### Summary

Students are introduced to Hooke's law as well as stress-strain relationships. First they learn the governing equations, then they work through several example problems, first individually, then as a class. Through the lesson's two-part associated activity, students 1) explore Hooke's law by experimentally determining an unknown spring constant, and then 2) apply what they've learned to create a strain graph depicting a tumor using Microsoft Excel®. After the activities, the lesson concludes with a stress-strain quiz to assess each student's comprehension of the concepts.

### Engineering Connection

### Educational Standards

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### Pre-Req Knowledge

### Learning Objectives

- Explain the stress and strain concepts and the relationship between them.
- Explain Hooke's law and apply it to analyze springs.
- Use Microsoft Excel® to make a simple strain plot.
- Relate stress and strain to the unit's engineering challenge.

### Introduction/Motivation

### Lesson Background and Concepts for Teachers

Legacy Cycle Information

*Research and Revise*phase of the legacy cycle. Students begin to learn the basic concepts required for creating a strain graph to depict cancerous tissue. Following this lesson, have students revise their initial thoughts and at the conclusion of the associated activity, students should have the skills necessary to

*Go Public*with a solution. But before

*Going Public*, have students complete the Stress, Strain and Hooke's Law Quiz as part of the

*Test Your Mettle*phase of the legacy cycle. The quiz serves as a formative assessment while the next lesson's

*Go Public*phase provides a summative assessment.

Lecture Information

**F= -k * Δx**where

**Δx**is the distance a spring has been stretch,

**F**is the restoring force exerted by the spring and

**k**is the spring constant which characterizes elastic properties of the spring's material. This law is valid within the elastic limit of a linear spring, when acting along a frictionless surface.

**σ = E* ε**. We will now explore the measures of stress and strain.

**σ = F/A**where average stress,

**σ**, equals force,

**F**, acting over area,

**A**. The SI unit for stress is pascals (Pa) which is equal to 1 Newton per square meter. The Psi is an alternative unit which expresses pounds per square inch. The units of stress are equal to the units of pressure which is also a measure of force per unit area.

**σ = E* ε**, where

**σ**represents stress,

**ε**represents strain and

**E**represents Young's modulus of elasticity. Using this means of inferring stress, strain is a geometrical measure of deformation and Young's modulus is a measure used to characterize the stiffness of an elastic material. Strain does not carry a unit but the units of Young's modulus are Pa.

**ε = Δ**

*l/l*_{0}where strain,

**ε**, is change in

**divided by initial length ,**

*l***.**

*l*_{0}*You are required to SHOW ALL WORK. Useful constants that are provided in a table below. (assume given constants have three significant figures (SF). Please also note that the relationships we have just discussed are given below.*

- steel

- Young's module: 200x10
^{9}E(Pa)

- cast iron

- Young's module: 100x10
^{9}E(Pa)

- concrete

- Young's module: 20.0x10
^{9}E(Pa)

*F=m*a σ=F/A ε = Δl/l*_{0}σ = E* ε F= -k * Δx- A 3340 N ball is supported vertically by a 1.90 cm diameter steel cable. Assuming the cable has a length of 10.3 m, determine the stress and the strain in the cable.
- Consider an iron rod with a cross-sectional area of 3.81 cm2 that has a force of 66,700 N applied to it. Find the stress in the rod.
- A concrete post with a 50.8 cm diameter is supporting a compressive load of 8910 Newtons. Determine the stress the post is bearing.
- The concrete post in the previous problem has an initial height of 0.55 m. How much shorter is the post once the load is applied (in mm)?
- A construction crane with a 1.90 cm diameter cable has a maximum functioning stress of 138 MPa. Find the maximum load that the crane can endure.
- Consider Hooke's law as a simple proportionality where F is directly proportional to Δx. Therefore, we know the force stretching a spring is directly proportional to the distance the spring stretches. If 223 N stretches a spring 12.7 cm, how much stretch can we expect to result from a of 534 N?
- Figure 1 shows a column of fatty tissue, determine the strain in each of the three regions.

### Vocabulary/Definitions

stress: |
The physical pressure, pull or other force exerted on a system by another. A load, force, or system of forces producing a strain. The ratio of force to area. |

strain: |
Deformation of a body or structure as a result of an applied force. Stretch beyond the proper point or limit. |

radiologist: |
A medical specialist who examines photographs of tissues, organs, bones for use in the treatment of disease. |

### Associated Activities

- Applying Hooke's Law to Cancer Detection - Student groups explore Hooke's law by collecting displacement data for springs with unknown spring constants by adding various masses of known weight. After answering a series of application questions, they apply their new understanding to explore a tissue of known surface area. Then then apply the pertinent relationships to depict a cancerous tumor amidst normal tissue by creating a Microsoft Excel® graph.

### Attachments

### Assessment

Post-Introduction Assessment:

*Problem Set*: Have students complete the Stress, Strain and Hooke's Law Problem Set in class to gauge their comprehension. The final question of the problem set and the application questions from the associated activity serve as an assessment of students' understanding of the challenge. Use these questions as a means of testing whether students are applying their acquired knowledge toward solving the engineering challenge.

Post-Lesson Assessment:

*Quiz*: Administer the Stress, Strain and Hooke's Law Quiz as a formative post-lesson assessment, serving as part of the

*Test your Mettle*phase of the legacy cycle.

### References

Dictionary.com. Lexico Publishing Group,LLC. Accessed December 28, 2008. (Source of vocabulary definitions, with some adaptation) http://www.dictionary.com

### Contributors

Luke Diamond, Meghan Murphy

### Copyright

© 2013 by Regents of the University of Colorado; original © 2007 Vanderbilt Univerity

### Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University

### Acknowledgements

Last modified: August 31, 2015