Question

Find extension in wire of length ‘l’ area of cross section ‘A’, mass ${{M}_{x}}$ , Young’s modulus Y under its own weight?

Answer 1

Hint: Young’s modulus is defined as the ratio of stress to the longitudinal strain. Within the elastic limit, the ratio of stress to corresponding strain produced is called Young’s modulus for the material of a body and denoted by Y. The unit of young’s modulus is the same as the unit of stress since strain is a dimensionless quantity.
Formula used:
Young’s modulus is given by,
$Y=\dfrac{\dfrac{F}{A}}{\dfrac{\vartriangle l}{l}}$
F is the force applied
l is the original length
A is the area of cross section
and $\vartriangle l$ is the elongation.

Complete answer:
If F is the force applied, l is the original length, A is the area of cross section and $\vartriangle l$ is the elongation.
Then,
Young’s modulus,$Y=\dfrac{Stress}{Strain}$
where,
$Stress=\dfrac{Force}{Area}=\dfrac{F}{A}$
$Strain=\dfrac{\vartriangle l}{l}$
Hence, Young’s modulus is given by,
$Y=\dfrac{\dfrac{F}{A}}{\dfrac{\vartriangle l}{l}}$
$\Rightarrow Y=\dfrac{Fl}{A\vartriangle l}$
By rearranging the equation we get,
$\vartriangle l=\dfrac{Fl}{YA}$
Substitute $F={{M}_{x}}g$ in the above equation.
Thus, $\vartriangle l=\dfrac{{{M}_{x}}gl}{YA}$

Additional information:
Elasticity is that property of matter by virtue of which a body regains its original shape or size on removal of the deforming force. If the body completely regains its original shape or condition on the removal of the deforming forces, it is said to be perfectly elastic. While if it completely retains its modified form, it is said to be perfectly plastic.

Note:
As there are 3 types of strain there are three types of modulus of elasticity. They are Young ‘s modulus, Bulk modulus and Rigidity modulus. Here, Young’s modulus corresponds to longitudinal strain, Bulk modulus corresponds to volume strain and Rigidity modulus corresponds to shear modulus. The unit of young’s modulus is the same as the unit of stress since strain is a dimensionless quantity. This is the same in the case of other modulus also.