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

Question

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

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
A. resultant amplitude is `(1+sqrt(2))a`
B. phase of the resultant motion relative to the first is `90^(@)`
C. energy associated with the resultaing motion is `(3 + 2sqrt(2))` time the energy associated with any single motion
D. resulting motion is not simple harmonic

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - A::C
From susperposition principal
`y = y_(1) + y_(2) + y_(3)`
`= asin omegat + a sin(omegat + 45^(@)) + asin (omegat + 90^(@))`
`= a[sin omegat + sin (omegat +90^(@))] + a sin (omegat + 45^(@))`
`= 2asin(omegat + 45^(@))cos 45^(@) + a sin (omegat + 45^(@))`
`= (sqrt(2) + 1) a sin (omegat + 45^(@)) = A sin (omegat + 45^(@))`
Therefore, resultant motion is simple harmonic of amplitude. `A = (sqrt(2) + 1)` and which differ in phase by `45^(@)` relative to the first.
`:. (E_("resultant"))/(E_("single")) = (A/a)^(2) = (sqrt(2) + 1)^(2) = (3+2sqrt(2))`
`:. E_("resultant") = (3+2sqrt(2))E_("single")`