Built on Facts

Diavolo!

Posted by Matt Springer on August 31, 2009
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This guy is Allo Diavolo:

i-5ff7f98a415105893e51d7a970c0db5a-diavolo.png

He was a circus daredevil. At the dawn of the 20th century he worked on a number of stunts dressed in his ominous horned outfit. These days a lot of people, including me, have heard of him as an example in the pages of physics textbooks. In my case it was Halliday and Resnick, a standard (and good) freshman physics text. Diavolo did a trick where he looped the loop riding a bicycle. It’s pretty impressive:

i-52c3dd21fe75c49f7f159128dc25a711-loop.jpg

The breathless circus ad copy claims that this stunt is more or less certain death. It wasn’t, of course. I wouldn’t want to try it, but it’s certainly a physical possibility. How?

As it happens, it’s a basic fact of classical mechanics at an object in instantaneous uniform circular motion will undergo an acceleration. After all, an acceleration is just change in velocity, and velocity is not just a speed. It’s a speed in a particular direction. If the direction is changing, the velocity is changing. If the velocity is changing, there’s an acceleration. Newton was the guy who figured out acceleration hundreds of years ago. One of his famous laws of motion is that force equals mass times acceleration. Therefore if we know something about the forces and accelerations acting on Diavolo, we can say something about the requirements to do this trick without going splat.

An object in uniform circular motion is undergoing an acceleration equal to:

i-a35d3ff6ec1bc05d89156f4465f8cfa0-2f.png

The velocity is v, the radius of the circle is r. The bicycle’s motion isn’t quite uniform; the speed is changing along with the direction. But this is ok, our equation remains valid for circular motion as long as we’re interested in the component of the acceleration that’s perpendicular to the surface of the track. And that’s what we’re after. We want to see whether he stays on it or not. To do so, he needs that acceleration we wrote above.

There’s two forces acting on him. One is the force of gravity. It’s equal to mg, where m is his mass and g is the acceleration due to gravity at the earth’s surface. The other force is the so-called normal force, which is just the force the track exerts on him as it prevents him from ghosting through the surface. If his speed is just barely enough to prevent a fall, the normal force will be zero. Therefore the only force acting is the mg gravity, which will as Newton said be equal to the mass times acceleration:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

Do a little algebra jazz and find v:

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Halliday and Resnick estimate the radius of the loop at 2.7 meters, which plugged into the above equation give a minimum speed of about 5 m/s, or around 11.2 miles per hour at the top of the loop.

What makes it especially tough to do in practice is that this translates to a considerably higher speed at the bottom of the loop. There the normal force is going to be quite high, and it’s the normal force that we think of and feel as the “g-force”. Keeping balanced and steering while experiencing these powerful and rapidly changing forces is quite the trick.

How about we make this a quick little Monday exercise? Using the equations for kinetic energy and gravitational potential energy, estimate his speed at the bottom of the loop. From there, repeat the procedure we used above to find the g-force. (Hint: g-force is N/m). Good luck!

(Slight personal ad: It’s the beginning of a new semester, and that means it’s time to be violated by horrible textbook prices! As always, one good way to wave some money is to buy online. If do do so via Amazon.com, consider getting there via the link on the left sidebar. At zero extra cost to you, your purchase will help support me financially. For science!)

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Comments

  1. #1 rob
    August 31, 2009

    how about an extra credit problem. say the great Diavolo doesn’t use the peddles on the bike–he just coasts down the ramp and through the loop. if the height of the top of the ramp is H, how high can the top of the loop be for Diavolo to make it through the loop without falling off?

  2. #2 Tim Gaede
    August 31, 2009

    I got 11.5 m/s (25.7 m.p.h) and 5g acceleration (feeling 6g)

  3. #3 Tim Gaede
    August 31, 2009

    Is there any particular book from Amazon you’d recommend?

  4. #4 rob
    August 31, 2009

    yeah, i got 11.5 m/s at the bottom of the loop too.

  5. #5 Carl Brannen
    August 31, 2009

    I think the most death defying stunt (from that part of the 20th century before OSHA) was the one where two single-seat planes, fly in formation. They lock their controls, then go out on the wings and jump across, thereby changing planes in midair. This was an Ormer Locklear stunt. And you thought that modern jets are dangerous.

  6. #6 Robert
    August 31, 2009

    According to a New York Times article on April 2, 1902, Diavolo’s spiral ellipse was 26 feet high and 18 feet wide. The ramp down to the spiral was 100 feet long, 3 feet wide, and had an angle of 45 degrees. The bicycle did not have pedals. Diavolo eventually was killed performing the stunt at a subsequent event.

    So that would be 8 meters for radius, but since they used the word “ellipse” we don’t know what the exact shape was. Or the Times could have been wrong.

  7. #7 mrcreosote
    August 31, 2009

    @Robert

    26 feet high = 8 meter diameter = 4 meter radius

  8. #8 Robert
    August 31, 2009

    Sorry about the diameter/radius goof!

    The New York Times had a photograph of Diavolo upside down in the loop. It’s hard to make an exact measurement, but I believe the radius (not diameter!) is around 3 meters which is not too far from what Resnick and Halliday said, especially if you calculate radius based on the cyclist’s center of mass.

    The picture is here: http://query.nytimes.com/mem/archive-free/pdf?res=9C04E1D71E3BE631A2575BC2A96E9C946597D6CF

    The story with the picture is about a man named Conn Baker who also performed the stunt after Diavolo died. Baker broke a leg and two ribs before doing it successfully.

  9. #9 Ed
    September 1, 2009

    I seem to remember this sort of problem in one of my physics classes. But it seemed to me that they said it worked better with a parabola (or maybe an ellipse?)rather than a circle. It was something to do with not needing the speed quite so high. If you do the calculations with a different conic section do you get significantly different results – i.e. slower speeds still making it around?

  10. #10 Tim Gaede
    September 1, 2009

    Roller coaster loops today often have a clothoid shape, which means that the curvature is more blunt at the bottom so that the “g-forces” the riders feel will be more even throughout the loop. Perhaps this makes the ride less nauseating while remaining thrilling. Incidentally, there is at least one roaster that has the riders lie down so that when they experience the strongest “g-forces” through loops, they are directed “backward” rather than “up” resulting in a supposedly better sensation.

    My ‘puter confirms the 11.5 m/s figure and gives me an elapsed time of 1.06 seconds. I did not have to compute the normal force to figure those out because it did not alter the speed and the trajectory was constrained.