Q.Two wires of the same material and the same cross section are stretched on a sonometer in succession. Length of one wire is 60 cm and that of the other is 30 cm. An unknown load is applied to the first wire and second wire is loaded with 1.5 kg. If both the wires vibrate with the same fundamental frequencies, calculate the unknown load.

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 …