Kilogram

The kilogram ($\mathrm{kg}$) is the SI base unit for Mass.

Mass measures inertia: how strongly an object resists changes in motion. It is not the same as weight. Weight is a Force caused by gravity and is measured in Newtons:

$$W=mg$$

The modern SI defines the kilogram by fixing the Planck constant exactly:

$$h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$$

Since

$$1\mathrm{J}=1\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}$$

we have

$$\mathrm{J}\mathrm{s}=\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-1}$$

Once the Metre and Second are defined, fixing $h$ fixes the kilogram. In practice, this is realised with precision experiments such as a Kibble balance, which compares mechanical power to electrical power.

Useful mechanics links:

$$F=ma\Rightarrow 1\mathrm{N}=1\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}$$ $$E_k=\frac{1}{2}m{v}^2\Rightarrow \mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}=\mathrm{J}$$

Andrew-friendly check: if a formula contains mass, ask whether it is describing inertia, momentum, energy, or gravitational force. The unit trail usually reveals which one.

Common mistake: kilograms are not “amount of matter” in a vague everyday sense; in mechanics, treat them as the unit that scales acceleration response.

Quick check

Mass is not weight. A kilogram is a unit of mass; weight is a force. Near Earth’s surface, a $1\mathrm{kg}$ mass has weight approximately

W=mgpprox 9.8\,\mathrm{N}.

So if a problem says “weight in kg”, mentally translate carefully: it probably means mass, not force.