Ch 11: Equilibrium & Elasticity

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 11: Equilibrium & Elasticity

Problem 26

Chapter 11, Problem 26

Two circular rods, one steel and the other copper, are joined end to end. Each rod is 0.750 m long and 1.50 cm in diameter. The combination is subjected to a tensile force with magnitude 4000 N. For each rod, what are (a) the strain and (b) the elongation?

Verified step by step guidance

First, understand that strain is a dimensionless quantity defined as the change in length divided by the original length. It can be expressed as \( \text{strain} = \frac{\Delta L}{L_0} \), where \( \Delta L \) is the change in length and \( L_0 \) is the original length.

Next, recognize that the elongation \( \Delta L \) can be found using Hooke's Law, which relates the force \( F \), the original length \( L_0 \), the cross-sectional area \( A \), and Young's modulus \( E \) of the material: \( \Delta L = \frac{F L_0}{A E} \).

Calculate the cross-sectional area \( A \) of the rods using the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the rod. Given the diameter is 1.50 cm, convert it to meters and find the radius.

For each rod, use the given tensile force \( F = 4000 \) N, the calculated cross-sectional area \( A \), and the respective Young's modulus \( E \) for steel and copper to find the elongation \( \Delta L \) using the formula \( \Delta L = \frac{F L_0}{A E} \).

Finally, calculate the strain for each rod using the formula \( \text{strain} = \frac{\Delta L}{L_0} \), where \( \Delta L \) is the elongation found in the previous step and \( L_0 = 0.750 \) m is the original length of each rod.

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Key Concepts

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

Stress and Strain

Stress is the force applied per unit area on a material, while strain is the deformation or displacement it experiences due to this stress. Strain is a dimensionless quantity calculated as the change in length divided by the original length. Understanding these concepts is crucial for determining how materials respond to forces.

Young's Modulus

Young's Modulus, or the modulus of elasticity, is a measure of a material's ability to withstand changes in length when under lengthwise tension or compression. It is defined as the ratio of stress to strain in the linear elasticity regime of a uniaxial deformation. This property is essential for calculating the strain and elongation of materials when subjected to forces.

Material Properties of Steel and Copper

Steel and copper have distinct mechanical properties, including different Young's Moduli, which affect how they deform under stress. Steel typically has a higher Young's Modulus than copper, meaning it is stiffer and less prone to deformation. Understanding these properties is necessary to calculate and compare the strain and elongation in each rod when subjected to the same tensile force.

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