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Chapter 7: Problem 933

There is no change in the volume of a wire due to change in its length on stretching. What is the possion's ratio of the material of the wire $\ldots \ldots \ldots$ (A) \(+0.5\) (B) \(-0.50\) (C) \(0.25\) (D) \(-0.25\)

Short Answer

Expert verified
The correct Poisson's ratio for the case of no change in volume of a wire due to change in its length on stretching is \(\sigma = 0\). However, this option is not available in the given choices. There might be an error in the provided choices or the question itself.

Step by step solution

01

Understand the given information

We are given that the volume of the wire does not change when its length changes. So, the product of the length, cross-sectional area, and Poisson's ratio should not change.
02

Represent the variables

Let the initial length of the wire be \(L_0\), the initial diameter be \(D_0\), and the volume be \(V_0\). Also, let the final length of the wire be \(L_1\), the final diameter be \(D_1\), and the Poisson's ratio be \(\sigma\).
03

Relate the volume to length and diameter

We know that the volume of the wire is the product of its length and cross-sectional area. Lets assume that the wire has a cylindrical shape, then the cross-sectional area could be represented as \(A_0 = \frac{\pi D_0^2}{4}\). Thus, we have: \[V_0 = L_0 A_0\] Since the volume of the wire does not change, its final volume is also \(V_0\). Therefore: \[V_0 = L_1 A_1\] Where \(A_1 = \frac{\pi D_1^2}{4}\). Now, we have: \[L_0 A_0 = L_1 A_1\]
04

Calculate longitudinal and transverse strains

Longitudinal strain is defined as the ratio of change in length to the original length: \[\epsilon_L = \frac{L_1 - L_0}{L_0}\] Transverse strain is defined as the ratio of change in diameter to the original diameter: \[\epsilon_T = \frac{D_1 - D_0}{D_0}\]
05

Write the equation for Poisson's ratio

Poisson's ratio is defined as the ratio of transverse strain to longitudinal strain: \[\sigma = -\frac{\epsilon_T}{\epsilon_L}\]
06

Express Poisson's ratio in terms of volume change

From steps 3 and 4, we have: \[\frac{L_1 - L_0}{L_0} = -\frac{D_1 - D_0}{D_0} \times \sigma\] Using the result from Step 3, we can write: \[\frac{L_1 - L_0}{L_0} = -\frac{L_0}{L_1} \times \sigma\]
07

Solve for Poisson's ratio

Now, we can rearrange the equation from Step 6 to find \(\sigma\): \[\sigma = -\frac{(L_1 - L_0)L_1}{L_0^2}\] Since there is no change in volume, we have: \[L_1 = L_0\] Therefore, Poisson's ratio is: \[\sigma = -\frac{(L_1 - L_0)L_0}{L_0^2} = -\frac{0}{L_0} = 0\] From the given choices, none of them match the correct value of Poisson's ratio. It's possible that there's a mistake in the given choices or the question is flawed.

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