Question

A particle is subjected to two SHM's $X_1=a_1\sin \omega t$ and $X_2=a_2\sin (\omega t+\varphi )$ simultaneously where $a_1=3.0cm,a_2=4.0cm$. Find resultant amplitude if the phase difference $\varphi$ has values

(a) ${\theta }^{\circ }$

(b) ${60}^{\circ }$

(c) ${90}^{\circ }$.

Answer 1

Resultant amplitude $a=\sqrt{a_1^2+a_2^2+2a_1{a}_2\cos \varphi }$

(a) If $\varphi =\theta$

$a=\sqrt{a_1^2+a_2^2+2a_1{a}_2\cos \theta }=\sqrt{25+24\cos \theta }$

(b)$\varphi ={60}^{\circ },\cos \left(60\right)=\frac{1}{2}$

$a=\sqrt{25+12}=\sqrt{37}$

(c))$\varphi ={90}^{\circ },\cos \left(90\right)=0$

$a=\sqrt{25}=5$