Question

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

Answer 1

The energies of the two notes are given as equal so by comparing the two energies and plugging the given values of frequencies that are given to us in the form of a relationship, we can find the relationship between the amplitudes of the two notes.
Formula Used:
E$\propto$ f$\times {{A}^{2}}$

Complete answer:
Given,
The energies of the two notes A and B are equal.
Let the energies be ${{E}_{A}}$ and ${{E}_{B}}$
According to question, ${{E}_{A}}$= ${{E}_{B}}$
Now we have,
Frequency of A, ${{f}_{A}}$ = 4*frequency of B, ${{f}_{B}}$
To find: The amplitude of B compared to that of A
We have,
Energy of a note is directly proportional to its frequency and the energy is directly proportional to the square of the amplitude of the note.
For note A,
${{E}_{A}}\propto {{f}_{A}}{{A}^{2}}_{A}$ ………. (1)
For note B,
${{E}_{B}}\propto {{f}_{B}}{{A}_{B}}^{2}$ ………….(2)
Where f represents the frequency and A represents the amplitude.
Comparing (1) and (2), since the two energies are equal,
${{E}_{A}}$= ${{E}_{B}}$
$\Rightarrow {{f}_{A}}A_{A}^{2}={{f}_{B}}A_{B}^{2}$
$\begin{aligned} & \Rightarrow 4{{f}_{B}}{{A}_{A}}^{2}={{f}_{B}}A_{B}^{2} \\\ & \Rightarrow {{A}_{B}}=2{{A}_{A}} \\\ \end{aligned}$
Therefore, the amplitude of A as compared to the amplitude of B is double.

The correct answer is (A) double.

Additional Information:
The energy of a note is dependent on both the frequency and the amplitude. The energy is directly proportional to both of these, that is, the more the frequency or the more the amplitude, the more is the energy.

Note:
In this problem, care must be taken while noting down the conditions given in the question and while comparing the quantities. Since energy is dependent on both amplitude and frequency, both play a role in determining the final relationship between the amplitude of note A and B.