Question

Derive the expression for young's modulus of a wire in terms of its radius.(3 marks) ​

Answer 1

Derivation of Young's modulus of a wire

Explanation:

Let's assume a wire of length 'L' having cross-sectional area 'A' having radius 'r.' when force 'F' is applied on end the change in length is ΔL.

As shown in the given figure,

from the definition, we know that,

longitudinal stress = longitudinal force/area = F/A

now, the area is equal to A = πr²

thus longitudinal stress = F/πr²

and similarly, longitudinal strain = L/ ΔL

now, the young modulus is the ratio of longitudinal stress and strain

Young modulus = longitudinal stress/ longitudinal strain

on putting values

$Y = \dfrac{F\times \delta L}{\pi r^{2} \times L}$

Thus, the expression of young's modulus of a wire in terms of the radius is shown above.