Question

A musical instrument P produce sound waves of frequency n and amplitude A. Another musical instrument produces sound waves of frequency $n/4$. The waves produced by P and Q have equal energies. If the amplitude of waves produced by P is $A_{p}$, the amplitude of waves produced by Q will be}

• A. $2A_{p}$
• B. $4A_{p}$
• C. $6A_{p}$
• D. $9A_{p}$

Hint:

To maintain equal energy, if frequency decreases, amplitude must increase proportionally.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem states that two sound waves from instruments P and Q have the same energy. We need to find the relationship between their amplitudes given their frequencies.

Step 2: Key Formula or Approach:
The energy (\( E \)) of a sound wave is proportional to the square of its frequency (\( n \)) and the square of its amplitude (\( A \)):
\[ E \propto n^2 A^2 \] Since energies are equal:
\[ n_p^2 A_p^2 = n_q^2 A_q^2 \]

Step 3: Detailed Explanation:
Given:
Frequency of P, \( n_p = n \)
Amplitude of P, \( A_p = A_p \)
Frequency of Q, \( n_q = \frac{n}{4} \)
We need to find Amplitude of Q, \( A_q \).
Equating energies:
\[ (n)^2 (A_p)^2 = \left( \frac{n}{4} \right)^2 (A_q)^2 \] \[ n^2 A_p^2 = \frac{n^2}{16} A_q^2 \] Canceling \( n^2 \) from both sides:
\[ A_p^2 = \frac{A_q^2}{16} \] \[ A_q^2 = 16 A_p^2 \] Taking the square root:
\[ A_q = 4 A_p \]

Step 4: Final Answer:
The amplitude of waves produced by Q will be \( 4 A_p \).