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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 450, then :
(A) the resultant amplitude is (1 + √2) a
(B) the phase of the resultant motion relative to the first is 900.
(C) the energy associated with the resulting motion is (3 + 2√3) times the energy associated with any single motion
(D) the resulting motion is not simple harmonic

(I.I.T 1999)

Radhika Batra , 11 Years ago
Grade 11
anser 1 Answers
Kumar

Last Activity: 6 Years ago

Let simple harmonic motions be represented by 
y1=asin(ωtπ4)
y2=asinωt
 and 
y3=asin(ωt+π4)
. On superimposing, resultant SHM will be
y=a [sin(ωtπ4)+sinωt+sin(ωt+π4)]
=a [2sinωtcosπ4+sinωt]
=a [2sinωt+sinωt]=a (1+2)sinωt
Resultant amplitude =
(1+2)a
 Energy is S.H.M. µ (Amplitude)2 \ 
EResultantESingle=(Aa)2=(2+1)2=(3+22)
Þ 
EResultant=(3+22)ESingle
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