Question: You are given four tuning forks. The fork with the lowest frequency oscillates at 500 Hz. By striking two tuning forks at a time,you can produce the following beat frequencies, 1, 2, 3, 5, 7, and 8Hz. What are the possible frequencies of the other three forks?(There are two sets of answers)

1. The fork with lowest frequency is$f_4=500\text{Hz}$
2. The beat frequencies are$\text{1Hz,2Hz,3Hz,5Hz,7Hz,and8Hz}$
3. There are two sets of answer.

Formula is as follow:

The beat frequency is,

.$f_{beat}=f_1-f_2$

where, f is frequency.

Let,$f_1{f}_{2,}{f}_{3}and{f}_4$be the frequencies of the corresponding fork 1, fork 2, fork 3, and fork 4respectively, so that,$f_1>f_2>f_3>f_4$

.

Then, the beat frequency is,

$f_{beat}=f_1-f_4$

Thus, the highest frequency f₁ of the corresponding fork is given by,

$$\begin{gathered} f_1=f_{beat}+f_4 \\ {f}_1=8+500 \\ {f}_1=508Hz \end{gathered}$$

Similarly, the beat frequency is,

$f_{beat}=f_2-f_4$

Thus, the frequency f₄ of the corresponding fork is given by,

$$\begin{gathered} f_2=f_{beat}+f_4 \\ {f}_2=7+500 \\ {f}_2=507Hz \end{gathered}$$

Similarly, the beat frequency is,

$f_{beat}=f_3-f_4$

Thus, the frequency f₄ of the corresponding fork 3 is given by,

$$\begin{gathered} f_3=f_{beat}+f_4 \\ {f}_3=5+500 \\ {f}_3=505Hz \end{gathered}$$

Now, for second set, also $f_1>f_2>f_3>f_4$and

The beat frequency is,

$f_{beat}=f_1-f_4$

Thus, the highest frequency of the corresponding fork is given by,

$$\begin{gathered} f_1=f_{beat}+f_4 \\ {f}_1=8+500 \\ {f}_1=508H_z \end{gathered}$$

Similarly, the beat frequency is,

$f_{beat}=f_2-f_4$

Thus, the frequency f₂ of the corresponding fork is given by,

role="math" localid="1661435994772" $$\begin{gathered} f_2=f_{beat}+f_4 \\ {f}_2=3+500 \\ {f}_2=503\text{Hz} \end{gathered}$$

Similarly, the beat frequency is,

$f_{beat}=f_3-f_4$

Thus, the frequency of the corresponding fork 3 is given by,

$$\begin{gathered} f_3=f_{beat}+f_4 \\ {f}_3=1+500 \\ {f}_3=501\text{Hz} \end{gathered}$$

Hence, the possible frequencies of other three forks are,$$\begin{gathered} \text{f1=508Hz,f2=507Hz,andf3=505Hz,or,} \\ \text{f1=508Hz,f2=503Hz,andf3=501Hz,} \end{gathered}$$.

Thus, by using the formula for beat frequency,$f_{beat}$this question can be solved.