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angular momentum
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Definition/Summary
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Angular momentum (or "moment of momentum") is position "cross" momentum:  .
Angular momentum of a rigid body equals moment of inertia tensor "times" angular velocity:  .
Net torque on a rigid body equals rate of change of angular momentum:  . In particular, when net torque is zero, angular momentum is constant.
Angular momentum is measured relative to a point, and is additive in the sense that the angular momentum of a body is the sum (or integral) of the angular momentum of its parts, measured relative to the same point:  .
Surprisingly, angular momentum of a rigid body is not generally parallel to (instantaneous) angular velocity: this is obvious from the formula  . For this reason, angular velocity is not generally constant (there is precession), even when angular momentum is.
Angular momentum is a vector (strictly, a pseudovector), with dimensions of mass times distance squared per time ( ), and measured in units of kg m²/s (or N.m.s or J.s). |
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Equations
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Angular momentum (point mass):
Angular momentum (rigid body):
Fundamental equation of motion:
Rotational kinetic energy:
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Recent forum threads on angular momentum
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Breakdown
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Physics
> Classical Mechanics
>> Newtonian Dynamics
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Extended explanation
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Angular velocity:
A rigid body has an instantaneous axis of rotation, and an instantaneous angular velocity  parallel to that axis.
The position of the axis depends on the (inertial) frame of reference, but  itself does not.
Where the axis passes through the body, all points on the axis are instantaneously stationary.
Each point  on the body has instantaneous velocity  , where  is any point on the axis.
Centre of mass:
The centre of mass of a rigid body with density  and total mass  is the point  (fixed in the body but not fixed in space) such that  , ie 
Ordinary (linear) momentum and orbital angular momentum:
The ordinary (linear) momentum is   .
The orbital angular momentum is the angular momentum calculated as if the whole body was concentrated at its centre of mass:  .
Spin:
The spin is the angular momentum measured relative to the centre of mass:  .
Spin is not additive, since the spins of the parts of a rigid body are measured relative to the different centres of mass of each part. For example, if a "dumbell" of two uniform spheres rigidly joined by a light rod rotates about an axis, each sphere has angular momentum parallel to the axis, as measured relative to its own centre of mass, but not generally as measured relative to the other sphere's centre of mass, and so the total spin is not generally parallel to the axis, even though the individual spins always are.
Therefore angular momentum equals spin plus orbital angular momentum:
 
 
Fundamental equation of motion (rotational):
"Crossing" Newton's second law,  ,
with the position vector of each infinitesimal part of a rigid body (and cancelling out the torques of the internal reaction forces between the parts, and since  ) gives:
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net torque = rate of change of angular momentum.
Precession:
When net torque is zero, a rigid body has constant angular momentum:  .
However, angular momentum is parallel to angular velocity only when it is parallel to a principal axis of the body.
In all other cases, the angular velocity vector ( ) will rotate about the (fixed) angular momentum vector ( ): this is precession. See moment of inertia.
Moment of inertia tensor:
A tensor converts one vector to a different vector. The two vectors are parallel only if they are eigenvectors of the tensor.
The moment of inertia tensor converts the angular velocity vector of a rigid body into the angular momentum vector:  .
The moment of inertia tensor is a symmetric tensor. Its eigenvectors are the principal axes of the rigid body, and its eigenvalues are the principal moments of inertia, and so in a coordinate system aligned with the those axes, its matrix is diagonal:
Published lists of moments of inertia always include only moments of inertia about principal axes.
In any other coordinate system, its matrix is:
where  ,  , and so on.
The words "moment of inertia" usually refer to the diagonal elements of the tensor, the moments of inertia about particular axes.
The tensor Î is fixed in the body, but is not generally fixed in space, and so is not generally constant in the fundamental equation of motion  .
However, Î is constant in Euler's equations ( etc), which are the fundamental equations of motion relative to coordinates fixed in the body. See Euler's equations. |
Commentary
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