Table 0.2 Commonly used derived units.
| Quantity | Unit name | Symbol and base units |
|---|---|---|
| Area | square metre | m2 |
| Volume | cubic metre | m3 |
| Velocity (speed) | metre per second | m s−1 |
| Acceleration | metre per second squared | m s−2 |
| Density | kilogram per cubic metre | kg m−3 |
| Concentration | mole per cubic metre | mol m−3 |
| Energy | joule | J = kg m2 s−2 |
| Force | newton | N = J m−1 = kg m s−2 |
| Pressure | pascal | Pa = N m−2 = kg m−1 s−2 |
| Frequency | hertz | Hz = s−1 |
It is very important when carrying out calculations to keep track of the units, as you will be expected to quote the units of your answers. Units of each value in a calculation multiply or cancel with each other, as shown in the example here:
Acceleration is defined as the change in velocity (speed) divided by the time taken and can be represented by this equation: acceleration = .
Replacing the quantities with their units, we obtain: acceleration = .
The unit can also be written as s−1, so the units for acceleration are m s−1 × s−1.
To multiply s−1 by s−1, we simply add the −1 superscripts: acceleration = m s−1 × s−1 = m s−2.
If we know the speed of an object and wish to determine how far it travels in a certain time, we multiply the speed by the time: distance = speed × time.
Inserting the units for speed and time gives us the unit for distance = m s−1 × s.
To multiply s−1 by s, again we add the superscripts: distance = m s−1 × s = m. This gives us the answer for distance in the correct units of metres. You should always check your answer when doing calculations to make sure that the value you obtain and the units are sensible.
Worked Example 0.1
Determine the derived units for the following quantities using the definitions given:
1 density =
2 entropy =
3 pressure =
4 power =
Solution
1 density = = kg m−3
2 entropy = = = kg m2 s−2 K−1
3 pressure = = = = kg m s−2 × m−2 = kg m−1 s−2
4 power = = = = kg m2 s−2 × s−1 = kg m2 s−3
0.3 Expressing large and small numbers using scientific notation
When studying science, you will likely come across numbers that are either extremely large or very small. It often isn't convenient (or practical) to write out the numbers ‘longhand’. An example of this is in using the constant for the speed of light in vacuum. The value of this constant is approximately three hundred million metres per second (m s−1), which can be written as 300 000 000 m s−1. This method of writing numbers is sometimes called non‐exponential form. A much simpler way of expressing this number is in exponential form, so 300 000 000 m s−1 is written as 3 × 108 m s−1. The number 108 represents one hundred million or 100 000 000. When is it written as × 108 this means there are eight zeroes after the 3. Some numbers are very small: for example, the Planck constant is 6.6 × 10−34 J s. The number 10−34 indicates there are 33 zeroes after the decimal point and before the number itself. Clearly, it would be very tedious to write the number out in full!
Take care!
3 × 102 = 300 (i.e. two zeroes after the number before the decimal point) 3 × 10−2 = 0.03 (i.e. just one zero after the decimal point before the number)
When a number is written in exponential form, it is called scientific notation. The standard format is that the number is written with one digit before the decimal point and the multiplier 10n is added after the number. The following examples show how numbers can be rewritten using scientific notation:
4 is written as 4 × 100.Any number raised to the power of zero is equal to 1, so 100 = 1.
42 is written as 4.2 × 101.It isn't usual to use the factors 100 and 101 unless specifically required to give an answer in scientific notation.
242 is written as 2.42 × 102.
2424 is written as 2.424 × 103.
To express numbers smaller than 1 using scientific notation, i.e. a decimal number, the number is converted to a number between 1 and 10 and the exponential factor 10–n used to indicate the position of the decimal point, as in these examples:
0.4 is written as 4 × 10−1.
0.042 is written as 4.2 × 10−2.
0.00242 is written as 2.42 × 10−3.
Box
