Three simle harmionic motions in the same direction having the same amplitude (a) and same period are superposed. If each differs in phase from the next by `45^@`, then.

Options

A. the resultant amplitude is `(1 + sqrt(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 + 2 sqrt(2))` times the energy associated with any single motion.
D. the resulting motion is not simple harmonic.

Answer

A, C

Explanation

From superposition principle
`y = y_1 + y_2 + y_3`
`= a sin omega t + a sin (omega t + 45^@) + a sin (omega + 90^@)`
=`a [sin omega t + sin (omega t + 90^@] + a sin (omega t + 45^@)`
=`2 a sin (omega t + 45^@) cos 45^@ + a sin (omega t + 45^@)`
=`(sqrt(2 + 1) a sin (omega t + 45^@)`
Therefore resultant motion is simple harmonic of amplitude `A = (sqrt (2 + 1) a`
and which differ in phase by `45^@` relative to the first.
Energy in (SHM) `prop (amplitude)^2 [because E = (1)/(2) mA^2 omega^2]`
:. `E_(resultant)/E_(single)= ((A)/(a))^2 = (sqrt 2 + 1)^2 = (3 +2 sqrt(2))`
:. `E_(resultant) = (3 +2 sqrt(2)) E_(single)`.