use SI base units to check the homogeneity of physical equations

SI Units – Checking the Homogeneity of Physical Equations

1. Why Homogeneity Matters

In physics every valid equation must be dimensionally homogeneous: each term on both sides of the equation must have the same SI base units. This provides a quick check for algebraic mistakes and helps in deriving new relationships.

2. The Seven SI Base Units

Quantity	Symbol	SI Base Unit	Unit Symbol
Length	ℓ	metre	m
Mass	m	kilogram	kg
Time	t	second	s
Electric current	I	ampere	A
Thermodynamic temperature	T	kelvin	K
Amount of substance	n	mole	mol
Luminous intensity	Iv	candela	cd

3. Common Derived Units (selected)

Quantity	Symbol	Derived Unit (SI)	Unit Symbol	Base‑unit expression
Velocity	v	metre per second	m s⁻¹	m s⁻¹
Acceleration	a	metre per second squared	m s⁻²	m s⁻²
Force	F	newton	N	kg m s⁻²
Energy	E	joule	J	kg m² s⁻²
Power	P	watt	W	kg m² s⁻³
Pressure	p	pascal	Pa	kg m⁻¹ s⁻²
Electric charge	Q	coulomb	C	A s
Voltage	V	volt	V	kg m² s⁻³ A⁻¹

4. Dimensional Symbols

When performing dimensional analysis we replace each quantity by its dimensional symbol:

• Length → $[L]$
• Mass → $[M]$
• Time → $[T]$
• Current → $[I]$
• Temperature → $[\Theta]$
• Amount of substance → $[N]$
• Luminous intensity → $[J]$

5. Procedure for Checking Homogeneity

1. Write the equation in symbolic form.
2. Replace each physical quantity with its dimensional symbol.
3. Combine the symbols using algebraic rules (e.g., $[v]=[L][T]^{-1}$).
4. Ensure that the dimensions on the left‑hand side (LHS) equal those on the right‑hand side (RHS).
5. If they differ, locate the algebraic error or missing factor.

6. Worked Examples

Example 1 – Newton’s Second Law

Equation: $F = m a$

Dimensions:

$$[F] = [M][L][T]^{-2},\qquad [m] = [M],\qquad [a] = [L][T]^{-2}$$

Since $[M][L][T]^{-2} = [M]\times[L][T]^{-2}$, the equation is homogeneous.

Example 2 – Kinetic Energy

Equation: $E_k = \frac{1}{2} m v^2$

$$[E_k] = [M][L]^2[T]^{-2},\qquad [m] = [M],\qquad [v]^2 = ([L][T]^{-1})^2 = [L]^2[T]^{-2}$$

Both sides give $[M][L]^2[T]^{-2}$ – the equation is dimensionally consistent.

Example 3 – Incorrect Form of the Wave Speed

Supposed equation: $v = \frac{f}{\lambda}$ (where $f$ is frequency, $\lambda$ wavelength)

$$[v] = [L][T]^{-1},\qquad [f] = [T]^{-1},\qquad [\lambda] = [L]$$

Right‑hand side dimensions: $[f]/[\lambda] = [T]^{-1}[L]^{-1}$, which is $[L]^{-1}[T]^{-1}$, not $[L][T]^{-1}$. The correct relation is $v = f\lambda$.

7. Common Pitfalls

• Forgetting that $1\ \text{N}=kg\,m\,s^{-2}$, not $kg\,m\,s^{-1}$.
• Mixing units (e.g., using cm with s without conversion).
• Dropping constants with dimensions (e.g., $k$ in Hooke’s law $F = -kx$ has units $N\,m^{-1}$).
• Assuming angles are dimensionless in all contexts – in SI they are treated as radian (dimensionless), but when combined with other quantities you must keep track of the context.

8. Practice Questions

1. Show that the period $T$ of a simple pendulum $T = 2\pi\sqrt{\frac{L}{g}}$ is dimensionally homogeneous.
2. Check the homogeneity of the equation for electric power $P = IV$.
3. Determine whether the expression $E = mc^3$ could represent an energy. Justify using dimensional analysis.

9. Suggested Diagram

10. Summary

Using SI base units to verify the homogeneity of physical equations is a powerful, quick‑check tool. By consistently applying dimensional symbols and the rules of algebra, you can spot errors, confirm derived formulas, and deepen your understanding of the relationships between physical quantities.