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Chapter 13: Problem 66

An approximately round tendon that has an average diameter of \(8.5 \mathrm{~mm}\) and is \(15 \mathrm{~cm}\) long is found to stretch \(3.7 \mathrm{~mm}\) when acted on by a force of \(13.4 \mathrm{~N}\). Calculate Young's modulus for the tendon.

Short Answer

Expert verified
To calculate the Young's modulus of a tendon with a diameter of 8.5 mm, a length of 15 cm, and an elongation of 3.7 mm when a force of 13.4 N is applied, we followed these steps: 1. Calculate the cross-sectional area: A = 56.75π mm². 2. Calculate the stress: Stress = 76.11 N/m². 3. Calculate the strain: Strain = 0.0247. 4. Calculate Young's modulus: Y = 3081.0 N/m² or 3.08 kN/m². Therefore, the Young's modulus of the tendon is approximately 3.08 kN/m².

Step by step solution

01

Calculate the Cross-Sectional Area

To find the stress, we need to calculate the tendon's cross-sectional area. Since the tendon is approximately round, we can use the formula for the area of a circle, which is \(A = \pi r^2\), where \(r\) is the radius. Given the diameter of \(8.5\mathrm{~mm}\), we can find the radius and then the area. First, find the radius: \(r = \frac{\text{diameter}}{2} = \frac{8.5\mathrm{~mm}}{2} = 4.25\mathrm{~mm}\) Next, calculate the cross-sectional area using the radius: \(A = \pi r^2 = \pi (4.25\mathrm{~mm})^2 = 56.75\pi\mathrm{~mm^2}\)
02

Calculate the Stress

Stress is the force acting on a material divided by its cross-sectional area. Using the force of \(13.4 \mathrm{~N}\) and the calculated cross-sectional area, we can find the stress: Stress = \(\frac{\text{Force}}{\text{Area}} = \frac{13.4 \mathrm{~N}}{56.75\pi\mathrm{~mm^2}} = \frac{13.4 \mathrm{~N}}{56.75\pi\mathrm{~mm^2}} \times \frac{1\mathrm{~N/m^2}}{1\mathrm{~N/mm^2}} = 76.11\mathrm{~N/m^2}\)
03

Calculate the Strain

Strain is the change in length of a material divided by its original length. Using the given elongation of \(3.7 \mathrm{~mm}\) and the original length of \(15\mathrm{~cm}\), we can find the strain: Strain = \(\frac{\Delta L}{L} = \frac{3.7\mathrm{~mm}}{15\mathrm{~cm}} = \frac{3.7\mathrm{~mm}}{150\mathrm{~mm}} = 0.0247\)
04

Calculate Young's Modulus

Now that we have values for stress and strain, we can calculate Young's modulus using the formula \(Y = \frac{\text{Stress}}{\text{Strain}}\): \(Y = \frac{76.11\mathrm{~N/m^2}}{0.0247} = 3081.0\mathrm{~N/m^2}\) or \(3.08\mathrm{~kN/m^2}\) The Young's modulus of the tendon is approximately \(3.08\mathrm{~kN/m^2}\).

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