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Relative Density Dimensional Formula

Dimensional formulas of physical quantitiesClass 9Class 11Updated 1 Nov 2026

[RD]=[ML−3][ML−3]=[M0L0T0][RD] = \frac{[ML^{-3}]}{[ML^{-3}]} = [M^{0} L^{0} T^{0}]
SI unit
no unit (a pure ratio)
Dimensions
[M⁰ L⁰ T⁰]
where
  • ρ density of the substance kg/m³
  • ρw density of water at 4 °C kg/m³

The relative density dimensional formula reads [M⁰ L⁰ T⁰]. Every power is zero, which means relative density is dimensionless and has no unit at all. Relative density is also called specific gravity.

It is defined as the density of a substance divided by the density of water at 4 °C, which is 1000 kg/m³. Both densities carry [ML⁻³], so mass and length cancel from top and bottom. Iron, with a density of 7870 kg/m³, has relative density 7.87, with nothing written after the number. Ways to calculate it from weights and volumes are shown under the relative density formula.

Quick facts

  • Dimensional formula

    [M⁰ L⁰ T⁰]

  • Defining formula

    RD=ρsubstanceρwaterRD = \frac{\rho_{substance}}{\rho_{water}}

  • SI unit

    None, it is a pure ratio

  • Other name

    Specific gravity

  • Reference

    Water at 4 °C, 1000 kg/m³

  • Type

    Scalar, dimensionless

Proof of the relative density dimensional formula

  1. Write the definition: RD=ρsρwRD = \frac{\rho_s}{\rho_w}.
  2. Density is mass per volume, so [ρ]=[M][L3]=[ML−3][\rho] = \frac{[M]}{[L^{3}]} = [ML^{-3}] for both the substance and water.
  3. Divide: [RD]=[ML−3][ML−3][RD] = \frac{[ML^{-3}]}{[ML^{-3}]}.
  4. Subtract powers: [M1−1L−3+3T0]=[M0L0T0][M^{1-1}L^{-3+3}T^{0}] = [M^{0}L^{0}T^{0}].

The same cancelling happens with the weight form used in experiments. Relative density equals the weight of a body divided by the weight of an equal volume of water. Both are forces with [MLT⁻²], so their ratio again has zero powers. The density bracket is the one building block you need.

Why relative density has no unit

A unit only survives when dimensions survive. Here they all cancel, so the kg/m³ on top cancels the kg/m³ below. The same is true in any unit system: in CGS, iron is 7.87 g/cm³ and water 1.00 g/cm³, and the ratio is still 7.87.

This leads to a handy shortcut. Because water is 1 g/cm³, the density of any substance in g/cm³ is numerically equal to its relative density. Multiply the relative density by 1000 to get the SI density in kg/m³.

Writing "RD = 7.87 kg/m³" is wrong. Either write RD = 7.87, or write density = 7870 kg/m³.

Worked example with Archimedes' principle

Example: a stone hangs from a spring balance that reads 6.0 N in air and 3.6 N once the stone is dipped completely in water. Work out its relative density, then its density.

Upthrust from the water: Wair−Wwater=6.0−3.6=2.4 NW_{air} - W_{water} = 6.0 - 3.6 = 2.4\ \text{N}.

RD=WairWair−Wwater=6.0 N2.4 N=2.5RD = \frac{W_{air}}{W_{air} - W_{water}} = \frac{6.0\ \text{N}}{2.4\ \text{N}} = 2.5. The newtons cancel, just as the zero powers predict.

Density: ρ=2.5×1000=2.5×103 kg/m3\rho = 2.5 \times 1000 = 2.5 \times 10^{3}\ \text{kg/m}^{3}.

Cross check: mass = 6.0/9.8 = 0.612 kg, and the stone's volume equals the water it pushes aside, 2.4/(1000 × 9.8) = 2.45 × 10⁻⁴ m³. Dividing gives 2.50 × 10³ kg/m³, so both routes agree.

Other ratios with zero powers

Common dimensionless quantities
QuantityRatio of
Relative densityDensity over density of water
StrainChange in length over original length
Refractive indexSpeed of light in vacuum over speed in medium
Coefficient of frictionFriction force over normal force
Angle in radiansArc length over radius
Relative permittivityPermittivity over ε₀

A dimensional check cannot detect a mistake in a pure ratio, since every such ratio has the same bracket. For relative density, check instead that your answer makes sense: anything that floats in water must have relative density below 1. See also bulk density, which, unlike relative density, keeps the unit kg/m³.

Errors with relative density dimensions

  • Giving relative density the bracket of density. Density is [ML⁻³]; the ratio is [M⁰ L⁰ T⁰].
  • Attaching a unit. No kg/m³ and no g/cm³ after a relative density.
  • Using the wrong reference. The standard is water at 4 °C, not air and not sea water.
  • Thinking dimensionless means zero. Mercury has relative density 13.6; it has no unit, not no value.

For a full list of quantities with and without dimensions, browse the dimensional formulas list, where every zero power bracket is flagged.

Frequently asked questions

What is the dimensional formula of relative density?

Every power in it is zero: [M⁰ L⁰ T⁰]. Relative density is density divided by density, so mass and length cancel and nothing remains.

Does relative density have a unit?

None at all. Two quantities with identical units are divided, so the units cancel. Write the bare number, for example 0.92 for ice.

Are relative density and specific gravity the same?

Yes. Both names mean the density of a substance compared with the density of water at 4 °C, and both are dimensionless.

How do you get density from relative density?

Multiply by the density of water. In SI that means multiplying by 1000 kg/m³, so relative density 2.7 for aluminum gives 2700 kg/m³.

Why is relative density the same in SI and CGS?

Since the dimensions cancel, the conversion factors for mass and length cancel too. The ratio does not depend on the unit system.