A point mass is subjected to two simultaneously sinusoidal displacements in x-direction x1(t) = A sin ωt and x2(t) = A sin . Adding a third sinusoidal displacement x3(t) = B sin (ωt + φ) brings the mass to a complete rest. The values of B and φ are :

Engineering

Physics

Superposition of Simple Harmonic Motion

Question

A point mass is subjected to two simultaneously sinusoidal displacements in x-direction x1(t) = A sin ωt and x₂(t) = A sin $(ωt + \frac{2 π}{3})$ . Adding a third sinusoidal displacement x₃(t) = B sin (ωt + φ) brings the mass to a complete rest. The values of B and φ are :

$\sqrt{2} A, \frac{3 π}{4}$

$A, \frac{4 π}{3}$

$\sqrt{3} A, \frac{5 π}{6}$

$A, \frac{π}{3}$

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Here $ϕ = π + θ$

A cos 30º = B sin $θ \Rightarrow Bsinθ = \frac{\sqrt{3} A}{2}$ and A sin 30º + B cos $θ = A \Rightarrow Bcosθ = \frac{A}{2}$

Solving above, B = A and $θ = 60 º = \frac{π}{3} .$ Hence $f=240º= \frac{4 π}{3}$