Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: are normal reaction and tension conservative ?



>For homework, come up with an example along the following lines:
> -- We have an object at the lower end of a long rope.
> -- I move the upper end of the rope.
> -- The result is spectacularly non-conservative.


At 07:23 PM 7/1/01 +0530, Chetan wrote:
Suppose a man is pulling a body of mass 'm' vertically up with an
accerleration 'a'.
...
After time 't' the man suddenly stops pulling. The body then rises
through some height and returns to the position at the moment the pull in
the string became zero

That'll do it.

(i am not sure about the non-stretchy part)

Right. If we disregard the stretchiness of the rope in this case, it
becomes very hard to analyze the bouncing and jouncing that the body will
undergo. It might bounce forever. More realistically, we should assume a
slight amount of stretchiness and internal friction, so that the bouncing
will eventually die out.

========================

Here is another solution to the problem, that does not require letting the
string go slack, and which has significant technological applications.

Consider the apparatus shown in
http://www.monmouth.com/~jsd/physics/dpa.gif

The body is initially swinging slightly. Every time it passes through the
middle, I pull up on the handle, doing work against centrifugal
force. Every time it reaches either end of its swing, I let down on the
handle, doing no work against centrifugal force, since the object is nearly
stopped. At the middle of the next half-cycle, I pull up on the handle
again.....

This is an extremely powerful way of pumping energy into the oscillatory mode.

You can demonstrate this for yourself with a playground swingset. Stand on
the seat. At the end of each half-cycle, lower yourself by bending your
knees. At the middle of each half-cycle, raise yourself.....

Be careful! This works so well that in a few cycles you can drive the
swing to a dangerous amplitude. Also: The taller the swing the more
interesting it is.

This is called _parametric_ pumping. The name arises because the length of
the rope is usually considered "just" a parameter in the problem, distinct
from the main dynamical variable (the angle of displacement along the
swinging motion). But in this case we are fiddling with the length parameter.

Note the categorical distinction:
1) For the usual way of driving a swing, you sit on the seat and wiggle
your feet in tune with the swing's natural frequency (omega). The
amplitude converges exponentially to some dynamic equilibrium.
2) For parametric pumping, you move up and down at 2*omega. The
amplitude grows exponentially, running away from equilibrium. It's like
negative damping.

The ramifications of this are numerous, practical, and beautiful.
*) For present purposes, it is one more illustration of the fact that a
moving constraint is not conservative.
*) There are other ramifications that we can discuss later if anybody is
interested:
-- Low noise amplifier
-- Quantum nondemolition voltmeter
-- Conservation of phase space; Bogoliubov transformation.
-- Ideal amplifier, ideal quantum measurement device, analyzable in
detail from first principles.

Let me thank you all for being so encouraging.

Thanks to you, and congratulations for asking such an interesting question.