Dimensional Consistency of Equations

CBSE Class 11 Physics: Chapter 1 – Units and Measurements (Section 1.6.1)

📘 1.6.1 – Dimensional Consistency of Equations

⚙️ Principle of Dimensional Homogeneity

The principle of homogeneity states:

An equation is dimensionally consistent (or homogeneous) if the dimensions of each term on both sides of the equation are the same.

Mathematically:

$$\text{If } A = B + C + D \quad \Rightarrow \quad [A] = [B] = [C] = [D]$$

✅ Only like dimensions can be added, subtracted, or equated.

📏 Why Check Dimensional Consistency?

• It helps verify correctness of derived equations.

• It ensures all terms are physically compatible.

• It provides a preliminary check before experimental validation.

• It is independent of unit systems, saving effort in unit conversions.

❗ Limitation: A dimensionally correct equation may not always be physically correct, but a dimensionally incorrect one is certainly wrong.

✅ Example: Kinematic Equation

$$x = x_0 + v_0 t + \frac{1}{2} a t^2$$

Where:

• $x$: displacement

• $x_0$: initial position

• $v_0$: initial velocity

• $a$: acceleration

• $t$: time

Check Dimensions:

• $[x] = [x_0] = \text{Length} = [L]$

• $[v_0 t] = [L T^{-1}] \cdot [T] = [L]$

• $\left[\frac{1}{2} a t^2 \right] = [L T^{-2}] \cdot [T^2] = [L]$

✅ All terms have dimension $[L]$ → dimensionally consistent.

🎯 Dimensional Check for:

$$\frac{1}{2} m v^2 = m g h$$

Where:

• $m$: mass $[M]$

• $v$: velocity $[L T^{-1}]$

• $g$: acceleration $[L T^{-2}]$

• $h$: height $[L]$

🔍 Left Side:

$$\left[\frac{1}{2} m v^2\right] = [M] \cdot [L T^{-1}]^2 = [M] \cdot [L^2 T^{-2}] = [M L^2 T^{-2}]$$

🔍 Right Side:

$$[m g h] = [M] \cdot [L T^{-2}] \cdot [L] = [M L^2 T^{-2}]$$

✅ Both sides = $[M L^2 T^{-2}]$ → dimensionally correct equation

📌 This is the equation of conservation of mechanical energy for a body under gravity.

⚠️ Special Note on Functions

• Arguments of functions like sin, cos, tan, log, exp must be dimensionless.

• For example:

  • $\sin(\theta)$, where $\theta$ is angle in radians (a pure number: $[1]$)

  • $\exp(x)$, $\log(x)$ must have $x$ as dimensionless

🔁 Difference Between Units and Dimensions

• Units: Based on a chosen standard (like meter, second)

• Dimensions: Inherent nature of a quantity (like Length $[L]$, Mass $[M]$)

Testing dimensions is easier and more general than testing units.

✅ Key Takeaways

• An equation is dimensionally consistent if all its terms share the same dimensions.

• A dimensionally inconsistent equation is always wrong.

• Dimensionally consistent ≠ always physically valid.

• Functions like sin, log, exp only accept dimensionless inputs.

• Helps derive or verify formulas without performing actual experiments.