Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by 45o, then:(i) the resultant amplitude is (1+2)a(ii) the phase of the resultant motion relative to the first is 900(iii) the energy associated with the resulting motion is 3+22 times the energy associated with any single motion(iv) the resulting motion is not simple harmonic

Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by 45^o, then:
(i) the resultant amplitude is $(1 + \sqrt{2}) a$
(ii) the phase of the resultant motion relative to the first is 90⁰
(iii) the energy associated with the resulting motion is $(3 + 2 \sqrt{2})$ times the energy associated with any single motion
(iv) the resulting motion is not simple harmonic

A
(i, iii)
B
(i, ii)
C
(i, iv)
D
(iii, iv)

Solution:

Resultant $y = y_{1} + y_{2} + y_{3}$

$y = a \sin (ωt - 45^{o}) + a \sin ωt + a \sin (ωt + 45^{o}) \Rightarrow y = (1 + \sqrt{2}) a \sin ωt$
So the phase of the resulting motion relative to the first (1) is 45°.
$As Energy \propto {(Amplitude)}^{2} So \frac{E_{R}}{E_{S}} = {[\frac{(1 + \sqrt{2}) a}{a}]}^{2} = 3 + 2 \sqrt{2}$

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