A vibratory motion is represented by $x = 2A\,\cos \omega t + A\,\cos \,\left( {\omega t + \frac{\pi }{2}} \right) + A\,\cos \,\left( {\omega t + \pi } \right)$ $ + \frac{A}{2}\,\cos \left( {\omega t + \frac{{3\pi }}{2}} \right)$. The resultant amplitude of the motion is
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$\mathrm{x}=2 \mathrm{Acos} \omega \mathrm{t}+\mathrm{Acos}\left(\omega \mathrm{t}+\frac{\pi}{2}\right)+\mathrm{Acos}(\omega \mathrm{t}+\pi)+$
$\frac{\mathrm{A}}{2} \cos \left(\omega \mathrm{t}+\frac{3 \pi}{2}\right)$
$=2 \mathrm{Acos} \omega \mathrm{t}-\mathrm{A} \sin \omega \mathrm{t}-\mathrm{Acos} \omega \mathrm{t}+\frac{\mathrm{A}}{2} \mathrm{sin} \omega \mathrm{t}$
$=$ $Acos\omega t$ $-\frac{\mathrm{A}}{2} \mathrm{sin} \omega \mathrm{t}$
The amplitude of the resultant motion is
$A_{R}=\sqrt{(A)^{2}\left(-\frac{A}{2}\right)^{2}}=\frac{\sqrt{5} A}{2}$
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