One type of steel has density $7800 k g / m^{3}$ and will break if the tensile stress exceeds ${7.0 ×10}^{8} {N/m}^{2}$ . You want to make a guitar string using 4.0g of this type of steel. While in use, the guitar string must be able to withstand a tension of 900N without breaking. Determine the maximum length the string can have (in m)

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answer is 0.4.

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Detailed Solution

The breaking stress is $\frac{900 N}{π r^{2}} = 7.0 \times 10^{8} n / m^{2},$
Solving for ‘r’ gives the minimum radius $r = \sqrt{\frac{900 N}{π (7.0 \times 10^{8} N / m^{2})}} = 6.4 \times 10^{- 4} m .$
The mass and density are fixed $(ρ = \frac{M}{π r^{2} L})$ so the minimum radius gives the maximum length
$L = \frac{M}{π r^{2} ρ} = \frac{4.0 \times 10^{- 3} k g}{π {(6.4 \times 10^{- 4} m)}^{2} (7800 k g / m^{3})} = 0.40 m$