Question

A source of sound vibrating at frequency \(500\ \text{Hz}\) is rotated in a circle of radius \(\frac{100}{\pi }\ \text{cm}\) at a constant angular speed of \(5\) revolutions per second. A listener is in the plane of the circle. The maximum and minimum frequency of sound heard are \(\{\)Speed of sound on air \(=332\ \text{m/s}\}\)

• A. \(500\ \text{Hz}\) and \(480\ \text{Hz}\)
• B. \(516\ \text{Hz}\) and \(485\ \text{Hz}\)
• C. \(416\ \text{Hz}\) and \(490\ \text{Hz}\)
• D. \(520\ \text{Hz}\) and \(500\ \text{Hz}\)

Hint:

The source speed on the circle is omega times r; the highest frequency is heard when it moves directly towards the listener and the lowest when it moves directly away.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Doppler effect for a moving source and a stationary listener: \(f'=f\frac{v}{v\mp v_{s}}\).

Step 2: Calculation
Radius \(r=\frac{100}{\pi}\ \text{cm}=\frac{1}{\pi}\ \text{m}\) and \(\omega=2\pi\times 5=10\pi\ \text{rad/s}\), so \(v_{s}=\omega r=10\ \text{m/s}\).

Step 3: Compare the options
Maximum: \(f_{max}=500\times\frac{332}{332-10}\approx 516\ \text{Hz}\). Minimum: \(f_{min}=500\times\frac{332}{332+10}\approx 485\ \text{Hz}\). This matches the second option.

Final Answer:
The maximum and minimum frequencies are about \(516\ \text{Hz}\) and \(485\ \text{Hz}\). \[ \boxed{\text{(B) }516\ \text{Hz and }485\ \text{Hz}} \]