📏 Lesson 8: Dimensional Analysis & Unit Algebra

Treating units like variables — your ultimate error checker

🎯 Learning Objectives

Perform dimensional analysis to verify equations and catch errors
Convert units systematically using conversion factors
Derive correct formulas by matching dimensions
Use units as a debugging tool during problem-solving

🔑 Key Concepts

The Fundamental Idea

Units obey the same algebra as variables. You can multiply, divide, cancel, and raise them to powers — just like x, y, z.

If an equation is physically correct, the units on both sides must match exactly.

💡 Superpower: When you're stuck on a physics problem, write down the units of what you know and what you need. The unit path often reveals the formula!

Base SI Units

Quantity	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Current	ampere	A
Temperature	kelvin	K
Amount	mole	mol

Derived Units (Memorize These!)

Quantity	Formula	SI Unit	Name
Velocity	m/s	m/s	m/s
Acceleration	m/s²	m/s²	m/s²
Force	kg·m/s²	N	newton
Energy/Work	kg·m²/s²	J	joule
Power	kg·m²/s³	W	watt
Pressure	kg/(m·s²)	Pa	pascal
Momentum	kg·m/s	kg·m/s	—
Frequency	1/s	Hz	hertz

📐 Worked Examples

Example 1: Checking an Equation

Is v² = v₀² + 2ax dimensionally consistent?

[v²]          = (m/s)² = m²/s²
[v₀²]         = (m/s)² = m²/s²
[2ax]         = (m/s²)(m) = m²/s²
✓ All terms have m²/s² — consistent!

Example 2: Deriving a Formula

Period T of a pendulum depends on length L and gravity g. Find the relationship.

Assume T ∝ Lᵃ gᵇ (a, b unknown). [T] = s, [L] = m, [g] = m/s².

s = mᵃ (m/s²)ᵇ = mᵃ⁺ᵇ s⁻²ᵇ
Match powers:
m: 0 = a + b
s: 1 = -2b → b = -1/2
Then a = 1/2
T ∝ √(L/g)  ✓ Matches known formula!

Example 3: Unit Conversion

Convert 60 mph to m/s. (1 mile = 1609 m, 1 hour = 3600 s)

60 mi/hr × (1609 m / 1 mi) × (1 hr / 3600 s)
= 60 × 1609 / 3600 m/s
= 26.8 m/s

Example 4: Catching an Error

Student writes: KE = ½mv (forgot the square). Check units:

[KE] = J = kg·m²/s²
[½mv] = (kg)(m/s) = kg·m/s
MISMATCH! kg·m²/s² ≠ kg·m/s
→ Formula is WRONG. Must be KE = ½mv².

🏋️ Practice Problems

Check dimensional consistency: v = v₀ + at, x = v₀t + ½at², v² = v₀² + 2ax
Find dimensions of: G (gravitational constant), k (spring constant), ε₀ (permittivity)
Convert: 100 km/h to m/s; 1 atm (101325 Pa) to lb/in²; 1 hp (746 W) to ft·lb/s
Use dimensional analysis to guess: centripetal acceleration a_c = ? (depends on v, r)
Use dimensional analysis to guess: period of mass-spring T = ? (depends on m, k)
Drag force: F_d = ½ρv²C_dA. Find dimensions of ρ, C_d, A.
Reynolds number: Re = ρvL/μ. Show it's dimensionless. What's dimension of μ?

✅ Answers

Click to reveal answers

1) All consistent (m/s, m, m²/s²) 2) [G]=m³/kg·s², [k]=kg/s², [ε₀]=C²/N·m² 3) 27.8 m/s, 14.7 psi, 550 ft·lb/s

4) a_c ∝ v²/r (only combination giving m/s²) 5) T ∝ √(m/k) 6) [ρ]=kg/m³, [C_d]=1, [A]=m² 7) Re dimensionless, [μ]=kg/m·s (dynamic viscosity)

🎉 Algebra Review Complete!

You've mastered the algebraic foundation for Physics 101. The skills you've practiced — formula manipulation, solving systems, quadratics, proportions, and dimensional analysis — will be used in every single physics lesson.

Next up: Trigonometry — for vectors, components, waves, and oscillations.

🚀 Start Trigonometry →