Sonometer: Laws of Vibrating Strings
Loading simulation…
flagWhat you'll discover
- arrow_forwardState the laws of length and tension for vibrating strings
- arrow_forwardUse a paper rider to detect resonance between wire and fork
- arrow_forwardVerify fL = constant at fixed tension
- arrow_forwardRelate the sonometer equation f = (1/2L)√(T/μ) to the data
The vibrating string equation
A stretched wire of vibrating length L, tension T and mass per unit length μ has fundamental frequency f = (1/2L)√(T/μ). Three laws fall out: the law of length (f ∝ 1/L), the law of tension (f ∝ √T) and the law of mass (f ∝ 1/√μ). The sonometer — a wire over two movable bridges on a hollow sounding box — is built to test them.
For the law of length you keep the load fixed and find the resonant lengths for several forks: each product f × L should come out the same number. Doubling the frequency must exactly halve the length, like fretting a guitar string at its midpoint to jump an octave.
The paper rider trick
The wire’s vibration is far too small to see. The classic detector is a tiny inverted-V paper rider straddling the wire midway between the bridges. Strike the fork, press its stem on the sounding box, and adjust the bridge in millimetre steps. Off resonance the rider sits still; at resonance the wire suddenly vibrates with large amplitude and the rider dances and is flung off.
Why the midpoint? The fundamental mode has its antinode there, so the rider feels the maximum motion. Approach resonance from both sides and take the mean of the two bridge positions where the rider just jumps — that brackets the true resonant length.
Tension, errors, precautions
The tension is supplied by a hanging load: T = Mg, and you must include the hanger’s own mass. For the law of tension, fix one fork and find the resonant length at several loads — L should grow as √T, so a graph of L² against T is a straight line.
Error sources: friction at the pulley makes the true tension differ from Mg; the wire may be kinked or non-uniform (μ varies); the bridges must be knife-sharp so the vibrating length is well defined; and the fork loses frequency if struck violently. Precautions: strike gently on rubber, press the stem firmly on the box for good energy transfer, and keep the rider tiny so it does not load the wire.