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:
||9.8 m/s/s for Earth
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
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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