SonometerJEE Main

Concept

The sonometer demonstrates the three laws of vibrating strings: frequency is inversely proportional to length, proportional to the square root of tension, and inversely proportional to the square root of mass per unit length. Every stringed instrument is tuned by exactly these three knobs.

Key formula

Derivation

Wave speed on a string comes from Newton's law on a curved element: $v = \sqrt{T/\mu}$.

Both ends (bridges) are nodes → $L = n\lambda/2$, so $f = nv/2L$. The three laws are just this formula read three ways: $f \propto 1/L$ (law of length), $f \propto \sqrt T$ (tension), $f \propto 1/\sqrt\mu$ (mass).

Scenarios to explore

• Sonometer — Laws of vibrating strings — length, tension & mass density.

Real-world applications

• Guitar frets shorten L; tuning pegs change T; thicker bass strings raise μ.
• Piano bass strings are wrapped to boost μ without losing flexibility.
• Sonometer + tuning fork measures an unknown frequency (unison method).

JEE exam tips

• Percentage game: Δf/f = −ΔL/L + ½ΔT/T − ½Δμ/μ — one line answers most sonometer MCQs.
• A rider (paper) on the wire jumps off at resonance — classic detection trick.
• Same wire, halved length at same T → octave (f doubles).

Common mistakes

• f ∝ T instead of √T — quadrupling tension only doubles pitch.
• Using total mass instead of mass per unit length.
• Forgetting both bridges are nodes (L = λ/2 for the fundamental, not λ).

Exam traps to avoid

• Doubling the hanging MASS doubles T only if the wire is light and pulley frictionless.
• μ = πr²ρ for a cylindrical wire — thicker wire of the same material: f ∝ 1/r.