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 `(l+sqrt2)`
B. the phase of the resultant motion relative to the first is `90^(@)`
C. the energy associated with the resulting motion is `(3+2sqrt2)` times the energy associated with any single motion
D. the resulting motion is not simple harmonic

Answer

AC

Explanation

`Y=Y_(1)+Y_(2)+Y_(3)`
=`asinomegat+asin(omegat+45^(@))+asin(omegat+90^(@))`
=`a{sinomegat+sin(omegat+90^(@))}+asin(omegat+45^(@))`
=`2asin(omegat+45^(@))cos45^(@)+asin(omegat+45^(@))`
=`(sqrt2+1)asin(omegat+45^(@))=Asin(omegat+45^(@))`
Therefore, resultant motion is simple harmonic of amplitude `A(sqrt2+1)a` and which differs in phase by `45^(@)` relative to the first.
Energy in `SHM prop (amplitude)^(2)`
`[E=(1)/(2)mA^(2)omega^(2)]`
therefore `(E_("resultant"))/(E_("single"))=((A)/(a))^(2)=(sqrt2+1)^(2)=(3+2sqrt2)` therefore `E_("resultant")=(3+2sqrt2)E_("single")`.