∙Achilles tendon. The Achilles tendon, which connects the...

$\bullet$ Achilles tendon. The Achilles tendon, which connects the calf muscles to the heel, is the thickest and strongest tendon in the body. In extreme activities, such as sprinting, it can be subjected to forces as high as 13 times a person's weight. According to one set of experiments, the average area of the Achilles tendon is $78.1 \mathrm{mm}^{2},$ its average length is $25 \mathrm{cm},$ and its average Young's modulus is 1474 MPa. (a) How much tensile stress is required to stretch this muscle by 5.0$\%$ of its length? (b) If we model the tendon as a spring, what is its force constant? (c) If a 75 kg sprinter exerts a force of 13 times his weight on his Achilles tendon, by how much will it stretch?

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Key Concepts

Spring Constant

The spring constant is a measure of a spring’s resistance to deformation, defined as the ratio of the force applied to the displacement produced. In the context of materials, this concept is extended by relating geometric dimensions and material properties, such as Young's modulus, to determine an effective force constant.

Hooke's Law

Hooke's Law states that, within the elastic limit, the deformation of an object is directly proportional to the force applied to it. This principle is fundamental in understanding the behavior of elastic materials and forms the basis for modeling them as springs.

Tensile Stress and Strain

Tensile stress is defined as the force applied per unit area on a material, while tensile strain is the relative change in length (deformation) experienced by the material. These concepts form the basis for understanding how materials respond to loads, particularly within their elastic limits.

Young's Modulus

Young's Modulus is a material property that quantifies the stiffness of a material by relating tensile stress to tensile strain in the elastic region. It provides a measure of how much a material will deform under a given load and is essential in calculations involving elastic deformation.

Transcript

00:01 In part a, we want to figure out what the stress is.

00:04 So recall that the young's modulus is defined as stress over strain.

00:11 So stress over strain.

00:15 And now we can determine that a 5 % elongation, which is given in the problem, 5 % elongation implies that delta l over l0 is equal to 0 .05, since delta l is the change in length and l0 is the original length.

00:35 So 5 % elongation implies that the ratio is equal to this.

00:42 And so this is just the strain.

00:45 And we know young's modulus, so we can find the stress by solving for it.

00:49 So stress is equal to young's modulus times strain.

00:54 And this is equal to the strain.

00:57 And we're given young's modulus.

00:58 So we can solve this to get 7 .4 times 10 to the 7 pascals.

01:06 And that's the answer to part a.

01:11 In part b, we want to figure out what the spring constant is if this can be modeled as a spring.

01:17 And so now we recall that the stress is also written as ft over the cross -sectional area.

01:26 We can solve this for ft since we just solved for the stress and we know a.

01:31 And we get that ft is equal to 5 .76 times 10 to the third newton's.

01:40 This will be the force that we plug in into hook's law in order to find the spring constant...

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