Ch 11: Equilibrium & Elasticity

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 11: Equilibrium & Elasticity

Problem 27

Chapter 11, Problem 27

A circular steel wire 2.00 m long must stretch no more than 0.25 cm when a tensile force of 700 N is applied to each end of the wire. What minimum diameter is required for the wire?

Verified step by step guidance

First, identify the relevant formula for the problem. The formula for the elongation of a wire under tensile stress is given by:

\frac{ΔL}{L} = \frac{F}{A E}

, where

ΔL

is the change in length,

L

is the original length,

F

is the force applied,

A

is the cross-sectional area, and

E

is the Young's modulus of the material.

Next, rearrange the formula to solve for the cross-sectional area

A

A = \frac{F L}{E ΔL}

. This will allow you to find the area needed to ensure the wire stretches no more than the specified amount.

Substitute the known values into the equation. The force

F

is 700 N, the original length

L

is 2.00 m, and the change in length

ΔL

is 0.25 cm (which should be converted to meters). The Young's modulus

E

for steel is approximately 2.1 x 10¹¹ N/m².

Calculate the cross-sectional area

A

using the substituted values. This will give you the minimum area required to ensure the wire does not stretch beyond the specified limit.

Finally, use the formula for the area of a circle

A = \frac{π d^2}{4}

to solve for the diameter

d

. Rearrange the formula to find

d

and substitute the calculated area to find the minimum diameter required.

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Key Concepts

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

Young's Modulus

Young's Modulus is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It is a constant for a given material and is used to predict how much a material will deform under a given load. In this problem, it helps determine how much the steel wire will stretch when a force is applied.

Tensile Stress and Strain

Tensile stress is the force applied per unit area of a material, while tensile strain is the deformation or elongation per unit length. These concepts are crucial for understanding how materials respond to forces, and they are used to calculate the change in length of the wire when a specific force is applied.

Cross-sectional Area of a Wire

The cross-sectional area of a wire is important in determining how it will respond to tensile forces. It is calculated using the diameter of the wire, and a larger area means the wire can withstand more force without stretching. In this problem, finding the minimum diameter involves ensuring the wire's cross-sectional area is sufficient to limit its elongation under the given force.

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Textbook Question

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