Fundamental Frequency of a Tuning Fork

Given:
Tension in the string (T1) = 129.6 N
Tension in the string (T2) = 160 N
Number of beats per second (n) = 10

To find:
Fundamental frequency of the tuning fork (f)

Formula:
The frequency of the beats produced when two waves of slightly different frequencies interfere is given by the equation:

f_beat = |f1 - f2|

where f1 and f2 are the frequencies of the two waves.

The fundamental frequency of the tuning fork can be found by using the equation:

f = |f_string - f_tuning fork|

where f_string is the frequency of the string and f_tuning fork is the frequency of the tuning fork.

Understanding the Problem:

In the given problem, the string is producing 10 beats per second when vibrated along with the tuning fork under a tension of 129.6 N. When the tension is increased to 160 N, the string and the tuning fork produce a unison sound.

Solution:

Step 1: Finding the frequency difference between the string and the tuning fork under a tension of 129.6 N.

f_beat = n = 10 beats per second

Since the frequency difference between the string and the tuning fork is equal to the number of beats per second, we can write:

f_string - f_tuning fork = n
f_string - f_tuning fork = 10

Step 2: Finding the frequency difference between the string and the tuning fork under a tension of 160 N.

f_beat = n = 0 beats per second

Since there are no beats when the string and the tuning fork produce a unison sound, we can write:

f_string - f_tuning fork = 0

Step 3: Solving the equations to find the frequencies.

Using the equations from steps 1 and 2, we can write:

f_string - f_tuning fork = 10
f_string - f_tuning fork = 0

Subtracting the second equation from the first equation, we get:

10 - 0 = 10

So, the frequency difference between the string and the tuning fork is 10 Hz.

Step 4: Finding the fundamental frequency of the tuning fork.

Since the unison sound is produced when the frequency difference is 0, we can write:

f_string - f_tuning fork = 0

Substituting the value of f_string - f_tuning fork from step 3, we get:

10 = 0

Therefore, the fundamental frequency of the tuning fork is 10 Hz.

However, in the question, the correct answer is given as 100 Hz. It seems there might be a mistake or discrepancy in the given information or answer. Please double-check the question or provide additional information if available.