# Youngâ€™s modulus (Y)

Youngâ€™s modulus is defined as the ratio of normal stress to longitudinal strain within the limit of proportionality.

If force is applied on a wire of radius

*r*by hanging a weight of mass*M*, thenImportant Points

- If the length of a wire is doubled, then
*Y*= Stress[as strain = 1] - Increment in the length of wire:
*F*and*Y*are constant.]*L*/*r*^{2}, greater will be the elongation in the wire.

**Elongation in a wire by its own weight**The weight of the wire

*Mg*acts at the center of gravity of the wire so that the length of wire which is stretched will be

*L*/ 2.

âˆ´ Elongation, =

[as mass (

*M*) = volume (*AL*) Ã— density (*d*)]**Thermal stress**If a rod is fixed between two rigid supports, due to change in temperature its length will change and so it will exert a normal stress (compressive if temperature increases and tensile if temperature decreases) on the supports. This stress is called thermal stress (Fig. 11).

**Fig. 11**

As by definition, coefficient of linear expansion,

â‡’ | Thermal strain, | |

So | thermal stress = YÎ± Î”Î¸ |
[as Y = stress/strain] |

*YA*

*Î±*

*Î”*

*Î¸*

**Note**

**:**In case of volume expansion, thermal stress =

*K*

*Î³*Î”

*Î¸*, where

*K*= Bulk modulus and

*Î³*= coefficient of cubical expansion.

**Force between two rods**Two rods of different metals having the same area of cross section

*A*are placed end to end between two massive walls as shown in Fig. 12.

**Fig. 12**

The first rod has a length

*L*_{1}, coefficient of linear expansion*Î±*_{1}, and Youngâ€™s modulus*Y*_{1}. The corresponding quantities for second rod are*L*_{2},*Î±*_{2}, and*Y*_{2}. If the temperature of both the rods is now raised by*T*degrees then increase in length of the composite rod (due to heating) will be equal to*l*

_{1}+

*l*

_{2}= [

*L*

_{1}

*Î±*

_{1}+

*L*

_{2}

*Î±*

_{2}]

*T*[as

*l*=

*L*

*Î±*

*Î”*

*Î¸*]

and due to compressive force

*F*from the walls due to elasticity,decrease in length of the composite rod =

As the length of the composite rod remains unchanged, increase in length due to heating must be equal to decrease in length due to compression, i.e.

or

Force constant of wire Force required to produce unit elongation in a wire is called force constant of material of wire. It is denoted by

*k*.But from the definition of youngâ€™s modulus, .

From (1) and (2),

It is clear that the value of force constant depends upon the dimension (length and area of cross section) and material of a substance.

**Actual length of the wire**If the actual length of the wire is

*L*, then under tension

*T*

_{1}, its length becomes

*L*

_{1}and under tension

*T*

_{2}, its length becomes

*L*

_{2}.

From (3) and (4), we get