Question

The dimensional formula of modulus of elasticity is:
A). $\left[ {{M}^{0}}{{L}^{-1}}{{T}^{2}} \right]$
B). $\left[ {{M}^{0}}L{{T}^{-2}} \right]$
C). $\left[ ML{{T}^{-2}} \right]$
D). $\left[ M{{L}^{2}}{{T}^{-2}} \right]$

Answer 1

Hint: To obtain the dimensions of elasticity, we must know about stress and strain. Basically, the dimensional formula of elasticity is the same as dimension of stress because the strain is a dimensionless quantity. Stress can be expressed as force per unit area.

Complete step by step answer:
For dimension of elasticity HOOKE’S is very important
HOOKE’s law gives that within the limit of elasticity stress is directly proportional to the strain.
Stress is the internal restoring force acting per unit area of deforming body is called stress.$\text{stress = }\dfrac{\text{restoring force}}{\text{area}}$
Its unit is $N{{m}^{-2}}$ or Pascal.
It is a tensor quantity.
Strain can be defined as the fractional change in configuration of the object's shape.
From Hooke's law we can write,
$\text{stress = }k\times \text{ strain}$
k is modulus of elasticity
Hence,
$k=\dfrac{\text{stress}}{\text{strain}}$
Where stress is force per unit area. So, we can write,
$\text{stress = }\dfrac{\text{force}}{\text{area}}\text{ = }\dfrac{\text{mass }\times \text{ acceleration}}{\text{area}}$
Hence strain is dimensionless quantity
Therefore, K has a dimension of stress.
Now, dimension of mass is $M$
Dimension of acceleration is, $L{{T}^{-2}}$
Dimension of area is, ${{L}^{2}}$
The dimension of stress is $=\dfrac{M\times L{{T}^{-2}}}{{{L}^{2}}}=M{{L}^{-1}}{{T}^{-2}}$
And strain is just a number and hence it is dimensionless.
The dimension of modulus of elasticity is $M{{L}^{-1}}{{T}^{-2}}$
The correct option is (A).
Additional information:
Stress is of two types:
Normal stress: If deforming force is applied normal to the area, then the stress is called normal stress
If there is an increase in length, then stress is called tensile stress
If there is a decrease in length, then stress is called compression stress
Tangential stress: if deforming force is applied tangentially, then the stress is called tangential stress
According to change in configuration the strain is of three type:
$\text{longitudinal strain = }\dfrac{\text{change in length}}{\text{original length}}$
$\text{volumetric strain = }\dfrac{\text{change in volume}}{\text{original volume}}$
Shearing strain is the angular displacement of the plane perpendicular to the fixed surface
Elasticity: It is that property of the object by virtue of which it regains its original configuration after the removal of the deforming force.

Note: Modulus of elasticity $=\dfrac{\text{stress}}{\text{strain}}$
We must know that the dimension of stress or modulus of elasticity $\left[ M{{L}^{-1}}{{T}^{-2}} \right]$ is same as that of dimension of pressure.
When we want to obtain the dimension of any derived physical quantity always break it into the constituent fundamental quantities. Then it will be easy to get the answer.