Sonometer: Linear Density μ
A string of unknown μ and fixed length. Measure the fundamental f₁ at several tensions and recover μ from a linear fit of f₁² versus T.
Goal
Determine the hidden linear density μ from f₁ = (1/(2L)) √(T/μ). A plot of f₁² versus T has slope 1/(4 L² μ), so μ = 1/(4 L² · slope).
Equipment
- Sonometer / monochord
- Tension hanger
- Frequency meter
- Fixed bridges (known L)
Experiment
Theory
Wave speed on a taut string is c = √(T/μ). The fundamental of a string fixed at both ends is f₁ = c/(2L) = (1/(2L)) √(T/μ). Squaring: f₁² = T / (4 L² μ). The bench hides μ and c; you only read a frequency meter after each pluck.
Procedure
- The vibrating length L is fixed and known. Linear density μ is hidden. You only change the tension T.
- Pluck / record; the frequency meter logs f₁ with small readout noise.
- The notebook computes f₁². Repeat for at least 6 tensions spread from about 40 N to 160 N.
- Fit f₁² versus T; μ = 1/(4 L² · slope). Compare with the reference.
Conclusion
The fitted linear density agrees with the hidden μ within tolerance. Main uncertainties: frequency-meter noise, assuming an ideal flexible string, and a perfectly fixed length L.