Question

Dimensional formula for stress is
A. \[{{\text{M}}^0}{\text{L}}{{\text{T}}^{ - 2}}\]
B. \[{{\text{M}}^0}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}\]
C. \[{\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}\]
D. \[{\text{M}}{{\text{L}}^2}{{\text{T}}^{ - 2}}\]

Answer 1

Hint: Stress is equal to the applied force per unit area of the cross-section. The unit of force is \[{\text{kg}}\,{\text{m/}}{{\text{s}}^2}\] and the unit of area is \[{{\text{m}}^2}\]. Using this, express the dimensions of force and area of cross section and equate it with the assumed dimensions of stress.

Formula used:
Stress, \[\sigma = \dfrac{F}{A}\]
Here, F is the force and A is the area of cross-section.

Complete step by step answer:
To answer this question, let’s define the stress. Stress is the applied force per unit area of cross-section. Therefore, we can express the stress as,
\[\sigma = \dfrac{F}{A}\]
Here, F is the force and A is the area of cross-section.

We know that the unit of force is \[{\text{kg}}\,{\text{m/}}{{\text{s}}^2}\] and the unit of area is \[{{\text{m}}^2}\]. Now, let’s express the dimensions of the both sides of the above equation as follows,
\[{{\text{M}}^x}{{\text{L}}^y}{{\text{T}}^z} = \dfrac{{{\text{ML}}{{\text{T}}^{ - 2}}}}{{{{\text{M}}^0}{{\text{L}}^2}{{\text{T}}^0}}}\]
\[ \Rightarrow {{\text{M}}^x}{{\text{L}}^y}{{\text{T}}^z} = {\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}\]

Equating the powers on the both sides, we get,
\[x = 1\] , \[y = - 1\] and \[z = - 2\]
Therefore, substituting these powers in the left hand side which is the dimensions of stress, we get,
\[\therefore\sigma = {\text{M}}{{\text{L}}^{ - 1}}{{\text{T}}^{ - 2}}\]

So, the correct answer is option C.

Note: Since the dimensional formula for stress is the same as the dimensional formula for pressure, students can recall the dimensional formula for pressure to answer this question. The unit of stress is pascal. There are different types of stress such as normal stress, tensile stress but the dimensional formula for all these quantities is the same. The other quantity that we often use is the strain. The strain is dimensionless.