a swing is modelled as a pendulum - pi.math.cornell.edu
TRANSCRIPT
![Page 1: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/1.jpg)
![Page 2: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/2.jpg)
A swing is modelled as aPendulum
![Page 3: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/3.jpg)
The position of the pendulum is given by the angle x:
![Page 4: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/4.jpg)
The position of the pendulum is given by the angle x:
It’s angular velocity is given by x’
x’
![Page 5: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/5.jpg)
The “state” of the pendulum is given by the pair (x, x’)
![Page 6: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/6.jpg)
![Page 7: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/7.jpg)
![Page 8: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/8.jpg)
![Page 9: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/9.jpg)
![Page 10: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/10.jpg)
![Page 11: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/11.jpg)
![Page 12: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/12.jpg)
![Page 13: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/13.jpg)
![Page 14: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/14.jpg)
First method for pumping:
Standing rider squats at A and C, and stands at B and D
![Page 15: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/15.jpg)
![Page 16: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/16.jpg)
Physics of why it works:Conservation of angular momentum
Angular momentum = L^2 x’
(L^2 before squat) (x’ before squat) =(L^2 after squat) (x’ after squat)
x’ after = (L^2 before/L^2 after) x’ before
![Page 17: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/17.jpg)
![Page 18: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/18.jpg)
![Page 19: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/19.jpg)
Second method for pumping:Seated rider rotates body at A and C.
![Page 20: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/20.jpg)
![Page 21: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/21.jpg)
Physics of why it works:
Rotating body adds a burst of angular momentum to the system, call it M.
At A and C, the angular velocity is zero.
The result of M is to make x jump a constant amount.
![Page 22: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/22.jpg)
![Page 23: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/23.jpg)
![Page 24: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/24.jpg)
Comparison of the two types of pumping
![Page 25: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/25.jpg)
x’after = x’before * K, where K>1
x after = x before + Mwhere M>0
![Page 26: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/26.jpg)
![Page 27: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/27.jpg)
In one case we multiply by a number K>1.
In the other case we add a number M>0.
Which is better?
![Page 28: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/28.jpg)
ConclusionStanding pumping multiplies the current amplitude by a factor larger than 1.
Seated pumping adds a fixed quantity to the current amplitude.
Thus for small amplitudes, seated pumping is better.But for larger amplitudes, standing pumping is better
So start off by seated pumping, then switch to standing pumping when the amplitude gets large enough.
![Page 29: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/29.jpg)
ConclusionStanding pumping multiplies the current amplitude by a factor larger than 1.
Seated pumping adds a fixed quantity to the current amplitude.
Thus for small amplitudes, seated pumping is better.But for larger amplitudes, standing pumping is better
So start off by seated pumping, then switch to standing pumping when the amplitude gets large enough.
![Page 30: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/30.jpg)
ConclusionStanding pumping multiplies the current amplitude by a factor larger than 1.
Seated pumping adds a fixed quantity to the current amplitude.
Thus for small amplitudes, seated pumping is better.But for larger amplitudes, standing pumping is better
So start off by seated pumping, then switch to standing pumping when the amplitude gets large enough.
![Page 31: A swing is modelled as a Pendulum - pi.math.cornell.edu](https://reader033.vdocuments.us/reader033/viewer/2022052601/628cdb6687954617190d722e/html5/thumbnails/31.jpg)
Reference
Modeling the Pumping of a SwingS.Wirkus, R.Rand, A.RuinaThe College Mathematics Journal 29:266-275 (1998)