A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a 16.016.016.0-cm strip of the donated aorta reveal that it stretches 3.753.753.75 cm when a 1.501.501.50-N pull is exerted on it. If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is 1.141.14 cm, what is the greatest force it will be able to exert there?

Ch 06: Work & Kinetic Energy

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 06: Work & Kinetic Energy

Problem 33b

Chapter 6, Problem 33b

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a $16.0$ -cm strip of the donated aorta reveal that it stretches $3.75$ cm when a $1.50$ -N pull is exerted on it. If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is $1.14$ cm, what is the greatest force it will be able to exert there?

Verified step by step guidance

Step 1: Understand the problem. The problem involves the elastic properties of a material, which can be analyzed using Hooke's Law. Hooke's Law states that the force exerted by a spring or elastic material is proportional to its extension, given by the formula:

F = k x

, where

F

is the force,

k

is the spring constant (or stiffness), and

x

is the extension.

Step 2: Calculate the spring constant

k

of the aortal material using the given data. From the problem, the force

F = 1.50

N and the extension

x = 3.75

cm (convert to meters:

x = 0.0375

m). Rearrange Hooke's Law to solve for

k

k = F / x

Step 3: Substitute the known values into the formula for

k

. This will give you the stiffness of the aortal material in units of N/m.

Step 4: Use the calculated spring constant

k

to determine the greatest force the material can exert when stretched by 1.14 cm (convert to meters:

x = 0.0114

m). Apply Hooke's Law again:

F = k x

Step 5: Substitute the value of

k

from Step 3 and the new extension

x = 0.0114

m into the formula to calculate the maximum force

F

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Key Concepts

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

Hooke's Law

Hooke's Law states that the force exerted by a spring (or elastic material) is directly proportional to the amount it is stretched or compressed, as long as the elastic limit is not exceeded. Mathematically, it is expressed as F = kx, where F is the force, k is the spring constant, and x is the displacement from the equilibrium position. This principle is crucial for understanding how materials behave under stress and is applicable to the aorta's elastic characteristics.

Elastic Limit

The elastic limit is the maximum extent to which a material can be deformed elastically (i.e., without permanent deformation). Beyond this limit, the material may undergo plastic deformation, meaning it will not return to its original shape. Understanding the elastic limit is essential for the surgeon to ensure that the aorta can withstand the forces it will encounter without failing.

Young's Modulus

Young's Modulus is a measure of the stiffness of a material, defined as the ratio of stress (force per unit area) to strain (proportional deformation in length). It provides insight into how much a material will stretch or compress under a given load. In this context, calculating Young's Modulus for the aorta material will help determine the maximum force it can exert when stretched to a specific distance.

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Textbook Question

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a $16.0$ -cm strip of the donated aorta reveal that it stretches $3.75$ cm when a $1.50$ -N pull is exerted on it. What is the force constant of this strip of aortal material?

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