A particle moving with SHM from an extremity of the path towards centre is observed to be at distances x1, x2 and x3 from the centre at the ends of three successive seconds. The period of SHM is:

x=A cos ωt, then x1=A cos ω, x2=A cos 2ω,x3=A cos 3ω          x1+x32x2=A(cos ω+cos 3ω)2A cos 2ω                        =2A cos 2ω cos ω2A cos 2ω= cos ω                     ω=cos-1x1+x32x2 ⇒              T=2πcos-1x1+x32x2