1.4 Dimensional Analysis

Dimensional analysis is a powerful tool in physics, helping us understand relationships between physical quantities. It's all about breaking down measurements into their basic components of length, mass, and time.

By checking if equations have consistent dimensions on both sides, we can spot errors and gain insights. This technique guides experiments, simplifies complex problems, and even helps predict how things will behave in different situations.

Dimensional Analysis

Dimensions of physical quantities

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• Represent fundamental properties of physical quantities (length, mass, time)
• Expressed using symbols in square brackets $[[L](https://www.fiveableKeyTerm:L)]$, $[[M](https://www.fiveableKeyTerm:M)]$, $[[T](https://www.fiveableKeyTerm:T)]$
• Other dimensions include electric current $[[I](https://www.fiveableKeyTerm:I)]$, temperature $[\Theta]$, amount of substance $[[N](https://www.fiveableKeyTerm:N)]$, luminous intensity $[[J](https://www.fiveableKeyTerm:J)]$
• Velocity has dimensions of length divided by time $[LT^{-1}]$ (meters per second)
• Acceleration has dimensions of length divided by time squared $[LT^{-2}]$ (meters per second squared)
• Dimensional homogeneity requires all terms in a mathematical expression to have the same dimensions
  • Adding or subtracting quantities requires identical dimensions (cannot add length to mass)
  • Multiplying or dividing quantities results in the product or quotient of their dimensions (area is length times width)
• Base units are fundamental units that cannot be broken down further (e.g., meter, kilogram, second)
• Derived units are formed by combining base units (e.g., velocity in meters per second)

Dimensional consistency in equations

• Equations must have the same dimensions on both sides to be physically meaningful
• Checking dimensional consistency helps identify errors in equations (if dimensions do not match, the equation is incorrect)
• Dimensionless quantities have no dimensions and are represented by $[1]$ (angles, ratios, trigonometric functions)
• Dimensionless constants can be added to or multiplied with any quantity without affecting its dimensions ($\pi$, $e$)
• Example: $F = ma$ is dimensionally consistent because $[F] = [MLT^{-2}]$ and $[ma] = [M][LT^{-2}] = [MLT^{-2}]$
• Example: $v = v_0 + at$ is dimensionally consistent because $[v] = [LT^{-1}]$, $[v_0] = [LT^{-1}]$, and $[at] = [LT^{-2}][T] = [LT^{-1}]$
• Dimensional equations express the relationship between quantities in terms of their dimensions

Applications of dimensional analysis

• Determine the form of an unknown relationship between physical quantities
  1. Write the desired quantity as a product of powers of the relevant variables
  2. Equate the dimensions of both sides
  3. Solve for the unknown exponents
• Buckingham Pi Theorem reduces the number of variables needed to describe a physical system
  • Number of dimensionless groups (Pi groups) equals the number of variables minus the number of independent dimensions
• Rayleigh's method systematically applies dimensional analysis
  1. Express the target variable as a product of powers of the independent variables
  2. Equate the dimensions of both sides
  3. Solve for the exponents
• Guides experiments and provides insight into the behavior of physical systems
  • Identifies the most important variables and their relationships (Reynolds number in fluid dynamics)
  • Helps design scale models and similitude studies (wind tunnel testing)
• Example: pendulum period $T$ depends on length $L$, mass $m$, and gravitational acceleration $g$
  • $T = k L^a m^b g^c$ where $k$ is a dimensionless constant
  • $[T] = [L]^a [M]^b [LT^{-2}]^c = [T]$
  • Solving for $a$, $b$, and $c$ yields $T = k \sqrt{\frac{L}{g}}$, showing that mass does not affect the period
• Dimensional matrix can be used to organize and solve dimensional analysis problems systematically

Additional Concepts in Dimensional Analysis

• Physical constants often have dimensions and can be used in dimensional analysis (e.g., gravitational constant, speed of light)
• Unit conversion involves using dimensional analysis to convert between different units of measurement
• Dimensional analysis can be used to check the validity of equations and formulas in physics