Equations y_1 = A sin omega t and y_2 = frac{ | vedclass

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(D) The amplitude of the first motion is $A_1 = A$.The second equation is $y_2 = \frac{A}{2} \sin \omega t + \frac{A}{2} \cos \omega t$.We can rewrite

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$\sqrt{2}$

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(D) The amplitude of the first motion is $A_1 = A$.The second equation is $y_2 = \frac{A}{2} \sin \omega t + \frac{A}{2} \cos \omega t$.We can rewrite this by multiplying and dividing by $\sqrt{2}$:$y_2 = \frac{A}{\sqrt{2}} \left( \frac{1}{\sqrt{2}} \sin \omega t + \frac{1}{\sqrt{2}} \cos \omega t \right)$Using the identity $\sin(a+b) = \sin a \cos b + \cos a \sin b$,where $\cos 45^\circ = \sin 45^\circ = \frac{1}{\sqrt{2}}$:$y_2 = \frac{A}{\sqrt{2}} \sin(\omega t + 45^\circ)$.Thus,the amplitude of the second motion is $A_2 = \frac{A}{\sqrt{2}}$.The ratio of the amplitudes is $\frac{A_1}{A_2} = \frac{A}{A/\sqrt{2}} = \sqrt{2}$.

Equations $y_1 = A \sin \omega t$ and $y_2 = \frac{A}{2} \sin \omega t + \frac{A}{2} \cos \omega t$ represent $S.H.M.$ The ratio of the amplitudes of the two motions is

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