The dimensional formula for relative density is [M0L0T0], meaning it is a dimensionless quantity.
Understanding Relative Density and Dimensions
Relative density (also known as specific gravity) is the ratio of the density of a substance to the density of a reference substance, usually water for liquids and solids, or air for gases. Because it's a ratio of two densities, the units cancel out, making it a dimensionless quantity.
Dimensional Analysis Explained
Dimensional analysis is a powerful tool used to check the relationships between physical quantities by identifying their dimensions. The fundamental dimensions are usually mass (M), length (L), and time (T). Other dimensions like temperature (Θ), electric current (I), amount of substance (N), and luminous intensity (J) can also be included depending on the context.
Density, defined as mass per unit volume, has the dimensional formula of [ML-3].
Since relative density is calculated as:
Relative Density = (Density of Substance) / (Density of Reference Substance)
Then, dimensionally:
Relative Density = [ML-3] / [ML-3] = [M0L0T0]
This clearly shows that relative density is dimensionless. It's a pure number without any physical units associated with it.
Example
Consider a block of aluminum. Let's say its density is 2700 kg/m3, and the density of water (our reference substance) is 1000 kg/m3.
Relative Density of Aluminum = (2700 kg/m3) / (1000 kg/m3) = 2.7
The units (kg/m3) cancel out, leaving us with a dimensionless number, 2.7.
Importance of Dimensionless Quantities
Dimensionless quantities play crucial roles in physics and engineering. They are used in:
- Similarity analysis: Allowing the scaling of models to represent larger systems.
- Fluid mechanics: The Reynolds number, a dimensionless quantity, determines whether fluid flow is laminar or turbulent.
- Heat transfer: The Nusselt number is a dimensionless number that represents the ratio of convective to conductive heat transfer.
Because relative density is dimensionless, it is independent of the system of units used (SI, CGS, etc.). The value will remain the same, regardless.
