Dimensional Analysis and Units

1. Introduction

Dimensional analysis is a mathematical technique used to analyze the relationships between different physical quantities by identifying their fundamental dimensions. It is widely used in physics, engineering, and applied mathematics.

2. Units and Dimensions

Physical Quantities: Quantities used to describe physical phenomena (e.g., length, mass, time, etc.).
Fundamental Quantities:
- Length: [L]
- Mass: [M]
- Time: [T]
- Electric Current: [I]
- Temperature: [Θ]
- Amount of Substance: [N]
- Luminous Intensity: [J]
Derived Quantities: Expressed in terms of fundamental quantities (e.g., velocity [LT^-1], force [MLT^-2]).
Units: Standardized measurements of physical quantities.
- SI Units:
  - Length: meter (m)
  - Mass: kilogram (kg)
  - Time: second (s)
  - Electric Current: ampere (A)
  - Temperature: kelvin (K)
  - Amount of Substance: mole (mol)
  - Luminous Intensity: candela (cd)

3. Dimensional Formula

The expression that shows how a physical quantity is related to the fundamental dimensions is called its dimensional formula.

Velocity: [L][T]^-1
Force: [M][L][T]^-2
Energy: [M][L]²[T]^-2

4. Principle of Homogeneity of Dimensions

This principle states that:

In any valid physical equation, the dimensions of all terms on both sides of the equation must be the same.
Used to:
1. Verify the correctness of equations.
2. Derive relationships between quantities.

5. Applications of Dimensional Analysis

Checking the correctness of equations: Use the dimensional formula for all terms in the equation and ensure the dimensions are consistent on both sides.
Deriving relationships between physical quantities: Assume a relationship and substitute dimensional formulas to find exponents.
Converting units: Example: Convert speed from m/s to km/h.
```
1 m/s = 3.6 km/h
```
Estimation of orders of magnitude: Quickly estimate approximate values or trends using dominant terms.

6. Limitations of Dimensional Analysis

Does not provide numerical constants (e.g., π, e, k).
Cannot distinguish between dimensionless quantities.
Cannot determine functions (e.g., trigonometric, logarithmic) in equations.
Limited to algebraic relationships; does not work for non-dimensionalized equations.

7. Examples

Verify Equation of Motion: v = u + at
- Dimensions of v, u: [L][T]^-1
- Dimensions of at: [L][T]^-1
- Both sides have the same dimensions; equation is correct.
Derive Formula for Period of a Pendulum:
- Period (T) depends on length (l) and gravitational acceleration (g).
- Assume T = k * l^a g^b.
- Substitute dimensions: [T] = [L]^a[L][T]^-2b.
- Solve for a and b: a = 1/2, b = -1/2.
- Final relation: T ∝ √(l/g).

8. Frequently Used Dimensional Quantities

Quantity	Dimensional Formula
Speed/Velocity	[L][T]^-1
Acceleration	[L][T]^-2
Force	[M][L][T]^-2
Energy	[M][L]²[T]^-2
Power	[M][L]²[T]^-3
Pressure	[M][L]^-1[T]^-2
Charge	[I][T]
Electric Field	[M][L][T]^-3[I]^-1

9. Tips for Dimensional Analysis

Memorize fundamental dimensions and derived unit dimensions.
Break down complex units into base dimensions.
Apply the principle of homogeneity consistently.