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Biceps Muscle. 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 blength 0.200 \(\mathrm{m}\) m and cross- sectional area 50.0 \(\mathrm{cm}^{2} .\)

Short Answer

Expert verified
Young's modulus is 33,333.33 Pa under relaxed and 666,666.67 Pa under tensed conditions.

Step by step solution

01

Understanding Young's Modulus Equation

Young's modulus is a measure of the stiffness of a material. For a uniform cylinder, it is given by the formula:\[ Y = \frac{F}{A} \times \frac{L_0}{\Delta L} \]where \( F \) is the force applied, \( A \) is the cross-sectional area, \( L_0 \) is the original length, and \( \Delta L \) is the change in length (elongation).
02

Convert Units

Since the elongation \( \Delta L \) is given in centimeters, we need to convert it to meters to match the other units:\[ 3.0 \, \text{cm} = 0.03 \, \text{m} \]
03

Calculate Young's Modulus for Relaxed Muscle

Substituting the given values for the relaxed muscle: \( F = 25.0 \, \text{N} \), \( A = 50.0 \, \text{cm}^2 = 0.005 \, \text{m}^2 \), \( L_0 = 0.200 \, \text{m} \), \( \Delta L = 0.03 \, \text{m} \) into the formula:\[ Y_{\text{relaxed}} = \frac{25.0}{0.005} \times \frac{0.200}{0.03} = 33333.33 \, \text{Pa} \]
04

Calculate Young's Modulus for Tensed Muscle

Substituting the given values for the tensed muscle: \( F = 500 \, \text{N} \), \( A = 0.005 \, \text{m}^2 \), \( L_0 = 0.200 \, \text{m} \), \( \Delta L = 0.03 \, \text{m} \) into the formula:\[ Y_{\text{tensed}} = \frac{500}{0.005} \times \frac{0.200}{0.03} = 666666.67 \, \text{Pa} \]

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Key Concepts

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

Elasticity in Muscles
Elasticity refers to the ability of a material to return to its original shape after being stretched or compressed. Muscles, especially in humans, exhibit elasticity, which allows them to stretch and contract as needed. When you move, your muscles experience force and deformation. This elasticity helps muscles absorb the energy when stretched and then use it efficiently when contracting back.
In the context of the exercise, a biceps muscle demonstrates elasticity as it elongates under different forces. When the biceps elongate by 3.0 cm with a force of 25.0 N in a relaxed state, and with 500 N under maximum tension, it shows how it behaves at different levels of stress. This elasticity is crucial for flexibility, strength, and overall muscle function in movements and postures.
Force and Deformation
Force and deformation are key concepts in understanding how materials like muscles react to stress. Force refers to any interaction that, when unopposed, changes the motion of an object. Deformation, on the other hand, refers to the change in shape or size of an object due to applied force.
The biceps muscle in the exercise experiences deformation when a force is applied, elongated by 3.0 cm both in the relaxed and tensed states. The amount of deformation depends on the amount of force applied to the muscle. With greater force, the muscle undergoes more stress, but its ability to return to its original shape (elasticity) is a measure of its strength and functionality.
Physics of Materials
The physics of materials involves studying how materials behave under various conditions of stress or force. Known as material science, this area explores different properties, such as Young's Modulus, elasticity, tensile strength, and more.
Young's Modulus, which is discussed in the exercise, is a fundamental property that measures the stiffness of a material. By examining Young's Modulus, we understand how much a material will deform under a particular load. It is described by the formula \( Y = \frac{F}{A} \times \frac{L_0}{\Delta L} \).
  • \( F \): Force applied to the material.
  • \( A \): Cross-sectional area through which the force is applied.
  • \( L_0 \): Original length of the material before deformation.
  • \( \Delta L \): Change in length after force is applied.
The varied Young's Modulus values for the biceps muscle under relaxed and tensed states illustrate that muscle stiffness changes under different conditions. Such insights are essential in fields like biomechanics and rehabilitation, where understanding material properties can aid in developing treatments and training regimes.

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Most popular questions from this chapter

Flying Buttress. (a) A symmetric building has a roof sloping upward at \(35.0^{\circ}\) above the horizontal on each side. If each side of the uniform roof weighs \(10,000 \mathrm{N}\) , find the horizontal force that this roof exerts at the top of the wall, which tends to push out the walls. Which type of building would be more in danger of collapsing: one with tall walls or one with short walls? Explain. (b) As you saw in part (a), tall walls are in danger of collapsing from the weight of the roof. This problem plagued the ancient builders of large structures. A solution used in the great Gothic cathedrals during the 1200 s was the flying buttress, a stone support running between the walls and the ground that helped to hold in the walls. A Gothic church has a uniform roof weighing a total of \(20,000 \mathrm{N}\) and rising at \(40^{\circ}\) above the horizontal at each wall. The walls are 40 \(\mathrm{m}\) tall, and a flying buttress meets each wall 10 \(\mathrm{m}\) below the base of the roof. What horizontal force must this flying buttress apply to the wall?

A \(240-\mathrm{kg}, 50.0 \mathrm{cm}\) -long uniform bar has a small \(1.10-\mathrm{kg}\) mass glued to its left end and a small \(2.20-\mathrm{kg}\) mass glued to the other cad. You want to balance this system horizontally on a fulcrum placed just under its center of gravity. How far from the left end should the fulcrum be placed?

A petite young woman distributes her 500 \(\mathrm{N}\) weight equally over the heels of her high-heeled shoes. Each heel has an area of 0.750 \(\mathrm{cm}^{2}\) (a) What pressure is exerted on the floor by each heel? (b) With the same pressure, how much weight could be supported by two flat- bottomed sandals, each of area 200 \(\mathrm{cm}^{2} ?\)

A diving board 3.00 \(\mathrm{m}\) long is supported at a point 1.00 \(\mathrm{m}\) from the end, and a diver weighing 500 \(\mathrm{N}\) stands at the free end \((\text { (Fig. } 11.24) .\) The diving board is of uniform cross section and weighs 280 \(\mathrm{N}\) . Find (a) the force at the support point and (b) the force at the left-hand end.

A specimen of oil having an initial volume of 600 \(\mathrm{cm}^{3}\) is subjected to a pressure increase of \(3.6 \times 10^{5} \mathrm{Pa},\) and the volume is found to decrease by 0.45 \(\mathrm{cm}^{3} .\) What is the bulk modulus of the material? The compressibility?

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