To be able to convert from one type of unit to another, we need to know the equivalence between both units. We can convert between units as long as both use the same basic units because they must measure the same physical property.
Consistency in converting units
Consistency is needed when converting units. We cannot convert time to length, but we can translate time to frequency because both use time as a base. We can also convert power units to watts as energy per second units, and so on. Let’s look at these two examples in more detail.
We want to convert the oscillation of a pendulum in time to its frequency. The period (T ), expressed in seconds, is the time it takes to complete one cycle of an oscillation. The frequency (f ) is the number of occurrences of a repeating event per unit of time and is measured in hertz. The formula to convert from period to frequency is \(ƒ = \frac{1}{T}\).
The inverse of the value ‘x’ for T in seconds gives us the value ‘Y’ in Hertz.
\[\frac{1}{x[seconds]} = Y[Hertz]\]
If the pendulum takes 3.2 seconds to come and go, we need to divide 1 by 3.2 seconds.
\(frecuency = \frac{1}{3.2 \space seconds}\)
This gives us 0.3125 [Hertz].
Let’s say we have a machine that consumes 60 watts of power each second. We want to convert power consumed to energy per second. The equation that links power, energy, and time is:
\[P = \frac{E}{t}\]
Here, P is the power in watts, E is the energy in joules, and t is the time it takes to consume or produce energy in seconds. If the machine’s consumption and energy production are measured each second, then 60 watts means 60 joules every second.
The relationship below expresses this better. Here, every watt unit is equivalent to a unit of Joules per second.
\[[watts] = \frac{[joules]}{1[second]}\]
Replace the ‘watts’ with 60:
\(60 \space watts = \frac{60 \space joules}{1 \space second} = 60 \space joules/second\)
Now let’s say we have a machine that produces 100 joules watts each minute. We want to know how much power is produced every second. We have to divide the amount of energy in joules by the number of seconds it took the machine to produce 100 watts.
\(\frac{100 \space joules}{60 \space seconds} = 1.67 \space joules/second\)
As we know, Joules per second is watts.
\(1.67 \space joules/second = 1.67 \space watts\)
Converting larger units to smaller units
To convert larger units to small ones, we need to multiply by a factor. If we wish to convert from different scales and units, we need to multiply by two factors to scale the number and convert between derived units.
Converting basic units from different scales
To convert between basic units from a smaller to a larger scale, we need to multiply by a factor. If A is ten times B, we need to multiply B by 10 to obtain A. Let’s look at some examples.
We want to convert 1,234 tons to kilograms. We know that a ton is 1000 kilograms, so we can carry this out by multiplying 1,234 by 1000. This gives us 1234 kilograms.
We want to convert 0.3 metres to millimetres. We know that 1 millimetre is equal to \(1 \cdot 10 ^ {-3}\) meters, so we need to divide 0.3 by \(1 \cdot 10 ^ {-3}\), which gives us 300 millimetres.
We can also convert from metres to millimetres by multiplying by 1000, as 1 metre equals 1000 millimetres.
Converting derived units from different scales
To convert between derived units and from a larger scale to a smaller one, we need to multiply by several factors. Consider the following example.
Convert 10 km/h to m/s.
Our calculations are more complex here. First, we need to convert 10 kilometres to metres. To convert kilometres to metres, we use the factor of \(1 \cdot 10 ^ 3\), giving us a velocity of 10000 [m/h].
\(10 \space km/h = 10 \cdot (1 \cdot 10^3) m/h = 1000 \space m/h\)
Now we need to convert from hours to seconds. This factor equals 3600, as 1 hour is equal to 60 minutes, and each minute to 60 seconds.
We must, therefore, divide 10000m by 3600s.
\(\frac{1000 \space m/h}{3600} = 2.8 \space m/s\)
The result is 2.8 m/s.
You can use a rule of thumb to calculate km/h to m/s just by dividing the number of km/h by 3.6.
If we do this at 10km/h, we obtain the same result:
\(\frac{10 \space km/h}{3.6} = 2.8 \space m/s\)
Converting smaller units to larger units
To convert smaller to larger units, we need to divide by a factor. As noted earlier, if we wish to combine conversion from different scales and units, we need to divide by two factors, one to scale the number and another to convert between derived units.
Converting basic units from different scales
To convert between basic units from a smaller to a larger scale, we need to divide by a factor. If, for instance, A is ten times larger than B, we need to divide A by 10 to obtain B. See the following two examples:
We want to convert 23.4 m to kilometres. As one kilometre is 1000 metres, we need to divide 23.4 by 1000, which gives us 0.023 kilometres.
We wish to convert 400 kelvin to megakelvin. The prefix ‘mega’ means \(1 \cdot 10 ^ 6\), so one megakelvin is one million kelvin. Dividing 400 kelvin by 1,000,000 gives us 0.0004 megakelvin.
Converting derived units from different scales
To convert between derived units from small to large scales, we need to use several factors. A more complex conversion from watts to kilonewton-metres per second is required, for instance, in the example below.
We have a machine that consumes 1300 watts. The prefix kilo is equivalent to \(1 \cdot 10 ^ 3\) in standard form. This means that we need to divide 1300 watts by \(1 \cdot 10 ^ 3\) to get kilowatts.\(1300 \space watts = 1.3 \space kilowatts\)
In a second step, we need to convert kilowatts to newton-metres per second. As 1 watt is equivalent to 1 newton-metre per second, this is straightforward. 1.3 kilowatts are equal to 1.3 kilonewton-metres per second.
\(1.3 \space kilowatts = 1.3 \space kiloNewtons \cdot m/s\)
Converting units from different systems
We might need to convert units from different systems, such as the imperial and SI systems. Converting temperature, volume, and length between imperial and SI units are three common operations. A simple way to convert between imperial and SI is by using weights. Multiplying the imperial or SI values by the correct weight gives us the value in the other unit system.
Figure 1. Gallons are a commonly used unit from the imperial system.
The following table lists the conversion weights for converting between the imperial and SI systems.
Imperial to SI | SI to imperial |
Imperial unit | Conversion weight | SI unit | Conversion weight |
1 gallon | 3.7854 litres | 1 litre | 0.264172 gallons |
1 mile | 1.60934 kilometres | 1 kilometre | 0.621371 miles |
1 foot | 0.3048 metres | 1 metre | 3.28084 feet |
1 pound | 0.453592 kilograms | 1 kilogram | 2.20462 pounds |
To convert between Fahrenheit and Celsius, we need to use the formulas below:
\(C^{0} = \frac{5(F^{0} - 32)}{9}\)
\(F^{0} = 1.8 C^0 + 32\)
Examples of converting between imperial and SI units
The conversion of units between systems is very common in everyday life, as the imperial system and the US Customary System of units (USCS) are still widely used. See the following examples.
The outside temperature is given as 32 degrees Fahrenheit. How much is that in Celsius?
\(C^0 = \frac{5(F^0 -32)}{9}\)
Replacing F0 with 32, we get:
\(C^0 = 0\)
It is a cold day!
You need to refuel your rental car during your holidays in the USA. The car is from Europe, and the tank has a capacity of 40 litres. The filling station sells the fuel in gallons, which cost $3.10 (USD). How much would it cost you to fill up the tank?
First, you need to convert the 40 litres to gallons applying the unit factor weight in the table above, which tells you that 1 litre is equivalent to 0.26 gallons.
\(40 \space liters = 40 \cdot (0.26 \space gallons) = 10.4 \space gallons\)
Then, you need to multiply this by the price of $3.10.
\(10.4 \cdot 3.1 [USD] = 32.24 [USD]\)
Converting Units - Key takeaways
- Converting units allows us to translate values from one type of physical quantity to another as, for instance, when we need to translate the work done by a device to the amount of energy per second used by that device to perform the work.
- Converting units between different scales helps us understand the scale of the values with which we are working as, for instance, when we convert the speed of an aeroplane from km/h to m/s.
- Unit conversion is present in all fields of science and technology.
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