Question

On the superposition of two harmonic oscillations represented by $x_1=asin(\omega t+{\varphi }_1)and{x}_2=asin(\omega t+{\varphi }_2)$ a resulting oscillation with the same time period and amplitude is obtained. The value of ${\varphi }_1-{\varphi }_2$ is

• A. ${120}^{\circ }$
• B. ${90}^{\circ }$
• C. ${60}^{\circ }$
• D. ${15}^{\circ }$

Answer 1

The correct option is A ${120}^{\circ }$

Resultant amplitude

$=\sqrt{a_1^2+a_2^2+2a_1{a}_{2}cos\Delta \varphi }$

$\Rightarrow a^2=2a^2+2a^{2}cos(f_1-f_2)$

$\Rightarrow cos(f_1-f_2)=-\frac{1}{2}$

$\Rightarrow f_1-f_2={120}^{\circ }or\frac{2\pi }{3}rad$