The Tautochrone Half-pipe

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It wasn't too long ago that I learned about the tautochrone and brachistochrone in my mechanics class. Looking back on it a few weeks later, it seemed remarkable to me that you could release two particles at different points on the curve and yet they will still reach its minima at the same time. Why should nature provide such a singular path? It defies the day-to-day common "sense" my mind developed!

Today the reality of the problem led to me to do nothing less than construct my own tautochrone. It's really quite simple to do with household materials. I didn't have to step out once for parts or tools.

Although a cycloid is more difficult to express mathematically than, say, a parabola, it's actually one of the easiest curves to generate with any precision. Anyone who had a Spirograph will know what I mean. Unfortunately, I got rid of my spirograph years ago, so I made do with what I had: an old CD, tape, and a pencil.

If you examine the picture you'll see that I attached the pencil to a couple of old 9V batteries for stability. The CD made a perfect generating circle for my cycloid because the resulting cycloid fit the box I used perfectly. CD's have a radius of about 60mm, so the resulting width is 2*pi*60mm = 37.7cm. We know this is the width from the generating equations for the cycloid:

x = r(t - sin(t))
y = r(1 - cos(t))

where x is the horizontal distance, r the radius of the generating circle, y the height, and t the angle traveled by the generating circle. (So from t = 0 to 2*pi gives one cycloidal hump) At t = 2*pi, x = 2*pi*r.

"So what?", you say, "This is just math!" With that, I'll show you some pictures now. (Right click and select "view image" if you want to see them in detail.)

As you can see, I place my finger inside the CD and roll it against the side of the box. The CD must roll without slipping!

The big cycloid traced by the pencil is rather beautiful, to a nerd like me. Of course, a cycloid drawn on paper doesn't help a marble roll down a slope, so I next drew a cycloid on the side of my box. You can probably see it pretty easily.

Cutting it was the next step, which involved a sharp knife and some patience. Knives don't cut curves without care.

After that it was a matter of cutting some standard poster board to width and length and taping it to the sides of my cycloidal form, so that the surface fits the curve.

It's a cycloidal (tautochrone) half-pipe! The motion under gravity by this surface was fully what the equations were telling me (Just think what the chemical equations for TNT tell me!) . Two marbles released simultaneously from two different heights hit the bottom at the same time. Since the time to reach the bottom is a constant, you may wonder what that is. It's just 246 milliseconds. (The wikipedia article explains why.) You may also want to see it for yourself, so here's a video. Bravo!

Beware, the file is 8mb in size, and I'm not clued in enough to compress it. tautodrop.avi

So, if you ever feel the (strange!) urge to investigate the tautochrone, this is your guide! I think it's a marvelous demonstration of motion.

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