Question

Three simple harmonic motions in the same line having amplitude (a) and time period T are superposed. If each Differ in phase from the next by ${45}^{\circ }$, then

• A. the resultant motion is simple harmonic
• B. the phase of the resultant motion relative to the first is ${45}^{\circ }$
• C. the resultant amplitude is $(2+\sqrt{2})a$
• D. the energy associated with the resulting motion is $(3+\sqrt{2})$ times the energy associated with any single motion.

Answer 1

Answer: (A), (B), (D)

The correct options are
A the resultant motion is simple harmonic
B the phase of the resultant motion relative to the first is ${45}^{\circ }$
D the energy associated with the resulting motion is $(3+\sqrt{2})$ times the energy associated with any single motion.


Resultant amplitude :
$A=(a+acos{45}^{\circ })\sqrt{2}=a(\sqrt{2}+1)$
Net energy $\alpha A^2=a^2(\sqrt{2}+1{)}^2=(3+2\sqrt{2})a^2$