Question

The poisson ratio for the material of a wire is $0.4$. When a force is applied on the wire, longitudinal strain is $\dfrac{1}{100}$. The percentage change in the radius of the wire is
(1). $0.1$
(2). $0.8$
(3). $0.4$
(4). $0.2$

Answer 1

Poisson’s ratio describes the deformations that take place in a body when a deforming force or stress is applied to it. It is the ratio of relative change in area to relative change in length. Longitudinal strain is the relative change in length. Substituting the corresponding values, we can calculate change in radius.

Formula used:
$\nu =\dfrac{\dfrac{\Delta d}{d}}{\dfrac{\Delta l}{l}}$
$strain=\dfrac{\Delta l}{l}$

Complete step-by-step solution:
The poisson’s ratio is the measure of deformation in a body when a force is applied to it. It is given by-
$\nu =-\dfrac{\dfrac{\Delta d}{d}}{\dfrac{\Delta l}{l}}$ - (1)
$\nu$ is the poisson’s ratio
Here, $\Delta d$ is the change in diameter while $d$is the initial diameter
$\Delta l$ is the change in length while $l$ is the initial length

The negative sign indicates that when there is an increase in length, the area decreases and vice versa.

Strain is the measure of deformation of a dimension of a body. Longitudinal strain is described as the ratio of change in length to the initial length. It is given by-
$strain=\dfrac{\Delta l}{l}$ - (2)

Therefore, using eq (1) and eq (2), we get,
$\nu =\dfrac{\dfrac{\Delta d}{d}}{strain}$

Given, longitudinal strain= $\dfrac{1}{100}$, $\nu =0.4$. Substituting in the above equation, we get,
$\begin{aligned} & \nu =\dfrac{\dfrac{\Delta d}{d}}{\dfrac{1}{100}} \\\ & \Rightarrow \nu =100\dfrac{\Delta d}{d} \\\ & \Rightarrow 0.4=100\dfrac{\Delta d}{d} \\\ & \therefore \dfrac{\Delta d}{d}=\dfrac{0.4}{100} \\\ \end{aligned}$
We know that-
$2r=d$

Substituting $d$ we get,
$\begin{aligned} & \dfrac{\Delta d}{d}=\dfrac{0.4}{100} \\\ & \Rightarrow \dfrac{\Delta r}{r}=\dfrac{0.4}{100} \\\ \end{aligned}$

For percentage change-
\eqalign{ & \Rightarrow \dfrac{{\Delta r}}{r} = \dfrac{{0.4}}{{100}} \cr & \Rightarrow \dfrac{{\Delta r}}{r}{\text{ }}\% {\text{ }} = \dfrac{{0.4}}{{100}} \times 100 \cr & \therefore \dfrac{{\Delta r}}{r}{\text{ }}\% {\text{ }} = 0.4 \cr}

Therefore, the percentage change in the radius is $0.4$. Hence the correct option is (3).

Note:
The force that causes deformations in a body is called stress. Stress is the force applied per unit area. Strain is dimensionless as it is the ratio between two same units. There are two other types of strain; shearing strain and volumetric strain. Shearing strain is the change in the angle between two lines which were initially perpendicular. Volumetric strain is the ratio of change in volume to the initial volume.