Question

A tuning fork C produces 8 beats per second with another tuning fork D of frequency $340{\text{Hz}}$. When the prongs of the tuning fork C are filed a little, the number of beats produced per second decreases to 4. Find the frequency of the tuning fork C before filing its prongs.

Answer 1

Hint:When the prongs of a tuning fork are filed, the frequency of that tuning fork will increase. Here, it is mentioned that the beats produced per second decreases when tuning fork C is filed. The beats produced per second refer to the difference between the frequencies of the two tuning forks.

Formula used:
-The frequency of the beats produced is given by, ${f_{beat}} = {f_1} - {f_2}$ where ${f_1}$ and ${f_2}$ are the frequencies of the two sound waves created by two tuning forks.
Complete step by step answer.
Step 1: List the parameters known from the question.
The beat frequency when the forks C and D were sounded together is given to be 8 beats per second.
As the tuning fork C of unknown frequency is filed, the beat frequency becomes 4 beats per second.
The frequency of the tuning fork D is given to be ${f_D} = 340{\text{Hz}}$
Let ${f_C}$ be the unknown frequency of the tuning fork C.
Step 2: Using the expression for beat frequency, we can find the frequency ${f_C}$ before and after filing.
The frequency of the beats produced is given by, ${f_{beat}} = {f_D} - {f_C}$ ------- (1)
From equation (1), we can express the frequency of fork C as ${f_C} = {f_D} + {f_{beat}}$ .
But we know the frequency of a sound wave can be ${f_C} = {f_D} + {f_{beat}}$ -------- (2)
or ${f_C} = {f_D} - {f_{beat}}$ ------- (3).

Before filing
We consider the case where no filing is added to the tuning fork C.
Substituting for ${f_D} = 340{\text{Hz}}$ and ${f_{beat}} = 8$ in equations (2) and (3) we get, ${f_C} = 340 + 8 = 348{\text{Hz}}$ or ${f_C} = 340 - 8 = 332{\text{Hz}}$
So the two possible frequencies of fork C before filing are ${f_C} = 348{\text{Hz}}$ or ${f_C} = 332{\text{Hz}}$.

After filing
We now consider the case where filing is added to the tuning fork C. The number of beats produced becomes 4. And the frequency ${f_C}$ of the fork C increases.
Now if ${f_C} = 348{\text{Hz}}$, then on filing, it would become ${f_C} > 348{\text{Hz}}$. Then the relation for the beat frequency ${f_{beat}} = {f_C} - 340 = 4$ suggests that ${f_C} = 344{\text{Hz}}$, but this is impossible as $344{\text{Hz}} < 348{\text{Hz}}$. So, the frequency of the tuning fork C will not be $348{\text{Hz}}$ i.e., ${f_C} \ne 348{\text{Hz}}$ .
Thus the frequency of fork C before filing is ${f_C} = 332{\text{Hz}}$ .

Note:When we consider ${f_C} = 332{\text{Hz}}$, we know that after filing the frequency of the tuning fork C must increase i.e., ${f_C} < 332{\text{Hz}}$. Then the relation of beat frequency, ${f_{beat}} = 340 - {f_C} = 4$ implies that ${f_C} = 336{\text{Hz}}$ i.e., the frequency of the fork C has increased from $332{\text{Hz}}$ to $336{\text{Hz}}$ on filing. While we calculate the beat frequency, we subtract the smaller frequency from the bigger one so that the beat frequency remains positive.