Question

The displacement of an oscillating particle is given by x= a sin ωt + b cos ωt where a, b and ω are constants. Prove that the particle performs a linear S.H.M. with amplitude A = \(\sqrt{a^2+b^2}\)

Answer 1

x= a sin ωt + b cos ωt

Let a = A cos ∅ and b = A sin ∅, so that

A² = a² + b² and tan ∅ = b/a

 x=A cos ∅ sin ωt + A sin ∅ cos ωt

 x = A sin (ωt+∅)

which is the equation of a linear SHM with amplitude  A = \(\sqrt{a^2+b^2}\)   and phase constant ∅ = tan^-1(b/a) as required.