Worked example — Converting compound units

Problem

Converting compound units. What is a quantity of $36\;\text{km/h}$ in metres per second? And one of $7{,}5\;\text{g/cm}^3$ in kilograms per cubic metre?

Solution

Step 1 — From km/h to m/s. We substitute $1\;\text{km} = 10^3\;\text{m}$ and $1\;\text{h} = 3600\;\text{s}$:

$36\;\text{km/h} = \frac{36\cdot 10^3\;\text{m}}{3600\;\text{s}} = 10\;\text{m/s}$

$\ev{36\;\text{km/h} = 10\;\text{m/s}}$

Rule of thumb: to go from km/h to m/s, divide by $3{,}6$ (indeed $10^3/3600 = 1/3{,}6$).

Step 2 — From g/cm³ to kg/m³. We substitute $1\;\text{g} = 10^{-3}\;\text{kg}$ and $1\;\text{cm} = 10^{-2}\;\text{m}$, so $1\;\text{cm}^3 = (10^{-2})^3\;\text{m}^3 = 10^{-6}\;\text{m}^3$:

$7{,}5\;\text{g/cm}^3 = \frac{7{,}5\cdot 10^{-3}\;\text{kg}}{10^{-6}\;\text{m}^3} = 7{,}5\cdot 10^{3}\;\text{kg/m}^3$

$\ev{7{,}5\;\text{g/cm}^3 = 7{,}5\cdot 10^{3}\;\text{kg/m}^3}$

Rule of thumb: it always holds that $1\;\text{g/cm}^3 = 1000\;\text{kg/m}^3$, so you just multiply by a thousand.

Linked notes

Topics: Grandezze fisiche e misura Skills: Conversione di unità Methods: Metodo dei fattori di conversione