Question

The period T of a pendulum with length L meters that makes a maximum angle of θ₀ with the vertical is

The vertical is:

T= 4\sqrt{\frac{L}{9}}\int _0^{\frac{\pi }{2}}\frac{dx}{\sqrt{1-k^2sin^2x}}

where k=sin((1/2)θ₀) and g=9.8 m/sec² in the acceleration due to gravity.

(a) Find the first four terms of a series expansion for T by expanding the integrand using the binomial series and integrating term by term (your answer will include L, g, k). You may use the following integration fact:

The integration fact is:

\int _0^{\frac{\pi }{2}}sin^2\left(x\right)dx=\frac{1\cdot 3\cdot 5...\left(2n-1\right)}{2\cdot 4\cdot 6...\left(2n\right)}\frac{\pi }{2}

(b) The period is often approximated by T ~ 2*pi*(sqrt(L/g)). What is the relative error between this approximation and the one found in part (a) for a pendulum with L=1 meter and θ₀=20°? What if θ₀=60°?

Answer 1