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. 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 motionis `(3+ 2 sqrt(2))` times the energy associated 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 t+90^@)`
=`a{sin omegat+ sin(omegat + 90^@)}+a sin(omega t+ 45^@)`
=`2a sin(omegat +45^@)cos 45^@ + a sin(omega t+ 45^@)`
=`(sqrt(2)+1)a sin (omegat + 45^@) = A sin(omegat + 45^@)`
Therefore resultant motion is simple harmonic of amplitude `A = (sqrt(2) +)a` and which differs in phase by `45^@` relative to the first.
Energy in `SHM prop ("amplitide")^(2)`
`[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")`.