Question

Two collinear harmonic oscillations are made to superpose. If their displacements are represented as: x1 = 2a cos ωt and x2 = 3a cos (ωt +ϕ) Calculate the amplitude of the resultant oscillation when the phase difference is (i) 2π, (ii) 0, (iii) π/2 and (iv) π

Answer 1

i)

$$2a \cos ωt+ 3a \cos (ωt +2π)=5a \cos ωt$$

$$A=5a$$

ii)

$$2a \cos ωt+ 3a \cos (ωt +0)=5a \cos ωt$$

$$A=5a$$

iii)

$$2a \cos ωt+ 3a \cos (ωt +0.5π)=2a \cos ωt- 3a \sin (ωt)$$

$$A=a\sqrt{2^2+3^2}=\sqrt{13}a\approx 3.6a$$

iv)

$$2a \cos ωt+ 3a \cos (ωt +π)=-a \cos ωt$$

$$A=a$$