Q.A sonometer wire of length 50 cm is stretched by keeping weights equivalent of 3.5 kg. The fundamental frequency of vibration is 125 Hz. Determine the linear density of the wire.

The Sonometer: What It Is and Why It Matters

Imagine a guitar string. Pluck it, and you hear a note. Press your finger down at a different fret — the vibrating part of the string gets shorter — and the note gets higher. Tighten the tuning peg, and the note rises again. Swap the string for a thicker one, and the pitch drops. That's the entire physics of a sonometer, stripped down to its bones.

A sonometer is simply a laboratory version of that guitar string. It's a long, hollow wooden box with a thin wire stretched tightly over two fixed bridges. You can change three things about the wire: how long the vibrating segment is (by moving a third, movable bridge), how tight the wire is (by hanging weights on one end), and what kind of wire you use (different materials or thicknesses). A small, light paper rider placed on the wire helps you see when the wire is vibrating strongly — it dances or falls off at resonance.

The Core Relationship: Frequency and Its Dependence

The sonometer exists to verify one central formula. For a stretched string, the fundamental frequency $f$ (the lowest note it can produce) is:

where:

• $L$ is the vibrating length of the string (in metres)
• $T$ is the tension in the string (in newtons)
• $\mu$ is the linear mass density — mass per unit length of the string (in kg/m)

This isn't a random equation. It comes from the wave equation for a string fixed at both ends. The wave speed on a string is $v = \sqrt{T/\mu}$, and the fundamental standing wave has a wavelength $\lambda = 2L$. Since $v = f\lambda$, you get $f = v/(2L) = (1/2L)\sqrt{T/\mu}$.

What the Sonometer Actually Shows You

You can test each variable one at a time, keeping the others constant.

Length: Move the movable bridge to change $L$. Pluck the wire and find the tuning fork that matches its pitch. You'll discover that $f \propto 1/L$ — halve the length, double the frequency. That's why guitar frets get closer together as you go up the neck.

Tension: Hang different weights on the end of the wire. More weight means more tension. You'll find $f \propto \sqrt{T}$. To double the frequency, you need four times the tension.

Linear density: Use wires of different thicknesses or materials. A thicker wire has larger $\mu$, so $f \propto 1/\sqrt{\mu}$. Heavy strings on a piano are thick and produce low notes; thin strings produce high notes.

How the Experiment Runs

You place a tuning fork of known frequency near the wire. Adjust the movable bridge until the wire vibrates most violently in sympathy with the fork — that's resonance. At that point, the wire's fundamental frequency matches the fork's frequency. Measure the length $L$ between the bridges. Repeat for different tensions or different wires, and you can verify the formula experimentally.

The paper rider is your indicator. When the wire vibrates strongly, the rider jumps off. No rider movement means you're off-resonance.

The Big Picture

The sonometer isn't just a lab instrument. It's the physical embodiment of why stringed instruments work the way they do. Every time a guitarist bends a string (changing tension), or a violinist presses a finger down (changing length), or a piano uses different gauge wires for bass and treble notes, they're using the exact same physics the sonometer demonstrates. The formula $f = \frac{1}{2L}\sqrt{T/\mu}$ governs them all.

The sonometer experiment is a standard part of the NCERT Class 11 Physics Waves chapter and its associated practical syllabus, and 'sonometer formula f = 1/2L √(T/μ)' or 'sonometer experiment important questions class 11 physics' are common searches for board revision. This length-tension-density relationship is also occasionally tested in JEE Main physics as a direct numerical.