use SI base units to check the homogeneity of physical equations

SI Units & Homogeneity Check

In physics we need to make sure that every side of an equation has the same units. This is called dimensional homogeneity. Think of it like mixing a recipe – you can’t put a cup of sugar and a meter of flour together and expect a cake!

The 7 SI Base Units

Checking Homogeneity: A Step‑by‑Step Guide

1. Write down the equation you want to check.
2. Replace each physical quantity with its SI base units.
3. Combine the units on each side of the equation.
4. Make sure the exponents of each base unit match on both sides.

📐 Example 1: Newton’s Second Law – \$F = m a\$

Units:

• \$F\$ → \$\mathrm{kg\,m\,s^{-2}}\$ (newton)
• \$m\$ → \$\mathrm{kg}\$
• \$a\$ → \$\mathrm{m\,s^{-2}}\$ (metre per second squared)

Right side: \$\mathrm{kg} \times \mathrm{m\,s^{-2}} = \mathrm{kg\,m\,s^{-2}}\$, which matches the left side. ??

Homogeneous!

⚖️ Example 2: Kinetic Energy – \$E = \tfrac{1}{2} m v^2\$

Units:

• \$E\$ → \$\mathrm{J}\$ (joule)
• \$m\$ → \$\mathrm{kg}\$
• \$v\$ → \$\mathrm{m\,s^{-1}}\$

Right side: \$\tfrac{1}{2} \times \mathrm{kg} \times (\mathrm{m\,s^{-1}})^2 = \tfrac{1}{2} \times \mathrm{kg} \times \mathrm{m^2\,s^{-2}} = \mathrm{kg\,m^2\,s^{-2}}\$ which is exactly a joule. ??

Homogeneous!

Common Pitfalls

• Mixing SI units with other systems (e.g., using pounds or inches).
• Forgetting to square or cube units when raising to a power.
• Assuming a dimensionless constant (like 2 or 0.5) can change the unit balance.

Quick Reference Cheat Sheet