How To Pump A Swing

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How To Pump A Swing?

• Tareq Ahmed Mokhiemer
• Physics Department

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Contents

• Introduction to the swing physics and different
  pumping schemes
• Pumping a swing from a standing position
• Qualitative understanding
• Pumping from seated position
• Qualitative understanding
• Conclusion

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What is meant by pumping a swing ?

• Repetitive change of the riders position and/or
  orientation relative to the suspending rod.

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How to get a swing running from a standing
position?
By standing and squatting at the lowest point
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• The motion of the child is a modeled by the
  variation with r with time ? r(t)
• This is equivalent to a parametric oscillator

• Let

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• By scanning against pumping frequencies
• amplification was found to occur at 2
• Constant pumping frequencie ? Succession of
  amplification and attenuation.

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Another (naïve) model for r(t)
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No Amplification !!
Expected result !
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A more realistic model
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• For each initial velocity ? a threshold for the
  steepness of r(t) at ?0.

Unexpected result !!
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How does pumping occur physically?

• Two points of view
• The conservation of angular momentum
• Conservation of energy

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conservation of angular momentum
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Conservation of energy
At the highest point
At the mid-point Gravitational force
Centrifugal force
Only gravitational force
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Another scheme for pumping from a standing
position
The swinger pumps the swing by leaning forward
and backward while standing
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Pumping a swing from a seated position
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Potential Energy
kinetic energy
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The equation of motion
M I1g Sin(f(t)) N g Sin(f(t)?(t))-I1f(t)
I2 (f(t) ?(t))-2 I2 N ?(t) Cos(?(t))-I1 N
?(t) Cos(?(t))0
A Surprise
The oscillation grows up linearly!!
T(t) is either 0.5 rad when f is gowing or -0.5
when f is decreasing
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The growth rate is proportional to the steepness
of the frequency of the swingers motion
T(t) is changes between 0.7 rad and -0.7 rad
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A special case
The Lagrangian reduces to
And the equation of motion is
A driven Oscillator.
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T(t) changes sinusoidally
Pumping occurs at approximately the natural
frequency not double the frequency.
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Pumping from a seated position

• More efficient in starting the swing from rest
  position

• With the same frequency of the swinger motion,
  the oscillation grows faster in the seated
  pumping.

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Conclusion
Standing position
Seated position

• Exponential growth
• Parametric Oscillator

• Linear growth
• Driven Oscillator
• Efficient in starting the swing from rest