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Two simple harmonic motions are represented by
$\begin{aligned} \mathrm{y}_{1} &=5[\sin 2 \pi \mathrm{t}+\sqrt{3} \cos 2 \pi \mathrm{t}] \\ \text { and } \quad \mathrm{y}_{2} &=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right) \end{aligned}$
The ratio of their amplitudes is
Options:
$\begin{aligned} \mathrm{y}_{1} &=5[\sin 2 \pi \mathrm{t}+\sqrt{3} \cos 2 \pi \mathrm{t}] \\ \text { and } \quad \mathrm{y}_{2} &=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right) \end{aligned}$
The ratio of their amplitudes is
Solution:
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Verified Answer
The correct answer is:
$2: 1$
$\mathrm{y}_{1}=5[\sin 2 \pi \mathrm{t}+\sqrt{3} \cos 2 \pi \mathrm{t}]$
$=10\left[\frac{1}{2} \sin 2 \pi \mathrm{t}+\frac{\sqrt{3}}{2} \cos 2 \pi \mathrm{t}\right]$
$=10\left[\cos \frac{\pi}{3} \sin 2 \pi t+\sin \frac{\pi}{3} \cos 2 \pi t\right]$
$=10\left[\sin \left(2 \pi t+\frac{\pi}{3}\right)\right]$
$\Rightarrow \mathrm{A}_{1}=10$
Similarly, $\quad \mathrm{y}_{2}=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right)$
$\Rightarrow \quad \mathrm{A}_{2}=5$
Hence, $\quad \frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}$
$=10\left[\frac{1}{2} \sin 2 \pi \mathrm{t}+\frac{\sqrt{3}}{2} \cos 2 \pi \mathrm{t}\right]$
$=10\left[\cos \frac{\pi}{3} \sin 2 \pi t+\sin \frac{\pi}{3} \cos 2 \pi t\right]$
$=10\left[\sin \left(2 \pi t+\frac{\pi}{3}\right)\right]$
$\Rightarrow \mathrm{A}_{1}=10$
Similarly, $\quad \mathrm{y}_{2}=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right)$
$\Rightarrow \quad \mathrm{A}_{2}=5$
Hence, $\quad \frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}$
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