1cUnits and Dimensional Analysis
Master SI base units, metric prefixes, and the powerful technique of dimensional analysis for unit conversion and equation checking.
Unit errors have caused real disasters — NASA lost the Mars Climate Orbiter in 1999 because one team used imperial units and another used SI. Dimensional analysis is your built-in error-checker for every calculation.
Lesson Overview
Every physical quantity must be expressed in units. In this lesson you will learn the seven SI base units, how to use prefixes to scale them, and how dimensional analysis — tracking units like algebraic quantities — lets you convert between units and check whether equations are physically valid.
Key Concepts
SI Base Units
Seven fundamental units: metre, kilogram, second, ampere, kelvin, mole, candela
Derived Units
Units formed by combining base units (e.g., m/s, kg·m/s², J = kg·m²/s²)
SI Prefixes
Multipliers: kilo- (10³), centi- (10⁻²), milli- (10⁻³), micro- (10⁻⁶), nano- (10⁻⁹)
Conversion Factor
A fraction equal to 1 that changes units without changing the quantity
Dimensional Analysis
Using unit cancellation to convert quantities or verify equations
Dimensional Homogeneity
Both sides of a valid equation must have the same dimensions
Convert 72 km/h to m/s.
Convert 5.4 × 10⁻³ kg to milligrams (mg).
Check whether the equation v = at is dimensionally consistent, where v is velocity (m/s), a is acceleration (m/s²), and t is time (s).
A room is 15 feet long. Convert this to metres. (1 ft = 0.3048 m)
Express 1 year in seconds using dimensional analysis.
Convert 250 cm to metres.
Hint: Use the conversion factor: 1 m = 100 cm. Set it up so centimetres cancel.
Convert 90 km/h to m/s.
Hint: Convert km → m (×1000) and h → s (÷3600) simultaneously.
Is the equation F = mv²/r dimensionally consistent? (F in N = kg·m/s², m in kg, v in m/s, r in m)
Hint: Substitute dimensions on the right side and simplify. Compare to the left side.
How many microseconds are in 3.5 milliseconds?
Hint: 1 ms = 10⁻³ s; 1 μs = 10⁻⁶ s. Find the ratio.
A car travels 400 miles. Convert to kilometres. (1 mile = 1.609 km)
Hint: Multiply by the conversion factor with miles in the denominator.
Key Vocabulary
SI Base Unit
One of the seven fundamental units from which all other units are derived.
Example: The metre (m) is the SI base unit of length.
Dimensional Analysis
A method of converting units by multiplying by conversion factors arranged so unwanted units cancel.
Example: Converting 60 km/h to m/s: 60 × (1000/3600) = 16.7 m/s.
Conversion Factor
A ratio equal to 1 that expresses the equivalence between two units.
Example: (100 cm / 1 m) is a conversion factor because 100 cm = 1 m.
Derived Unit
A unit expressed as a combination of SI base units.
Example: The newton (N) = kg·m/s² is a derived unit for force.
Interactive Practice — 5 Questions
What is the SI base unit of mass?
Which prefix means 10⁻⁶?
Convert 3.5 km to metres.
Dimensional analysis is used to:
The unit of speed, m/s, is an example of a:
Independent Practice
List the seven SI base units with their symbols and the physical quantity each measures.
Convert 0.045 km to centimetres using dimensional analysis. Show all conversion factors.
A sprinter runs 100 m in 9.58 s. Express her average speed in km/h.
Check whether the equation KE = ½mv² is dimensionally consistent. (KE in joules = kg·m²/s²)
★ The gravitational constant G has units of N·m²/kg². Express G entirely in SI base units (kg, m, s) and verify this is consistent with Newton's law of gravitation F = Gm₁m₂/r².
ChallengeCommon Mistakes
Forgetting to convert both parts of a compound unit (e.g., converting km but not h when finding m/s).
Always convert every unit in the expression. For km/h → m/s, multiply by (1000 m/km) AND by (1 h/3600 s).
Inverting the conversion factor (e.g., multiplying by h/1000 m instead of 1000 m/h).
Set up the conversion factor so the unit you want to eliminate is in the denominator and cancels.
Assuming grams (g) is the SI base unit of mass.
The SI base unit of mass is the kilogram (kg), not the gram.
Math Tips
Write out every unit at every step. Units that appear in both numerator and denominator cancel, just like numbers. If your final units are wrong, you made an error in setting up the conversion factors.
To check dimensional homogeneity, replace every symbol with its dimensions in square brackets: [v] = m/s, [a] = m/s², [t] = s, [m] = kg, [F] = kg·m/s².