Substitution, Unit Checking, and Significant Figures

Once an equation is selected, the mechanical process of solving it begins. Carelessness in this phase causes the majority of engineering calculation errors.

1. Substitution

Substitution is the process of replacing the abstract variables in an equation with the actual numbers and units from your "Given" data.

• Best Practice: Substitute numbers with their units directly into the equation before reaching for a calculator. For example, instead of writing $F = 5 \times 10$, write $F = 5\text{ kg} \times 10\text{ m/s}^2$.
• Why: This visually confirms you have placed the right numbers in the right places.

2. Unit Checking (Dimensional Analysis)

In engineering, a number without a unit is meaningless. Unit checking is the practice of multiplying and canceling the units in an equation just like you do with the numbers.

• If you are calculating velocity ($v = d/t$), and you substitute distance in meters and time in seconds, your final unit must mathematically resolve to meters per second (m/s).
• The Golden Rule: If the units do not cancel out to produce the correct final unit (e.g., if you calculate a force and the units result in kg/m instead of Newtons), your equation or substitution is wrong. You must stop and fix it. Do not just append the correct unit to the end of a wrong calculation.

3. Significant Figures (Precision)

A calculator will happily divide 10 by 3 and give you 3.333333333. In engineering, writing down that many decimal places implies a level of manufacturing precision that is impossible or impossibly expensive to achieve.

Significant figures represent the actual meaningful precision of a number.

• If you measure a pipe with a standard tape measure as 15 cm long, you only have two significant figures. You do not know if it is 15.000 cm or 15.123 cm.
• Rule of Computation: The final result of a calculation cannot be more precise than the least precise measurement used to calculate it. If you multiply 15 cm by 3.14159, your answer should be rounded to reflect the precision of the "15" (e.g., 47 cm, not 47.12385 cm).