CP5: Frequency of a vibrating string
Welcome to one of the most musical chapters in your Physics course! In Core Practical 5, we explore the physics behind stringed instruments like guitars and violins. By the end of these notes, you will understand how the pitch (frequency) of a string changes when you change its length, how tight it is, or how heavy the string is. Don't worry if the formulas look a bit intimidating at first—we will break them down step-by-step!
1. The Big Picture: What are we investigating?
The goal of this practical is to investigate how the frequency \(f\) of the first harmonic (the simplest standing wave pattern) is affected by three specific factors:
- Length (\(l\)): The distance between the vibration generator and the bridge.
- Tension (\(T\)): How tightly the string is stretched (usually by hanging masses).
- Mass per unit length (\(\mu\)): How "heavy" or "thick" the string is.
Did you know? This experiment is a direct application of the standing wave theory you learned in Unit 2. If you need a refresher on the basics of waves, you can cross-reference "CP4: Speed of sound in air" for more on wave behavior.
2. The Underlying Theory
To understand the experiment, we combine two important equations from your syllabus:
- The wave equation: \(v = f\lambda\)
- The speed of a transverse wave on a string: \(v = \sqrt{\frac{T}{\mu}}\)
When a string vibrates at its first harmonic (fundamental frequency), the length of the string \(l\) is exactly half a wavelength. Therefore, \(\lambda = 2l\).
By substituting these into each other, we get the "Master Equation" for this practical:
\(f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}\)
Key Takeaway:
- If \(l\) increases, \(f\) decreases (Longer strings produce lower notes).
- If \(T\) increases, \(f\) increases (Tighter strings produce higher notes).
- If \(\mu\) increases, \(f\) decreases (Thicker strings produce lower notes).
3. Experimental Setup
To perform this experiment, you typically need:
- A signal generator connected to a vibration generator.
- A length of string or wire passed over a pulley.
- Masses hanging from the end of the string to provide tension \(T = mg\).
- A wooden bridge to define the vibrating length \(l\).
- A metre rule to measure length.
Step-by-Step Procedure:
- Set up the apparatus with a fixed mass (tension) and a fixed length.
- Slowly increase the frequency on the signal generator from 0 Hz until the string vibrates with maximum amplitude in the first harmonic (one "loop" with nodes at the ends and an antinode in the middle).
- Record the frequency \(f\) from the signal generator.
- Repeat the process by changing one variable at a time (either \(l\), \(T\), or \(\mu\)) while keeping the others constant.
4. Measuring the Variables Accurately
In Unit 3, you are often tested on your choice of instruments and how to reduce uncertainty.
Mass per unit length (\(\mu\)):
To find \(\mu\), you measure the total mass of a long piece of string using a balance and its total length using a metre rule. Then calculate \(\mu = \frac{mass}{length}\). To be even more precise, you could use a micrometer screw gauge (resolution \(0.01 \text{ mm}\)) to check the diameter of the string at various points to ensure it is uniform.
Tension (\(T\)):
Tension is calculated as \(T = mg\). Make sure the masses are labeled correctly and check them with a balance if you suspect they are inaccurate.
Length (\(l\)):
Use a metre rule (resolution \(1 \text{ mm}\)). Ensure you measure the distance from the vibration generator to the bridge, not the whole string!
Quick Tip: Always do a zero check on your digital instruments (like a balance or micrometer) to avoid systematic errors!
5. Data Analysis and Graphing
To confirm the relationship \(f = \frac{1}{2l} \sqrt{\frac{T}{\mu}}\), we usually plot graphs to look for straight lines through the origin.
Investigation A: Frequency vs Length
If we keep \(T\) and \(\mu\) constant, \(f \propto \frac{1}{l}\).
Plot a graph of \(f\) on the y-axis against \(\frac{1}{l}\) on the x-axis.
Expected Result: A straight line through the origin. The gradient would be \(\frac{1}{2} \sqrt{\frac{T}{\mu}}\).
Investigation B: Frequency vs Tension
If we keep \(l\) and \(\mu\) constant, \(f \propto \sqrt{T}\).
Plot a graph of \(f^2\) on the y-axis against \(T\) on the x-axis.
Expected Result: A straight line through the origin. The gradient would be \(\frac{1}{4l^2\mu}\).
6. Uncertainties and Practical Improvements
Physics is never perfect! Here are things to look out for in exam questions:
- Inconsistent Readings: If you repeat a measurement and get a value that is very different, identify it as an anomaly and check your setup. Always take repeat readings and calculate a mean to reduce the effect of random errors.
- Node Positioning: It can be hard to judge exactly where the node is. Using a sharp wooden bridge helps define the length \(l\) more clearly.
- Parallax Error: When measuring length with the metre rule, look directly over the markings to avoid misreading them.
- Uncertainty Calculation: Remember that for a single reading, uncertainty is half the resolution. For repeat readings, it is half the range.
7. Health and Safety
Safety is a common Unit 3 question topic. For this experiment, consider:
1. Eye Protection: Strings are under tension and can snap. Wear safety goggles to protect your eyes.
2. Falling Masses: If the string snaps, the heavy masses could fall on your feet. Place a "padded catch box" or a tray of sand beneath the masses to catch them safely.
8. Summary Table for Quick Review
Relationship: \(v = f\lambda\) and \(v = \sqrt{\frac{T}{\mu}}\)
Independent Variable: Length, Tension, or Mass per unit length
Dependent Variable: Frequency (\(f\))
Control Variables: The two variables you are NOT currently testing
Graph for \(l\): \(f\) vs \(1/l\) (Straight line through origin)
Graph for \(T\): \(f^2\) vs \(T\) (Straight line through origin)
Don't worry if this seems tricky at first! Just remember that all we are doing is checking if the math matches the music. Keep practicing those graph gradients, and you'll be a pro in no time!