Understanding the Dimension of Stress

Stress is defined as the force acting per unit area. It is a measure of the internal forces that neighbouring particles of a continuous material exert on each other.

The formula for stress is given by:

\(\text{Stress} = \frac{\text{Force}}{\text{Area}}\)

Dimensional Formula of Force

Force is given by Mass multiplied by Acceleration.

• Dimension of Mass is \([M^1]\).
• Dimension of Acceleration is \([L^1 T^{-2}]\). This is derived from Velocity (\([L^1 T^{-1}]\)) divided by Time (\([T^1]\)).

Therefore, the dimensional formula of Force is \([M^1 L^1 T^{-2}]\).

Dimensional Formula of Area

Area is a measure of two-dimensional space, typically length multiplied by width.

The dimensional formula of Area is \([L^2]\).

Deriving the Dimension of Stress

To find the dimension of stress, we substitute the dimensions of Force and Area into the stress formula:

\(\text{Dimension of Stress} = \frac{\text{Dimension of Force}}{\text{Dimension of Area}}\)

\(\text{Dimension of Stress} = \frac{[M^1 L^1 T^{-2}]}{[L^2]}\)

Using the rules of exponents, when dividing terms with the same base, we subtract the exponents:

\(\text{Dimension of Stress} = [M^1 L^{1-2} T^{-2}]\)

\(\text{Dimension of Stress} = [M^1 L^{-1} T^{-2}]\)

Comparing with Options

The derived dimensional formula for stress is \([M^1 L^{-1} T^{-2}]\). We compare this with the given options:

Option 1: \([M^1 L^{-1} T^{-2}]\)

Option 2: \([M^{-1} L^{-1} T^{-2}]\)

Option 3: \([M^{-1} L^1 T^{-2}]\)

Option 4: \([M^{-1} L^{-1} T^{2}]\)

Our calculated dimension \([M^1 L^{-1} T^{-2}]\) matches the dimension given in the first option.