MHT CET · Physics · Waves and Sound
The equation of the wave is \(\mathrm{Y}=10 \sin \left(\frac{2 \pi \mathrm{t}}{30}+\alpha\right)\) If the displacement is \(5 \mathrm{~cm}\) at \(\mathrm{t}=0\) then the total phase at \(\mathrm{t}=7.5 \mathrm{~s}\) will be \(\left(\sin 30^{\circ}=0.5\right)\)
- A \(\frac{\pi}{3} \mathrm{rad}\)
- B \(\frac{\pi}{2} \mathrm{rad}\)
- C \(\frac{2 \pi}{5} \mathrm{rad}\)
- D \(\frac{2 \pi}{3} \mathrm{rad}\)
Answer & Solution
Correct Answer
(D) \(\frac{2 \pi}{3} \mathrm{rad}\)
Step-by-step Solution
Detailed explanation
\(\begin{array}{ll} & \text { At } \mathrm{t}=0,10 \sin \left(\frac{2 \pi \times 0}{30}+\alpha\right)=5 \\ \therefore \quad & 10 \sin \alpha=5 \\ \therefore \quad & \sin \alpha=0.5 \\ \therefore \quad & \alpha=\frac{\pi}{6}\end{array}\)
Thus, at \(\mathrm{t}=7.5 \mathrm{~s}\),
\(
10 \sin \left(\frac{2 \pi \times 7.5}{30}+\alpha\right)=10 \sin \left(\frac{\pi}{2}+\frac{\pi}{6}\right)
\)
\(\therefore \quad\) The total phase is \(\frac{2 \pi}{3}\).
Thus, at \(\mathrm{t}=7.5 \mathrm{~s}\),
\(
10 \sin \left(\frac{2 \pi \times 7.5}{30}+\alpha\right)=10 \sin \left(\frac{\pi}{2}+\frac{\pi}{6}\right)
\)
\(\therefore \quad\) The total phase is \(\frac{2 \pi}{3}\).
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