Angular Momentum

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An Angular Momentum is a rotational motion analog of the linear momentum.

References

2016

  • (Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/angular_momentum Retrieved:2016-2-8.
    • In physics, angular momentum (less often moment of momentum or rotational momentum) is a physical quantity corresponding to the amount of rotational motion of an object, taking into account how fast a distribution of mass rotates about some axis. It is the rotational analog of linear momentum. For example, a conker twirling around on a short chord has a lower angular momentum compared to twirling a large heavy sledgehammer at high speed. For a conker of a given mass, increasing the length of chord and angular speed of twirling increases the angular momentum of the conker, and for a fixed angular speed and length, a heavier conker has a larger angular momentum than the lighter conker. In principle, the conker and sledgehammer could have the same angular momentum, despite the differences in their masses and sizes.

      Symbolically, angular momentum it is denoted L, J, or S, each used in different contexts. The definition of angular momentum for a point particle is a pseudovector, L = r×p, the cross product of the particle's position vector r (relative to some origin) and its momentum vector p = mv. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object (how fast it rotates about an axis) via the moment of inertia I (which depends on the shape and distribution of mass about the axis of rotation). However, while ω always points in the direction of the rotation axis, the angular momentum L may point in a different direction depending on how the mass is distributed.

      Angular momentum is additive; the total angular momentum of a system is the (pseudo)vector sum of the angular momenta. For continua or fields one uses integration. The total angular momentum of anything can always be split into the sum of two main components: "orbital" angular momentum about an axis outside the object, plus "spin" angular momentum through the centre of mass of the object.

      Angular momentum is important because it is connected to the symmetry of rotation, and is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. Torque can be defined as the rate of change of angular momentum, analogous to force. 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 falling cat problem, and precession of tops and gyros. Applications include the gyrocompass, control moment gyroscope, inertial guidance systems, reaction wheels, flying discs or Frisbees and Earth's rotation to name a few. In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is.

      In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the “spin” of elementary particles does not correspond to literal spinning motion.

2005

[math]\displaystyle{ L = m\;v\;r sin \theta }[/math]
or more formally by the vector product
[math]\displaystyle{ \vec{L} = \vec{r} \times \vec{p} }[/math]
The direction is given by the right hand rule which would give [math]\displaystyle{ vec{L} }[/math] the direction out of the diagram. For an orbit, angular momentum is conserved, and this leads to one of Kepler's laws. For a circular orbit, L becomes
[math]\displaystyle{ L = mvr }[/math]
The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity. It is analogous to linear momentum and is subject to the fundamental constraints of the conservation of angular momentum principle if there is no external torque on the object. Angular momentum is a vector quantity. It is derivable from the expression for the angular momentum of a particle.
[math]\displaystyle{ \vec{L}=\vec{I}\times{\vec{\omega}} }[/math]
Angular momentum and linear momentum are examples of the parallels between linear and rotational motion. They have the same form and are subject to the fundamental constraints of conservation laws, the conservation of momentum and the conservation of angular momentum .

1996

[math]\displaystyle{ \vec{L}=\vec{r}\times\vec{p} }[/math]
where [math]\displaystyle{ \vec{r} }[/math] is the radius vector and [math]\displaystyle{ p }[/math] is the momentum. If the velocity is perpendicular to the radius vector, then [math]\displaystyle{ L=mvr }[/math]

1963

  • (Feynman et al., 1963) ⇒ Richard P. Feynman, Robert B. Leighton and Matthew Sands (1963, 1977, 2006, 2010, 2013) "The Feynman Lectures on Physics": New Millennium Edition is now available online by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer ⇒ http://www.feynmanlectures.caltech.edu/
    • QUOTE: Although we have so far considered only the special case of a rigid body, the properties of torques and their mathematical relationships are interesting also even when an object is not rigid. In fact, we can prove a very remarkable theorem: just as external force is the rate of change of a quantity [math]\displaystyle{ p }[/math], which we call the total momentum of a collection of particles, so the external torque is the rate of change of a quantity [math]\displaystyle{ L }[/math] which we call the angular momentum of the group of particles.
(...)Like torque, angular momentum depends upon the position of the axis about which it is to be calculated.
(...)Now we shall go on to consider what happens when there is a large number of particles, when an object is made of many pieces with many forces acting between them and on them from the outside. Of course, we already know that, about any given fixed axis, the torque on the ith particle (which is the force on the ith particle times the lever arm of that force) is equal to the rate of change of the angular momentum of that particle, and that the angular momentum of the ith particle is its momentum times its momentum lever arm. Now suppose we add the torques [math]\displaystyle{ \tau_i }[/math] for all the particles and call it the total torque [math]\displaystyle{ \tau }[/math]. Then this will be the rate of change of the sum of the angular momenta of all the particles [math]\displaystyle{ L_i }[/math], and that defines a new quantity which we call the total angular momentum [math]\displaystyle{ L }[/math]. Just as the total momentum of an object is the sum of the momenta of all the parts, so the angular momentum is the sum of the angular momenta of all the parts. Then the rate of change of the total [math]\displaystyle{ L }[/math] is the total torque:
[math]\displaystyle{ \tau=\sum\tau_i=\sum \frac{dL_i}{dt}=\frac{dL}{dt} }[/math]
Now it might seem that the total torque is a complicated thing. There are all those internal forces and all the outside forces to be considered. But, if we take Newton’s law of action and reaction to say, not simply that the action and reaction are equal, but also that they are directed exactly oppositely along the same line (Newton may or may not actually have said this, but he tacitly assumed it), then the two torques on the reacting objects, due to their mutual interaction, will be equal and opposite because the lever arms for any axis are equal. Therefore the internal torques balance out pair by pair, and so we have the remarkable theorem that the rate of change of the total angular momentum about any axis is equal to the external torque about that axis!
[math]\displaystyle{ \tau=\sum\tau_i=\tau_{ext}=dL/dt }[/math]
Thus we have a very powerful theorem concerning the motion of large collections of particles, which permits us to study the over-all motion without having to look at the detailed machinery inside. This theorem is true for any collection of objects, whether they form a rigid body or not.
One extremely important case of the above theorem is the law of conservation of angular momentum: if no external torques act upon a system of particles, the angular momentum remains constant.
A special case of great importance is that of a rigid body, that is, an object of a definite shape that is just turning around. Consider an object that is fixed in its geometrical dimensions, and which is rotating about a fixed axis. Various parts of the object bear the same relationship to one another at all times. Now let us try to find the total angular momentum of this object. If the mass of one of its particles is [math]\displaystyle{ m_i }[/math], and its position or location is at [math]\displaystyle{ (x_i,y_i) }[/math], then the problem is to find the angular momentum of that particle, because the total angular momentum is the sum of the angular momenta of all such particles in the body. For an object going around in a circle, the angular momentum, of course, is the mass times the velocity times the distance from the axis, and the velocity is equal to the angular velocity times the distance from the axis:
[math]\displaystyle{ L_i=m_iv_ir_i=m_ir^2_i\omega, }[/math]
or, summing over all the particles [math]\displaystyle{ i }[/math], we get
[math]\displaystyle{ L=I\omega }[/math]
where
[math]\displaystyle{ I=\sum_im_ir^2_i }[/math]
This is the analog of the law that the momentum is mass times velocity. Velocity is replaced by angular velocity, and we see that the mass is replaced by a new thing which we call the moment of inertia [math]\displaystyle{ I }[/math], which is analogous to the mass.
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