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Re: [Phys-l] starting a swing



I have often wondered if the following scenario is analogous to pumping yourself higher on a swing:

(This started as a test question that I wrote to include conservation of angular momentum and simple harmonic motion in one problem.)

A clown is wearing a harness that allows him to hang from a wire and twist as a torsional pendulum. He does this with a small amplitude, with arms outstretched.
Then, as he passes through the equilibrium position, he rapidly pulls his arms in. Like an ice-skater, his angular velocity jumps up, just at the moment that it was a maximum. This gives a boost to his kinetic energy and thus to the max displacement.

Then, at the maximum displacement, he re-extends his arms. Since this occurred at the moment angular velocity was zero, there is no change in angular velocity at that moment. But like a piston in a thermodynamic cycle, he is ready to repeat the process and he can cyclically drive his energy and amplitude ever higher.

So my question is: when you are pumping yourself higher on a swing, can we think of it as strategically changing your moment of inertia at the right time?


________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker [jsd@av8n.com]
Sent: Saturday, February 07, 2009 7:12 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] starting a swing

On 02/07/2009 01:57 PM, Stefan Jeglinski wrote:
I'm thinking about the mechanism by which one *starts* a playground
swing. I would like to think about it qualitatively, from both an
energy and a force standpoint.

OK.

But let's keep in mind that starting is not the whole
story.

If you just sit in a swing and move your legs back and forth at the
knees, not much happens. To start, one lays back and moves their legs
forward, more or less straightening them. The lay-back and legs-up
drops the center of mass. This drop in the potential energy is
accompanied by an increase in kinetic energy, hence the initial
motion.

The vertical motion of the CM is insignificant twice
over in this situation
-- the CM doesn't move much, and
-- even if it did, it wouldn't be useful for *starting*
the swing.

But what are the forces for this initial motion? It would have to be
tension in the rope/chain, no?

Yes ... but force is a vector, and here the angle is at
least as important as the magnitude. Think about the
initial angular momentum ... it has to come from somewhere.

By laying back, the rope is bent at an
angle about the location where your hands are, and this changes the
tension in the rope, correct? This same bent requires by simple
geometry that the seat be higher than its original position - not a
problem as long as the CM has been lowered.

Imagine a motorized remote-control yo-yo. If you suddenly
start the motor, it will try to climb the string. It will
climb up one *side* of the string. The string will no longer
be vertical. This will produce a component of the force (aka
tension) in a horizontal direction. As seen from the left
side, lowering your back is a clockwise motion, and raising
your legs is also a clockwise motion, so you are in fact
starting to climb the ropes, thereby changing the angle of
the dangle and producing a sideways force.

Now compare this analysis:

http://www.hk-phy.org/articles/swing/swing_e.html

This article does not refer to starting a swing, but seems to analyze
the dynamics once a swing is going. It seems that it can now be kept
going simply by changing the CM. I don't have a swing handy to test
this, but my memory seems to serve me that this is qualitatively
correct also.

That is absolutely not a way of *starting* a swing.

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

Here's the deal:

There are two methods of _driving_ a swing after it is started.

a) One involves moving your back and legs at frequency f, the
natural frequency of the swing. This produces a sideways force
varying with frequency f. The magnitude of the force is essentially
independent of the amplitude of the motion, so the amplitude
converges exponentially toward a dynamic equilibrium level.

b) The other method involves standing on the swing and moving
vertically at frequency 2f. You stand up at the bottom, at the
middle of each half-cycle, doing work against centrifugal force
(in your frame). You squat down at the top of the swing, at
the end of each half-cycle, neither doing nor undoing any work
against centrifugal force, since you are not moving.

This is called a degenerate parametric amplifier. The "parameter"
here is the length of the pendulum, i.e. the distance from the
pivot to the CM.

For constant length of stroke (roughly the length of your legs)
you do *more* work per cycle when the amplitude is large, because
the centrifugal force is large. This means the amplitude does
not converge to any equilibrium, dynamic or otherwise. It
diverges exponentially.

Again, this is completely useless for *starting*.

If you try this, beware that it is very powerful, and becomes
more powerful as the amplitude increases, and you can get yourself
into trouble, perhaps more easily than you might imagine.

It is more fun on a very tall swing, because it takes longer and
you can build up more energy before it becomes dangerously nonlinear.
Once upon a time I built a 50 foot tall swing in my back yard.
Good for parties attended by lots of physicists. Requires strict
adult supervision.

Parametric amplifiers have played an important role in the history
of quantum measurement, because unlike a 741 op-amp or a one-transistor
class-A amplifier, you can analyze every detail of the parametric
amplifier. You can understand how it works at the level of second
quantized ladder operators: a† + a.

I can say a lot more about this if anybody is interested.
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