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Math 21a Supplement on Torque and Angular Momentum
According to Newton, the position vector,
r
(t), of a particle changes with time, t, under the
influence of a force vector
F
according to the rule
m
r
´´ =
F
.
(1)
Here, m is the mass of the particle.
The
momentum
of the particle is, by definition, the vector
p
≡
m
r
´.
Thus, momentum is
mass times velocity.
The
angular
momentum
is defined to be the vector
L
≡
m
r
×
r
´ =
r
×
p
.
(2)
Note that
L
is perpendicular to both the position vector
r
and the velocity vector
r
´.
In some sense, angular momentum measures the deviation from motion on a straight line.
Indeed if
L
= 0, then the velocity vector
r
´ is proportional to the position vector
r
, and this implies
that the particle travels along a straight line (but maybe back and forth).
To see that the condition
r
´
being proportional to
r
means straight line motion, differentiate the unit vector
r
/r under this
assumption to see that the latter is constant.
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This note was uploaded on 02/18/2012 for the course MATH 310 taught by Professor Gregoryj.galloway during the Fall '11 term at University of Miami.
 Fall '11
 GregoryJ.Galloway
 Math, Multivariable Calculus

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