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 enegry 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 isn wt +a sin (wt +45^(@)) +a sin (wt +90^(@))`
`=a [sin wt +sin (wt +90^(@))] +a sin (wt +45^(@))`
`=2a sin (wt +45^(@) +a sin (wt+45^(@))`
`=A sin (wt +45^(@))`
Therefore, resultant motio 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)`
`[E =(1)/(2)mA^(2) omega^(@)] (E_("resultant"))/(E_("single")) =((A)/(a))^(2) =(sqrt(2)+1)^(2) =(3+2sqrt(2))`
`:. E_("resultant") = (3+2sqrt(2))E_("single")`