The Normal Component of Acceleration
The wire constrains the bead to follow a particular
path. In a sense, the wire is acting on the bead
through what we call the normal component of acceleration.
The normal component of acceleration's responsibility
is simply to change the direction of motion of the bead.
The normal component depends only on the curvature
k of the wire and the speed v of the
bead at the particular moment and is given by
normal component = k * v^2
On a related note, the magnitude of the acceleration
of an object moving at a uniform speed v along
a circle of radius r is given by
magnitude of acceleration = v^2/r
The radius r and the curvature k are
related by the following equation:
r = 1/k
For a general curve, r is the radius of the circle
that best fits the curve at the point in question. It is
sometimes called the "osculating" circle where "osculate"
comes from the Latin word that means "to kiss."
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