2. Consider a pendulum with equation of motion:

*+0² sin(x) = 0,

which oscillates with amplitude 'a'. Define time 't' appropriately and write the system in non-

dimensional form as:

x" + sin(x)=0. Then, for the non-dimensional system,

i) Show that when it has a periodic motion, its period, T, is given by 4K(B), where

B-sin²(a/2) and

px/2

K(B) = ² do/√(1-B sin² $).

ii) Show that K(B) has the power series expansion K(B) [1 + (1/2)2B +

(1*3/2*4)2³²+...] /2, for |ß<<1. Then, deduce that the period for motion of small amplitude

'a' is given by T=2x[1 + a²/16+11a4/3072 +0(ab)].

=