Two simple harmonic motions are given by $$x1=a2sin$$⁡ω$$t+3a2cos$$⁡ωt and $$x2=asin$$⁡ω$$t+acos$$⁡ωt above, the minimum phase difference between the two simple harmonic motions is (in$$ $$degrees)

Question

Two simple harmonic motions are given by $$x1=a2sin$$⁡ω$$t+3a2cos$$⁡ωt and $$x2=asin$$⁡ω$$t+acos$$⁡ωt above, the minimum phase difference between the two simple harmonic motions is (in$$ $$degrees)

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Amplitude of the first simple harmonic motion $$isA1=a22+3a22=aAmplitude$$ of the second motion $$isA2=a2+a2=2a$$∴ $$A1A2=12We$$ can rewrite the two motions $$asx1=a12sin$$⁡ω$$t+32cos$$⁡ω$$t=acos$$⁡$$60$$∘$$sin$$⁡ω$$t+$$$$sin$$⁡$$60$$∘$$cos$$⁡ωtor  $$x1=asin$$⁡ω$$t+60$$∘   ........(1)and$$  $$x2=a212sin$$⁡ω$$t+12cos$$⁡ω$$t=a2cos$$⁡$$45$$∘$$sin$$⁡ω$$t+$$$$sin$$⁡$$45$$∘$$cos$$⁡ωtor  $$x2=a2sin$$⁡ω$$t+45$$∘ ........$$(2)From$$ $$(1) and (2) it follows that the amplitudes are a and $$a2$$ (a$$ result obtained $$above) the phases are $$60$$ and $$45$$°. Therefore, phase difference $$=$$ $$60$$°$$-$$ $$45$$° $$=$$ $$15$$°

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