Four simple harmonic vibrations:
${y_1} = 8\,\cos\, \omega t;\,{y_2} = 4\,\cos \,\left( {\omega t + \frac{\pi }{2}} \right)$ ;
${y_3} = 2\cos \,\left( {\omega t + \pi } \right);\,{y_4} = \,\cos \,\left( {\omega t + \frac{{3\pi }}{2}} \right)$ ,
are superposed on each other. The resulting amplitude and phase are respectively;
Medium
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$\cos \theta=\hat{i}$
$\sin \theta=\hat{j}$
Result at $=8 \hat{j}-4 \hat{i}-2 \hat{j}+i$
$=-3 \hat{i}+6 j$
Magnitude $=\sqrt{45}$
$\tan \theta=\frac{1}{2}$
$\theta=\tan ^{-1}\left(\frac{1}{2}\right)$
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