Angular Momentum

Syntax

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Overview

In physics, the angular momentum of a particle about an origin is a vector quantity related to rotation, equal to the mass of the particle multiplied by the cross product of the position vector of the particle with its velocity vector. The angular momentum of a system of particles is the sum of that of the particles within it
Angular momentum is an important concept in both physics and engineering, with numerous applications. 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. Rotational symmetry of space is related to the conservation of angular momentum as an example of Noether's theorem. The conservation of angular momentum explains many phenomena in nature.

Equations

To finish off our comparison of translational (straight-line) and rotational motion, let's consider the rotational equivalent of momentum, which is angular momentum. For straight-line motion, momentum is given by p = mv. Momentum is a vector, pointing in the same direction as the velocity. Angular momentum has the symbol L, and is given by the equation:

Angular momentum is also a vector, pointing in the direction of the angular velocity.

In the same way that linear momentum is always conserved when there is no net force acting, angular momentum is conserved when there is no net torque. If there is a net force, the momentum changes according to the impulse equation, and if there is a net torque the angular momentum changes according to a corresponding rotational impulse equation.

Angular momentum is proportional to the moment of inertia, which depends on not just the mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation. This leads to some interesting effects, in terms of the conservation of angular momentum.

A good example is a spinning figure skater. Consider a figure skater who starts to spin with their arms extended. When the arms are pulled in close to the body, the skater spins faster because of conservation of angular momentum. Pulling the arms in close to the body lowers the moment of inertia of the skater, so the angular velocity must increase to keep the angular momentum constant.

Parallels between straight-line motion and rotational motion
Let's take a minute to summarize what we've learned about the parallels between straight-line motion and rotational motion. Essentially, any straight-line motion equation has a rotational equivalent that can be found by making the appropriate substitutions (I for m, torque for force, etc.).

Example - Falling down
You've climbed up to the top of a 7.5 m high telephone pole. Just as you reach the top, the pole breaks at the base. Are you better off letting go of the pole and falling straight down, or sitting on top of the pole and falling down to the ground on a circular path? Or does it make no difference?

The answer depends on the speed you have when you hit the ground. The speed in the first case, letting go of the pole and falling straight down, is easy to calculate using conservation of energy:

In the second case, also apply conservation of energy. If you have negligible mass compared to the telephone pole, just work out the angular velocity of the telephone pole when it hits the ground. In this case we use rotational kinetic energy, and the height involved in the potential energy is half the length of the pole (which we can call h), because that's how much the center of gravity of the pole drops. So, for the second case:

For a uniform rod rotating about one end, the moment of inertia is 1/3 mL2. Solving for the angular velocity when the pole hits the ground gives:

For you, at the end of the pole, the velocity is h times the angular velocity, so:

So, if you hang on to the pole you end up falling faster than if you'd fallen under the influence of gravity alone. This also means that the acceleration of the end of the pole, just before the pole hits the ground, is larger than g (1.5 times as big, in this case), which is interesting.

Which way do these angular variables point, anyway?
Displacement is a vector. Velocity is a vector. Acceleration is a vector. As you might expect, angular displacement, angular velocity, and angular acceleration are all vectors, too. But which way do they point? Every point on a rolling tire has the same angular velocity, and the only way to ensure that the direction of the angular velocity is the same for every point is to make the direction of the angular velocity perpendicular to the plane of the tire. To figure out which way it points, use your right hand. Stick your thumb out as if you're hitch-hiking, and curl your fingers in the direction of rotation. Your thumb points in the direction of the angular velocity.

If you look directly at something and it's spinning clockwise, the angular velocity is in the direction you're looking; if it goes counter-clockwise, the angular velocity points towards you. Apply the same thinking to angular displacements and angular accelerations.

A circular disk of mass “M” and radius “R” is rotating with angular velocity “ω” about its vertical axis. When two small objects each of mass “m” are gently placed on the rim of the disk, the angular velocity of the ring becomes (a)Mω
(M+4m)
(b)Mω
(M+2m)
(c)Mω
(M
2
+4m)
(d)Mmω
(M+4m)

:

Two disks of moments of inertia I1 and I2 and having angular velocities ω1 and ω2 respectively are brought in contact with each other face to face such that their axis of rotation coincides. The situation is as shown in the figure just before the contact. What is the angular velocity of the combined system of two rotating disks, if they acquire a common angular velocity?

Figure 1: Two disks are rotating about a common axis. Two rotating disks

(a)I1+I2
ω1+ω2
(b)I1ω1+I2ω2
ω1+ω2
(c)I1ω1+I2ω2
I1+I2
(d)I1I2ω1ω2
I1+I2

Conservation of Angular Momentum

In a closed system angular momentum is constant. This conservation law mathematically follows from continuous directional symmetry of space (no direction in space is any different from any other direction). See Noether's theorem.
(The cross-product of velocity and momentum is zero, because these vectors are parallel.)

It is assumed that internal interaction forces obey Newton's third law of motion in its strong form, that is, that the forces between particles are equal and opposite and act along the line between the particles.

In orbits, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit

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).

The conservation of angular momentum in Earth–Moon system results in the transfer of angular momentum from Earth to Moon (due to tidal torque the Moon exerts on the Earth). This in turn results in the slowing down of the rotation rate of Earth (at about 42 nsec/day), and in gradual increase of the radius of Moon's orbit (at ~4.5 cm/year rate).

Applications

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External Links

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Joke

The answer to the problem was 'log(1+x)'. A student copied the answer from the good student next to him, but didn't want to make it obvious that he was cheating, so he changed the answer slightly, to 'timber(1+x)'

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