Question

When a metal wire is stretched by a load, the fractional change in its value \[\dfrac{{\Delta V}}{V}\] is proportional to?
A. \[ - \dfrac{{\Delta l}}{l}\]
B. \[\dfrac{{\Delta l}}{l}\]
C. \[\sqrt {\dfrac{{\Delta l}}{l}} \]
D. None of the above.

Answer 1

Hint: We know that the work done per unit volume by the wire is given as:
\[\dfrac{{Stress \times Strain}}{2}\] which is equal to elastic potential energy i.e. \[\dfrac{{F \times L}}{2}\]. From the above two equations, we get the relation between the Volume and Length of material solid.


Formula used:
The volume $V$ can be given by the equation:
\[V = \dfrac{{\pi \times {d^2} \times l}}{4}\]
Here, $d$ is the diameter of the wire and $l$ is the length of the wire.

Complete step by step solution:
As we mentioned the above given formula i.e.
\[V = \dfrac{{\pi \times {d^2} \times l}}{4}\]
Further, this can be broken down into derivative forms as,
\[\dfrac{{\Delta V}}{V} = \dfrac{{2 \times \Delta d}}{d} + \dfrac{{\Delta l}}{l}\]
From the above, we can see the Poisson’s ratio and the above equation can be rewritten as:
\[\dfrac{{d \times \Delta V}}{V} = \dfrac{{(1 - 2\sigma \times \Delta l)}}{l}\]

Poisson’s ratio helps in the measurement of the deformation that occurs in a material. It occurs in the direction that is perpendicular to the direction of the applied force. Also, we know the relation between diameter and Poisson’s ratio:
\[\dfrac{{\Delta d}}{d} = \dfrac{{( - \sigma \times \Delta l)}}{l}\]
As per the above equation, we can clearly see that relationship between the Volume and length of the wire is given as:
\[\dfrac{{\Delta V}}{V} = \dfrac{{ - \Delta l}}{l}\]

Hence, option A is the correct answer

Note: With the help of the equation for elastic potential energy, we got the relationship between the \[\dfrac{{\Delta V}}{V} = \dfrac{{ - \Delta l}}{l}\] and make sure Poisson's ratio is mentioned for the specific material that has been asked for. Most materials have Poisson’s ratio that ranges from 0.0 to 0.5 , and materials that are soft, for example rubber, have a ratio of 0.5 . Solid substances on the other hand have Poisson’s ratio that ranges from 0.2 to 0.3.