The displacement of the motion of a particle is represent by a equation $Y = ASinωt + BCosωt$ . The motion of the particle is

SHM with amplitude $\sqrt{A^{2} + B^{2}}$

SHM with amplitude A+B

SHM with amplitude A

Oscillatory but not in SHM

Solution

SHM with amplitude $\sqrt{A^{2} + B^{2}}$

A linear combination of sine and cosine functions

$y = Asinωt + Bcosωt$ ....1

$Let A = α cosϕ . . . . 2 B = α sinϕ . . . . 3$

then equation (1) becomes

$y = acosϕsinωt + asinϕcosωt y = asin (ωt + ϕ) . . \sin (α + β) = sinαcosβ + cosαsinβ$

it is clear that the equation number (2) in simple harmonic motion with the amplitude $α$ squaring and adding (2) and(3),we get

$A^{2} + B^{2} \leq α^{2} (\cos^{2} ϕ + \sin^{2} ϕ) α = \sqrt{A^{2} + B^{2}}$

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