Substitution, Unit Checking, and Significant Figures

Once an equation is selected, the mechanical process of solving it begins. Carelessness during this stage is the cause of most errors in engineering calculations.

1. Substitution

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

• Best Practice: Substitute the number and its unit directly into the equation before picking up 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 that you have placed the correct numbers in the correct positions.

2. Unit Checking (Dimensional Analysis)

In engineering, numbers without units are meaningless. Unit checking is the practice of multiplying and canceling units in an equation exactly as you do with numbers.

• If you are calculating velocity ($v = d/t$) and you input distance in meters and time in seconds, your final units must mathematically process out to meters per second (m/s).
• Golden Rule: If the units do not cancel out to yield the correct final unit (e.g., if you are calculating Force, but your resultant unit is kg/m instead of Newtons), your equation or your substitution is wrong. You must stop and fix it. Never just append the correct unit to the end of a flawed calculation.

3. Significant Figures

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 either physically impossible or prohibitively 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.
• Calculation Rule: The final result of a calculation can be no more precise than the least precise measurement used in the calculation. If you multiply 15 cm by 3.14159, your answer should be rounded to reflect the precision of "15" (e.g., 47 cm, not 47.12385 cm).