Pea and spoon effect.

The pea and spoon effect is what happens when you load a pea on one end of a spoon, lay the spoon perpendicular to a table edge, and smack the other end, sending the pea hurtling across the room, much to the displeasure of your parents. Other ways of doing this are pencils loaded on rulers at school and large rocks in trebuchets for castle sieges. You know, kid stuff.

The mathematics are quite simple. First you start with the Lagrange-Euler equations:

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Where L is the energy equation for the entire system.

So for a ball rolling on a plane, where the table is assumed to have negligible inertia, and thus no impact on the energy equation, it can be written as:

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where I is the moment of inertia, or 2/5*m*r^2.

We can simplify:

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where kt=(1/(1+2/5)).

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This yields in the end:

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The last term in the last equation is the interesting one, as this shows why the ball is launched into the air.

An interesting side-effect of this term is the fact that when the table tilts rapidly to the left, the ball initially rolls to the right. In fact, it's even more complicated than that. The ball moves to the left, but it rolls to the right.

How is this? Well, this term comes from the upward motion of the table pushing on the ball. As the table pushes upward, it also pivots underneath the ball. So the ball, which is moving upward, now is further from the origin than before, because it's the same x-distance, but now it's got the y-axis component, too. This means that along the tilted table surface, the only way the ball could be further than before is either by A) slipping or B)rolling. In our case, this movement is definitely rolling. (If the movement were sufficiently quick, you might be able to provoke slipping of the ball/table interface. I'm not certain how fast you'd have to rotate, but it'd be pretty darned fast.)