A vibratory motion is represented by $x = 2\,A\,\cos \,\omega t + A\,\cos \,\left( {\omega t + \frac{\pi }{2}} \right) + A\,\cos \,\left( {\omega t + \pi } \right) + \frac{A}{2}\,\cos \,\left( {\omega t + \frac{{3\pi }}{2}} \right)$ The resultant amplitude of motion is
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which leads to
so resultant amplitude is
$=\sqrt{\frac{A^{2}}{4}+A^{2}}=\sqrt{\frac{5 A^{2}}{4}}=\frac{\sqrt{5} A}{2}$
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