"Two simple harmonic motions are represented by the equations ${x_1} = 5\sin \left( {2\pi t + {\pi \over 4}} \right)$ and ${x_2} = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t)$. The amplitude of second motion is ................ times the amplitude in first motion."

Question

Accepted Answer

"$${x_2} = 5\sqrt 2 \left( {{1 \over {\sqrt 2 }}\sin 2\pi t + {1 \over {\sqrt 2 }}\cos 2\pi t} \right)\sqrt 2 $$

$= 10\sin \left( {2\pi t + {\pi \over 4}} \right)$

$\therefore$ ${{{A_2}} \over {{A_1}}} = {{10} \over 5} = 2$"

Two simple harmonic motions are represented by the equations

${x_1} = 5\sin \left( {2\pi t + {\pi \over 4}} \right)$ and ${x_2} = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t)$. The amplitude of second motion is ................ times the amplitude in first motion.

Solution

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Two simple harmonic motions are represented by the equations ${x_1} = 5\sin \left( {2\pi t + {\pi \over 4}} \right)$ and ${x_2} = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t)$. The amplitude of second motion is ................ times the amplitude in first motion.

Solution

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Two simple harmonic motions are represented by the equations

${x_1} = 5\sin \left( {2\pi t + {\pi \over 4}} \right)$ and ${x_2} = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t)$. The amplitude of second motion is ................ times the amplitude in first motion.