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Grade 11Electric Current

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)

Profile image of Radhika Batra
12 Years agoGrade 11
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1 Answer

Profile image of Kumar
7 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