1.4 Dimensional Analysis - University Physics Volume 1

Learning Objectives

• Find the dimensions of a mathematical expression involving physical quantities.
• Determine whether an equation involving physical quantities is dimensionally consistent.

Key Concepts

• Dimensions express a physical quantity's dependence on base quantities as a product of symbols representing these base quantities.
• Base Quantities include Length (L), Mass (M), Time (T), Current (I), Thermodynamic Temperature, Amount of Substance (N), and Luminous Intensity (J).
• Dimensions follow algebraic rules, similar to units.

Dimension Examples

• Length: L or L$^1$
• Mass: M or M$^1$
• Time: T or T$^1$
• Area: L$^2$
• Volume: L$^3$
• Speed: L/T or LT$^{-1}$
• Density: M/L$^3$ or ML$^{-3}$

Dimensional Consistency Rules

1. Every term in an expression must have the same dimensions.
2. Arguments of standard mathematical functions (e.g., trigonometric, logarithmic) must be dimensionless.

• Dimensional Consistency: An equation must be dimensionally consistent to be correct as a physical law.

Examples

Using Dimensions to Remember Equations

• Example: For area vs. circumference of a circle:
  • $r^2$ (Area) vs. $2 \pi r$ (Circumference)
  • Check dimensions to determine which matches area: $[r^2] = L^2$ (area) vs. $[2\pi r] = L$ (length)

Checking Equations for Dimensional Consistency

• Equation (a): $s = vt + 0.5at^2$
  • $[s] = L$, $[vt] = L$, $[0.5at^2] = L$ – Consistent
• Equation (b): $s = vt^2 + 0.5at$
  • $[s] = L$, $[vt^2] = L^2T$, $[at] = LT^{-1}$ – Not Consistent
• Equation (c): $v = \sin(at^2/s)$
  • Argument of sin must be dimensionless. $[at^2/s] = 1$, but $[v] = LT^{-1}$ – Not Consistent

Significance

• Dimensional analysis helps verify equations, check for errors, and recall formulations.

Additional Considerations

• Calculus and Dimensions
  • Derivatives and integrals can be dimensionally analyzed:
    • Derivative: $[dv/dt] = [v][t]^{-1}$
    • Integral: $[\int v dt] = [v][t]$

Check Your Understanding

• Volume vs. Surface Area: For a sphere, $\frac{4}{3}\pi r^3$ (Volume) vs. $4\pi r^2$ (Surface Area).
• Is $v = at$ dimensionally consistent? Yes, as it aligns with $[v] = LT^{-1}$ and $[at] = LT^{-1}$.

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Note: Always confirm any mathematical function inputs are dimensionless to ensure accuracy.