Simple Harmonic Motion 5 Question 19

24. Three simple harmonic motions in the same direction having the same amplitude and same period are superposed. If each differ in phase from the next by 45, then

(1999, 3M)

(a) the resultant amplitude is (1+2)a

(b) the phase of the resultant motion relative to the first is 90

(c) the energy associated with the resulting motion is (3+22) times the energy associated with any single motion

(d) the resulting motion is not simple harmonic

Analytical & Descriptive Questions

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Solution:

  1. From superposition principle

y=y1+y2+y3=asinωt+asin(ωt+45)+asin(ωt+90)=a[sinωt+sin(ωt+90)]+asin(ωt+45)=2asin(ωt+45)cos45+asin(ωt+45)=(2+1)asin(ωt+45)=Asin(ωt+45)

Therefore, resultant motion is simple harmonic of amplitude

A=(2+1)a

and which differ in phase by 45 relative to the first.

Energy in SHM ( amplitude )2[E=12mA2ω2]

Eresultant Esingle =A2a=(2+1)2=(3+22)Eresultant =(3+22)Esingle 



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