Dimensional Analysis in Physics

Dimensional Analysis

• Dimensional analysis is a method to check formulae and equations using units (dimensions).
• It involves converting measurable quantities into their base units using SI Base units.

Ignoring Units

• A radian is a ratio, so it doesn't have a unit and can be ignored in dimensional analysis.
• Ratios such as coefficients of friction (μ) and restitution (e) can also be ignored.
• Trigonometric ratios (like sin, cos, or tan) and logarithms (log) can be ignored in dimensional analysis.

Conducting Dimensional Analysis

• Replace variables with their units, ignoring constants and unitless terms.
• Only add or subtract the same units.
• Multiply and divide different units to simplify the expression or equation.

Example: Area of a Rectangle

• The formula for area of a rectangle is length × width.
• The base unit of length and width is m, so the unit of area is m².

Base Units of Force

• The formula for force is F = ma, where mass has a base unit of kg and acceleration has a base unit of ms⁻².
• The base unit of force is kg × ms⁻² = kgm/s².

Exercises

• Find the dimensions of:
  • Momentum = Mass × Velocity
  • Kinetic Energy = ½ × mass × velocity²
  • Potential Energy = mass × gravity × height
  • Electric Potential = Current × resistance

Validating Equations

• Dimensional analysis can be used to validate equations.
• A formula is dimensionally valid if the units of the left-hand side (LHS) are equal to the units of the right-hand side (RHS).

Worked Example: v = u - gt

• The formula v = u - gt is dimensionally valid because:
  • LHS = v = ms⁻¹
  • RHS = ms⁻¹ - ms⁻² × s = ms⁻¹ - ms⁻¹ = ms⁻¹
  • RHS = LHS

Finding the Formula for Maximum Range

• The formula for maximum range R of a projectile on a horizontal plane involves initial speed u and acceleration due to gravity g.
• The formula can be found using dimensional analysis, and it does not include dimensionless constants.