Dimensional Analysis Study Notes

Chapter 1: Dimensional Analysis

Generalities

• The term dimensional analysis is more comprehensive than just units of measurement. It encompasses fundamental concepts related to dimensions in physics and measurements.

Dimension Definition

1. Dimension

   • A dimension is a fundamental property that describes physical quantities.

   • Mass (dimension) is distinct from its unit of measurement (kilogram).

   • Example: Measurement of distance utilizes various units (feet, meters, kilometers), each representing the dimension of length.

Dimensional Analysis

2. Definition

   • Dimensional analysis is a method used in physics and sciences to verify relationships between physical quantities by determining their dimensions.

   • Key points:

     • Quantities on both sides of a mathematical equation must possess the same dimension.

     • All quantities must belong to the same system (base).

   • Important Note: Dimensional analysis does not guarantee the validity of physical laws; that is determined through mathematical analysis.

   • Example: The areas of a square and a triangle share the same dimension (𝐿²) but are not equal.

Fundamental Units

3. Seven Fundamental Units

   • Dimensions are classified into seven fundamental categories:

     • Length (𝐿) = [𝑙]

     • Time (𝑇) = [𝑡]

     • Mass (𝑀) = [𝑚]

     • Temperature (𝜃) = [𝜃]

     • Electric current (𝐼) = [𝐼]

     • Amount of substance (𝑁) = [𝑛]

     • Luminous intensity (𝐽) = [𝐽]

Expressing Dimensions

4. Expression of Physical Quantity

   • The dimension of a physical quantity (G) is expressed mathematically as:
     $[𝐺] = 𝐿^𝑎 M^𝑏 T^𝑐 I^𝑑 J^𝑒 𝜃^𝑓 N^𝑔$

   • Variables (a, b, c, d, e, f, g) denote the powers associated with fundamental dimensions.

   • Dimensionless Quantities: Certain quantities, like plane angle and solid angle, are independent of basic dimensions and are considered dimensionless (e.g., sin x, cos x).

   • Notable Values:

     • [Numeric value] = 1

     • [angle] = 1

     • [cos α] = [sin α] = [tan α] = [cot α] = [ln x] = [e^x] = 1

     • Dimensionless quantities are referred to as pure numbers.

Properties of Dimensional Equations

5. Properties

   • Avoid including a unit system when writing a dimensional equation.

   • If $[G] = 1$, then G is constant but could have a unit, such as $[2 ext{π}] = 1$ (Radian or degrees).

   • Equations involving dimensions must satisfy the following:

     • If $G = A imes B$, then:
       $[𝐺] = [𝐴][𝐵]$

     • If $G = A B$, then $[𝐺] = [𝐴][𝐵]$

     • If $G = A^n$, then $[𝐺] = [𝐴]ⁿ$

     • Here, n is dimensionless.

     • Dimensions cannot be added or subtracted directly: $G = A ± B
       ightarrow [G] = [A] = [B]$.

   • Important Notes:

     • A heterogeneous (non-homogeneous) equation is necessarily false.

     • A homogeneous equation is not necessarily true.

     • In mechanics, the primary quantities needed are mass (l) and time (t).

     • Denote quantities symbolically within square brackets (e.g., [𝐺]).

Examples of Dimensional Analysis

6. Dimensional Equations

   • Example 1: For displacement ($s$), given by $s = l^2$, the dimension is:
     $[s] = [l²] = [l]^2 = 𝐿²$

   • Example 2: For velocity ($v$), defined as $V = x/t$, its dimension becomes:
     $[v] = [x][t]^{-1} = L T^{-1}$

   • Example 3: For force ($F = ma$):
     $[F] = [m][a] = M L T^{-2}$

Derived Quantities

7. Categories of Physical Quantities

   • Physical Quantities: Any measurable property, like mass, time, area, volume, energy, density, and pressure.

   • Types of Physical Quantities:

   1. Scalar Quantities: Described by magnitude only (e.g., mass, temperature).

   2. Vectorial Quantities: Described by both magnitude and direction (e.g., velocity, force).

Measurement Methods

8. Measuring Physical Processes

   • Two primary methods exist:

     • Direct Measurement (e.g., using a ruler for distance).

     • Calculation using mathematical relationships (e.g., calculating speed using $ν = rac{x}{T}$).

International System of Units

9. Unit Standards

   • The value of a physical quantity is expressed in relation to standard units known as units.

   • The MKSA International System was adopted in 1946, establishing fundamental units:

     • Length = Meter (m)

     • Mass = Kilogram (kg)

     • Time = Second (s)

     • Current Intensity = Ampere (A)

     • Temperature = Kelvin (K)

     • Amount of Substance = Mole (Mol)

     • Luminous Intensity = Candela (Cd)

   • Derived Units: Area (m²), Velocity (m/s), Force (kg m/s²).

Alternative Unit Systems

10. CGS System

    • The CGS system (Centimeter, Gram, Second) is less common, established in 1847, where:

      • Distance = Centimeters (cm)

      • Force = Dynes (Dyne)

      • Energy = Ergs (Erg).

Dimensional Consistency Examples

11. Example Analysis

    • Velocity:
      $[v] = [x][t]^{-1} = L T^{-1}$

    • Acceleration:
      $[a] = [v][t]^{-1} = L T^{-2}$

    • Force:
      $[F] = [m][a] = M L T^{-2}$

    • Charge Quantity:
      $[Q] = [i][t] = IT$

    • Kinetic Energy:
      $E = rac{1}{2}mv^2 <br>ightarrow [E] = M L^{2} T^{-2}$

Exercises on Dimensional Consistency

12. Exercise 01: Analyze the change in displacement with acceleration and time:
    $x = rac{1}{2} a t^2$

    • Left side dimension:
      $[x] = L$

    • Right side dimension:
      $[ rac{1}{2}][a][t^2] = L T^{-2} T^2 = L$

    • The equation is dimensionally correct.

13. Exercise 02: Deriving the period in relation to length and gravitational force:
    $T = k l^x g^y$

    • Identify:

      • Since $[T] = T$, $[l] = L$, and constant K is dimensionless:

      • $[T] = [k][l]^x[g]^y$ we express:

      • Given the relation for gravitational force ($
        ho = mg$):

      • Using the dimensions:

      • $[g] = [<br>ho] m$

      • Therefore, by substituting dimensions, we derive:
        $[T] = L^x (LT^{-2})^y$

    • Solving yields:

      • $x + y = 0$, and $-2y = 1$ leading to:

      • $y = - rac{1}{2}, ext{ and } x = rac{1}{2}$.

    • Finally, the relationship for the period is:
      $T = k rac{l^{1/2}}{g^{1/2}}$

Chapter 1: Dimensional Analysis - Conclusion

• Comprehensive and exhaustive understanding of dimensional analysis, units, and derived quantities fosters a deeper appreciation for the mathematical relationships in physics and related fields.