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 motion is not simple harmonic
B. the resultant amplitude is `(sqrt2+1)a`
C. the phase difference between the second SHM and the resultant motion is zero.
D. the energy in the resultant motion is three times the energy in each separate SHM.

Answer

B, C

Explanation

Let the simple harmonic motions be given by
`x_1=asin(2pi(t)/(T))` .(i)
`x_2=asin(2pi(t)/(T)+(pi)/(4))`. (ii)
and `x_3=asin((2pi(t)/(T))+(pi)/(2))` (iii)
Then the resultant periodic motion, by the principle of superposition is given by
`x=x_1+x_2+x_3`
`=a(sin(2pit)/(T))+asin((2pit)/(T)+(pi)/(4))+asin((2pit)/(T)+(pi)/(2))`
`=a[sin((2pit)/(T))+asin((2pit)/(T)+(pi)/(2))]+asin((2pit)/(T)+(pi)/(4))`
`=2asin((2pit)/(T)+(pi)/(4))cos((pi)/(4))+asin((2pit)/(T)+(pi)/(4))`
`=a(sqrt2+1)sin((2pit)/(T)+(pi)/(4))`
which is a simple harmic motion with an amplitude `a(sqrt2+1)` and phase angle `(pi)/(4)` and the same period, it has the same phase as second SHM.
The energy of resultant motion is proportional to
`[a(sqrt2+1)]^2` (i.e.,) `[a^2(2+1+2sqrt2)]=(3+2sqrt2)s^2` which is greqter than three time the energy of each separate SHM.