Four simple harmonic vibrations:y_1 = 8 cos(o | vedclass

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(A) Represent the vibrations as phasors:$y_1 = 8 \cos(\omega t) \implies \vec{A}_1 = 8\hat{i}$$y_2 = 4 \cos(\omega t + \frac{\pi}{2}) = -4 \sin(\omega

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$\sqrt{45}$ and $	an^{-1}(1/2)$

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(A) Represent the vibrations as phasors:$y_1 = 8 \cos(\omega t) \implies \vec{A}_1 = 8\hat{i}$$y_2 = 4 \cos(\omega t + \frac{\pi}{2}) = -4 \sin(\omega t) \implies \vec{A}_2 = 4\hat{j}$$y_3 = 2 \cos(\omega t + \pi) = -2 \cos(\omega t) \implies \vec{A}_3 = -2\hat{i}$$y_4 = 1 \cos(\omega t + \frac{3\pi}{2}) = 1 \sin(\omega t) \implies \vec{A}_4 = -1\hat{j}$The resultant vector $\vec{A} = \vec{A}_1 + \vec{A}_2 + \vec{A}_3 + \vec{A}_4 = (8-2)\hat{i} + (4-1)\hat{j} = 6\hat{i} + 3\hat{j}$.Amplitude $A = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45}$.Phase $\phi = 	an^{-1}(\frac{A_y}{A_x}) = 	an^{-1}(\frac{3}{6}) = 	an^{-1}(1/2)$.

Four simple harmonic vibrations:
$y_1 = 8 \cos(\omega t)$;
$y_2 = 4 \cos(\omega t + \frac{\pi}{2})$;
$y_3 = 2 \cos(\omega t + \pi)$;
$y_4 = 1 \cos(\omega t + \frac{3\pi}{2})$,
are superposed on each other. The resulting amplitude and phase are respectively:

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Four simple harmonic vibrations:$y_1 = 8 \cos(\omega t)$;$y_2 = 4 \cos(\omega t + \frac{\pi}{2})$;$y_3 = 2 \cos(\omega t + \pi)$;$y_4 = 1 \cos(\omega t + \frac{3\pi}{2})$,are superposed on each other. The resulting amplitude and phase are respectively: