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
The function ##f## is defined as follows:
\begin{equation*}
f(t) =
\begin{cases}
1, \text{ when } 2k < t < (2k+1),\\
0, \text{ when } t = k,\\
2, \text{ when } (2k-1) < t < 2k, & k \in \mathbb{Z}\\
\end{cases}
\end{equation*}
What is the period ##T## of the function ##f##...
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A thin, uniform, 3.8kg bar, 80cm long has two 2.5 kg balls glued on at either end. It is supported horizontally by a thin, horizontal, frictionless axle passing through its centers and perpendicular to the bar. Suddenly the right hand ball becomes detached and falls off...
Homework Statement
A 2.9-kg particle P is located at [(r)\vec] = 3.3 m [^(x)] + 1.8 m [^(y)] from the origin of the x-y coordinate system shown in the Figure. It moves with a velocity of [(v)\vec] = −4.1 m/s [^(x)] + 2.6 m/s [^(y)]. A force, [(F)\vec] = 2.7 N [^(x)] + 1.4 N [^(y)] acts on the...
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...
Hello.
Let's have two electrons with same orbital quantum number li and these electrons are in antiparallel; one electron has magnetic quantum number mi = a and and other electron has mi = -a (but we don't know which one has ml = a as we're in coupled representation to talk about total angular...
I can see how it would be conserved for the situation of a star turning into a white dwarf since the object is just contracting. Just like the classic ice skater example.
But what about a super nova? Say a star with spin up goes supernova and that the remaining black hole also has spin up but...
Homework Statement
We consider a pendulum of length L hanging along the z-axis with a mass (taken to be unity) at the end, attached to an arm of length R free to rotate about the z-axis but restricted to the xy-plane. The system is completely described by the angle of the pendulum rod with...
To radiate energy, the Poynting vector must not drop faster than with the inverse square of the distance. Under what circumstances can EM angular momentum be emitted to the vacuum of space (i.e. without being recovered via inductive coupling) and yet not lead to energy losses through radiation...
Homework Statement
A stationary, axisymmetric, spacetime has two Killing vector fields [ξt, ξφ] corresponding to translation along t or φ directions. A particle of unit mass moving in this spacetime has a four-velocity u = γ[ξt + Ωξφ].
(i) Explain why we can interpret this as a particle moving...
Homework Statement
Homework Equations
I= sum m r2
L= r p
or
L=I W
The Attempt at a Solution
I= m1 r12 + m2 r22
I= 5.20 (0.9)2+ 2.20(0.9)2= 5.994 kg.m2
Then I used the second equation of second momentum
L(Angular momentum) = I W
L= 5.994 x 4.60
In the solutions sheet, he used the first...
Homework Statement
(This is a problem I myself created, so it may sound a bit trivial/stupid.) A particle of mass m in the xy plane has velocity v and a radius vector r with respect to some origin. After some time Δt, the same particle has velocity v and a radius vector r' with respect to the...
Homework Statement
A shaft is turning at 65.0 rad/s at time zero. Thereafter, its angular acceleration is given by α = -10 rad/s 2 - 5t rad/s 3 where t is the elapsed time. (a) Find its angular speed at t = 3.00 s. (b) How far does it turn in the 3.00 s seconds?
t=3 seconds
wi= 65.0 rad/s
α =...
I wonder what would the angular momentum vector look like for these gears.
As it rotates, there is no clear direction on where the angular momentum vector is pointing. This object is symmetric.
Here's the video about these.
Homework Statement
A record player rotates normally at a rate of 18 rev/m.
It takes 70 seconds for it to slow down to a stop when you turn it off.
Homework Equations
Calculate the magnitude of its angular acceleration.
The Attempt at a Solution
answer key says the correct answer should be...
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...
For Q11(b), what is the relation between the angular acceleration ##\alpha## of the bottom right cylinder and its horizontal acceleration ##a_x##?
I get ##\alpha=\frac{a_x}{\sqrt{3}R}##, which is half the given answer (7.84) below.
After the bottom right cylinder rolls around the top cylinder...
So a light particle is orbiting a massive particle by gravity.
We take both particles as spot particles.
The light particle makes an eccentric orbit where maximum radius of the orbit equals 2 and minimum radius equals 1.
I suppose the mass of the massive particle such that the speed of the...
Homework Statement
To lower himself from a balcony an 80kg stuntman grabs a rope connected to a 400kg cylinder with a 1.2m diameter that is free to rotate about its axis of symmetry. What is the stuntman's acceleration as he falls?
Homework Equations
I missed this on a homework assignment. I...
Homework Statement
Two masses have same mass m, both of them are tied by a string and put in a hollow cylinder.
When the cylinder start to rotate until the angular speed is ω.
The tension in the string is T and the string breaks, both masses move away from the axis in the hollow cylinder...
Homework Statement
Here is a copy of the pdf problem set {https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU} the problem in question is problem number 1 which asks you to prove the orthonormality of the spherical Harmonics Y_1,1 and Y_2,1.
Homework Equations
Y_1,1 =...
Homework Statement
The motion of a particle moving in a circle in the x-y plane is described by the equations: r(t)=3.15, Θ(t)=8.86t
Where Θ is the polar angle measured counter-clockwise from the + x-axis in radians, and r is the distance from the origin in m.
1. Calculate the x-coordinate...
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
"The flywheel rotates with angular velocity of w=0.005theta^2 rad/s. Determine the angular acceleration after it has rotated 20 revolutions.
Homework Equations
I thought the equations were all but self-evident using the problem description! (See below.)
alpha=dw/dt
The...
Homework Statement
At first I needed to calculate the angular resolution of a telescope (diameter 1m, for visible light) so I used θ=1.22λ/D and got 4.88x10^-7 rad. Now I am asked: "If we wished to use this telescope to image the moon, what is the closest distance two objects can be to be...
Homework Statement
Homework Equations
Kinematic theorems:
The Attempt at a Solution
Ok, I'm trying to use another angle, phi, and find some relation betwen phi and theta so I can derive the equation for the angular velocity of rod B with respect to S, but I'm stuck at that point. How...
Consider a flat 2D rigid body rotating about an axis perpendicular to the body passing through a point P that is
(1) in the same plane as the body and
(2) different from the body's center of mass (CM).
In this case does Theorem 7.1 (eqn 7.9) still apply?
In the last step of the derivation of...
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##...
Hi
I have seen an example of commutator of the Parity operator of the x-coordinate , Px and angular momentum in the z-direction Lz calculated as [ Px , Lz ] ψ(x , y) = -2Lz ψ (-x , y)
I have tried to calculate the commutator without operating on a wavefunction and just by expanding...
So Imagine a ball moving off a frictionless cliff with a velocity v to the right, at the bottom of the cliff the surface has friction.
So after the collision, the x component of the velocity would be reduced, and the angular momentum would be fixed clockwise because the frictional impulse...
Homework Statement
Given this diagram, the problem is to find an expression for β/ΘE in terms of X/ΘE and Y/ΘE.
Homework Equations
β = Θ – α(Θ)
Dsβ = DsΘ – Dlsα'(Θ)
The Attempt at a Solution
I really only need help starting this problem. In my textbook and every document I can find online...
Homework Statement
Let ##\left|\psi\right\rangle## be a non-degenerate stationary state, i.e. an eigenstate of the Hamiltonian. Suppose the system exhibits symmetry for time inversion, but not necessarily for rotations. Show that the expectation value for the angular momentum operator is zero...
Homework Statement
You are asked to measure the moment of inertia of a large wheel for rotation about an axis perpendicular to the wheel at its center. You measure the diameter of the wheel to be 0.600 m. Then you mount the wheel on frictionless bearings on a horizontal frictionless axle at...
Homework Statement
At a fair, Hank and Finn play with a horizontal 5.4 m long bar able to rotate about a pole going through its exact center. Hank pushes with 32 N at one end of the bar and Finn pushes with 18 N in the opposite direction at the other end. (Assume both forces are always...
Hi!
I don't know if I'm in the right forum of this site but I'm trying anyway. I was just wondering if someone could explain how the step- and impulse response is related to an angular position (of e.g. a spacecraft )? Just a little about the theory since I am trying to actually understand...
Some observed neutron stars rotate hundreds of times per second. Speeds at the surface of these stars are as much as 15% the speed of light. These huge speeds are generated because angular momentum is conserved when a large rotating pre-super nova star collapses into a neutron star.
The...
Homework Statement
A uniform thin rod of Length L and mass M is pivoted at one end is held horizontally and then released from rest. Assuming the pivot to be frictionless, find
a) Angular velocity of the rod when it reaches its vertical position
b) The force exerted by the pivot at this time...
Homework Statement
I was in lab and we used a mass hanger and disk and plate attached to pulley to and a motion sensor to measure moment of inertia. Angular acceleration was recorded along with other values such as linear velocity and position and all that.
The question is:
the hanging mass...
My textbook says that for a central force at the origin, the angular momentum is constant, because the derivative rxF is zero since F points radially outwards so it is in the same direction as r. Ok, but what about the angular momentum about a point other than the origin, or the angular momentum...
Homework Statement
Assume no friction for 1 - 6
1. Draw a free body diagram of the fly wheel (from above), and a free body diagram of the weight (from side).
2. What force appears in both diagrams?
3. What is the relationship between the torque on the flywheel and the tension in the string...
Homework Statement
Homework Equations
v=rω
conservation of energy
The Attempt at a Solution
I don't know whether it is correct or not, and I am quite confused in part b
it asks the angular speed of the pulley
can I say that v=rω since the pulley and box A,B have a common velocity?
so is ω...
Homework Statement
Gear 2 in the figure below is driven at a speed of 80 rpm in the CCW direction viewed from the right end. Gear 4 meshes with a fixed ring gear (gear 7) and with gear 5. Find the angular velocity of gear 5. N2= 16 T, N3= 32 T, N4= 30 T, N5= 22 T and N7= 77 T.
Homework...
ηϖ1. Homework Statement
Homework Equations
I=½MR2
PE=mgh
The Attempt at a Solution
The first thing that jumped out at me was "uniform cylinder" so I went ahead and calculated the moment of inertia for the cylinder and got I=½(4.4)(.4)2 = .352 and held onto that.
Then, I calculated the...
Homework Statement
http://imgur.com/koz4PpI
Homework EquationsThe Attempt at a Solution
I was able to calculate the net torque on the pulley as 55.88 and the Inertia as .352. Those could be wrong, but that's as far as I could get. I really have no idea. Any kind of help/explanation would be...
In the image above, a centroid with radius 1 is depicted. F1 is pointing directly at point A (which is the center of the circle), and F2 is pointing directly at point B. The radius for finding the torque would be the perpendicular between the center of the object and the force vector, so r1...
Homework Statement
https://holland.pk/uptow/i4/ae8d3da6c3ce3cad10eb98dd3208a955.png
Homework Equations
τ=I⋅α
The Attempt at a Solution
I want to discuss in part (a) of the question
Tension in 5 kg block string=mg=5*9.8=49N
Taking moment at the pivot, a net torque acting on 15 kg post is...
OK, I understand that 'spin' is just an unfortunate name some someone gave to a particular quantum property and that particles do not actually spin. I also accept that I may never be able to understand what spin is without also understanding the mathematics involved.
However, I am also told...
Hi Folks,
Problem Statement
How would one use the conservation of angular momentum to explain the attached picture?
The rod is held fixed horizontally..the person holds on to the cork and then let's go...apparently the glass is saved due to this conservation...
Relevant Equations
Momentum...