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Angular Momentum
Angular momentum
From Wikipedia, the free encyclopedia.
In
physics, angular momentum intuitively measures how much the
linear momentum is directed around a certain point called the
origin; the
moment of
momentum. Since angular momentum depends upon the origin of choice, one must
be careful when discussing angular momentum to specify the origin and not to
combine angular momenta about different origins.
The traditional mathematical definition of the angular
momentum of a particle about some origin is:
-
L = r×p
where L is the angular momentum of the particle, r
is the position of the particle expressed as a displacement vector from the
origin, and p is the linear momentum of the particle. If a system
consists of several particles, the total angular momentum about an origin can be
gotten by adding (or integrating) all the angular momenta of the constituent
particles.
For many applications where one is only concerned about
rotation around one axis, it is sufficient to discard the vector nature of
angular momentum, and treat it like a scalar where it is positive when it
corresponds to a counter clock-wise rotations, and negative clock-wise. To do
this, just take the definition of the cross product and discard the unit vector,
so that angular momentum becomes:
-
L = |r||p|sinθ
where θ is the angle between r and p measured
from r to p; an important distinction because without it, the sign
of the cross product would be meaningless. From the above, it is possible to
reformulate the definition to either of the following:
-
L = ±|p||rperpendicular|
where rperpendicular is called the lever
arm distance to p. The easiest way to conceptualize this is to
consider the lever arm distance to be the distance from the origin to the line
that p travels along. With this definition, it is necessary to consider
the direction of p (pointed clock-wise or counter clock-wise) to figure
out the sign of L). Equivalently:
-
L = ±|r||pperpendicular|
where pperpendicular is the component of
p that is perpendicular to r. As above, the sign is decided base on
the sense of rotation.
In analogy to
Newton's second law for linear momentum, we have the following
law about angular momentum:
-
where τ is the net
torque about the origin. This implies that angular momentum is a
conserved quantity as long as there is no net torque applied to the
particle. What's more, this conservation can be generalized to a system of
particles under most conditions so that:
-
where τexternal is any torque applied to
the system of particles.
The conservation of angular momentum is used extensively in
analyzing what is called central force motion. In central force motion,
two bodies form an isolated system not influenced by outside forces, and the
origin is placed somewhere on the line between the two bodies. Since any force
the bodies exert on each other must be directed along this line, there can be no
net torque, with respect to the afore-mentioned origin, on either body. Thus,
angular momentum is conserved. 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.
In modern (late
20th century) theoretical physics, angular momentum is described using an
different formalism. Under this formalism, angular momentum is the
2-form
Noether charge associated with rotational invariance (As a result, angular
momentum isn't defined for general curved spacetimes, unless it happens to be
asymptotically rotationally invariant). For a system of point particles without
any intrinsic angular momentum, it turns out to be
-
(Here, the
wedge product is used.).
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