A piano string 1.5m long is made of steel of density 7.7× 103kgm- 3 and Young's modulus 2× 1011Nm- 2. It is maintained at a tension which produces an elastic strain of 1 % in the string. The fundamental frequency of transverse vibrations of string is

Question

A piano string 1.5m long is made of steel of density 7.7× 103kgm- 3 and Young's modulus 2× 1011Nm- 2. It is maintained at a tension which produces an elastic strain of 1 % in the string. The fundamental frequency of transverse vibrations of string is (A) 85Hz (B) 170Hz (C) 340Hz (D) 310Hz

Tardigrade Admin · Accepted Answer

Y=(F . l/A ⋅ Δ l)⇒ F=YA.(Δ l/l) =2× 1011× (1/100)× A n=(1/2 × 1 . 5)√(F/A d)=(1/3)√(2 × 109 A/ A × 7 . 7 × 103) n=170Hz

Q. A piano string $1.5 m$ long is made of steel of density $7.7 \times 1 0^{3} k g m^{- 3}$ and Young's modulus $2 \times 1 0^{11} N m^{- 2} .$ It is maintained at a tension which produces an elastic strain of $1%$ in the string. The fundamental frequency of transverse vibrations of string is

$85 Hz$

$170 Hz$

$340 Hz$

$310 Hz$

$Y = \frac{F . l}{A \cdot Δ l} \Rightarrow F = Y A . \frac{Δ l}{l}$
$= 2 \times 1 0^{11} \times \frac{1}{100} \times A$
$n = \frac{1}{2 \times 1.5} \frac{F}{A d} = \frac{1}{3} \frac{2 \times 1 0 ^{9} A}{A \times 7.7 \times 1 0 ^{3}}$
$n = 170 Hz$

Q. A piano string 1.5m long is made of steel of density 7.7×103kgm−3 and Young's modulus 2×1011Nm−2. It is maintained at a tension which produces an elastic strain of 1% in the string. The fundamental frequency of transverse vibrations of string is

85Hz

170Hz

340Hz

310Hz