q_e}{4\pi\epsilon_0}\frac{r - r'}{|r - r'|^3}$$
Magnetic field of a magnetic charge...
So, I was reading my textbook in the section regarding net torque, and they gave an example of a seesaw with one person at each end, and they said that there is a net external torque due to the force of gravity on each person. I completely understand that; however, when I was reading another...
Given the figure, how can i arrive to this formula knowing that angular momentum is conserved?
I know that p = mv and L = p x r. So the initial momentum will be L1 = mV x R and the final momentum will be L2 = mv x r.
I am not sure how R will equal to b since the distance between the...
Statement of the problem :
"Using the definition L = r ##\times## p, prove that the direction of L is constant for an alpha (##\alpha##) particle whose scattering is shown in the diagram below. "
Relevant equations :
We are aware that the scattering takes place via a central force F = F(r)...
Homework Statement
A system has a ball and a uniform rod. The rod is rotating about point X on a frictionless table until it strikes the ball. The rod stops and the ball moves away.
Variables:
Rod's mass: m1
Ball's mass: m2
Rod's original angular velocity: ω
Ball's final velocity: v
Rod's...
Homework Statement
Not an actual homework problem but a discussion that came up in class while we were learning about torque.
A tall box is sliding across a surface with friction f, mass m, and velocity v. What equations would you use to figure out if the box would tip over while sliding to a...
Homework Statement
(Problems/diagrams referenced are attached as images.)
Homework Equations
Net torque about an origin = time derivative of the angular momentum vector about the same origin.
The Attempt at a Solution
I've solved these problems before, but I'm now looking back at them and...
I know that total magnetic moment of an electron (I am not sure if it is magnetic moment of electron or atom, please clarify this) is sum of magnetic moment caused by orbital motion and spin angular momentum.
So,
Total magnetic moment = Orbital magnetic moment + spin magnetic moment
Do I have...
Homework Statement
A point particle travels in a straight line at constant
speed, and the closest distance it comes to the origin of coordinates
is a distance l. With respect to this origin, does the particle
have nonzero angular momentum? As the particle moves along its
straight-line path...
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...
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 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...
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
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 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...
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
A 25kg child is spinning on a merry-go-round of mass 150kg and radius 2m at a constant angular velocity of 1rev/s. The child slowly walks to the center of the merry-go-round. Treat the child as a point mass and the merry-go-round as a uniform solid disk, and neglect friction...