Question

An 80-cm-long, 1.0-mm-diameter steel guitar string must be tightened to a tension of 2000 N by turning the tuning screws. By how much is the string stretched?

Short answer

The string is stretched by$0.010m$.

Step-by-step solution

Given information

The length of the string is $L=80cm=0.80m$.

The radius of the string is role="math" localid="1650298194619" $r=\frac{1mm}{2}=0.5mm=0.0005m$

The tension in the string is $T=2000N$

Young's modulus is$Y=20\times {10}^{10}N/m^2$

Elongation of the string

Young's modulus is given by $Y=\frac{{\displaystyle \raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$A$}\right.}}{{\displaystyle \raisebox{1ex}{$∆L$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}}$.

The elongation of the string is

role="math" localid="1650298181390" $$\begin{gathered} ∆L=\frac{T}{A}\frac{L}{Y} \\ =\frac{2000}{\mathrm{\pi }{\left(0.0005\right)}^2}\frac{0.80}{20\times {10}^{10}} \\ =0.010m \end{gathered}$$

Therefore, the string is stretched by$0.010m$.