Replacing a Monochord with a Violin — Monochord Lab Adaptation for a 13-year-old

Quick summary

You can use a violin instead of a monochord to study how string frequency depends on length, tension and mass per length. A violin is more complicated than a simple monochord because it has a bridge, body resonance and you usually bow the string. But if you control the variables (for example, pluck the string and use a mute or cloth to reduce body resonance), you can still test the same physics and get good data.

What is the basic physics? (short and simple)

The frequency f of a vibrating string is given by:

f = (1 / 2L) * sqrt(T / μ)

• L = vibrating length of the string (meters)
• T = tension in the string (newtons)
• μ = linear mass density (mass per length, kg/m)

This means if tension and μ stay the same, frequency is approximately proportional to 1/L. If you change tension, frequency changes like the square root of T.

How a violin differs from a monochord

• Violin strings are fixed at nut and bridge; the vibrating length is the distance between nut and bridge (called the scale length).
• The violin body resonates and amplifies sound — it adds extra peaks in the sound and affects loudness and decay.
• Bowing adds continuous excitation and can change harmonic content; plucking (pizzicato) is closer to a monochord pluck.
• Different violin strings have different μ values (G, D, A, E strings are not identical).

What you will need

• Violin (student instrument is fine) and bow or will use plucking
• Ruler or caliper to measure vibrating length L (mm or cm ruler okay)
• Capo, rubber wedge, or a finger to shorten the vibrating length; small soft cloth or rubber mute to damp the body
• Smartphone with a tuner or spectral-analysis app (apps like 'Spectrogram', 'SpectrumView', 'Tunable', or free Audacity on a computer for FFT)
• Notebook or spreadsheet for data
• Optional: spare piece of the same string to measure mass per meter (weigh a known length on a precise scale)

Step-by-step experiment (simple plan)

1. Measure the open-string vibrating length L0 (from nut to bridge). Record it.
2. Put a soft cloth over the violin body or use a mute to reduce resonance so you are mostly hearing the string vibration (this makes data cleaner).
3. Choose one string to study (A or D are good). Pluck the open string with the same strength each trial (rest your hand on the same point each time so force is more consistent).
4. Use the tuner/spectrum app to measure the fundamental frequency f0. Repeat the pluck 3–5 times and average the measured f0. Record uncertainties (range or standard deviation).
5. Shorten the vibrating length to a set of lengths (for example: 100%, 90%, 80%, 70% of L0). Use a capo or press lightly with finger on the string at the chosen point so the string vibrates between the finger and bridge—measure the new vibrating length L each time.
   • At each length, pluck and measure frequency 3–5 times and average.
6. Optional: If you can change tension safely (by tuning pegs), you can also test how frequency changes with tension. Change tuning in small steps and measure frequency. Be careful not to overtighten or break a string.
7. Record all data in a table: L (m), 1/L (1/m), measured f (Hz), uncertainty.

How to analyze

Primary test: If tension and μ are constant, plot f (vertical axis) vs 1/L (horizontal axis). According to the formula, f should be proportional to 1/L, so the points should lie close to a straight line through the origin.

Other checks:

• Plot f vs 1/L and find slope. The slope equals (1/2) * sqrt(T/μ).
• If μ is known, you can rearrange to estimate T: T = μ * (2Lf)^2.
• If you cannot measure μ, you can measure the ratio of frequencies when length changes: f1/f2 = L2/L1 (if T and μ constant).

Sample calculation (approximate example)

Example: open A string measured f = 440 Hz, scale length L = 0.328 m. First compute wave speed v = 2 L f = 2 * 0.328 * 440 = 288.6 m/s.

If the string's linear mass μ were 0.0007 kg/m (typical order of magnitude for a steel string), tension would be:

T = μ * v^2 = 0.0007 * (288.6)^2 ≈ 58 N (this is an example; real μ varies).

The important thing is the method: measure L and f, compute v = 2Lf, and if μ is known compute T; otherwise test proportional relations between f and 1/L.

Tips to get good data

• Pluck near the middle of the string to get a strong fundamental (plucking near the bridge emphasizes higher harmonics).
• Keep plucking force as consistent as possible. Using a small pick helps with consistency.
• Use a mute or cloth to reduce the violin body's resonance if the body makes the measured peak messy.
• Measure lengths carefully (the point where the finger touches becomes a new node—measure from that point to the bridge).
• If using a tuner app, look at the spectral display to confirm you are reading the fundamental frequency, not a harmonic.

Sources of error and why results may differ from a simple monochord

• Body resonance changes the sound and can hide the true fundamental or add extra peaks.
• The bridge and nut are not perfect fixed points; small differences in contact can change effective length.
• If you bow instead of pluck, the excitation method changes harmonic content and frequency detection can be harder.
• Tension is harder to control precisely on a violin than on a lab monochord with a weight pulley system.
• Strings are not perfect ideal strings; wound strings and stiffness cause slight inharmonicity (overtones are not exact multiples of the fundamental).

Safety

• Do not overtighten strings — they can snap and fly off. Turn pegs slowly and stop if you feel strong resistance.
• Keep fingers and eyes away if a string breaks. Wear safety glasses if an adult wants to test extreme tensions (not recommended for this lab).
• Handle the bow carefully (do not point or swing it near others).

What to report

Include: your procedure, table of L and f (averaged values and uncertainties), plot of f vs 1/L, a short discussion comparing your results to the prediction f ∝ 1/L, and a paragraph on sources of error and how the violin differs from a textbook monochord.

Short troubleshooting

• If the app shows multiple strong peaks, try muting the body more or pluck more softly.
• If frequencies are noisy, make sure the phone is close to the instrument and in a quiet room.
• If you cannot change length with a capo, use a finger lightly to stop the string at the desired point (press just enough to create a clear pitch).

If you want, tell me which string you plan to use, the approximate scale length of your violin, and whether you have a tuner app; I can suggest a specific set of lengths to test and help you plan the data table and graph.