PhysicsLAB Resource Lesson
Vibrating Systems - Simple Pendulums

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When the applied frequency generated by an outside source matches the natural frequency of a vibrating system, resonance occurs. Resonance states are not limited to interference conditions. For example, pendulums have a natural period of
  • L represents the pendulum's length and
  • g represents the gravitational field strength, or the local acceleration due to gravity which is commonly taken as 9.8 m/sec2
Let's consider a pendulum having a length 1 meter. Its period would be equal to
approximately 2 seconds. Therefore it has a natural frequency of
approximately f = ½ = 0.5 hertz. 
If you were pushing a small child on a swing that was 1 meter long, you could push him once during each swing, that is once each 2 seconds; or you could push every other swing, once each 4 seconds; or push every third swing, once each 6 seconds. 
Expressing these time intervals in terms of frequency, you could push at the following frequencies:
f = 1/2 = 0.50 hertz
f = 1/4 = 0.25 hertz
f = 1/6 = 0.17 hertz
That is, you can "drive" the pendulum's motion by applying an external forced vibration that either matches the pendulum's natural frequency or equals a sub-multiple of its natural frequency. In any of these instances, you could witness the swing's resonance by noticing its amplitude increasing with each push.
 What would be first three time intervals at which you could push a small child in a swing that has a length of 2.25 meters that would result in resonance?

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