Oscillations

JEE Main level, numerical answer, medium

27 of 194

Question 27 of 194

Two waves executing simple harmonic motion travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the 3\sqrt{3} times of amplitude of individual motions. The phase difference between the two motions is _____ (degree)

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Answer: 60

When two waves of the same frequency superimpose, the resultant amplitude is given by
Aresultant=A12+A22+2A1A2cos⁡ϕ{A}_{resultant}=\sqrt{{A}_{1}^{2}+{A}_{2}^{2}+2{A}_{1}{A}_{2}\cos \phi }
⇒3A=A2+A2+2A2cos⁡ϕ\Rightarrow \sqrt{3}A=\sqrt{{A}^{2}+{A}^{2}+2{A}^{2}\cos \phi }
⇒3A2=2A2+2A2cos⁡ϕ\Rightarrow 3{A}^{2}=2{A}^{2}+2{A}^{2}\cos \phi
⇒cos⁡ϕ=12\Rightarrow \cos \phi =\dfrac{1}{2}
∴ϕ=60∘∴\phi =60^{\circ}
∴∴Phase difference =60=60 degree

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