# Elasticity

Elasticity:

The property of matter by virtue of which it regains its original configuration after removing the deforming force is called elasticity. All objects in nature are elastic and no effects in nature are rigid or plastic.

Elastic after Effect:

The delay of material to regain the original position after removing the deforming force is called elastic after effect. All objects in nature exhibit thus property in some extent. But, quarty has less value of elastic after effect. So, the pointers of moving coil galvanometer or voltmeter are made of quarty but not of steel or aluminum.

Poisson’s ratio:

The ratio of lateral strain is to the longitudinal strain is called poisson's ratio and it is denoted by sigma'$\sigma$'. If $\beta$ be the lateral strain and $\alpha$ be the longitudinal strain, then,

$\alpha = \frac{\beta }{\sigma }$

$= \frac{{\frac{{\Delta {\rm{D}}}}{{\rm{D}}}}}{{\frac{{\rm{e}}}{{\rm{l}}}}}$

The Poisson's ratio has not units and dimensions because it is the ratio of two. The ratio of change in diameter is to the original diameter is called lateral strain whereas; the ratio is change in length as the original length.

Modulus:

It is defined as the ratio of the tangential stress to the shear strain, within the elastic limit. Thus

ɧ = ${\rm{tangential}}\frac{{{\rm{stress}}}}{{{\rm{shearing\:stress}}}}$

Compressibility:

Compressibility (C) is the reciprocal of bulk modulus of elasticity is called compressibility,

I.e.  ${\rm{C}} = \frac{1}{{\rm{K}}}$

Stress:

The deforming force per unit area of cross section is called stress. Since, deforming force and restoring force are equal in magnitude, we also can write as follows:

The restoring force per unit area of cross section area is also called stress. If 'F' be the restoring force acting on an area A, then

Stress = $\frac{{\rm{F}}}{{\rm{A}}}$

The S.I. unit of stress is Newton /m2 (${\rm{N}}/{{\rm{m}}^2}$

If the deforming force is acting perpendicular to an area then the stress produced is called normal stress. There are two systems of normal stress. They are:

Tensile Normal stress:

The tensile normal stress is that stress in which there is increase in length of the wire or object then it is called compressive normal stress.

Tangential stress:

If the deforming force is applied parallel to a surface, then the stress produced is called tangential stress. Generally, thus stress is used to just shift the area if the surface.

Hooke's law:

It states that, within the elastic limit, the deforming force is directly proportion to the extension produced i.e.

${\rm{F}} \propto {\rm{e}}$ ------ (i) $\to$ where F is the deforming force and e is the extension produced.

Hooke's law can be verified experimentally by using vernier apparatus as follows:

Now, two wires A and B are taken in which wire A is connected to main scale and wire B is connected to venire scale. The wire A is reference wire whereas the wire B is the experimental wire. Initially, equal weights are kept on scale pans S1 and S so that the both wires A and B become from kinks and then become ready for experiment. In this condition, the main scale reading and vernier scale reading l0 is found.

Now, a load of 0.5 kg is added on scale pan S1 and it is left for two minutes. After that main scale reading and venire scale reading are noted. The difference of this reading and previous reading gives the extension produced. This process is repeated by taking other weights of 1kg, 1.5kgs, 2 kgs....... and then the corresponding extension are noted.

Modulus of elasticity:

Within the elastic unit, it is found that the stress is directly proportional to the strain. i.e.

stress$\propto {\rm{strain}}$

or, $\frac{{{\rm{stress}}}}{{{\rm{strain}}}} = {\rm{constant}} = {\rm{E}}$

Where E is the modulus of elasticity and its unit is ${\rm{N}}/{{\rm{m}}^2}$.

Young’s modulus:

It is defined as the ratio of normal stress to the longitudinal strain, within the elastic limit. Thus,

${\rm{y}} = \frac{{{\rm{normal\: stress}}}}{{{\rm{longitudinal\: strain}}}}$

Bulk modulus:

It is defined as the ratio of normal stress to the volumetric strain, within the elastic limit. Thus,

${\rm{K}} = \frac{{{\rm{normal\: stress}}}}{{{\rm{volumetric\: strain}}}}$

Modulus of rigidity:

It is defined as the ratio of the tangential stress to the shear strain, within the elastic limit. Thus

ɧ = ${\rm{tangential}}\frac{{{\rm{stress}}}}{{{\rm{shearing\: stress}}}}$

The elastic energy stored in a stretched wire is = $\frac{1}{2}$ of the product of deforming force and the extension produced:

When a wire is stretched by a deforming force, it does work on the wire. This work is stored in the wire in the form of potential energy. The energy stored in a stretched wire due to the deforming force or restoring force is called elastic energy stored and the elastic energy stored in a wire can be found as follows

Let's take a wire and initial length l and area of cross section 'A' in which a deforming force 'F' is applied on it. This deforming force produces an extension 'x' in the wire. If 'y' be the young's modulus of elasticity of the material of the wire, then

y = $\frac{{{\rm{F*l}}}}{{{\rm{A*x}}}}$

or, F = $\frac{{{\rm{yAx}}}}{{\rm{l}}}$ ------(i)

If the deforming force is acting continuously on the wire, then, this force does work. If 'dx' be the small displacement or extension produced on the wire. By this, deforming force, then, the small work done by this deforming force is

dw = Fx dx

or, dw = $\frac{{{\rm{yAx}}}}{{\rm{l}}}{\rm{*dx}}$ --------(ii)

If the deforming force is acting continuously on the wire, then, it does a large amount of work like the small work. Then, the total work done in stretching the wire to an extension 'e' is obtained by collecting or integrating the small work as

w = $\mathop \smallint \limits_0^{\rm{e}} {\rm{dw}}$

or, w = $\mathop \smallint \limits_0^{\rm{e}} \frac{{{\rm{yAx*dx}}}}{{\rm{l}}}$

or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\mathop \smallint \limits_0^{\rm{e}} {\rm{xdx}}$

or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\left[ {\frac{{{{\rm{x}}^{1 + 1}}}}{{1 + 1}}} \right]_0^{\rm{e}}$

or, w = $\frac{{{\rm{yA}}}}{{\rm{l}}}\left[ {\frac{{{{\rm{x}}^2}}}{2}} \right]_0^{\rm{e}}$

or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}\left[ {{{\rm{x}}^2}} \right]_0^{\rm{e}}$

or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}\left[ {{{\rm{e}}^2} - 0} \right]$

or, w = $\frac{{{\rm{yA}}}}{{2{\rm{l}}}}{\rm{*}}{{\rm{e}}^2}$

or, w = $\frac{1}{2}{\rm{*}}\left( {\frac{{{\rm{yAe}}}}{{\rm{l}}}} \right){\rm{*l}}$

or, w = $\frac{1}{2}{\rm{*F*e}}$ ---------(iii)

The work done in equation (iii) is stored in the wire in the form of potential energy. So, the potential energy stored in a stretched wire is given by,

E = W = = $\frac{1}{2}{\rm{*F*e}}$

or, E = $\frac{1}{2}{\rm{*F*e}}$ ---------(iv)

Hence, the elastic energy stored in a stretched wire is = $\frac{1}{2}$ of the product of deforming force and the extension produced.

Go Top