Question

A steel wire of length $1.5m$ has density 8.8 x 10${}^3$ kg/m${}^3$ and Young's modulus 2.2 x 10${}^{11}$ N/m${}^2$. It is subjected to a tension which produces an elastic strain of $1$%. Its fundamental frequency of vibration must be :

• A. $256Hz$
• B. $178Hz$
• C. $167Hz$
• D. $200Hz$

Answer 1

Answer: (C)

$167Hz$

Stress, $\frac{F}{A}=\frac{Y\Delta L}{L}$

Frequency, $f=\frac{1}{2L}\sqrt{\frac{F}{\mu }}=\frac{1}{2L}\sqrt{\frac{AY\Delta L}{L\left(A\rho \right)}}$

$=\frac{1}{2\times 1.5}\sqrt{\frac{2.2\times {10}^{11}\times {10}^{-2}}{8.8\times {10}^3}}$

$=\frac{1}{2\times 3}\times {10}^3=\frac{{10}^3}{6}\approx 167Hz$