JEE Advanced 2023 · Paper 2 · Physics Q14

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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 $4\sqrt2\ \mathrm{m\,s^{-1}}$ on a circular path around $O$ , as shown in the figure. There are three points $P$ , $Q$ and $R$ 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: The speed of sound in air is $324\ \mathrm{m\,s^{-1}}$ ]

A circle centred at O, where source S1 sits. Points on the circle: R at the top, P at the bottom (diametrically opposite), Q at the right end of the horizontal diameter. Source S2 is on the circle between P and Q, moving anticlockwise (arrow near Q pointing up towards R). Dashed vertical and horizontal diameters through O.

When only $S_2$ is emitting sound and it is at $Q$ , the frequency of sound measured by the detector in Hz is _____.

Your answer

Answer and worked solution

Official answer: 648

Velocity at Q. Q is the rightmost point of the circle. Moving anticlockwise, $S_2$ there has velocity $4\sqrt2$ m/s straight up, towards R.

Line of sight. The direction from Q to the detector at P (bottom) is $\frac1{\sqrt2}(-1,-1)$ . The component of the source velocity along it is
$4\sqrt2\cdot\left(-\tfrac1{\sqrt2}\right)=-4\ \text{m/s},$
so the source is receding from P at 4 m/s.

$f=656\times\frac{324}{324+4}=656\times\frac{324}{328}=648\ \text{Hz}.$

Topics:Doppler effect Uniform circular motion

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JEE Advanced 2023 · Paper 2, Physics question 14