Question

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

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

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.

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 \(\sigma\).

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: \[L_0 A_0 = L_1 A_1\]

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: \[\epsilon_T = \frac{D_1 - D_0}{D_0}\]

Write the equation for Poisson's ratio

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

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: \[\frac{L_1 - L_0}{L_0} = -\frac{L_0}{L_1} \times \sigma\]

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.

Key concepts

Elasticity

Elasticity is a key property of materials that describes their ability to return to their original shape after being deformed. When a material like a wire is stretched or compressed, it can either revert to its initial form or stay altered once the force is removed. This is determined by the material's elastic limit. Within this limit, the wire or other materials are seen as elastic.
In scenarios involving wires, including this exercise, elasticity becomes crucial because it dictates how the wire responds under stress. Elasticity also connects directly to concepts like Poisson's Ratio, which helps us understand how materials behave in different dimensions when under stress.

Volume Conservation

Volume conservation is a principle stating that the volume of an object remains constant, even when its shape changes due to applied forces. In this problem, the wire's volume does not change even though its length changes when stretched.
This concept is essential because it helps in deducing the Poisson's ratio of a material. When a wire stretches and its volume remains the same, it implies that as the wire gets longer, its cross-sectional area reduces. Mathematically, this means that the product of the initial and final dimensions remains the same.

• Initial volume: Given by initial length times cross-sectional area.
• Final volume: Again, length times cross-sectional area, remaining constant.

This unmoving volume helps us explore relationships between length change, area change, and material properties such as Poisson's ratio.

Longitudinal Strain

Longitudinal strain occurs when there's a change in length of a material subjected to stress. It's defined as the ratio of the change in length to the original length of the material. This simple formula, \(\epsilon_L = \frac{L_1 - L_0}{L_0}\), is significant because it quantifies deformation in materials, especially when combined with the aspect of Poisson's ratio.
In our exercise, understanding longitudinal strain is important as it relates to the wire's stretching. A material's response to longitudinal strain can affect its cross-section and thus influence volume conservation. Longitudinal strain, paired with transverse strain, reveals critical insights into how materials contract or expand in different directions under stress.