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

Options

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

Answer

A, C

Explanation

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")`