Question

Two wires, each 1 m long and of the same diameter, have densities 8 × 10³ kg/m³ and 2 × 10³ kg/m³ and are stretched by tensions 196 N and 49 N respectively. Compare their fundamental frequencies.

Answer 1

Data: L₁ = L₂ = 1 m, d₁ = d₂ (∴  r₁ = r₂ ), ρ₁ = 8 × 10³ kg/m³, ρ₂ = 2 × 10³ kg/m³, T₁ = 196 N, T₂ = 49 N

Fundamental frequency, n = `1/(2L) sqrt(T/m)`

= `1/(2L) sqrt(T/(πr^2 ρ))`

= `1/(2Lr) sqrt(T/(πρ))`

∴ `n_1 = 1/(2L_1r_1) sqrt(T_1/(πρ_1))` and `n_2 = 1/(2L_2r_2) sqrt(T_2/(πρ_2))`

∴ `n_1/n_2 = L_2/L_1 xx r_2/r_1 sqrt(T_1/T_2 xx ρ_2/ρ_1)`

= `1 xx 1xx sqrt(196/49 xx (2 xx 10^3)/(8 xx 10^3)) = 1`

∴ Their fundamental frequencies are the same.