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Question:
Two simple harmonic motion, are represented by the equations $\mathrm{y}_{1}=10 \sin \left(3 \pi t+\frac{\pi}{3}\right)$
$\mathrm{y}_{2}=5(\sin 3 \pi \mathrm{t}+\sqrt{3} \cos 3 \pi \mathrm{t})$
Ratio of amplitude of $y_{1}$ to $y_{2}=x: 1$. The value of $x$ is
Solution:
$y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{3}\right) \Rightarrow$ Amplitude $=10$
$y_{2}=5(\sin 3 \pi t+\sqrt{3} \cos 3 \pi t)$
$y_{2}=10\left(\frac{1}{2} \sin 3 \pi t+\frac{\sqrt{3}}{2} \cos 3 \pi t\right)$
$y_{2}=10\left(\cos \frac{\pi}{3} \sin 3 \pi t+\sin \frac{\pi}{3} \cos 3 \pi t\right)$
$\mathrm{y}_{2}=10 \sin \left(3 \pi \mathrm{t}+\frac{\pi}{3}\right) \Rightarrow$ Amplitude $=10$
So ratio of amplitudes $=\frac{10}{10}=1$
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