Physics Puzzle: Finding The Frequency Of Tuning Forks

Nov 9, 2025 by Admin 54 views

Hey guys! Let's dive into a fun physics problem. We're going to solve a puzzle involving tuning forks and a vibrating string. This problem is super interesting because it combines the concepts of sound waves, frequency, and resonance. The core of this challenge involves understanding how the frequency of a vibrating string changes with its length and how it interacts with the frequency of a tuning fork. So, grab your physics hats, and let's get started.

We will examine the principles of sound wave superposition. Understanding wave superposition helps us analyze the interference between sound waves from the tuning forks and the string. The beats are a result of this interference, occurring when the frequencies of the two sound sources are close but not exactly the same. The phenomenon of beats is a direct result of the superposition principle, where waves combine and create an interference pattern. The rate at which these beats occur is equal to the absolute value of the difference between the frequencies of the two sound sources. We'll use this crucial concept to find the frequencies of the tuning forks. Let’s carefully analyze each part of the problem to understand how we can calculate the frequency of the tuning forks. By examining the conditions, such as the initial beat frequency, the length of the string, and how changing the length of the string affects the resonance, we can systematically solve the problem.

The challenge presented involves a fascinating interplay of physics concepts: the vibration of strings and the behavior of sound waves. To successfully solve this, we must first recognize the relationship between the length of the string and its natural frequencies of vibration. When we pluck a string, it vibrates at multiple frequencies, known as harmonics, with the fundamental frequency being the first harmonic. The frequency of the fundamental harmonic is inversely proportional to the length of the string. That means that as the string gets shorter, the frequency goes up. This is a crucial concept to apply to the problem. The problem gives us a key piece of information: initially, the string and the tuning fork produce beats. Beats happen when two sound waves with slightly different frequencies interfere with each other. The number of beats per minute is the difference between the two frequencies. This understanding is key to unraveling the puzzle. Let's delve into the details: We're given that when we simultaneously sound tuning forks and a string with a length of 87 cm, we hear 90 beats per minute. This means that the difference between the frequency of the tuning forks and the frequency of the string is 90 beats per minute or 1.5 Hz (since there are 60 seconds in a minute). Then, the string's length is shortened by 0.3 cm, and the sounds from both sources match and merge. This crucial piece of information tells us that the string’s frequency has been changed to match the tuning fork's frequency.

Setting Up the Equations

Alright, let's break this down further. Our first step is to establish the equations. The frequency of a vibrating string is determined by: f = (v / 2L), where f is the frequency, v is the wave speed on the string, and L is the length of the string. However, we don't know the wave speed 'v', so let's represent the frequency of the tuning fork as 'f_fork' and the initial frequency of the string (when its length is 87 cm) as 'f_string1'. The beat frequency is the difference between these two frequencies:

|f_fork - f_string1| = 1.5 Hz (since 90 beats/minute = 1.5 Hz)

When the string is shortened to 86.7 cm, the frequencies match. Therefore, the frequency of the string is now equal to the frequency of the tuning fork: f_string2 = f_fork. The frequency of the string is inversely proportional to its length. So, if we denote the initial length of the string as L1 (87 cm) and the new length as L2 (86.7 cm), we get:

f_string1 = v / (2 * L1)
f_string2 = v / (2 * L2)
f_fork = f_string2

Since the wave speed 'v' remains constant in the problem, we can set up a relationship between the initial and final frequencies of the string. We have two equations here, one using the beat frequency and the other using the string length relationship. We can use these equations to solve for the frequency of the tuning forks. It's time to put on our detective hats and piece together these clues.

Now, let's delve deeper into the core principles of wave phenomena, specifically wave superposition and resonance. Wave superposition is the most important concept in understanding how waves combine when they occupy the same space. When two or more waves overlap, the resulting wave is the sum of the individual waves. This can result in constructive interference (where the waves add up, increasing the amplitude) or destructive interference (where the waves cancel each other out, decreasing the amplitude). Resonance occurs when an object vibrates at its natural frequency due to an external force or another vibrating object. In our problem, the string resonates with the tuning forks, and the condition for resonance occurs when the string's frequency matches the tuning fork's frequency. This understanding is crucial for the second part of the problem, where we're told that the tones of both sources match and merge. The concept of beats also plays a critical role here. Beats occur due to the superposition of waves with slightly different frequencies. The beat frequency is equal to the difference between the frequencies of the two waves, which is why we heard 90 beats per minute at the beginning of the problem. This phenomenon provides us with the key information to establish the initial relationship between the string's frequency and the tuning fork's frequency. By setting up the equations and considering the principles of wave phenomena, we are well-equipped to solve the problem and understand the final frequency of the tuning forks.

Solving for the Frequency

Okay, guys, it's time for some calculations. Let's denote the frequency of the tuning fork as f. Initially, the difference between the tuning fork frequency and the string frequency (f_string1) is 1.5 Hz. So, we have |f - f_string1| = 1.5 Hz. When the string is shortened, its frequency changes to f, so:

f_string1 = f ± 1.5 Hz

We know that the frequency of a string is inversely proportional to its length, so:

f_string1 / f = L2 / L1
f_string1 / f = 86.7 / 87
f_string1 = f * (86.7 / 87)

Now, substitute this value of f_string1 into the initial equation:

|f - f * (86.7 / 87)| = 1.5 Hz
f * |1 - (86.7 / 87)| = 1.5 Hz
f * (0.3 / 87) = 1.5 Hz
f = (1.5 * 87) / 0.3 Hz
f = 435 Hz

So, the frequency of the tuning fork is 435 Hz. Now we found the answer! The key to this problem was connecting the concepts of string frequency and beat frequency. By understanding how the string's frequency changes with length and by recognizing that the beat frequency is the difference between the frequencies, we were able to solve it.

Conclusion

In conclusion, we successfully solved the physics puzzle and found the frequency of the tuning forks to be 435 Hz! This problem showcases how seemingly simple phenomena, such as the vibration of a string, can involve complex physics principles. We’ve seen how frequency, wavelength, and resonance come into play, and how a slight change in the length of the string can bring everything into harmony. The beauty of physics problems lies in the fact that they often require you to connect different concepts and think critically. It is a fantastic practice for developing problem-solving skills! Keep exploring the world of physics, and you’ll discover even more fascinating phenomena.

So, whether you're a physics enthusiast or just curious, understanding how to approach and solve this type of problem can be quite rewarding. By methodically breaking down the given information, setting up equations, and applying the relevant physics principles, we can determine the frequency of the tuning forks.