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Unit Analysis and Dimensional Reasoning

24 min Engineering Math

Unit Analysis

Engineering calculations live or die by units. Dimensional analysis is a powerful technique for checking formulas and converting quantities.

Dimensional Analysis

Every term in a valid equation must have the same dimensions. If the dimensions don't match, the equation is wrong.

Base SI Units

Example 1

Verify $F = ma$ dimensionally.

$[F] = $ N $= $ kg·m/s². $[ma] = $ kg × m/s² = kg·m/s². ✓ Consistent.

Example 2

Convert 72 km/h to m/s.

$72 \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 20$ m/s.

Example 3

Energy has units kg·m²/s² (joules). Is $E = mv$ dimensionally correct?

$[mv] = $ kg · m/s — this is momentum, not energy. Wrong.

$E = \frac{1}{2}mv^2$: $[mv^2] = $ kg · m²/s² ✓

1. Convert 5 miles to kilometers (1 mi ≈ 1.609 km).

2. What are the SI units of pressure (force/area)?

3. Is $v = \sqrt{2gh}$ dimensionally correct?

4. Convert 1 hour to seconds.

5. What are the dimensions of power (energy/time)?

6. Convert 100 cm³ to m³.

Show Answer Key

1. $5 \times 1.609 = 8.045$ km

2. Pa = N/m² = kg/(m·s²)

3. $\sqrt{(m/s^2)(m)} = \sqrt{m^2/s^2} = m/s$ ✓

4. 3600 s

5. W = J/s = kg·m²/s³

6. $100 \times 10^{-6} = 10^{-4}$ m³