A 15-cm-long tendon was found to stretch 3.7 mm by a force of 13.4 N. The tendon was approximately round with an average diameter of 8.5 mm. Calculate Young’s modulus of this tendon.

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Step 1: Recall the formula for Young's modulus, which is defined as $ E = \frac{\sigma}{\epsilon} $, where $ \sigma $ is the stress and $ \epsilon $ is the strain. Stress is given by $ \sigma = \frac{F}{A} $, and strain is given by $ \epsilon = \frac{\Delta L}{L_0} $.

Step 2: Calculate the cross-sectional area $ A $ of the tendon, assuming it is approximately circular. The formula for the area of a circle is $ A = \pi r^2 $, where $ r $ is the radius. Convert the diameter of 8.5 mm to meters ($ r = \frac{8.5}{2} \times 10^{-3} $ m) and substitute into the formula.

Step 3: Compute the strain $ \epsilon $ using the formula $ \epsilon = \frac{\Delta L}{L_0} $. Here, $ \Delta L $ is the elongation (3.7 mm converted to meters) and $ L_0 $ is the original length of the tendon (15 cm converted to meters).

Step 4: Calculate the stress $ \sigma $ using the formula $ \sigma = \frac{F}{A} $, where $ F $ is the applied force (13.4 N) and $ A $ is the cross-sectional area calculated in Step 2.

Step 5: Substitute the values of $ \sigma $ and $ \epsilon $ into the formula for Young's modulus $ E = \frac{\sigma}{\epsilon} $ to determine the modulus of elasticity for the tendon.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Young's Modulus

Young's modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It quantifies how much a material will deform under a given load, providing insight into its elastic properties. A higher Young's modulus indicates a stiffer material that deforms less under stress.

Tensile Stress

Tensile stress is the force applied per unit area of a material, typically measured in pascals (Pa). It is calculated by dividing the applied force by the cross-sectional area of the material. In the context of the tendon, it helps determine how much force is exerted on the tendon relative to its size.

Tensile Strain

Tensile strain is the measure of deformation representing the displacement between particles in a material when subjected to tensile stress. It is a dimensionless quantity calculated as the change in length divided by the original length. In this case, it indicates how much the tendon stretches relative to its original length when a force is applied.

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