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Chapter 7: Problem 888

A copper wire and a steel wire of same diameter and length are connected end to end a force is applied, which stretches their combined length by $1 \mathrm{~cm}$, the two wires will have......... (A) different stresses and strains (B) the same stress and strain (C) the same strain but different stresses (D) the same stress but different strains

Short Answer

Expert verified
(C) the same stress but different strains

Step by step solution

01

Understanding stress and strain

Stress is a measure of the force applied to an object relative to its cross-sectional area, and strain is a measure of the relative change in length of an object when subjected to stress. Mathematically, we can express stress as: \[ \text{stress} = \frac{\text{force}}{\text{area}} \] and strain as \[ \text{strain} = \frac{\text{change in length}}{\text{original length}}. \] In this case, we have two wires with the same diameter and length, so their initial cross-sectional areas will be equal.
02

Understanding Young's modulus

Young's modulus is a material property that describes the relationship between stress and strain for a material when undergoing elastic (reversible) deformation. For a given material, the Young's modulus is defined as the ratio of stress to strain: \[ E = \dfrac{\text{stress}}{\text{strain}}. \] In this exercise, we have two different materials (copper and steel), so they will have different Young's moduli (denoted by \(E_{\text{Cu}}\) and \(E_{\text{s}}\), respectively).
03

Applying force and calculating strain

When a force is applied, it stretches the combined length of the two wires by 1 cm. It means that the total strain, the sum of the individual strains of the copper and steel wires, will be: \[ \text{total strain} = \text{strain}_{\rm Cu} + \text{strain}_{\rm s}. \] Using the definition of strain, we can write: \[ \dfrac{\Delta L_{\text{total}}}{L} = \dfrac{\Delta L_{\text{Cu}}}{L_{\text{Cu}}} + \dfrac{\Delta L_{\text{s}}}{L_{\text{s}}}, \] where \(\Delta L_{\text{total}}\), \(\Delta L_{\text{Cu}}\), and \(\Delta L_{\text{s}}\) are the changes in lengths of the combined wires, the copper wire, and the steel wire, respectively, and \(L\) is the original length of the wires.
04

Calculating stress using Young's modulus

We can use Young's modulus to relate the stress and strain experienced by each material. For copper and steel, we can write: \[ \text{stress}_{\text{Cu}} = E_{\text{Cu}} \times \text{strain}_{\text{Cu}} \] and \[ \text{stress}_{\text{s}} = E_{\text{s}} \times \text{strain}_{\text{s}}. \]
05

Determining the relationship between stress and strain

Now we have all the information needed to determine the relationship between the stress and strain experienced by copper and steel wires. Recall that the original cross-sectional areas are equal for both wires. Since the same force is applied to both materials, the stress in each material will also be equal: \[ \text{stress}_{\text{Cu}} = \text{stress}_{\text{s}}. \] However, since copper and steel have different Young's moduli, their strains will be different. Therefore, we can conclude that the copper and steel wires will have: (C) the same strain but different stresses

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Most popular questions from this chapter

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