Frequency of Tuning Fork Measured by Sonometer Guide

A tuning fork produces a nearly pure musical tone: when struck, its two prongs vibrate back and forth at one fixed frequency. That frequency is stamped on the fork (for example 512 Hz), but it can be measured independently with a sonometer, a single stretched wire whose pitch is tuned until it sings in unison with the fork.

It turns an invisible vibration into a length on a ruler: slide a bridge until wire and fork resonate, apply the stretched-string formula, and the frequency is found with no electronics.

Key Concepts

• Resonance (unison): the wire's natural frequency exactly matches the fork's, driving the wire into large visible vibration.
• Vibrating length (l): the distance between the two bridges of the sonometer that is actually free to vibrate, in metres.
• Tension (T): the stretching force on the wire, usually supplied by a hanging mass M so that T = Mg, in newtons.
• Linear mass density (μ): the wire's mass per unit length in kg/m, found from the mass of a measured length of the same wire.
• Fundamental mode: the wire vibrates as a single loop with nodes at the bridges and an antinode in the middle.

Formulas

Frequency of a stretched wire: f = (1 / 2l) √(T / μ). At resonance the wire's frequency equals the fork's, so f_fork = (1 / 2l) √(T / μ). Variables: l in metres, T = Mg in newtons (M = hanging mass in kg, g = 9.8 m/s²), μ in kg/m. This expresses the three laws of vibrating strings: f is inversely proportional to length, proportional to the square root of tension, and inversely proportional to the square root of linear density. Assumptions: uniform wire, sharp bridges, freely vibrating prongs, fundamental mode.

Worked Example

A steel sonometer wire has linear density μ = 0.50 g/m (5.0 × 10⁻⁴ kg/m) and carries a 4.0 kg hanger mass. When a tuning fork is sounded, the wire resonates most strongly with a vibrating length of 40.0 cm.

1. Find the tension: T = Mg = 4.0 × 9.8 = 39.2 N.
2. Convert the length: l = 0.400 m, so 1 / (2l) = 1 / 0.800 = 1.25 m⁻¹.
3. Compute T / μ = 39.2 / (5.0 × 10⁻⁴) = 78400, and √78400 = 280 (since 280² = 78400).
4. Multiply: f = 1.25 × 280 = 350 Hz.
5. So the fork's frequency is 350 Hz; repeating with different tensions and plotting f against 1/l confirms the inverse-length law.

Common Misconceptions

• Striking harder does not change frequency: it only makes the tone louder; pitch is set by the fork's size and material.
• Touching the prongs kills the tone, not the frequency. Damping removes energy and silences the fork, but the brief sound it made still had the same pitch.
• Beats mean you are close, not there: the true resonance point is where beats vanish and the wire's vibration is strongest.

Real-World Applications

• Tuning pianos, guitars and orchestral instruments against a reference fork.
• Calibrating acoustic sensors and checking the frequency response of microphones.
• Vibration testing of strings, cables and membranes in engineering labs.
• Medical tuning forks (128 Hz, 512 Hz) are used in hearing and nerve-conduction tests.

The sonometer turns pitch into geometry: tune the wire until fork and wire sing as one, measure the length, and the stretched-string formula gives the frequency.