JEE Advanced 2023 Paper 2 · Q14 · Sound Waves
PARAGRAPH I — $S_1$ and $S_2$ are two identical sound sources of frequency $656$ Hz. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2}$ m·s$^{-1}$ on a circular path around $O$. There are three points $P, Q, R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$. [Given: speed of sound $= 324$ m·s$^{-1}$] When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in Hz is ___.
Reveal answer + step-by-step solution
Correct answer:648
Solution
At $Q$, the velocity of $S_2$ makes angle $45°$ with the line $QP$ (since $Q$ is equidistant from $P$ and $R$, the direction from $Q$ to $P$ is perpendicular to $OQ$, while $S_2$'s velocity is tangential to the circle). The component of $S_2$'s velocity along $S_2 \to P$: $v_{\text{rad}} = 4\sqrt{2}\cos 45° = 4$ m/s, moving AWAY from detector. Doppler: $f' = f\cdot v/(v + v_s) = 656\cdot 324/(324 + 4) = 656\cdot 324/328 = 648$ Hz.
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