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