what is geometrical and physical meaning of angular momentum?

Question

do you know any physical significanse of these physical determinants like angular momentum and moment of inertia?please share your views.

Answer 1

Angular momentum is a property of a physical system that is a constant of motion[1] (is time-independent and well-defined) in two situations: (i) The system experiences a spherical symmetric potential field. (ii) The system moves (in quantum mechanical sense) in isotropic space. In both cases the angular momentum operator commutes with the Hamilton operator of the system. By Heisenberg's uncertainty relation this means that the angular momentum can assume a sharp value simultaneously with the energy (eigenvalue of the Hamiltonian).

An example of the first situation is an atom whose electrons only feel the Coulomb field of its nucleus. If we ignore the electron-electron interaction (and other small interactions such as spin-orbit coupling), the orbital angular momentum l of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherical symmetric electron-nucleus interactions. The individual electron angular momenta l(i) commute with this Hamiltonian. That is, they are conserved properties of this approximate model of the atom.

An example of the second situation is a rigid rotor moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum.

These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with spin, does not have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well.

In general the conservation of angular momentum implies full rotational symmetry (described by the groups SO(3) and SU(2)) and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to add these momenta to a total angular momentum of the combined system—a conserved property of the total system. The building of eigenstates of the total conserved angular momentum from the angular momentum eigenstates of the individual subsystems is referred to as angular momentum coupling.

Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schrödinger equation.

As an example we consider two electrons, 1 and 2, in an atom (say the helium atom). If there is no electron-electron interaction, but only electron nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, l(1) and l(2), are conserved. However, if we switch on the electron-electron interaction depending on the distance d(1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave d(1,2) invariant. In such a case neither l(1) nor l(2) is a constant of motion but L = l(1) + l(2) is. Given eigenstates of l(1) and l(2), the construction of eigenstates of L (which still is conserved) is the coupling of the angular momenta of electron 1 and 2.

In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, e.g. its spin and its orbital angular momentum.

Reiterating slightly differently the above: one expands the quantum states of composed systems (i.e. made of subunits like two hydrogen atoms or two electrons) in basis sets which are made of direct products of quantum states which in turn describe the subsystems individually. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary z axis). The subsystems are therefore correctly described by a set of l, m quantum numbers (see angular momentum for details). When there is interaction between the subsystems, the total Hamiltonian contains terms that do not commute with the angular operators acting on the subsystems only. However, these terms do commute with the total angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as angular momentum coupling terms, because they necessitate the angular momentum coupling.



In physics, the angular momentum of an object rotating about some reference point is the measure of the extent to which the object will continue to rotate about that point unless acted upon by an external torque.

In particular, if a point mass rotates about an axis, then the angular momentum with respect to a point on the axis is related to the mass of the object, the velocity and the distance of the mass to the axis.

Angular momentum is important in physics because it is a conserved quantity: a system's angular momentum stays constant unless an external torque acts on it. Torque is the rate at which angular momentum is transferred in or out of the system. When a rigid body rotates, its resistance to a change in its rotational motion is measured by its moment of inertia.

Angular momentum is an important concept in both physics and engineering, with numerous applications. For example, the kinetic energy stored in a massive rotating object such as a flywheel is proportional to the square of the angular momentum.

Conservation of angular momentum also explains many phenomena in sports and nature.

If a planet is found to rotate slower than expected, then astronomers suspect that the planet is accompanied by a satellite, because the total angular momentum is shared between the planet and its satellite in order to be conserved.

The conservation of angular momentum is used extensively in analyzing what is called central force motion. if the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant. Constant angular momentum is extremely useful when dealing with the orbits of planets and satellites, and also when analyzing the Bohr model of the atom.

The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of mass of her body closer to the axis she decreases her body's moment of inertia. Because angular momentum is constant in the absence of external torques, the angular velocity (rotational speed) of the skater has to increase.

The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron stars and black holes) when they are formed out of much larger and slower rotating stars (indeed, decreasing the size of object 104 times results in increase of its angular velocity by the factor 108)
In quantum mechanics the orbit and spin of a particle can interact through spin-orbit interaction. Or two separate particles, each with a well-defined angular momentum, may interact, for instance, by Coulomb forces. In both cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually still is. The procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. Angular momentum coupling in atoms is of importance in atomic spectroscopy. Angular momentum coupling of electron spins is of importance in quantum chemistry. Also in the nuclear shell model angular momentum coupling is ubiquitous.

Answer 2

"when a mass contracts, its rotation increases." seems like that is saying, when a apple falls off a tree, it accelerates towards the ground. A Newtonian physics student might ask (because students are annoying and they like to ask tough questions) "If the apple fell and the ground wasn't there, would it just keep going faster and faster forever?" well, we know that can't happen. Even Newton could explain why it wouldn't. Einstein would swing at a different pitch, but the answer would be the same. "when a mass contracts, its rotation increases." Does it? An assumption of conservation of Angular Momentum IMPLIES that happening, yet at what point does Einstein start beating on the door and remind us that what is being conserved is energy, NOT rotation? A smaller mass moving at relativistic speeds would gain energy through collapse, yet not increase rotation by much, the closer you get to c. As Yogi Berra might have said, if he were a Physics theorist... "It's Tau Zero all over again."

Answer 3

angular momentum around a point or a line is the cross product of momentum and distance to that point or line.

This is very significant in classical mechanics and in quantum mechanics because it is a conserved quantity. Conserved quantities are great in QM , because they help us describe systems. In classical mechanics, conservation of angular momentum is equivalent to Keplers "equal area law".

Moment of inertia comes into play a lot in astrophysics with galaxy and star and black hole dynamics. A more terrestrial example where it comes in is skaters spinning. A skater puts her arms out and spins, and then puts her arms up. She thereby decreases moment of inertia. Conservation of angular momentum thereby forces her rotational velocity to shoot up and she spins like crazy.

Answer 4

By definition Momentum is velocity x mass.

In case of Angular momentum mass is equivalent to I the inertia. velocity is angular velocity.

So it is.

Angular momentum = Angular velocity x Inertia(about the applicable axis).

All linear equations are good in angular motion when Mass is substituted by Inertia and other enties are substituted by Angular entities.(example, velocity by angular velocity,distance by angle in radians ......).

Answer 5

I'm keen to know what the intended projectile would be?