Question 234: Waves

Question

A musical instrument ' P ' produce 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 ' $\mathrm{A}_{\mathrm{p}}$ ', the amplitude of waves produced by ' $Q$ ' will be

Accepted Answer

The energy of a vibrating object (oscillation) is given by the formula:
$$ E = \frac{1}{2}m\omega^2A^2 $$

Here, $E$ is energy, $m$ is mass, $\omega$ is angular frequency, and $A$ is amplitude.
Since $\omega = 2\pi n$ (where $n$ is the frequency), the formula becomes:
$$ E \propto n^2A^2 $$

This means energy is directly related to the square of frequency and the square of amplitude.
The problem says both $P$ and $Q$ have the same energy, so:
$$ n_P^2A_P^2 = n_Q^2A_Q^2 $$
We know $n_Q = \frac{n_P}{4}$. Substitute this value:
$$ n_P^2A_P^2 = \left(\frac{n_P}{4}\right)^2 A_Q^2 $$
This gives:
$$ n_P^2 A_P^2 = \frac{n_P^2}{16}A_Q^2 $$
Multiply both sides by $16$:
$$ 16n_P^2 A_P^2 = n_P^2A_Q^2 $$
Divide both sides by $n_P^2$:
$$ 16A_P^2 = A_Q^2 $$
Take the square root of both sides:
$$ 4A_P = A_Q $$
So, the amplitude of $Q$ must be $4$ times the amplitude of $P$ ($A_Q = 4A_P$).

Answer

$2 A_p$

Answer

$4 A_p$

Answer

$6 A_p$

Answer

$9 A_p$

Waves

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