Question

The dimensions of stress are equal to
A. Force
B. Pressure
C. Work
D. \[\dfrac{1}{{{\text{Pressure}}}}\]

Answer 1

Hint: Recall the definition of stress and find the unit and dimension of stress. Identify the physical quantity which has the same unit as that of the stress. If the units are the same, the dimensions will also be the same.

Complete answer:
We know the definition of stress. It is defined as the force acting on a body per unit area of the body. It is denoted by the letter \[\sigma \]. Therefore, we can express the stress as,
\[\sigma = \dfrac{F}{A}\]
Here, F is the force and A is the area.

We know that the unit of force is \[{\text{kg}}\,{\text{m}}\,{{\text{s}}^{ - 2}}\] and unit of area is \[{{\text{m}}^2}\]. Therefore, the unit of stress would be,
\[\sigma = \dfrac{{{\text{kg}}\,{\text{m}}\,{{\text{s}}^{ - 2}}}}{{{{\text{m}}^2}}} = {\text{kg}}\,{{\text{m}}^{ - 1}}\,{{\text{s}}^{ - 2}}\]
Now, we express the dimensions of stress as,
\[\sigma = \left[ {{{\text{M}}^1}\,{{\text{L}}^{ - 1}}\,{{\text{T}}^{ - 2}}\,} \right]\]
We know that the pressure is also defined as applied force per unit area. Therefore, we can express the pressure P as follows,
\[P = \dfrac{F}{A}\]
Therefore, we have dimensions of the term \[\dfrac{F}{A}\] are \[\left[ {{{\text{M}}^1}\,{{\text{L}}^{ - 1}}\,{{\text{T}}^{ - 2}}\,} \right]\].
Thus, we can say that dimensions of stress are the same as dimensions of pressure.

So, the correct answer is option (B).

The unit of work is \[{\text{kg}}\,{{\text{m}}^2}{{\text{s}}^{ - {\text{2}}}}\] and the unit of stress is \[{\text{kg}}\,{{\text{m}}^{ - 1}}{{\text{s}}^{ - {\text{2}}}}\]. Therefore, the work and stress cannot have the same dimensions. Thus, the option (C) is incorrect.

Additional information:
The fundamental unit of stress is \[{\text{kg}}\,{{\text{m}}^{ - 1}}{{\text{s}}^{ - 2}}\] and S.I derived unit of stress is Pa or pascal. There are mainly two types of stress: normal stress and shearing stress. When the direction of applied force to deform the body is applied perpendicular to the body, then it is known as normal stress. When the direction of applied force to deform the body is parallel to the cross-sectional area of the body, the stress is known as shearing stress.


Note:To answer such types of questions, students can find the physical quantity which has the same unit. In this question, we know that the unit of stress is Pa and the unit of pressure is also Pa. Therefore, most of the time, if the units are the same, the dimensions of both the physical quantities are the same.