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#10. Principles of the Magnetic
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Circular Motion![]()
Because the force is perpendicular to the velocity, it can
only change the direction of motion, not its speed or
energy. Because no energy is needed to keep up the motion, it can (in
principle) persist indefinitely. In addition, however, there will also exist a small force
parallel to the axis, repelling the particle away from the tip of
the cone. That added force gradually slows down the particle's advance
down the axis and finally reverses it, causing it to "mirror" and bounce
back.
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It turns out that the product T x E, the period T times the energy E,
is almost a constant. It is not an exact constant, like total
energy in a system, but if the rate of change is slow enough, e.g. if the string
is pulled rather slowly, it comes very close.
The motion of electrons and ions spiraling around magnetic field
lines is also periodic. While the period of a pendulum changes when
its string gets longer or shorter, that of a spiraling ion or electron
changes as it moves into regions where the magnetic field is weaker or stronger.
Just as for a pendulum the product T x E stays very nearly constant, so here
too, a certain quality, an "adiabatic invariant," is almost kept at a
constant value. From that constancy it is possible to deduce the "mirroring" of
particle and many other properties of their motion.
That is somewhat similar to the case of the lowered support, but
the calculation gives a diferent rate. With the lowered support, work is
also done--but that happens when the weight is pulled in from the sides to
swing in a shorter arc, not at the bottom of the swing.
The process described here is related to the way children
"pump" a swing to make it go higher. The child moves arms, legs and body
in a way that works against the centrifugal force, and the energy invested
in overcoming this force ends up producing a more energetic swinging
motion.
(This is a highly simplified explanation and
assumes that from the point of view of the child in the swing, nature
behaves exactly the same as anywhere else, only a centrifugal force is
added. The actual situation can be more complicated.)
Furthr Explorations
The "Exploratorium" science museum in San Francisco has a small swing
(too small to carry a person) that can be "pumped" from the outside. The
seat of the swing, instead of hanging by two ropes or chains, is attached
to the axis by two smooth parallel rods.
Above the regular swing seat is a second seat, with two wide
holes threaded by the two rods. Under normal circumstances, the second
seat will drop to the bottom, on top of the regular one. However, a rope
is connected to its middle, going over the bar from which the swing hangs
and down again, and a person standing next to the swing can pull that
string or let it go, making the second seat rise along the rods or fall
down again.
With your hand, you set the swing moving with moderate motion.
Now, by pulling the rope or letting go at suitable times in the
oscillation, you can easily "pump up" the motion. You only need to pull
the swing up when it passes the lowest part of its motion, and let
it down again at the extreme ends of its motion, when for a brief
instant it is at rest. Next Stop: #10H. Motion
of Trapped Radiation--History
Author and Curator: Dr. David
P. Stern
Adiabatic Invariants
There
exists a different and somewhat more abstract manner of reaching the same
result. The period T of rotation, the time required by the particle for one
circuit around its guiding field line, becomes shorter as the particle
approaches the tip of the cone. After all, the total speed of the
particle is unchanged, its rotation speed nearly so, while the distance covered
by one circuit gets shorter and shorter near the tip.
In the theory of motions, this is an example of a periodic
motion whose period gradually decreases. The best-known periodic motion is
the back-and-forth swing of a pendulum, say of a weight suspended by a string
(drawing). The shorter the string, the shorter the time of each swing
("period"), which goes like the square root of the length. One can replace the
support point with a pulley wheel, which is gradually lowered and its string
shortened (ignore the word "pull" which is explained further below). The
bottom of the swing stays in the same height, but the period gets shorter and
shorter.
Note on the above illustration
Many books give this example but state that the string is pulled up,
over the wheel, while the pendulum is swinging. This is a more complex
situation. As the pendulum swings, it generates a centrifugal force, and
the pull on the string, besides lifting the weight to a higher average
position (which increases the potential energy), also has to overcome the
resistance of the centrifugal force. That requires an extra input of
energy from the force pulling the string, and since energy has to go
somewhere, it makes the swing of the pendulum more vigorous.
Last updated 25 November 2001 Back to the Index Page
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Mail to Dr.Stern:
education("at"
symbol)phy6.org
Co-author: Dr.
Mauricio Peredo
Spanish translation by J.
Méndez
Re-formatted 9-28-2004