Speed of a transverse wave on a straight wire (mass 6.0 g, length 60 cm and area of cross-section 1.0 mm²) is 90 ms⁻¹. If the Young's modulus of wire is 16 × 10¹¹ Nm⁻², the extension of wire over its natural length is :

Solution

Physics sir · 4 steps

The wave speed tells you the tension, and Young's modulus turns that tension into an extension.

1. Mass per length. μ = m/L = (6.0 × 10⁻³ kg)/(0.60 m) = 0.01 kg m⁻¹.
2. Tension from speed. On a stretched wire v = √(T/μ), so T = μ v² = 0.01 × 90² = 81 N.

   Δ L = (T L)/(A Y)

3. Put in numbers. With A = 1.0 mm² = 1.0 × 10⁻⁶ m², Δ L = (81 × 0.60)/(1.0 × 10⁻⁶ × 16 × 10¹¹) = 48.6/(1.6 × 10⁶) ≈ 3.0 × 10⁻⁵ m.
4. So. The extension is about 0.03 mm, which is option (a).

The idea to keep. A wave on a wire and a stretched wire share one quantity: the tension. Find T from v = √(T/μ), then use Y = TL/(A Δ L). Convert grams, cm and mm² to SI first.

Answer from the NTA final answer key. Solution by PrepWiser.