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

Question

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

Infinity Learn · Accepted Answer

Resultant $$y=y1+y2+y3$$                $$y=a$$ $$sin$$ ω$$t-45o+a$$ $$sin$$ ω$$t+a$$ $$sin$$ω$$t+45o$$ ⇒       $$y=1+2a$$ $$sin$$ ωt  So the phase of the resulting motion relative to the first (1) is $$45$$°. As       Energy ∝$$(Amplitude)2$$ So       $$ERES=1+2aa2=3+22$$

Detailed Solution

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