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Concepts Of Physics MCQ Edition [Volume 1]
Exam
Concepts Of Physics MCQ Edition [Volume 1]
Subject
Physics
Chapter
Sound Waves

Concepts Of Physics MCQ Edition [Volume 1] · Physics · Sound Waves

A bat flies at a speed of 6 m/s between two parallel walls while...

A bat flies at a speed of 6 m/s6 \text{ m/s} between two parallel walls while emitting an ultrasonic wave of frequency 4.5×104 Hz4.5 \times 10^4 \text{ Hz}. Determine the two frequencies heard by the bat, as well as the beat frequency between them. The speed of sound is 330 m/s330 \text{ m/s}.

Options

  1. A4.58×104 Hz4.58 \times 10^4 \text{ Hz}, 4.42×104 Hz4.42 \times 10^4 \text{ Hz}, 1640 Hz1640 \text{ Hz}
  2. B4.67×104 Hz4.67 \times 10^4 \text{ Hz}, 4.34×104 Hz4.34 \times 10^4 \text{ Hz}, 3270 Hz3270 \text{ Hz}
  3. C4.58×104 Hz4.58 \times 10^4 \text{ Hz}, 4.42×104 Hz4.42 \times 10^4 \text{ Hz}, 3270 Hz3270 \text{ Hz}
  4. D4.67×104 Hz4.67 \times 10^4 \text{ Hz}, 4.34×104 Hz4.34 \times 10^4 \text{ Hz}, 1640 Hz1640 \text{ Hz}

Correct answer

B. 4.67×104 Hz4.67 \times 10^4 \text{ Hz}, 4.34×104 Hz4.34 \times 10^4 \text{ Hz}, 3270 Hz3270 \text{ Hz}

Step-by-step solution

The bat acts as both a moving source and a moving observer.

For the wall the bat is approaching, the frequency of the reflected wave heard by the bat is given by the Doppler effect formula:

f_1 = f \left(\dfrac{v + v_b}{v - v_b} \right)

Substituting v=330 m/sv = 330 \text{ m/s}, vb=6 m/sv_b = 6 \text{ m/s}, and f=4.5×104 Hzf = 4.5 \times 10^4 \text{ Hz}:

f_1 = 4.5 \times 10^4 \left(\dfrac{330 + 6}{330 - 6} \right) =4.5 \times 10^4 \left(\dfrac{336}{324} \right) =4.5 \times 10^4 \left(\dfrac{28}{27} \right) \approx4.67×104 Hz4.67 \times 10^4 \text{ Hz}

For the wall the bat is moving away from, the frequency of the reflected wave heard by the bat is:

f2=f(v−vbv+vb)f_2 = f \left( \dfrac{v - v_b}{v + v_b} \right)

f2=4.5×104(330−6330+6)=4.5×104(324336)=4.5×104(2728)≈4.34×104 Hzf_2 = 4.5 \times 10^4 \left( \dfrac{330 - 6}{330 + 6} \right) = 4.5 \times 10^4 \left( \dfrac{324}{336} \right) = 4.5 \times 10^4 \left( \dfrac{27}{28} \right) \approx 4.34 \times 10^4 \text{ Hz}

The beat frequency is the difference between the two frequencies heard by the bat:

fbeat=f1−f2=4.5×104f_{\text{beat}} = f_1 - f_2 = 4.5 \times 10^4\left( \dfrac{28}{27} - \dfrac{27}{28} \right)<br><br>f_{\text{beat}} = 4.5 \times 10^4 \left( \dfrac{784 - 729}{756} \right) =45000 \times 55756≈\dfrac{55}{756} \approx3274 \text{ Hz} \approx 3270 \text{ Hz}
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