Question 322: Waves

Question

A musical instrument ' $P$ ' produces sound waves of frequency ' $n$ ' and amplitude ' $A$ '. Another musical instrument ' $Q$ ' produces sound waves of frequency $\frac{\mathrm{n}}{4}$. 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

Accepted Answer

To determine the amplitude of the sound waves produced by instrument $ Q $, we begin by considering the relationship between the energy of oscillations, frequency, and amplitude.
The energy $ E $ of oscillations is given by the formula:
$ E = \frac{1}{2} m \omega^2 A^2 = \frac{1}{2} m (2 \pi n)^2 A^2  $
where $\omega = 2 \pi n$.
Thus, we have:
$ E \propto n^2 A^2 $
Given that the energies of the waves produced by instruments $ P $ and $ Q $ are equal, we set up the equation:
$ n_1^2 A_1^2 = n_2^2 A_2^2 $
This leads to:
$ n_1 A_1 = n_2 A_2 $
Solving for the ratio of the amplitudes, we get:
$ \frac{A_2}{A_1} = \frac{n_1}{n_2} = \frac{n}{n/4} = 4 $
Therefore, the amplitude of the waves produced by $ Q $ is:
$ A_2 = 4 A_1 $
Given $ A_1 = A $ and $ A_2 = A_p $, we conclude:
$ A_p = 4A $

Answer

$2 \mathrm{~A}_{\mathrm{P}}$

Answer

$4 \mathrm{~A}_{\mathrm{P}}$

Answer

$6 \mathrm{~A}_{\mathrm{p}}$

Answer

$9 \mathrm{~A}_{\mathrm{P}}$

Waves

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