$kg m^{-3}$
Explanation
Density is a physical quantity defined as mass per unit volume. In the International System of Units (SI), derived units are constructed from the seven base units. The calculation of the unit for density requires the base unit for mass and the derived unit for volume.
Detailed Analysis
- Formula for Density: $\text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)}$
- SI Unit of Mass: The kilogram ($kg$).
- SI Unit of Volume: The cubic meter ($m^3$), derived from the base unit of length (meter).
- Derivation: Substituting the units into the formula:
$\text{Unit of Density} = \frac{kg}{m^3} = kg m^{-3}$
Analysis of Incorrect Options:
- $kg m^{-2}$ $kg m^{-2}$: This represents surface density (mass per unit area).
- $g cm^{-3}$ $g cm^{-3}$: This is the unit of density in the CGS (Centimetre-Gram-Second) system, not the SI system.
- $N m^{-3}$ $N m^{-3}$: This represents specific weight (weight per unit volume), where Newton ($N$) is the unit of force.
Key Takeaway: The SI unit of density is a derived unit obtained by dividing the base unit of mass ($kg$) by the derived unit of volume ($m^3$), resulting in $kg m^{-3}$.