question_answer

What will be the displacement equation of the simple harmonic motion obtained by combining the motions? \[{{x}_{1}}=2\,\sin \omega \,t\], \[{{x}_{2}}=4\,\sin \left( \omega \,t+\frac{\pi }{6} \right)\] and\[{{x}_{3}}=6\,\sin \left( \omega \,t+\frac{\pi }{3} \right)\]

A)  \[x=10.25\sin \,(\omega \,t+\phi )\]

B)  \[x=10.25\sin \,(\omega \,t-\phi )\]

C)  \[x=11.25\sin \,(\omega \,t+\phi )\]

D)  \[x=11.25\sin \,(\omega \,t-\phi )\]

Correct Answer: C

Solution :

The resultant equation is                 \[x=A\sin (\omega \,t+\phi )\] \[\Sigma {{A}_{x}}=2+4\cos {{30}^{o}}+6\cos {{60}^{o}}=8.46\] and        \[\Sigma {{A}_{y}}=4\sin {{30}^{o}}+6\cos {{30}^{o}}=7.2\] \[\therefore \] \[A=\sqrt{{{(\Sigma {{A}_{x}})}^{2}}+{{(\Sigma {{A}_{y}})}^{2}}}\]                 \[=\sqrt{{{(8.46)}^{2}}+{{(7.2)}^{2}}}=11.25\] and        \[\tan \phi \frac{\Sigma {{A}_{y}}}{\Sigma {{A}_{y}}}=\frac{7.2}{8.46}=0.85\] \[\Rightarrow \] \[\phi ={{\tan }^{-1}}(0.85)={{40.4}^{o}}\] Thus, the displacement equation of combined motion is                 \[x=11.25\sin \,\,(\omega \,t+\phi )\] where, \[\phi ={{40.4}^{o}}\]