Question

Two vibrating strings of the same material but of lengths Land 2L have radii 2r and r respectively. They are stretched under the same tension. Both the string vibrate in their fundamental modes. The one of length L with frequency Ʋ₁ and the other with frequency Ʋ₂. What is the ratio (Ʋ₁ / Ʋ₂) ? 

(A) 8 

(B) 2 

(C) 1 

(D) 4

Answer 1

The correct option is (C) 1.

Explanation:

Fundamental frequency with both ends fixed is Ʋ = (1 / 2ℓ) √(T/μ) 

For same Tension, Ʋ ∝ {1 / (ℓ√μ)} (1) 

Where μ = mass per unit length of wire = ρ ∙ A 

∴ μ = ρ(πr²) 

i.e. μ ∝ r² 

∴ √μ ∝ r 

from (1) Ʋ ∝ (1 / rℓ) 

∴ (Ʋ₁ / Ʋ₂) = {(r₂ℓ₂) / (r₁ℓ₁)} 

Given r₁ = 2r, r₂ = r, ℓ₁ = L, ℓ₂ = 2L 

∴ (Ʋ₁ / Ʋ₂­) = (r / 2r) × (2L / L) = 1