Question

The frequency of A note is 4 times that of B note. The energies of two notes are equal, then the amplitude of B note as compared to that of A will be-
A. Double
B. Equal
C. Four times
D. Eight times

Answer 1

In this question, we have a relation between amplitude and energy of wave. Also we know that the energy is directly dependent on the amplitude and the angular frequency. If amplitude and angular frequency increases then the energy will also increase.

Formula Used:
$E = {\omega ^2}{A^2}$,
Here $E$ is the energy and $A$ is the amplitude.

Complete step by step answer:
As we know that,
$E = {\omega ^2}{A^2}$
Here E is the energy and A is the amplitude. And
$\omega = 2\pi f$, here $f$ is frequency
If we substitute the value of $\omega$then we get-
$E = {A^2}{f^2}$ ($2\pi$is constant here)
According to question,
${A_A}^2{f_A}^2 = {A_B}^2{f_B}^2$ ------- (1)
Now, in this question
${f_A} = 4{f_B}$
$\Rightarrow\dfrac{{{f^2}_A}}{{{f^2}_B}} = 16$
Now substitute this value in equation (1), We get-
${A_A}^2.16 = {A_B}^2$
$\therefore{A_B} = 4{A_A}$
Here ${A_A}$ is the amplitude of note B and ${A_B}$ is the amplitude of note A.
So, we can conclude that the amplitude of note B is four times of the amplitude of note A.

Hence, option C is correct.

Note: There are two terms- node and Antinode. Node is a point along a standing wave where the value of amplitude is minimum and if we talk about antinode then it is a point along the standing wave where the wave will have the maximum amplitude.Amplitude is the maximum displacement from the equilibrium position of an object and frequency is the number of events per unit of time. The amount of energy in a wave can be defined by its amplitude and frequency.