Angular momentum


Angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.
The three-dimensional angular momentum for a point particle is classically represented as a pseudovector, the cross product of the particle's position vector and its momentum vector; the latter is in Newtonian mechanics. Unlike linear momentum, angular momentum depends on where this origin is chosen, since the particle's position is measured from it.
Angular momentum is an extensive quantity; that is, the total angular momentum of any composite system is the sum of the angular momenta of its constituent parts. For a continuous rigid body or a fluid, the total angular momentum is the volume integral of angular momentum density over the entire body.
Similar to conservation of linear momentum, where it is conserved if there is no external force, angular momentum is conserved if there is no external torque. Torque can be defined as the rate of change of angular momentum, analogous to force. The net external torque on any system is always equal to the total torque on the system; the sum of all internal torques of any system is always 0. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant.
The change in angular momentum for a particular interaction is called angular impulse, sometimes twirl. Angular impulse is the angular analog of impulse.

Examples

The trivial case of the angular momentum of a body in an orbit is given by
where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.
The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by
where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
Thus, for example, the orbital angular momentum of the Earth with respect to the Sun is about 2.66 × 1040 kg⋅m2⋅s−1, while its rotational angular momentum is about 7.05 × 1033 kg⋅m2⋅s−1.
In the case of a uniform rigid sphere rotating around its axis, if, instead of its mass, its density is known, the angular momentum is given by
where is the sphere's density, is the frequency of rotation and is the sphere's radius.
In the simplest case of a spinning disk, the angular momentum is given by
where is the disk's mass, is the frequency of rotation and is the disk's radius.
If instead the disk rotates about its diameter, its angular momentum is given by

Definition in classical mechanics

Just as for angular velocity, there are two special types of angular momentum of an object: the spin angular momentum is the angular momentum about the object's center of mass, while the orbital angular momentum is the angular momentum about a chosen center of rotation. The Earth has an orbital angular momentum by nature of revolving around the Sun, and a spin angular momentum by nature of its daily rotation around the polar axis. The total angular momentum is the sum of the spin and orbital angular momenta. In the case of the Earth the primary conserved quantity is the total angular momentum of the Solar System because angular momentum is exchanged to a small but important extent among the planets and the Sun. The orbital angular momentum vector of a point particle is always parallel and directly proportional to its orbital angular velocity vector ω, where the constant of proportionality depends on both the mass of the particle and its distance from origin. The spin angular momentum vector of a rigid body is proportional but not always parallel to the spin angular velocity vector Ω, making the constant of proportionality a second-rank tensor rather than a scalar.

Orbital angular momentum in two dimensions

Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, and treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum. Thus, where linear momentum is proportional to mass and linear speed
angular momentum is proportional to moment of inertia and angular speed measured in radians per second.
Unlike mass, which depends only on amount of matter, moment of inertia depends also on the position of the axis of rotation and the distribution of the matter. Unlike linear velocity, which does not depend upon the choice of origin, orbital angular velocity is always measured with respect to a fixed origin. Therefore, strictly speaking, should be referred to as the angular momentum relative to that center.
In the case of circular motion of a single particle, we can use and to expand angular momentum as reducing to:
the product of the radius of rotation and the linear momentum of the particle, where is the linear speed.
This simple analysis can also apply to non-circular motion if one uses the component of the motion perpendicular to the radius vector:
where is the perpendicular component of the motion. Expanding, rearranging, and reducing, angular momentum can also be expressed,
where is the length of the moment arm, a line dropped perpendicularly from the origin onto the path of the particle. It is this definition,, to which the term moment of momentum refers.

Scalar angular momentum from Lagrangian mechanics

Another approach is to define angular momentum as the conjugate momentum of the angular coordinate expressed in the Lagrangian of the mechanical system. Consider a mechanical system with a mass constrained to move in a circle of radius in the absence of any external force field. The kinetic energy of the system is
And the potential energy is
Then the Lagrangian is
The generalized momentum "canonically conjugate to" the coordinate is defined by

Orbital angular momentum in three dimensions

To completely define orbital angular momentum in three dimensions, it is required to know the rate at which the position vector sweeps out an angle, the direction perpendicular to the instantaneous plane of angular displacement, and the mass involved, as well as how this mass is distributed in space. By retaining this vector nature of angular momentum, the general nature of the equations is also retained, and can describe any sort of three-dimensional motion about the center of rotation – circular, linear, or otherwise. In vector notation, the orbital angular momentum of a point particle in motion about the origin can be expressed as:
where
  • is the moment of inertia for a point mass,
  • is the orbital angular velocity of the particle about the origin,
  • is the position vector of the particle relative to the origin, and,
  • is the linear velocity of the particle relative to the origin, and
  • is the mass of the particle.
This can be expanded, reduced, and by the rules of vector algebra, rearranged:
which is the cross product of the position vector and the linear momentum of the particle. By the definition of the cross product, the vector is perpendicular to both and. It is directed perpendicular to the plane of angular displacement, as indicated by the right-hand rule – so that the angular velocity is seen as counter-clockwise from the head of the vector. Conversely, the vector defines the plane in which and lie.
By defining a unit vector perpendicular to the plane of angular displacement, a scalar angular speed results, where
and
where is the perpendicular component of the motion, as above.
The two-dimensional scalar equations of the previous section can thus be given direction:
and for circular motion, where all of the motion is perpendicular to the radius.
In the spherical coordinate system the angular momentum vector expresses as

Analogy to linear momentum

Angular momentum can be described as the rotational analog of linear momentum. Like linear momentum it involves elements of mass and displacement. Unlike linear momentum it also involves elements of position and shape.
Many problems in physics involve matter in motion about some certain point in space, be it in actual rotation about it, or simply moving past it, where it is desired to know what effect the moving matter has on the point—can it exert energy upon it or perform work about it? Energy, the ability to do work, can be stored in matter by setting it in motion—a combination of its inertia and its displacement. Inertia is measured by its mass, and displacement by its velocity. Their product,
is the matter's momentum. Referring this momentum to a central point introduces a complication: the momentum is not applied to the point directly. For instance, a particle of matter at the outer edge of a wheel is, in effect, at the end of a lever of the same length as the wheel's radius, its momentum turning the lever about the center point. This imaginary lever is known as the moment arm. It has the effect of multiplying the momentum's effort in proportion to its length, an effect known as a moment. Hence, the particle's momentum referred to a particular point,
is the angular momentum, sometimes called, as here, the moment of momentum of the particle versus that particular center point. The equation combines a moment with a linear speed. Linear speed referred to the central point is simply the product of the distance and the angular speed versus the point: another moment. Hence, angular momentum contains a double moment: Simplifying slightly, the quantity is the particle's moment of inertia, sometimes called the second moment of mass. It is a measure of rotational inertia.
File:Moment of inertia examples.gif|thumb|right|Moment of inertia, and therefore angular momentum, is different for each shown configuration of mass and axis of rotation.
The above analogy of the translational momentum and rotational momentum can be expressed in vector form:
  • for linear motion
  • for rotation
The direction of momentum is related to the direction of the velocity for linear movement. The direction of angular momentum is related to the angular velocity of the rotation.
Because moment of inertia is a crucial part of the spin angular momentum, the latter necessarily includes all of the complications of the former, which is calculated by multiplying elementary bits of the mass by the squares of their distances from the center of rotation. Therefore, the total moment of inertia, and
the angular momentum, is a complex function of the configuration of the matter about the center of rotation and the orientation of the rotation for the various bits.
For a rigid body, for instance a wheel or an asteroid, the orientation of rotation is simply the position of the rotation axis versus the matter of the body. It may or may not pass through the center of mass, or it may lie completely outside of the body. For the same body, angular momentum may take a different value for every possible axis about which rotation may take place. It reaches a minimum when the axis passes through the center of mass.
For a collection of objects revolving about a center, for instance all of the bodies of the Solar System, the orientations may be somewhat organized, as is the Solar System, with most of the bodies' axes lying close to the system's axis. Their orientations may also be completely random.
In brief, the more mass and the farther it is from the center of rotation, the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by,
where is the radius of gyration, the distance from the axis at which the entire mass may be considered as concentrated.
Similarly, for a point mass the moment of inertia is defined as,
where is the radius of the point mass from the center of rotation, and for any collection of particles as the sum,
Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s or N⋅m⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity.. The units of angular momentum can be interpreted as torque⋅time. An object with angular momentum of can be reduced to zero angular velocity by an angular impulse of.
The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System.