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Question #42870

Difficulty: 8/10 | 🟢 Free

The Question

A sound source $S_1$ and a stationary observer $O$ are situated in a horizontal plane. $S_1$ moves on a circular track of radius $R$ centered at the origin $(0,0)$ . The observer $O$ is located at a distance $L$ from the origin along the $+x$ -axis, where $L \gg R$ . At $t=0$ , $S_1$ starts from rest at $(R,0)$ and moves counterclockwise with a constant angular acceleration $\alpha$ . $S_1$ continuously emits sound of frequency $f_0$ . A stationary source $S_2$ located at the origin also emits sound of frequency $f_0$ . The speed of sound in air is $c$ (assume $c \gg R\sqrt{\alpha}$ ). Find the average beat frequency recorded by the observer $O$ during the time interval corresponding to the first complete revolution of $S_1$ .

Options

A.
$\frac{2 f_0 R}{c} \sqrt{\frac{\alpha}{\pi}}$
B.
$\frac{\pi f_0 R}{c} \sqrt{\frac{\alpha}{\pi}}$
C.
$\frac{4 f_0 R}{c} \sqrt{\frac{\alpha}{\pi}}$
D.
$\frac{f_0 R}{c} \sqrt{\frac{\alpha}{\pi}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Beats and Frequency Difference|Non-Uniform Circular Motion|Mean vs RMS Values

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Doppler effect in non-uniform circular motion and the rigorous definition of average values.

A common trap is assuming that the average beat frequency is proportional to the average speed of the source ( $\langle v \rangle = R\sqrt{\pi\alpha}$ ), which incorrectly yields Option B.

Students must recognize that the instantaneous beat frequency is a non-linear function of time, requiring integration of the absolute frequency difference over the period.

The physical elegance of the problem lies in the transformation $v dt = R d\theta$ , which shows that the total number of beats depends only on the total distance covered along the line of sight ( $4R$ ), independent of the angular acceleration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Beats and Frequency Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free