Question

A wire of length \( L \), mass \( M \), density \( \rho \), radius \( R \) is stretched by a certain load. If \( r \) and \( \ell \) are the changes in radius and length respectively, then Poisson's ratio is

• A. \( \frac{Mr}{\pi R^3 \rho \ell} \)
• B. \( \frac{Mr}{\pi R^2 \rho \ell} \)
• C. \( \frac{Mr^2}{\pi R^3 \rho \ell} \)
• D. \( \frac{2 Mr^2}{\pi R^2 \rho \ell} \)

Hint:

Poisson's ratio is the ratio of the lateral strain to the longitudinal strain in a material under stress.

Answer 1

The Correct Option is A

Step 1: Understanding Poisson's ratio.
Poisson's ratio \( \nu \) is the ratio of lateral strain to longitudinal strain. The longitudinal strain is given by the change in length per original length, and the lateral strain is given by the change in radius per original radius. \[ \nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}} \] Step 2: Expression for Poisson's ratio.
In this case, we can express Poisson's ratio in terms of the material properties of the wire, its dimensions, and the changes in radius and length. Using the given data, the Poisson's ratio can be written as: \[ \nu = \frac{Mr}{\pi R^3 \rho \ell} \] Step 3: Conclusion.
Thus, Poisson's ratio is \( \frac{Mr}{\pi R^3 \rho \ell} \), which corresponds to option (A).