Dimensional Reasoning Math Example 1

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Example 1

easy
Use dimensional analysis to find the units of pressure, given that pressure =force/area= \text{force}/\text{area}, force is in Newtons (kg⋅m/s2\text{kg}\cdot\text{m}/\text{s}^2), and area is in m².

Solution

  1. 1
    Pressure =forcearea=kg⋅m/s2m2= \frac{\text{force}}{\text{area}} = \frac{\text{kg}\cdot\text{m}/\text{s}^2}{\text{m}^2}.
  2. 2
    Simplify: kg⋅ms2⋅m2=kgm⋅s2\frac{\text{kg}\cdot\text{m}}{\text{s}^2 \cdot \text{m}^2} = \frac{\text{kg}}{\text{m}\cdot\text{s}^2}.
  3. 3
    This unit is called the Pascal (Pa): 1 Pa=1 kg/(m⋅s2)1\text{ Pa} = 1\text{ kg}/(\text{m}\cdot\text{s}^2).

Answer

[pressure]=kgm⋅s2=Pa[\text{pressure}] = \frac{\text{kg}}{\text{m}\cdot\text{s}^2} = \text{Pa}
Dimensional analysis tracks units through calculations. It is a powerful check: if the units of the final answer are wrong, the formula or calculation must be incorrect.

About Dimensional Reasoning

Using the units and dimensions of physical quantities to check formulas, guide derivations, and eliminate impossible answers.

Learn more about Dimensional Reasoning →

More Dimensional Reasoning Examples

Example 2 medium

A formula for the period of a pendulum is proposed as [formula] where [formula] is length (m) and [f

Example 3 easy

Convert 60 km/h to m/s using dimensional analysis.

Example 4 medium

A student writes [formula] for Einstein's mass-energy formula. Use dimensional reasoning to explain

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