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 16.016.0-cm strip of the donated aorta reveal that it stretches 3.753.75 cm when a 1.501.50-N pull is exerted on it. What is the force constant of this strip of aortal material?

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 33a

Chapter 6, Problem 33a

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?

Verified step by step guidance

Step 1: Understand the problem. The force constant (k) is a measure of the stiffness of the material and is defined by Hooke's Law: $ F = k \Delta x $, where $ F $ is the applied force, $ \Delta x $ is the extension, and $ k $ is the force constant. We need to calculate $ k $ using the given values.

Step 2: Identify the given values. The applied force $ F $ is 1.50 N, the original length of the strip is 16.0 cm, and the extension $ \Delta x $ is 3.75 cm. Note that $ \Delta x $ is the change in length, not the total length.

Step 3: Convert units if necessary. Since the force constant $ k $ is typically expressed in $ \text{N/m} $, convert the extension $ \Delta x $ from centimeters to meters. Use the conversion factor: $ 1 \, \text{cm} = 0.01 \, \text{m} $. Thus, $ \Delta x = 3.75 \, \text{cm} \times 0.01 \, \text{m/cm} = 0.0375 \, \text{m} $.

Step 4: Rearrange Hooke's Law to solve for $ k $. The formula becomes $ k = \frac{F}{\Delta x} $. Substitute the values of $ F $ and $ \Delta x $ into the equation.

Step 5: Perform the calculation. Divide the force $ F $ (1.50 N) by the extension $ \Delta x $ (0.0375 m) to find the force constant $ k $. The result will be in $ \text{N/m} $.

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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 applied, k is the spring constant (or force constant), and x is the displacement from the equilibrium position. This principle is fundamental in understanding how materials deform under stress.

Elasticity

Elasticity is the property of a material that enables it to return to its original shape after being deformed by an external force. It is quantified by the material's elastic modulus, which measures the stiffness of the material. In the context of the aorta, understanding its elasticity is crucial for determining how it will behave under physiological conditions and how it can withstand the forces exerted by blood flow.

Force Constant

The force constant, often denoted as k, is a measure of the stiffness of an elastic material. It is defined as the ratio of the force applied to the displacement produced, as described by Hooke's Law. A higher force constant indicates a stiffer material that requires more force to achieve the same amount of stretch, which is particularly important in medical applications where material properties can affect the success of surgical repairs.

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Textbook Question

A $6.0$ -kg box moving at $3.0$ m/s on a horizontal, frictionless surface runs into a light spring of force constant $75$ N/cm. Use the work–energy theorem to find the maximum compression of the spring.

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Textbook Question

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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. 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?

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Textbook Question

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Is it reasonable that a $30$ -kg child could run fast enough to have $100$ J of kinetic energy?

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