Name: Poisson's Ratio
Uploaded: 2019-07-19
Duration: 9 min 46 s
Description: In this video, let us discuss Poisson's ratio, the most interesting and important measure of a property of a material. Watch this to explore.

Question

Define longitudinal strain, stress and Young modulus, write their units.

Accepted Answer

Longitudinal strain:

Within the elastic limit, it is the ratio of change in length to the original length.

Longitudinal strain $= \frac{Change in length}{Original length} = \frac{∆ L}{L}$

The longitudinal strain is possible only in solids.

The longitudinal strain is a pure ratio, so it is a unitless quantity.

When an external or deforming force is applied to a body, it gets deformed.

The internal restoring force acting on the unit area of cross-section of a deformed body is called the stress

$stress = \frac{F}{A}$

Where the area of cross-section.

$F =$ Force

Unit of stress $= \frac{unit of foce}{unit of area} = \frac{N}{m^{2}}$

Dimensional formula $= s t r e s s = \frac{F}{A} = \frac{{MLT}^{- 2}}{L^{2}} = {ML}^{- 1} T^{- 2}$

Kinds of stress:

Stress is of four kinds:

(i) Tensile stress.

(ii) Compressional stress.

(iii) Tangential or shearing stress.

(iv) Hydrostatic stress.

(i) Tensile stress:

It is the restoring force developed per unit cross-sectional area of a body when the length of the body increases in the direction of the deforming force. It is also called longitudinal stress.

Tensile stress $= \frac{F}{A}$

(ii) Compressional stress:

It is the restoring force developed per unit cross-sectional area of the body when the body is compressed i.e., when its length decreases under the action of deforming force.

(iii) Tangential stress:

When a deforming force acts tangentially to the surface of a body, it produces a change in the shape of the body. The tangential force applied per unit area is equal to the tangential stress.

(iv) Hydrostatic stress:

It a body is subjected to a uniform is subjected force from all sides, then the corresponding static stress.

Young modulus of elasticity:

Within the elastic limit. The ratio of longitudinal stress to the longitudinal strain is called Young's modulus of the material of the wire

As shown in fig. suppose a wire of length $l$ and corss-sectional area A suffers an increase in length $∆ l$ under a force $F$ acting along its length $l$ . Then Young's Modulus is

$Y = \frac{Longitudinal stress}{Longitudinal strain}$

$Y = \frac{\frac{F}{A}}{\frac{∆ l}{l}} = \frac{F}{A} \cdot \frac{l}{∆ l}$

If the wire has a circular cross-section of radius $r$ then

$Y = \frac{F}{{πr}^{2}} \cdot \frac{l}{∆ l}$ $F = m g (2^{n d} law of Newton)$

or $Y = \frac{M g l}{{πr}^{2} l}$

If $l = 1 m$ , $A = 1 m^{2}$ and $∆ m = 1 m$ then $Y = F$ Thus, Young's modulus of elasticity is equal to the force required to extend a wire of unit amount i.e., the force required to double the length of the wire.

Unit and dimensions of Y S.I. Unit $= N / m^{2}$

$Y = \frac{F}{A} \cdot \frac{l}{∆ l} = \frac{{MLT}^{- 2}}{L^{2}} = {ML}^{- 1} T^{- 2}$

Define longitudinal strain, stress and Young modulus, write their units.

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