Resonance and a Magic Trick

Magic tricks are cool. Especially when the trick is really physics. In this trick, I can make one of the four balls move more than the others. (When you watch the video, you will see why I am not a professional magician). You could set this up in a variety of ways. I state that if we (me and people around me) all work together with our mind and focus on the same ball, our brain waves can resonate with that ball and make it move. I let the people around me pick. In this video, I make the smallest two move.

So, what is the trick? The trick is not a trick. It is not resonance with brain waves, but it is resonance.

Each one of those balls, if displaced, will oscillate at a particular frequency. For a pendulum, this frequency is:


Where *g* is the local gravitational field (9.8 N/kg or 9.8 m/s²) and *l* is the length of the pendulum. So, each ball has a different length and thus a different frequency that it will swing back and forth at. If you drive (shake) that pendulum at the frequency that it naturally swings, the amplitude of its oscillation will get bigger. You have already seen this effect and used it. I know you have. If you have ever pushed a child (or adult) on a swing, you know that you can't just push them when ever you want. If you wait and push while they are the top of their swing, you will increase the amplitude of the motion. This is resonance.

So, in the magic trick, you just need to slightly shake the stick (like I said, I am not very good so you can probably notice that I am shaking it). If you shake at the frequency for one of the balls, its amplitude will increase. Well, how do you know what frequency? Do you have to calculate it before hand? No, just focus on the one you want to move and shake in sync with its motion.

To take this idea a little further than it needs to go (isn't that what I always do?), I decided to model this situation. Modeling a pendulum with a moving point isn't nearly as easy as an oscillating spring with an oscillating point. Both can be used to demonstrate resonance. In my model (created in [vpython](http://www.vpython.org)), I have 4 masses attached by springs to a moving wall. The four mass all have different masses. For a mass on a spring, the frequency of oscillation is:


In the case of the 4 masses with different length strings, the length changes and *g* does not. To make the situation similar, I will keep the spring constant (*k*) constant and change the mass.

(Hopefully I will remember to talk about the oscillation of a spring - it is awesome on so many different levels)

In this model, I have 4 different masses and each one has a particular "natural" frequency. If I shake the wall at the natural frequency of one of these masses (even if I shake it a little) the wall keeps pushing the mass at the right time making the amplitude larger. For the other masses, the pushes are not at the right time and they do not get bigger. Here is the position of the 4 masses when the wall oscillates at the frequency for mass 1:


The green line represents the position of mass 1. Note the black line is the position of the wall. It has a very small amplitude, but still produces significant motion in mass 1. The other masses still move, but they do not significantly increase in amplitude. What if I oscillate the wall at the frequency for mass 2?


And here are the graphs for resonance at the frequency for mass 3 and 4:


**The Trick**

So the trick is to move that stick at a frequency that is the same as one of the oscillators. If you move it just a little (small amplitude) maybe people won't notice and think you are Harry Potter or something.