If a wire having initial diameter of 2 mm produced the longitudinal strain of 0.1%, then the final diameter of wire will be (s = 0.5) (a) 2.002 mm (b) 1.998 mm (c) 1.999 mm (d) 2.001 mm

To find the final diameter of the wire after it experiences a longitudinal strain, we need to understand the relationship between longitudinal strain, diameter, and Poisson's ratio. In your case, the wire has an initial diameter of 2 mm and experiences a longitudinal strain of 0.1%. The Poisson's ratio (s) is given as 0.5. Let's break this down step by step.

Understanding Longitudinal Strain and Poisson's Ratio

Longitudinal strain (ε) is defined as the change in length (ΔL) divided by the original length (L). In this case, it is given as:

ε = 0.1% = 0.001 (as a decimal)

Poisson's ratio (s) is the ratio of the lateral strain (change in diameter) to the longitudinal strain. It is represented mathematically as:

s = - (lateral strain) / (longitudinal strain)

Given that the Poisson's ratio is 0.5, we can express the lateral strain as:

lateral strain = -s × longitudinal strain

Substituting the values, we have:

lateral strain = -0.5 × 0.001 = -0.0005

Calculating the Change in Diameter

To find the change in diameter (ΔD), we can relate lateral strain to the original diameter:

lateral strain = ΔD / D_initial

Rearranging this gives us:

ΔD = lateral strain × D_initial

Now, substituting the values:

ΔD = -0.0005 × 2 mm = -0.001 mm

Finding the Final Diameter

Now that we have the change in diameter, we can determine the final diameter:

D_final = D_initial + ΔD

Substituting in the values:

D_final = 2 mm - 0.001 mm = 1.999 mm

Conclusion

Therefore, the final diameter of the wire after undergoing a longitudinal strain of 0.1% is 1.999 mm. Among the choices provided, the correct answer is (c) 1.999 mm.