Unit conversion

Often the same quantity is expressed in units different from those we need, and a conversion is required. To do this safely we use the conversion factors method: the quantity is multiplied by fractions equal to $1$ — for example $1\;\text{km}/1000\;\text{m} = 1$ — chosen so as to cancel the unwanted units and leave only the desired ones.

The key idea is that multiplying by a fraction equal to $1$ does not change the physical value of the quantity, but changes the way it is written. By treating units as algebraic factors that simplify in the numerator and denominator, the method becomes almost automatic and drastically reduces the risk of errors.

Consider a tap dripping $30\;\text{cL}$ of water every $4\;\text{s}$: how many litres fill up in 3 days? It is best to first convert everything into consistent units. The time in seconds is

$3\;\text{days} = 3\cdot 24\cdot 60\cdot 60\;\text{s} = 259\,200\;\text{s}$

and the volume lost with each drop, in litres, is

$30\;\text{cL} = 0{,}3\;\text{L}.$

The ratio between volume and time is $0{,}3/4 = 0{,}075\;\text{L/s}$. So in 3 days one collects

$V = 0{,}075\cdot 259\,200 = 19\,440\;\text{L} \approx 19\;\text{m}^3,$

which is the equivalent of a $3\times 3\times 2$ metre closet full of water: a good reason not to overlook a dripping tap.

Key formula

$\begin{aligned} 1\;\text{L} &= 1\;\text{dm}^3 \\ 1\;\text{m}^3 &= 1000\;\text{L} \\ 1\;\text{day} &= 86\,400\;\text{s} \\ 1\;\text{km/h} &= \frac{1}{3{,}6}\;\text{m/s} \end{aligned}$

The last one is especially worth memorising: to go from km/h to m/s you divide by $3{,}6$, and vice versa you multiply by $3{,}6$. It is the most frequent conversion in kinematics.

Exercise generator

Practise conversions with the random generator below: pick the unit type (simple or composite, with optional powers), the notation and the number of significant figures, generate the exercises and try to solve them before revealing the solutions.

Collegamenti

Argomenti: Grandezze fisiche e misura Competenze: Conversione di unità Metodi: Metodo dei fattori di conversione

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