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Angular Momentum

Angular momentum is a vector quantity that represents the products of a body’s rotational inertia and rotational velocity. The angular momentum of a system of particles is the sum of angular momenta of individual particles. For rigid bodies, the angular momentum is expressed as the product of the body’s moment of inertia and its angular velocity ω in the following equation:


Angular momentum is conserved only when there is no net external torque acting on the system. An example of angular momentum is a spinning figure skater. The skater will spread out their arms to act against the angular momentum and slow down, or pull their arms in close to their chest to increase angular momentum.

Angular momentum, L, is about a given origin. It is defined as:

L=r x p

Where r is the position vector and p is the linear momentum. Angular momentum is the cross product. Therefore a right hand rule can be used. The right hand rule is where the thumb points in the direction of angular momentum, your hand is the direction of the position vector and your fingers are in the direction of linear momentum.

Muonic atoms and the size of the nucleus

Muonir atoms and the size of the nucleus. The negative muon (symbol pf) is a particle with the same charge as the electron but with a larger mass (m = 207 m,). High-speed muons are produced in violent nuclear collisions. These muons can be slowed down in matter and captured into orbits around the nuclei of atoms in the material.

Quantum Probability of Eigenvalue Measurement

A particle of spin-one and magnetic moment  is in a uniform ~B field of strength B. At t = 0, the component of the spin along an axis is at angle  with the ~B field direction is measured to be 1hbar. What is the probability that a measurement at time t(> 0 will yield the eigenvalues mhbar (with m = +1; 0, or -1)? Please

Ratio of Magnetic Moment to Angular Momentum

See the attached file. Suppose that an electron is a small spherical shell of mass m with a charge e spread over its surface. Show that the ratio of the magnetic moment to the angular momentum of such an electron would be e/2m, whether the electron is (a) moving in a circular orbit, or (b) spinning about a diameter. (experiment

Hurricane Pressure Change

For a hurricane of radius 330 km, the 'eye' is a circular area whose radius is about 10% that of the entire hurricane. If the air swirls around the 'eye' at 165 km/hr, and if the angular momentum of the air swirling in from the rim to the eye is relatively constant, then what is the pressure difference between the outer rim of

Matrix Representation and Operators

If the general angular momentum quantum number j is 1 there is a triplet of |j,m_j> states: |1 ,1>, |1,0> and |1,-1> In this case a matrix representation for the operators j_x j_y and j_z, can be constructed if we represent the |j,m_j> triplet by three component column vectors as follo


From the equations in the attached file develop an expression to help answer the following: Under the conditions of Fsed=Ffriction and given (1-pv) and w are constant, what physical properties of a particle would influence the sedimentation velocity of that particle moving through a sample cell under the influence of the abov

Physics: Earth's acceleration during eclipses; Twin sun system

1. Earth's acceleration during eclipses. What is the percentage change in the acceleration of Earth toward the Sun when the alignment of Earth, Sun, and Moon changes from an eclipse of the Sun (with the moon between Earth and Sun) to an eclipse of the Moon (Earth between Moon and Sun)? Assume that Earth's orbital path arou

Angular Acceleration Problems

Please give step-by-step solutions to the following problems. 1. A softball of mass 0.22 kg that is moving with a speed of 6.5 m/s collides head-on and elastically with another ball initially at rest. Afterward it is found that the incoming ball has bounced backward with a speed of 3.8 m/s. Calculate a) the velocity of the

Total spin state of two particles with spin 1 and spin 1/2

A. Consider a system of 2 particles: particle 1 has spin 1, and particle 2 has spin 1/2. Let S be the total angular momentum operator of the two particles, where the eigenvalues of S^2 and Sz are ħ^2s(s+1) and ħms, respectively. The particles are in the state s= 3/2 and ms = 1/2. Calculate the wave function |s = 3/2

Yo-yo speed and energy; angular speed of a bicycle wheel

A yo-yo has a rotational inertia of 1180 g·cm2 and a mass of 148 g. Its axle radius is 3.58 mm, and its string is 139 cm long. The yo-yo rolls from rest down to the end of the string. (a) What is the magnitude of its linear acceleration? (b) How long does it take to reach the end of the string? As it reaches the end of the stri

Three quantum numbers used to describe orbitals in atoms

1. What is the name of each of the three quantum numbers used to describe orbitals in atoms? what orbital characteristics does each quantum number describe? what are the limitations on the values of these quantum numbers? 2. What is the energy of the photon that is emitted when the state of an electron changes from n=4 to n=2 i

Energy Eigenstates of the Hamiltonian

See the attached files. 1. Operator Algebra. Evaluate the following expressions: See attached for equations Neutrino Oscillation made oversample. Neutrinos come in three varieties that we know of: the electron neutrino (V_e) the tau neutrino (V_T) and the muon neutrino (which is irrelevant to this problem). Nuclear fusi

Frictional force, car's airbag, height of the incline

1) Is it possible for the frictional force to increase the mechanical energy of a system? 2) How a car's airbag works in terms of momentum? AND in terms of energy? 3) A solid sphere, hoop, and disk (all with same mass and radius) are rolled up an incline with (potentially) different initial speeds. They each reach the same

The solution to Rotational Motion

A 62.99 kg woman stands at the rim of a horizontal turntable having a moment of inertia of 495 kg·m^2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) a

Ratational Motion: angular vel, acc, and centripetal force

1. There is an analogy between rotational and translational physical quantities. Identify the rotational term analogous to each of the following linear quantities. In each case, give the symbolic expression for the quantity, as well as its name. I filled in three of the lines as examples. (SEE ATTACHMENT for table to fill in)

The Hydrogen Atom and the Radial Angular Momentum

Hydrogen atom The radial probability density for an electron is r2R2(r). That means that the probability of finding an electron at a certain radius r within a radial thickness dr is dr* r2R2(r) for an infinitely thin shell and approximately r* r2avg R2(ravg) for a shell of finite thickness r. The quantity ravg is some average

Sample questions are demonstrated.

1. During the later portion of the swing phase of a walking stride the knee is extended from 35 degrees (initial) to 10 degrees (final) over a time period of 0.1 seconds. What was the angular velocity? 2. If an object achieves an angular acceleration of 12 rad/s2 from a moment (torque) of 300 Nm, what is the object's moment o

Momentum questions are embedded.

1. If a person pushed on a door with a force of 650 N and a moment arm of 0.75 meters, what would be the moment created? 2. A defensive lineman (mass = 88.5kg) is running at 12 m/s, and a linebacker (mass = 84kg) is running at 13.2 m/s. Determine which player has the greater linear momentum, and by how much. 3. The moment

Rotational motion: Rate of precession of a rotating wheel.

A wheel with mass less spokes has mass 1 kg and radius 10 cm and is mounted on one end of a mass less axle as figure. The axle rests on a pivot at a point 16 cm from the mounting point and 10 cm from the wheel. At the other end, a mass of 0.8 kg is attached. The wheel spins at an angular frequency of 10 rad/s. What is the

phonons and lattice vibrations

What is meant by the terms: (i) normal mode and (ii) phonon. Explain why phonons obey Planck-Bose/Einstein statistics. What is the difference between an acoustic mode, and optic mode? Quantized lattice vibrations are called phonons. When a phonon propagetes to a crystal lattice the atomic oscillators excited and vibrate as pe

Hydrogen Atom: Line Spectra and the Bohr Model

A singly ionized helium atom (He+) has only one electron in orbit about the nucleus. What is the radius of the ion when it is in the n = 3 excited state? I have an idea of how to do the problem, but that is with hydrogen. Please help and thank you very much!

Principle of Conservation of Momentum

Please see the attachment. 11. A 3.0-kg cart moving to the right with a speed of 1.0 m/s has a head-on collision with a 5.0-kg cart that is initially moving to the left with a speed of 2 m/s. After the collision, the 3.0-kg cart is moving to the left with a speed of 1 m/s. What is the final velocity of the 5.0-kg cart? (a

Do only part (d): Measuring Angular momentum A particle is in the state with wave function shi = 1/sqrt(2)[Y11 + Y1-1] (a) What value is obtained if L^2 is measured? (b) Does the particle have a definite value of Lz? (c) What are the probabilities of getting results h bar and - h bar and 0 for Lz? Are any other Lz results possible (d) Calculate <shi/Lz/shi> (e) Suppose that when Lz is measured the result h bar obtained. What is the wave function afterwards?

Measuring Angular momentum A particle is in the state with wave function shi = 1/sqrt(2)[Y11 + Y1-1] (a) What value is obtained if L^2 is measured? (b) Does the particle have a definite value of Lz? (c) What are the probabilities of getting results h bar and - h bar and 0 for Lz? Are any other Lz results possible

Tilted Gyroscope

A gyroscope consists of a flywheel of mass m, which has a moment of inertia I for rotation about its axis. It is mounted on a rod of negligible mass, which is supported at one end by a frictionless pivot attached to a vertical post, as shown in the diagram. The distance between the center of the wheel and the pivot is d. The whe

Rotation and Angular Momentum

See attachment please. Need FBD for each case. The 0.2 kg ball ( ball is sliding not rotating) and the supporting cord are revolving about the vertical axis on the fixed smooth conical surface with an angular velocity of W = 4 radians/sec. The green ball is held in position b = .3 m by the tension T in the yellow cord. If b

Inverted Pendulum with Counterweights - Equations of Motion

Could someone help me to derive the equations of motions for the system shown in the attach file. Basically it's a 2-dimensional "box" which should be stabilized on its rotating(pin) joint by adjusting counterweights m1 and m2 with linear motors. Counterweights m1 and m2 can move of speed v1 and v2 respectively. We can assume th

Questions Using Planck's Constant

Part 1 An electron microscope operates with a beam of electrons, each of which has an energy of 20 KeV. Use the uncertainty principle in the form delta(x)delta(p) (greater or equal to) h/2 to find the smallest size that such a device could resolve. Planck's constant is 1.0552 × 10^-34 J · s. Answer in units of pm. Part 2

Modern Physics

Modern Physics. See attached file for full problem description.