Recall Example 8.4, in which we analyzed a cable used to support an actor as he swung onto the stage. Suppose that the tension in the cable is 940 N as the actor reaches the lowest point. What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under

Question

Recall Example 8.4, in which we analyzed a cable used to support an actor as he swung onto the stage. Suppose that the tension in the cable is 940 N as the actor reaches the lowest point. What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under these conditions?

Accepted Answer

From the definition of Young’s modulus, we can solve for the required cross-sectional area. Assuming that the cross section is circular, we can determine the diameter of the wire. From Equation 12.6, we have

[latex] Y \equiv \frac{ ext { tensile stress }}{ ext { tensile strain }}=\frac{F / A}{\Delta L / L_{i}} [/latex] (12.6).

[latex] Y=\frac{F / A}{\Delta L / L_{i}} [/latex].

[latex] A=\frac{F L_{i}}{Y \Delta L}=\frac{(940 N )(10 m )}{\left(20 imes 10^{10} N / m ^{2} ight)(0.005 m )} [/latex].

[latex] =9.4 imes 10^{-6} m ^{2} [/latex].

[latex] ext { Because } A=\pi r^{2} ext {, the radius of the wire can be found from } [/latex].

[latex] r=\sqrt{\frac{A}{\pi}}=\sqrt{\frac{9.4 imes 10^{-6} m ^{2}}{\pi}}=1.7 imes 10^{-3} m =1.7 mm [/latex].

[latex] d=2 r=2(1.7 mm )=3.4 mm [/latex].

To provide a large margin of safety, we would probably use a flexible cable made up of many smaller wires having a total cross-sectional area substantially greater than our calculated value.

Question 12.6: Recall Example 8.4, in which we analyzed a cable used to sup...

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