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Chapter 11: Problem 26

A relaxed biceps muscle requires a force of \(25.0 \mathrm{~N}\) for an elongation of \(3.0 \mathrm{~cm} ;\) the same muscle under maximum tension requires a force of \(500 \mathrm{~N}\) for the same elongation. Find Young's modulus for the muscle tissue under each of these conditions if the muscle is assumed to be a uniform cylinder with length \(0.200 \mathrm{~m}\) and cross- sectional area \(50.0 \mathrm{~cm}^{2}\).

Short Answer

Expert verified
The Young's modulus for the muscle tissue in the relaxed state is \( Y_r = \frac{25.0/5.0 \times 10^{-3}}{0.03/0.2} N/m虏 \), and when under maximum tension, it is \( Y_t = \frac{500/5.0 \times 10^{-3}}{0.03/0.2} N/m虏 \).

Step by step solution

01

Determine the change in length

The elongation is given as 3.0 cm, which needs to be converted to meters because the length is given in meters. So \( \Delta L = 3.0 cm = 0.03 m \).
02

Convert cross-sectional area to appropriate units

The cross-sectional area is given as 50.0 cm虏, which needs to be converted to square meters: \( A = 50.0 cm虏 = 5.0 \times 10^{-3} m虏 \).
03

Calculate Young's modulus for relaxed state

Using the formula for Young's modulus \( Y_r = \frac{F/A}{\Delta L/L} \), and inserting values \( F = 25.0 N \), \( A = 5.0 \times 10^{-3} m虏 \), \( \Delta L = 0.03 m \), \( L = 0.2 m \), we find: \( Y_r = \frac{25.0/5.0 \times 10^{-3}}{0.03/0.2} N/m虏 \).
04

Calculate Young's modulus for maximum tension

Similarly for the state of maximum tension, the force \( F = 500 N \). Other parameters remain the same: \( Y_t = \frac{500/5.0 \times 10^{-3}}{0.03/0.2} N/m虏 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Elasticity of Muscle Tissue
When we look into how the body functions, specifically how muscles work, understanding elasticity is crucial. Muscle tissue ability to return to its original length after being stretched is referred to as elasticity. This attribute is vital in everyday movements and feats of athleticism alike.

When a muscle such as the biceps is elongated by an external force, it experiences both stress and strain. Stress is the force applied to the muscle per unit area, while strain is the relative change in the muscle's length. Young's modulus, a measure of the stiffness of an elastic material, is determined by the ratio of stress to strain under elastic conditions.

By conducting the exercise as outlined, the elasticity of the muscle tissue in both a relaxed state and under maximum tension can be revealed. In this case, the difference in the forces required for the same elongation highlights the varying elastic properties of the muscle under different states of tension.
Stress and Strain
Stress and strain are two fundamental concepts in the study of the mechanics of materials, including biological tissues. Stress is defined as the internal force exerted by an object per unit area, and strain is the measure of deformation representing the displacement between particles in the material body relative to a reference length.

In the context of muscle tissue, when a force is applied causing it to stretch, stress and strain develop within the tissue. In our exercise example, a biceps muscle elongates under a specific force, indicating strain, and this force divided by the muscle's cross-sectional area represents stress. Young's modulus is precisely determined from these values and allows us to compare the mechanical properties of muscle tissue under different conditions of tension.

The calculation steps outlined in the provided solution are essential for finding Young's modulus which quantifies the muscle tissue's response to stress and strain, exemplifying the muscle's elasticity.
Mechanical Properties of Biological Materials
Biological materials, such as muscle tissue, have unique mechanical properties that allow them to perform complex functions. These materials are often viscoelastic, meaning their mechanical response includes both viscous and elastic characteristics. They can endure stretching, compressing, and bending while still being capable of returning to their original shape.

The study of Young's modulus in muscle tissues exemplifies the analysis of mechanical properties 鈥� it is a parameter that represents the stiffness or rigidity of the material. A low Young's modulus indicates a material that is easily deformed under stress, while a high modulus signifies a stiff material.

Understanding these properties has crucial implications in fields ranging from biomechanics to medical device design. In the textbook exercise, determining Young's modulus in a relaxed versus a tensioned state of muscle provides valuable insights into how the muscle's mechanical behavior changes under different physiological conditions.

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Most popular questions from this chapter

You are a construction engineer working on the interior design of a retail store in a mall. A 2.00 -m-long uniform bar of mass \(8.50 \mathrm{~kg}\) is to be attached at one end to a wall, by means of a hinge that allows the bar to rotate freely with very little friction. The bar will be held in a horizontal position by a light cable from a point on the bar (a distance \(x\) from the hinge) to a point on the wall above the hinge. The cable makes an angle \(\theta\) with the bar. The architect has proposed four possible ways to connect the cable and asked you to assess them: $$ \begin{array}{lllll} \text { Alternative } & \text { A } & \text { B } & \text { C } & \text { D } \\\ \hline x(\mathrm{~m}) & 2.00 & 1.50 & 0.75 & 0.50 \\ \theta(\text { degrees }) & 30 & 60 & 37 & 75 \end{array} $$ (a) There is concern about the strength of the cable that will be required. Which set of \(x\) and \(\theta\) values in the table produces the smallest tension in the cable? The greatest? (b) There is concern about the breaking strength of the sheetrock wall where the hinge will be attached. Which set of \(x\) and \(\theta\) values produces the smallest horizontal component of the force the bar exerts on the hinge? The largest? (c) There is also concern about the required strength of the hinge and the strength of its attachment to the wall. Which set of \(x\) and \(\theta\) values produces the smallest magnitude of the vertical component of the force the bar exerts on the hinge? The largest? (Hint: Does the direction of the vertical component of the force the hinge exerts on the bar depend on where along the bar the cable is attached?) (d) Is one of the alternatives given in the table preferable? Should any of the alternatives be avoided? Discuss.

A uniform \(300 \mathrm{~N}\) trapdoor in a floor is hinged at one side. Find the net upward force needed to begin to open it and the total force exerted on the door by the hinges (a) if the upward force is applied at the center and (b) if the upward force is applied at the center of the edge opposite the hinges.

A steel cable with cross-sectional area \(3.00 \mathrm{~cm}^{2}\) has an elastic limit of \(2.40 \times 10^{8} \mathrm{~Pa}\). Find the maximum upward acceleration that can be given a \(1200 \mathrm{~kg}\) elevator supported by the cable if the stress is not to exceed one-third of the elastic limit.

The left-hand end of a uniform rod of mass \(2.00 \mathrm{~kg}\) andlength \(1.20 \mathrm{~m}\) is attached to a vertical wall by a frictionless hinge. The rod is held in a horizontal position by an aluminum wire that runs between the right-hand end of the rod and a point on the wall that is above the hinge. The cross-sectional radius of the wire is \(2.50 \mathrm{~mm}\) and the wire makes an angle of \(30.0^{\circ}\) with the rod. (a) What is the length of the wire? (b) An object of mass \(90.0 \mathrm{~kg}\) is suspended from the right-hand end of the rod. What is the increase in the length of the wire when this object is added? In your analysis do you need to be concerned that the lengthening of the wire means that the rod is no longer horizontal?

A ladder carried by a fire truck is \(20.0 \mathrm{~m}\) long. The ladder weighs \(3400 \mathrm{~N}\) and its center of gravity is at its center. The ladder is pivoted at one end (A) about a pin (Fig. E11.7); ignore the friction torque at the pin. The ladder is raised into position by a force applied by a hydraulic piston at \(C\). Point \(C\) is \(8.0 \mathrm{~m}\) from \(A,\) and the force \(\overrightarrow{\boldsymbol{F}}\) exerted by the piston makes an angle of \(40^{\circ}\) with the ladder. What magnitude must \(\boldsymbol{F}\) have to just lift the ladder off the support bracket at \(B ?\) Start with a free-body diagram of the ladder.
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