Understanding the Dimensional Formula for Stress

This explanation breaks down how to find the dimensional formula for stress. We'll explore the definitions and derive the formula step-by-step.

What is Stress?

In physics, stress is a measure of the internal forces that neighboring particles within a continuous material exert on each other. It's defined as the force acting perpendicularly on a unit area of a surface. Mathematically, it's expressed as:

$$ \text{Stress} = \frac{\text{Force}}{\text{Area}} $$

What are Dimensional Formulas?

A dimensional formula represents a physical quantity in terms of the fundamental physical quantities, usually mass (M), length (L), and time (T). It helps us understand the relationship between different physical quantities and is crucial for checking the consistency of equations.

Deriving the Dimensional Formula for Force

First, let's find the dimensional formula for force. Force is defined by Newton's second law as mass times acceleration:

$$ \text{Force} = \text{Mass} \times \text{Acceleration} $$

• The dimension for Mass is M.
• Acceleration is the rate of change of velocity, and its dimension is Length per Time squared ($L/T^2$), or $LT^{-2}$.

Therefore, the dimensional formula for Force is:

$$ [\text{Force}] = [M] \times [LT^{-2}] = MLT^{-2} $$

Deriving the Dimensional Formula for Area

Next, we need the dimensional formula for area. Area is typically calculated as length times width (or length squared):

$$ \text{Area} = \text{Length} \times \text{Width} $$

• The dimension for Length is L.
• The dimension for Width is also L.

Therefore, the dimensional formula for Area is:

$$ [\text{Area}] = [L] \times [L] = L^2 $$

Calculating the Dimensional Formula for Stress

Now we can combine the dimensions of force and area to find the dimensional formula for stress using the relationship:

$$ [\text{Stress}] = \frac{[\text{Force}]}{[\text{Area}]} $$

Substituting the derived formulas:

$$ [\text{Stress}] = \frac{MLT^{-2}}{L^2} $$

To simplify, we use the rules of exponents ($a^m / a^n = a^{m-n}$):

$$ [\text{Stress}] = ML^{1-2}T^{-2} $$

$$ [\text{Stress}] = ML^{-1}T^{-2} $$

Conclusion

The dimensional formula for stress, derived from the definitions of force and area, is $ML^{-1}T^{-2}$.