Calculating Sound Speed: C4 Recorder Frequency & Wavelength

Hey guys! Let's dive into a super interesting physics problem today. We're going to figure out how to calculate the speed of sound using some info about a recorder playing a C4 note. Sounds cool, right? This is a classic example of how physics concepts like frequency, wavelength, and speed are all connected. We'll break it down step-by-step so it's easy to understand. So, grab your thinking caps and let's get started!

Understanding the Fundamentals of Sound

Before we jump into the calculation, it's important to grasp the key concepts involved. Sound, as we know, travels in waves. These waves are basically vibrations that move through a medium, like air. Think of it like dropping a pebble into a pond – the ripples that spread outwards are similar to how sound waves propagate. Now, two crucial properties define these sound waves: frequency and wavelength.

• Frequency: Frequency is essentially how many wave cycles pass a specific point in one second. We measure frequency in Hertz (Hz), where 1 Hz means one cycle per second. So, a higher frequency means more waves are passing by each second, which translates to a higher-pitched sound. For example, the C4 note on a recorder has a frequency of 261 Hz, meaning 261 sound wave cycles reach your ear every second! That's pretty fast, huh?

• Wavelength: Wavelength, on the other hand, is the distance between two identical points on a wave, like the distance between two crests (the highest points) or two troughs (the lowest points). We typically measure wavelength in meters (m). Think of it as the physical length of one complete wave cycle. A longer wavelength corresponds to a lower-pitched sound, while a shorter wavelength corresponds to a higher-pitched sound.

The relationship between these two properties, frequency and wavelength, is what ultimately determines the speed of sound. This speed tells us how quickly the sound wave travels through the medium. And that's what we're going to calculate today!

The Formula That Connects Them All

Okay, so how do we actually calculate the speed of sound using frequency and wavelength? There's a nifty little formula that ties all these concepts together. This is a key formula in the world of sound and wave physics, and you'll probably see it again and again. Here it is:

Speed of Sound (v) = Frequency (f) × Wavelength (λ)

Where:

• v represents the speed of sound, usually measured in meters per second (m/s).
• f represents the frequency, measured in Hertz (Hz).
• λ (the Greek letter lambda) represents the wavelength, measured in meters (m).

This formula is super powerful because it shows us the direct relationship between these three important properties of sound. The faster the frequency or the longer the wavelength, the faster the speed of sound will be! Now that we have this formula in our toolbox, we're ready to tackle the problem.

Applying the Formula to Our Recorder Problem

Let's get back to our original problem: a C4 note on a recorder has a frequency of 261 Hz, and the wavelength of the sound in the air is 1.31 meters. We want to find the speed of the sound. We've got all the pieces of the puzzle, now we just need to plug them into our formula. Remember the formula?

v = f × λ

Here's what we know:

• Frequency (f) = 261 Hz
• Wavelength (λ) = 1.31 m

Now, let's substitute these values into the formula:

v = 261 Hz × 1.31 m

Time to do some multiplication! If you grab your calculator (or do it the old-fashioned way!), you'll find:

v = 341.91 m/s

So, the speed of sound in this case is approximately 341.91 meters per second. That's super fast! It means the sound wave travels almost 342 meters in just one second. No wonder sound seems to reach us almost instantly.

Analyzing the Result and Understanding the Context

Now that we've calculated the speed of sound, let's take a moment to think about what this result actually means. We found that the speed of sound is approximately 341.91 m/s. This value is quite close to the commonly accepted speed of sound in air at room temperature, which is around 343 m/s. This is a good sign! It tells us that our calculation is likely correct and that the given values for frequency and wavelength are consistent.

However, it's important to remember that the speed of sound isn't a constant. It can change depending on several factors, most notably the temperature of the air. Sound travels faster in warmer air and slower in colder air. This is because the molecules in warmer air have more energy and vibrate faster, allowing the sound waves to propagate more quickly. The humidity of the air can also have a slight effect, but temperature is the primary factor.

In our problem, we weren't given the temperature of the air. So, we've assumed it's close to room temperature. If the temperature were significantly different, the speed of sound would also be different. This is a crucial point to keep in mind when dealing with sound-related problems. Always consider the context and the factors that might influence the results.

Choosing the Correct Answer from the Options

Now, let's go back to the original question and the answer choices provided. We calculated the speed of sound to be approximately 341.91 m/s. The answer options were:

a. 200 m/s b. 300 m/s c. 340 m/s d. 370 m/s e. 400 m/s

Looking at these options, the closest value to our calculated result is 340 m/s. So, the correct answer is (c) 340 m/s. It's awesome when the math aligns perfectly with the options! This gives us even more confidence in our solution.

Why This Matters: Real-World Applications

Okay, so we've calculated the speed of sound for a C4 note on a recorder. But why is this actually important? Why do we even care about the speed of sound? Well, understanding the speed of sound has a bunch of real-world applications in various fields, from music and acoustics to engineering and even medicine!

• Music and Acoustics: In music, the speed of sound is crucial for designing instruments and concert halls. The size and shape of an instrument, like a recorder or a guitar, directly affect the frequencies it can produce. Similarly, the acoustics of a concert hall depend on how sound waves travel and reflect within the space. Architects and acousticians use their knowledge of the speed of sound to create spaces that have optimal sound quality.

• Engineering: Engineers need to consider the speed of sound when designing systems that involve sound waves, such as sonar systems used in submarines or ultrasound devices used in medical imaging. These systems rely on the precise timing of sound waves, and knowing the speed of sound is essential for accurate measurements and performance.

• Medicine: As mentioned, ultrasound is a powerful medical imaging technique. Ultrasound devices emit high-frequency sound waves that travel through the body and reflect off different tissues and organs. By measuring the time it takes for these waves to return, doctors can create images of the inside of the body. Again, the speed of sound is a critical factor in this process.

• Everyday Life: Even in our daily lives, understanding the speed of sound helps us make sense of the world around us. For example, we can estimate the distance of a lightning strike by counting the seconds between the flash and the thunder. Since light travels much faster than sound, the delay we experience is due to the time it takes for the sound of thunder to reach us. Each 5 seconds roughly corresponds to a mile.

Key Takeaways and Final Thoughts

Alright guys, we've covered a lot of ground today! We started with a seemingly simple question about the speed of sound for a C4 note on a recorder, and we've delved into the fundamental concepts of frequency, wavelength, and the relationship between them. We learned the important formula: v = f × λ, and we applied it to solve our problem. We also discussed how the speed of sound can be affected by factors like temperature and why understanding the speed of sound is crucial in various real-world applications.

So, what are the key takeaways from this exercise?

1. Sound travels in waves: These waves have properties like frequency and wavelength.
2. Frequency is the number of wave cycles per second (measured in Hz).
3. Wavelength is the distance between two identical points on a wave (measured in meters).
4. Speed of sound (v) = Frequency (f) × Wavelength (λ): This is the key formula to remember!
5. The speed of sound is not constant: It can be affected by factors like temperature.
6. Understanding the speed of sound has numerous real-world applications: From music to medicine, this concept is essential.

I hope this breakdown has helped you grasp the concepts and the calculations involved. Physics can seem a bit daunting at first, but when you break it down into smaller steps and relate it to real-world examples, it becomes much more approachable and even… fun! Keep exploring, keep asking questions, and keep learning. You've got this!