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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:

L=Iω

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.

### Angular Acceleration Calculations

The acceleration of point G has a magnitude of 15m/s^2 and is in the x direction. Point G is the center of the dis. The radius of the disk is R=0.5m and the instant shown theta=45 degrees and the angular speed of the disk is 7rad/s. Determine the magnitude of the angular acceleration of the disk. See attached file for more inf

### Magnitude of Angular Velocity

The rigid body shown in this diagram is moving such that the velocities of points A and B are the same. The magnitude of the velocity of point A is 25m/s and the distance between points A and B is 10m. Determine the magnitude of the angular velocity of the rigid body. Express your answer in rad/s and give you answer to 4

### Angular Speed, Mass Center, and Unit Vectors

At the instant shown, theta = 45 degrees, and the magnitude of the velocity of point G is v(G) = 16m/s and the angular speed of the disk is 7 rad/s. Point G is the mass center of the disk. The Radius of the disk if R = 0.5m. Determine the magnitude of the velocity of point C at the instant shown. The unit vector i, j, and are t

### Clebsch-Gordan coefficient

Explain what the "Clebsch-Gordan Coefficent" is.

### Coupled harmonic oscillators

Consider two coupled identical harmonic oscillators described by the Hamiltonian H= p1^2+p2^2/2m+1/2mw^2x1^2+1/2mw^2x2^2+gx1x2 1- What is the lowest energy of the system? 2- What is the ground state eigenfunction? 3-What is the energy and the eigenfunction for the first excited state?

### 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.

### Identifying Quantum Numbers For Electron

The principal quantum number, n, describes the energy level of a particular orbital as a function of the distance from the center of the nucleus. Additional quantum numbers exist to quantify the other characteristics of the electron. The angular momentum quantum number (?), the magnetic quantum number (m?), and the spin quantu

### 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 1hbar. What is the probability that a measurement at time t(> 0 will yield the eigenvalues mhbar (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

### Hamiltonian Eigenfunctions

1.The following function is one of the Hamiltonian eigenfunctions for the H atom. Is it also an eigenfunction of and ? If so, what are the corresponding eigenvalues? Please see the attached file for the full problem.

### solve angular speed based on moment of inertia

An ice skater has a moment of inertia of 5.0 kg-m^2 when her arms are outstretched. At this time, she is spinning at 3.0 revolutions per second. If she pulls her arms in, she decreases her moment of inertia to 2.0 kg-m^2. How fast is she spinning then? Please provide detailed explanation.

### 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

### Calculating circular velocity and forces

Please see the attached. I'm having a tough time understanding the simple elements of circular motion and the force vectors (horizontal, vertical and normal) that results from this motion. Please explain in simple terms if possible. See attached.

### Rotational Motion: Motion of clutch plates when engaged.

See attached file for proper format. A and B are two separate clutch plates. Plate A has a mass of 36 kg, a radius of gyration of 0.12 m and rotates at 660 rev min-1. Plate B has a mass of 45 kg, a radius of gyration of 0.16 m and is stationary. When the clutch plates are engaged, slipping ceases after 0.4 seconds and t

### 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

### Equation

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

### Why does distance make torque and work different

1. A torque is a product of a force and a distance (lever arm). Work is also the product of a force and a distance. Yet, "torque" and "work" are different. What is it about the distances that make "torque" and "work" different? 2. What is a gyroscope? What does the angular momentum has to do with a gyroscope? Provide examples

### 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 &#295;^2s(s+1) and &#295;ms, respectively. The particles are in the state s= 3/2 and ms = 1/2. Calculate the wave function |s = 3/2

### Determinig Allowed Radii for the Charged Particle

A Particle of charge q and mass m, moving with a constant speed v, perpendicular to a constant magnetic field B, follows a circular path. If in this case the angular momentum about the center of this circle is quantized so that mvr = 2nh, determine the allowed radii for the particle in terms of n, h, q, and B for n= 1,2,3....

### 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

### Angular Momentum and Velocity

Calculate the wavelength of the line emitted as n changes from 5 to 3 in the hydrogen atom. What is the angular momentum in the lowest energy state of the Bohr atom? What is the velocity of the electron in the state with n=1?

### Physics: Calculate the magnetic moment of the sphere

A sphere of radius a carries a uniform surface charge density sigma. The sphere is centered at the origin and rotates about the z-axis with constant angular velocity omega. Calculate the magnetic moment of the sphere.

### 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

### Dynamcis of Rotational Motion

A neutron star has an angular spped of 70.4 rad/s and underwent a "glitch" that increased its angular speed to Ω= Ω0+ Δ;Ω where ΔΩ/Ω0 = 2.01 x 10^-6. If the radius of the neutron star before the glitch was 11 km, by how much did its radius decrease in the glitch - the neutron is a uniform sphere. So far uniform sphere

### Physics: Moment of Inertia of disk B

Two disks are rotating about the same axis. Disk A has a moment of inertia of 8.86 kg·m2 and an angular velocity of +8.31 rad/s. Disk B is rotating with an angular velocity of -9.45 rad/s. The two disks are then linked together without the aid of any external torques, so that they rotate as a single unit with an angular velocit

### 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

### Angular Momentum and Torque

In the figure (please see the attachment), a 0.420 kg ball is shot directly upward at initial speed 37.5 m/s. What is its angular momentum about P, 2.20 m horizontally from the launch point, when the ball is at the following heights? i) at max height, ii) when it is halfway to the ground. Also find the torque on the ball