It is defined as the ratio of normal stress to longitudinal strain within limit of proportionality.

If force is applied on a wire of radius r by hanging a weight of mass M, then

Important points:-

(i) If the length of a wire is doubled,

Then longitudinal strain

Young’s modulus =

Þ    Y = stress                   [As strain = 1]

So young’s modulus is numerically equal to the stress which will double the length of a wire

(ii) Increment in the length of wire

So if same stretching force is applied to different wires of same material,   [As F and Y are constant]

i.e., greater the ratio ,greater will be the elongation in the wire.

(iii) Elongation in a wire by its own weight : The weight of the wire Mg act at the centre of gravity of the wire so that length of wire which is stretched will be L/2.

Elongation

[As mass (M) = volume (AL) × density (d)](iv) 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

As by definition, coefficient of linear expansion

Þ         thermal strain

So            thermal stress = Ya Dq                [As Y = stress/strain]

And tensile or compressive force produced in the body = YAa Dq

                            Where     K = Bulk modulus, g = coefficient of cubical expansion

                            Where     K = Bulk modulus, g = coefficient of cubical expansion

(v) Force between the 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 figure. The first rod has a length L₁, coefficient of linear expansion a₁ and young’s modulus Y₁. The corresponding quantities for second rod are L₂, a₂ and Y₂. If the temperature of both the rods is now raised by T degrees.

                            Where     K = Bulk modulus, g = coefficient of cubical expansion

(v) Force between the 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 figure. The first rod has a length L₁, coefficient of linear expansion a₁ and young’s modulus Y₁. The corresponding quantities for second rod are L₂, a₂ and Y₂. If the temperature of both the rods is now raised by T degrees.

Increase in length of the composite rod (due to heating) will be equal to

Increase in length of the composite rod (due to heating) will be equal to

(v) Force between the 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 figure. The first rod has a length L₁, coefficient of linear expansion a₁ and young’s modulus Y₁. The corresponding quantities for second rod are L₂, a₂ and Y₂. If the temperature of both the rods is now raised by T degrees.

Increase in length of the composite rod (due to heating) will be equal to

[As l = L a Dq]      

and due to compressive force F from the walls due to elasticity, decrease in length of the composite rod will be equal to

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

(vi) 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.

…..(i)

but from the definition of young’s modulus

…..(ii)        

from (i) and (ii)

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

(vii) Actual length of the wire : If the actual length of the wire is L, then under the tension T₁, its length becomes L₁ and under the tension T₂, its length becomes L₂.……(i)         and      

      ..…(ii)

From (i) and (ii) we get