1. What are Stress and Young's Modulus?

Stress (σ) is the force per unit area applied to a material, while Young's modulus (E) is a measure of the stiffness of a solid material, defined as the ratio of stress to strain.

2. Dimensional Analysis

Both stress and Young's modulus share the same dimensional formula:

Where:

• \( M \) — Mass dimension
• \( L \) — Length dimension
• \( T \) — Time dimension

Explanation: This dimensional formula arises from the fundamental definitions of stress (force/area) and Young's modulus (stress/strain, where strain is dimensionless).

3. Derivation of Dimensional Formulas

Stress Derivation: Stress = Force/Area = (Mass × Acceleration)/Area = [M][LT⁻²]/[L²] = [M L⁻¹ T⁻²]

Young's Modulus Derivation: Young's modulus = Stress/Strain = [M L⁻¹ T⁻²]/[dimensionless] = [M L⁻¹ T⁻²]

4. Using the Calculator

Tips: Enter the powers for mass (M), length (L), and time (T) dimensions. The calculator will verify if your input matches the standard dimensional formula for stress or Young's modulus.

5. Frequently Asked Questions (FAQ)

Q1: Why do stress and Young's modulus have the same dimensions?
A: Because Young's modulus is defined as stress divided by strain, and strain is a dimensionless quantity (length/length).

Q2: What are the SI units for stress and Young's modulus?
A: Both are measured in Pascals (Pa), which is equivalent to N/m² or kg/(m·s²).

Q3: Can dimensional formulas be used for unit conversion?
A: Yes, dimensional analysis is fundamental for converting between different unit systems and verifying equation consistency.

Q4: What other physical quantities share this dimensional formula?
A: Pressure, energy density, and modulus of rigidity all share the [M L⁻¹ T⁻²] dimensional formula.

Q5: Why is dimensional analysis important in physics?
A: It helps verify equations, derive relationships, convert units, and understand the fundamental nature of physical quantities.