Pendulum project update

[Alan Kay]

Hi Folks --

A good side point about pendulums is that the motion is harmonic only 
for small excursions, since harmonic motion is roughly the spring law 
which is proportional to x, and pendulums are proportional to sine x. 
Sine x and x are close to the same values only for small angles.

All these subtle details are reasons why we don't do pendulums with 
5th graders. Compare this to gravity near the surface of the earth 
where the accelleration is constant to about 1 part in a million 
(note that it isn't really constant because it is inversely 
proportional to the square of the distance and this is changing a 
little bit -- about 4 meters in the ball drop example).

Cheers,

Alan

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At 7:09 PM +0200 4/24/03, Andreas Raab wrote:
>Hi Bert,
>
>>  I guess you are a little bit blinded by all the math. It's really
>>  simple - the acceleration of the pendulum depends on its position. If
>>  it is vertical, you have zero acceleration (because the force of
>>  gravity is straight down and does not cause the pendulum to
>>  swing). If it is horizontal, you have the maximum acceleration,
>>  again because the force points straight down, but now this is
>>  exactly the direction to make the pendulum rotate.
>
>Exactly right.
>
>>  This trivially maps to an etoy (just increase the speed by the
>>  acceleration value), the only obstacle is to get the acceleration
>>  depending on the current angle. This is what the "weighing angles"
>>  discussion was all about. Either you do this (best for 5th graders I
>>  guess), or you "measure" it, like in the pendulum project I sent last
>>  week. It works fine without any trigonometry or square roots: You
>>  basically just take the _horizontal_ extent of a line that represents
>>  your pendulum. If the line is vertical, its horizontal extent
>>  is zero. If the line is horizontal, its horizontal extent is maximal.
>>  You still need a sign for the force, which you can get by checking
>>  the extent relative to the line's reference point.
>
>That's a little too abstract for my taste. One really needs to explain why
>the relation between gravity and pendulum is not strictly linear because
>otherwise you can do a very reasonable first order approximation of a
>pendulum by simply subtracting its direction from gravity's direction. Note
>that this fulfills all your constraints: It's zero if the pendulum points
>into the same direction as gravity and it increases up to 90 degrees.
>
>I've attached a project with this (wrong!) model to illustrate the fact. And
>my challenge is: Explain why this can't be correct (heh, heh ;-)
>
>Cheers,
>   - Andreas

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