Problem 56 - College Physics

Question

\(\bullet\) Heart repair. 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 \(\mathrm{cm}\) strip of the donated aorta reveal that it stretches 3.75 \(\mathrm{cm}\) when a 1.50 \(\mathrm{N}\) pull is exerted on it. (a) What is the force constant of this strip of aortal material? (b) If the maxi- mum distance it will be able to stretch when it replaces the aorta in the damaged heart is \(1.14 \mathrm{cm},\) what is the greatest force it will be able to exert there?

Step-by-Step Solution

Answer

a) 40.0 N/m, b) 0.456 N.

Step 1: Understand the Given Information

We have a 16.0 cm strip of aortal material that stretches 3.75 cm when a force of 1.50 N is applied. We need to find the force constant (or spring constant) for part (a).

Step 2: Recall Hooke's Law

Hooke's Law states that the force exerted by a stretchable material like a spring or an elastic strip is directly proportional to the displacement (stretch), formulated as \( F = kx \), where \( F \) is the force, \( k \) is the force constant, and \( x \) is the displacement.

Step 3: Calculate the Force Constant

From Hooke's Law, \( k = \frac{F}{x} \). Here, \( F = 1.50 \) N and \( x = 3.75 \) cm or 0.0375 m (since we should convert to meters for standard SI units). Thus, \( k = \frac{1.50}{0.0375} \approx 40.0 \text{ N/m} \).

Step 4: Part (a) Solution Conclusion

The force constant of the aortal material is 40.0 N/m.

Step 5: Determine Maximum Force Exertion

Now, for part (b), we need to calculate the maximum force exerted when the material stretches 1.14 cm. Using Hooke's Law again, and now \( x = 1.14 \) cm or 0.0114 m, the force is \( F = kx = 40.0 \times 0.0114 \approx 0.456 \text{ N} \).

Step 6: Part (b) Solution Conclusion

The greatest force the material will be able to exert in the damaged aorta location is approximately 0.456 N.

Key Concepts

Understanding Force Constant in Hooke's Law

Hooke's Law is a fundamental principle in physics that describes how materials respond to applied forces. When a material like a spring or an elastic strip is stretched, it exerts a force proportional to the amount of stretch. This is captured by the formula: \[ F = kx \] Here, \( F \) stands for the force applied, \( k \) is the force constant (also known as the spring constant), and \( x \) represents the stretch or displacement.The force constant \( k \) is a measure of how stiff or strong a material is in resisting deformation. The higher the value of \( k \), the less the material stretches for a given force. This constant is key for understanding material behavior under stress, such as how much a strip of aortal material can stretch or withstand forces, which is crucial in applications like medical surgeries where precision and reliability are paramount.

Exploring the Elasticity of Materials

Elasticity refers to a material's ability to return to its original shape and size after being deformed by an external force. In the context of physics, it's an important characteristic that determines how well materials can withstand stress and then recover. Key points about elasticity: - Elastic materials store energy when they are either stretched or compressed and release this energy when returning to their original shape. - The range over which a material remains elastic is called the elastic limit, beyond which permanent deformation occurs. - For the aortal material discussed in the exercise, understanding its elasticity helps ensure it functions properly in a biological setting without permanent stretch or damage. Measuring a material's elasticity involves observing how it reacts to forces, calculated through constants such as the force constant discussed in Hooke’s Law.

Spring Constant and Its Importance

The spring constant, symbolized as \( k \), is a vital concept in understanding how materials respond to forces. It essentially quantifies the stiffness of a spring or elastic material. Hooke’s Law uses the spring constant to establish a relationship between force and displacement in the formula \( F = kx \).Here's why the spring constant is important:- **Stiffness Measurement:** The spring constant measures how much force is needed to stretch or compress a material by a unit distance. Larger values of \( k \) indicate a stiffer material. - **Applications:** Knowing the spring constant is essential when designing anything that involves elasticity, like cardiovascular stents, where specific stretch capacities are required to accommodate varying pressures without damage.- **Predictions:** Engineers and scientists use the spring constant to predict how materials will behave under different conditions, ensuring safe and effective applications in technology and medicine. In the context of the heart repair scenario, the spring constant gives a detailed understanding of what to expect from the aortal material's performance.

Steps for Physics Problem Solving

Problem solving in physics can be methodical and rewarding if approached with a structured strategy. Here are some steps to effectively approach problems like those involving Hooke's Law and elasticity: - **Identify Known and Unknown Values:** Begin by clearly listing what information is given and what needs to be found. For instance, you're given a force and displacement and need to find the force constant. - **Recall Relevant Formulas:** Identify the relevant physics equations, such as Hooke’s Law, that relate the known values to the unknowns. - **Convert Units Appropriately:** Always ensure that units are converted to the standard system to avoid discrepancies in calculations. For example, converting centimeters to meters when using the International System of Units. - **Substitute Values and Solve:** Plug in the known values into the equations, and solve for the unknown variable. - **Verify and Reflect:** Check your answer for consistency with the problem context and reflect on the problem-solving process to reinforce understanding and skills. This strategy not only aids in arriving at the correct solutions but also helps develop deeper insights into physics concepts.