How To Pump A Swing?

1. How To Pump A Swing? Tareq Ahmed Mokhiemer Physics Department

2. 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

3. What is meant by “pumping a swing” ? Repetitive change of the rider’s position and/or orientation relative to the suspending rod.

4. How to get a swing running from a standing position? By standing and squatting at the lowest point

5. 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

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

9. Another (naïve) model for r(t)

10. No Amplification !! Expected result !

11. A more realistic model

13. For each initial velocity  a threshold for the steepness of r(t) at θ=0. Unexpected result !!

14. How does pumping occur physically? Two points of view • The conservation of angular momentum • Conservation of energy

15. conservation of angular momentum

18. Conservation of energy At the highest point At the mid-point: Gravitational force Centrifugal force Only gravitational force

20. Another scheme for pumping from a standing position The swinger pumps the swing by leaning forward and backward while standing

21. Pumping a swing from a seated position

22. Potential Energy = kinetic energy =

23. The equation of motion M I1g Sin(φ(t)) + N g Sin(φ(t)+θ(t))-I1φ’’(t) –I2 (φ’’(t)+ θ’’(t))-2 I2 N θ’’(t) Cos(θ(t))-I1 N θ’’(t) Cos(θ(t))==0 A Surprise The oscillation grows up linearly!! Θ(t) is either 0.5 rad when φ is gowing or -0.5 when φ is decreasing

24. The growth rate is proportional to the steepness of the frequency of the swinger’s motion Θ(t) is changes between 0.7 rad and -0.7 rad

25. A special case: The Lagrangian reduces to: And the equation of motion is A driven Oscillator.

26. Θ(t) changes sinusoidally Pumping occurs at approximately the natural frequency not double the frequency.

27. 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.

28. Conclusion Standing position Seated position • Exponential growth • Parametric Oscillator • Linear growth • Driven Oscillator • Efficient in starting the swing from rest