Question

The displacement of an oscillating particle is given by x = asinωt + bcosωt 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 = asinωt + bcosωt

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

A = a² + b² and tan φ = \(\cfrac{b}{a}\)

\(\therefore\) x = A cos φ sin ωt + A sin φ cos ωt

\(\therefore\) x = A sin (ωt + φ)

which is the equation of a linear SHM with amplitude A = \(\sqrt{a^2+b^2}\) and phase constant 

 as required.