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Chapter 10: Problem 2

A 91 -kg man's thighbone has a relaxed length of \(0.50 \mathrm{m},\) a cross- sectional area of \(7.0 \times 10^{-4} \mathrm{m}^{2},\) and a Young's modulus of \(1.1 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .\) By how much does the thighbone compress when the man is standing on both feet?

Short Answer

Expert verified
Question: Calculate the compression in length experienced by a man's thighbone when he is standing on both feet, given the following information: the man's mass is 91 kg, the length of the thighbone is 0.50 m, the cross-sectional area of the thighbone is \(7.0 \times 10^{-4} \mathrm{m}^{2}\), and the Young's modulus of the thighbone is \(1.1 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}\). Answer: The thighbone compresses by approximately \(2.898 \times 10^{-5}\) meters when the man is standing on both feet.

Step by step solution

01

Calculate the weight of the person

To calculate the weight of the person, we can use the formula: \(W = m \times g\) where W is the weight, m is the mass of the person (91 kg), and g is the acceleration due to gravity (9.81 m/s²). \(W = 91 \mathrm{kg} \times 9.81 \mathrm{m/s²} = 892.71 \mathrm{N}\)
02

Calculate the force exerted on one leg

The force exerted on one leg is half of the person's weight as the weight is equally distributed on both legs. Hence, the force on one leg is: \(F = \frac{W}{2} = \frac{892.71 \mathrm{N}}{2} = 446.355 \mathrm{N}\)
03

Calculate the stress

Stress is calculated by dividing the force exerted on the bone by its cross-sectional area. Here, Stress = \(\frac{F}{A}\) Stress = \(\frac{446.355 \mathrm{N}}{7.0 \times 10^{-4} \mathrm{m}^{2}} = 637650 \mathrm{N/m^{2}}\)
04

Calculate the strain

Using the formula for Young's modulus, we have: \(Y = \frac{\text{Stress}}{\text{Strain}}\) Here, we want to find the strain, so rearrange the formula: Strain = \(\frac{\text{Stress}}{Y}\) Strain = \(\frac{637650 \mathrm{N/m^{2}}}{1.1 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}} = 5.796818 \times 10^{-5}\)
05

Calculate the compression

Now that we have the strain, we can calculate the compression in the thighbone. Strain is the ratio of the compression (ΔL) to the initial length (L). Strain = \(\frac{\Delta L}{L}\) Here, we want to find the compression, so rearrange the formula: \(\Delta L = \text{Strain} \times L\) \(\Delta L = 5.796818 \times 10^{-5} \times 0.50 \mathrm{m} = 2.898409 \times 10^{-5} \mathrm{m}\) The thighbone compresses by approximately \(2.898 \times 10^{-5}\) meters when the man is standing on both feet.

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Most popular questions from this chapter

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