Re: [Phys-L] pendulum +- numerical methods -> large angle oscillations
I have my cal-based physics students measure some large angle periods. Of
course, there are large uncertainties because the damping is fairly large a
large starting angles, but they get a good comparison to the elliptical
integral correction factors (which I supply in table form along with the
mathematics describing how to develop the integral that must be solved
numerically). I have then do a range of starting angles. If you use a solid
rod or disk versus a string-mass simple pendulum you can have starting angles
larger than 90 degrees. That also gives them experience with physical pendula.
Plastic rulers with a hole close to the end work well. Three or four
oscillations gives a pretty decent period measurement so you can see a large
elliptical integral correction factor.
If you use photogates in "pendulum" mode and collect each period, you can
measure the periods well enough to calculate the non-harmonic deviation for 5,
10, 15 degree oscillations and compare those to the correction factors. If you
start at larger angles you can monitor the change in the period as the
amplitude decreases.
-> -----Original Message-----
-> From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of John
-> Denker
-> Sent: Tuesday, September 23, 2014 9:13 PM
-> To: Phys-L@Phys-L.org
-> Subject: Re: [Phys-L] pendulum +- numerical methods
->
-> Keep in mind that if all you want to know is how the nonlinearity affects the
-> pendulum, you can express that in terms of elliptic integrals. For not-too-
-> large angles the function is easy to evaluate. Any scientific math library
-> worthy of the name will have a canned function to do it.
-> See also figure 3 and the formulas at
-> https://en.wikipedia.org/wiki/Pendulum_%28mathematics%29#Arbitrary-
-> amplitude_period
-> or just
-> https://www.google.com/search?q=%22elliptic+integral%22
->