Q.State the laws of vibrating strings and explain how they can be verified using a sonometer.

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 …