Dimension Formula of Coefficient of Elasticity

Young's Modulus Dimensions:

\[ [E] = [M L^{-1} T^{-2}] \]

Dimension Formula:

[E] = M L^-1 T^-2

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1. What is Young's Modulus?

Young's modulus (E) is a measure of the stiffness of a solid material. It defines the relationship between stress (force per unit area) and strain (proportional deformation) in a material in the linear elasticity regime of a uniaxial deformation.

2. Understanding Dimensional Analysis

Dimensional analysis is a method used to understand the physical quantities in terms of their fundamental dimensions. The basic dimensions are:

\[ \text{Mass (M), Length (L), Time (T)} \]

All physical quantities can be expressed in terms of these fundamental dimensions with appropriate exponents.

3. Dimension Formula Derivation

Young's modulus is defined as:

\[ E = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]

Where:

Stress dimension: \( [F/A] = [M L^{-1} T^{-2}] \)
Strain is dimensionless: \( [\Delta L/L] = [1] \)
Therefore: \( [E] = [M L^{-1} T^{-2}] \)

Explanation: Since strain is a ratio of lengths, it is dimensionless. Stress has dimensions of pressure (force per unit area).

4. Using the Dimension Calculator

Instructions: Enter the exponents for mass (M), length (L), and time (T) dimensions. The calculator will generate the corresponding dimension formula. For Young's modulus, use M=1, L=-1, T=-2.

5. Frequently Asked Questions (FAQ)

Q1: What are the fundamental dimensions in physics?
A: The fundamental dimensions are Mass (M), Length (L), Time (T), Electric Current (I), Thermodynamic Temperature (Θ), Amount of Substance (N), and Luminous Intensity (J).

Q2: Why is strain dimensionless?
A: Strain is defined as the ratio of change in length to original length (ΔL/L). Since both numerator and denominator have length dimensions, they cancel out.

Q3: What is the SI unit of Young's modulus?
A: The SI unit is Pascal (Pa), which is equivalent to N/m² or kg/(m·s²).

Q4: Can dimensions be fractional?
A: In classical physics, dimensions are typically integer exponents. However, in some advanced physics contexts, fractional dimensions may appear.

Q5: How is dimensional analysis useful?
A: Dimensional analysis helps verify equations, derive relationships between physical quantities, and convert between different unit systems.