Question

An approximately round tendon that has an average diameter of \(3.5 \mathrm{~mm}\) and is \(15 \mathrm{~cm}\) long is found to stretch \(0.37 \mathrm{~mm}\) when acted on by a force of \(13.4 \mathrm{~N}\). Calculate Young's modulus for the tendon.

Short answer

Question: Calculate the Young's modulus for a tendon with an average diameter of 3.5 mm, a length of 15 cm, subjected to a force of 13.4 N that stretches it by 0.37 mm. Answer: To calculate the Young's modulus for the tendon, follow these steps: 1. Find the cross-sectional area of the tendon: A = π(1.75)^2 mm² 2. Calculate stress on the tendon: stress = 13.4 / (π(1.75)^2) N/mm² 3. Calculate strain on the tendon: strain = 0.37 / 150 4. Calculate Young's modulus of the tendon: Y = (13.4 / (π(1.75)^2)) / (0.37 / 150) N/mm² By solving for Y, you will find the Young's modulus for the tendon.

Step-by-step solution

Find the cross-sectional area of the tendon.

Since the tendon is approximately round, we can use the formula for the area of a circle to calculate its cross-sectional area. $$A = \pi r^2$$ where A is the cross-sectional area and r is the radius of the tendon. We are given the diameter, which is \(3.5 \mathrm{~mm}\), so the radius will be half of that. $$r = \frac{3.5}{2} \mathrm{~mm} = 1.75 \mathrm{~mm}$$ Now we can calculate the cross-sectional area. $$A = \pi (1.75)^2 \mathrm{mm^2}$$

Calculate stress on the tendon.

Stress is defined as the force acting on the tendon divided by its cross-sectional area. $$stress = \frac{F}{A}$$ where F is the force acting on the tendon, \(13.4 \mathrm{~N}\), and A is the cross-sectional area we calculated earlier. $$stress = \frac{13.4}{\pi(1.75)^2} \mathrm{\frac{N}{mm^2}}$$

Calculate strain on the tendon.

Strain is defined as the change in length divided by the original length of the tendon. $$strain = \frac{\Delta L}{L}$$ where \(\Delta L\) is the change in length, \(0.37 \mathrm{~mm}\), and L is the original length, \(15 \mathrm{~cm} = 150 \mathrm{~mm}\). $$strain = \frac{0.37}{150}$$

Calculate Young's modulus of the tendon.

We can now use the stress and strain values to find Young's modulus. $$Y = \frac{stress}{strain}$$ $$Y = \frac{\frac{13.4}{\pi(1.75)^2}}{\frac{0.37}{150}} \mathrm{\frac{N}{mm^2}}$$ Now, simply solve for Y to obtain the Young's modulus for the tendon.