Polynomial expression of Pendulum period with respect to angle (large)

[seboastien]

1. Homework Statement

The swinging period T(θ) for a small amplitude simple pendulum, is given by T, a constant for a constant length pendulum. If the initial angle θ is large, then the amplitude becomes large and the period needs to be corrected. The correction to the large amplitude period can be expressed as an even polynomial in initial angle θ. Find T(θ) for large amplitude pendulum

Homework Equations

Some even polynomial

The Attempt at a Solution

I assume that The function takes the form, T(θ) = (2pi*sqrt(l/g))(1+aθ^2 + bθ4 + cθ^6+...)
Where a,b,c... are constants. But I have no way of proving this, nor do I know how to find out what these constants are! Help!

 

[kuruman]

For a derivation of the amplitude correction see

http://webphysics.davidson.edu/alumni/BeKinneman/pendulum/report.htm

Note that the bottom line is an elliptic integral. In your case, I suggest that you expand the integrand as a polynomial in θ, then integrate each term of the polynomial separately.