KCET · Physics · Oscillations
Two simple harmonic motions are represented by
\(\begin{aligned} \mathrm{y}_{1} &=5[\sin 2 \pi \mathrm{t}+\sqrt{3} \cos 2 \pi \mathrm{t}] \\ \text {and}~ \mathrm{y}_{2} &=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right) \end{aligned}\)
The ratio of their amplitudes is
- A \(1: 1\)
- B \(2: 1\)
- C \(1: 3\)
- D \(\sqrt{3}: 1\)
Answer & Solution
Correct Answer
(B) \(2: 1\)
Step-by-step Solution
Detailed explanation
\(\mathrm{y}_{1}=5[\sin 2 \pi \mathrm{t}+\sqrt{3} \cos 2 \pi \mathrm{t}]\)
\(=10\left[\frac{1}{2} \sin 2 \pi \mathrm{t}+\frac{\sqrt{3}}{2} \cos 2 \pi \mathrm{t}\right]\)
\(=10\left[\cos \frac{\pi}{3} \sin 2 \pi t+\sin \frac{\pi}{3} \cos 2 \pi t\right]\)
\(=10\left[\sin \left(2 \pi t+\frac{\pi}{3}\right)\right]\)
\(\Rightarrow \mathrm{A}_{1}=10\)
Similarly, \(\mathrm{y}_{2}=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right)\)
\(\Rightarrow \mathrm{A}_{2}=5\)
Hence, \(\frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}\)
\(=10\left[\frac{1}{2} \sin 2 \pi \mathrm{t}+\frac{\sqrt{3}}{2} \cos 2 \pi \mathrm{t}\right]\)
\(=10\left[\cos \frac{\pi}{3} \sin 2 \pi t+\sin \frac{\pi}{3} \cos 2 \pi t\right]\)
\(=10\left[\sin \left(2 \pi t+\frac{\pi}{3}\right)\right]\)
\(\Rightarrow \mathrm{A}_{1}=10\)
Similarly, \(\mathrm{y}_{2}=5 \sin \left(2 \pi \mathrm{t}+\frac{\pi}{4}\right)\)
\(\Rightarrow \mathrm{A}_{2}=5\)
Hence, \(\frac{A_{1}}{A_{2}}=\frac{10}{5}=\frac{2}{1}\)
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