Ch. 12 - Static Equilibrium; Elasticity and Fracture

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo

Giancoli Douglas 5th edition

Ch. 12 - Static Equilibrium; Elasticity and Fracture

Problema 42

Capitolo 12, Problema 42

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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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.

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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.

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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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