studyblogDimensional Homogeneity:
Dimensional homogeneity is a fundamental principle in physics that requires the dimensions of each term in an equation to be the same on both sides. In other words, the physical units of measurement must be consistent and compatible on both sides of an equation. This principle is based on the fact that one cannot add or equate quantities that have different dimensions.
Here are the key aspects of dimensional homogeneity:
1. Equations and Dimensional Homogeneity:
- In any valid physical equation, the dimensions of each term on the left-hand side (LHS) must be the same as the dimensions of each term on the right-hand side (RHS).
2. Checking Dimensional Consistency:
- Dimensional analysis is often used to check the dimensional consistency of equations. This involves examining the dimensions of each term in an equation to ensure that they match.
3. Deriving Formulas:
- Dimensional analysis is also a useful tool for deriving formulas. By understanding the dimensions of the quantities involved, you can determine how they must be combined to yield a dimensionally consistent equation.
4. Importance in Physics:
- Dimensional homogeneity is a fundamental concept in physics that ensures the validity of equations and relationships between physical quantities.
| Dimension | SI unit | English unit |
|---|---|---|
| Mass [M] | kg (kilogram) | lbm (pound-mass) |
| Length [L] | m (meter) | ft (foot) |
| time [t] | s (second) | s (second) |
| Temperature [T] | K (kelvin) | R (rankine) |
| Amount of matter [N] | mol (mole) | mol (mole) |
| Electric current [I] | A (ampere) | A (ampere |
| Quantity of light [C] | cd (candela) | cd (candela) |
Dimensional Homogeneity in Equations
Rules about dimensions determine how equations are formulated. ‘Properly constructed’ equations representing general relationships between physical variables must be dimensionally homogeneous. For dimensional homogeneity, the dimensions of terms that are added or subtracted must be the same, and the dimensions of the right side of the equation must be the same as those of the left side. As a simple example, consider the Margules equation for evaluating fluid viscosity from experimental measurements:
The terms and dimensions in this equation are listed in Table 2.3. Numbers such as 4 have no dimensions; the symbol π represents the number 3.1415926536, which is also dimensionless. As discussed in Section 2.1.2, the number of radians per second represented by Ω has dimensions T–1, so appropriate units would be, for example, s–1. A quick check shows that Eq. (2.4) is dimensionally homogeneous since both sides of the equation have dimensions L–1MT–1 and all terms added or subtracted have the same dimensions. Note that when a term such as Ro is raised to a power such as 2, the units and dimensions of Ro must also be raised to that power.
Terms and Dimensions in Equation
| Term | Dimensions | SI units |
|---|---|---|
| μ (dynamic viscosity) | L−1MT−1 | pascal second (Pa s) |
| M (torque) | L2MT−2 | newton metre (N m) |
| h (cylinder height) | L | metre (m) |
| Ω (angular velocity) | T−1 | radians per second (s−1) |
| Ro (outer radius) | L | metre (m) |
| Ri (inner radius) | L | metre (m) |
Example
Let us take the simple equation F = ma.
Dimension of force is [M][L][T]−2.
The dimension of mass is [M].
Dimension of acceletation is [L][T]−2.
So, the dimension of mass times acceleration is [M][L][T]−2, which is the same as that of force. This equation is dimensionally homogeneous.
Tips for Understanding Dimensional Homogeneity:
- Learn and Memorize Units: Familiarize yourself with the units of commonly used physical quantities and memorize the dimensions associated with each unit.
- Practice Dimensional Analysis: Regularly practice dimensional analysis by checking the dimensional consistency of various equations.
- Apply Dimensional Analysis in Problem-Solving: Use dimensional analysis to derive and validate formulas, and apply it in solving physics problems.
Understanding dimensional homogeneity is crucial for maintaining the integrity of physical equations and ensuring that the relationships described by these equations are physically meaningful.
