Question

Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by ${45}^{\circ }$, then :

• A. The resultant amplitude is $(1+\sqrt{2})a$
• B. The phase of the resultant motion relative to the first is ${90}^{\circ }$
• 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 1

Answer: (A), (C)

The resultant amplitude is $(1+\sqrt{2})a$
The energy associated with the resulting motion is $(3+2\sqrt{2})$ times the energy associated with any single motion

Let the three simple harmonic oscillations be - here $A=a$

$x_1=Asinwt{x}_2=Asin(wt+\frac{\pi }{4})x_3=Asin(wt+\frac{\pi }{2})=Acoswt$

The resultant motion is $x=x_1+x_2+x_3$

$\therefore x=A[sinwt+sin(wt+\frac{\pi }{4})+coswt]$

$x=A[sinwt+sinwt\times \frac{1}{\sqrt{2}}+coswt\times \frac{1}{\sqrt{2}}+coswt]$

$⟹x=A(1+\frac{1}{\sqrt{2}})[sinwt+coswt]$

$⟹x=(\sqrt{2}+1)Asin(wt+\frac{\pi }{4})$

Thus the resultant motion is also a $SHM$ whose amplitude is equal to $(1+\sqrt{2})A$ and phase is ${45}^o$ relative to first.

Energy associated $SHM$ is directly proportional to square of the amplitude i.e $E\propto A^2$

Thus $\frac{E_{resultant}}{E_{single}}=\frac{\left[\right(\sqrt{2}+1)A{]}^2}{A^2}$

$\frac{E_{resultant}}{E_{single}}=3+2\sqrt{2}$