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The Physics of Resonance

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Playground Swings

A playground swing was one of Tesla's favorite examples of a resonant system. It's easy to measure its natural frequency. Each time the swing moves forward and then returns to its starting position counts as one cycle. Using a stop watch determine the length of time a swing needs to complete say 20 cycles. Divide 20 cycles by the time and you have the swings frequency in cycles per second or Hertz (Hz).

Since a swing is basically a pendulum it's possible to calculate its resonant or natural frequency using pendulum equations as follows:

1
f = (g/L)0.5
2p
where:
g = gravity constant
= 9.8 m/s/s for Earth
L = Length

Note that the natural frequency of the swing is not influenced by the mass of the person in it. In other words' it makes no difference whether a swing has a large adult or a small child in it. It will have the about the same natural frequency. Slight differences can be caused by slightly different locations of the person's center of mass. This is located about two inches below the navel. When people are sitting the center of mass is in about the same place relative to the seat of the swing regardless of whether the person is an adult or a child.

If a forcing function is applied to a swing at the natural frequency of the swing it will resonate. The amplitude of the swing will increase during each back and forth cycle. The forcing function can be provided by a second person pushing on the swing. In this case even a small child can make a large adult swing by pushing in sync with the swing's back and forth cycle. The forcing function can also be provided by the person in the swing. In this case the person in the swing shifts her center of mass very slightly by changing the position of her legs or torso. This creates a slight pushing force which makes the swing go higher and higher. It takes a very small force but it has to be timed perfectly. 

The big question is what keeps the swing from flying apart or spinning over the top of the swing's frame and subsequently killing its rider? After all, if it is a resonating system then it should be very dangerous to keep applying force in time with the swing's frequency.  The answer is fairly simple. The equation given above is only good for small angles. When the swing goes beyond a certain height it is no longer possible for the person in it to apply the necessary small force in sync with the natural frequency because the natural frequency changes. In other words the motion of the system is naturally limited.

 

Suggested Classroom Activities: Visit a playground and measure the natural frequency of a swing. It should make very little difference whether the person in it is large or small. Attach a flimsy piece of thread to the person in the swing. Instruct them not to assist in making the swing move and then attempt to make the swing resonate by pulling on the thread without breaking it. If the the force is applied in time with the natural frequency of the swing it will make the swing resonate.

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