Converting Mass Units with Dimensional Analysis
(a) Convert 2.50 kg to grams. (b) Convert 5.00 lb to kilograms, given 1 lb = 453.6 g. Show the unit cancellation in each step.
Given
Mass 1: 2.50 kg
Mass 2: 5.00 lb
Conversion factors: 1 kg = 1000 g; 1 lb = 453.6 g.
Find
2.50 kg expressed in grams, and 5.00 lb expressed in kilograms.
Principle
In dimensional analysis, a conversion factor is a fraction equal to 1 (like 1000 g / 1 kg) oriented so the unwanted unit cancels. Chaining factors converts through intermediate units — here pounds to grams to kilograms — and the surviving unit at the end must be the target unit, which self-checks the setup.
Formula
Solution
(a) Write 2.50 kg x (1000 g / 1 kg): the kg units cancel, leaving grams.
Multiply: 2.50 x 1000 = 2500, so 2.50 kg = 2500 g.
(b) First convert pounds to grams: 5.00 lb x (453.6 g / 1 lb) = 5.00 x 453.6 = 2268 g; the lb units cancel.
Then convert grams to kilograms: 2268 g x (1 kg / 1000 g) = 2.268 kg; the g units cancel.
Round to three significant figures: 2.27 kg.
Unit check
In each step the starting unit appears once in the numerator and once in the denominator, so it cancels exactly: kg/kg and lb/lb vanish, then g/g vanishes, leaving only the target unit (g in part a, kg in part b). The final units match what was asked for.
What this means physically
Dimensional analysis turns unit conversion into fraction multiplication where units cancel like numbers: 2.50 kg is 2500 g, and 5.00 lb is 2.27 kg (a bit over 2 kg, which makes sense since a pound is under half a kilogram). Checking that units cancel catches inverted factors before they corrupt a calculation.
Students often multiply by the conversion factor upside down — for example, computing 5.00 lb x (1 lb / 453.6 g), which gives a tiny meaningless number. The fix is to set up the factor so the starting unit cancels: if you begin with lb, lb must be in the denominator.