Unit 4 units and Uncertainties

4.1 Units, prefixes and scientific notation

Learning Outcomes

  1. Use SI units of all physical quantities appearing in the Content Statements.
  2. Use SI units of all physical quantities appearing in the Content Statements.
  3. Give answers to calculations to an appropriate number of significant figures.
  4. Check answers to calculations.
  5. Use prefixes (p, n, , m, k, M, G)
  6. Use scientific notation.

Aids to Understanding

Good Practice

In Higher Physics it is important we think before we write our answers down. For example if you have two pure number like 2 and 3 and divide them then we get 0.66666..recurring. However in physics we work with quantities or values that have units. So we would have 2.0 m / 3.0 m = 0.67 m/s. This is the correct answer to this calulation because our measured values do not justify an answer of greater accuracy despite what our calulator presents to us.

In this topics we discuss these matters more fully and a little bit about prefixes and scientific notation.

1) Units in Physics.

In the mathematics class we work with numbers, in the physics class we work with values. Each of the values we work with has a unit, for example 5.0 m or 10.0 s. Note we never (or should never) write 5 m or 10 s. The reason for this will be clear shortly but you should be starting to appreciate that in physics we work with values with units where in mathematics they mainly use unitless numbers; this means we need to treat them differently.

2) The International Standard

In physics we use the International Standard. This means instead of feet and pounds, we use meters and grams. Seconds, Coulombs and degrees celsius are also among the international standard of units. These units are fundamental in that they are not made up of other units and must be measured directly. Newtons for example are made from units of kilograms, metres and seconds (kgms-2) since F = ma.

3) Significant figures

Suppose you wish to measure the average speed of a car travelling down a slope; the measuring devices needed are a stopwatch and metre stick. Suppose the length of the slope was was 1.22 m and the time was 2.32 s. If we put these values in a calculator and divide the distance by the time we get 0.52586206896551724137931034482759 m/s. Now this is an incredibly accurate figure, but we never used an incredibly accurate stopwatch or metre stick. Since there are only three significant figures in the time and distance we can only have three significant figures in our answer. Thus our speed should be written as 0.526 m/s. If the time was 2.3 s and the length was 1.22 m then our answer should be taken taken to two significant figures only (since there are only two significant figures in the time) ie 0.53 m/s The moral of the story is do not write a value that has lots of figures. And if you are not sure keep the number of figures to three at the very most.

4) Calculations in Physics

This content statement is really a word of advice. If you have pressed multiple buttons on your calculator write the answer down. However it is wise to repeat the process because we can all mike typos.

5) Prefixes

If we have a large number of grams, say 10000.0g then we would normally write 10.0 kg. Similary, 0.001 m, would be written as 1.0 mm. The kg and mm are called prefixes. Prefixes increase in multiples of 103. Prefixes less than one are indicated by a smaller case letter, indexes greater than one are indicated by an upper case letter. The prefix kg is an exception to this rule. Some prefixes are listed below.
Prefix Magnitude
p - pico 10-12
n - nano 10-9
m - micro 10-6
m - milli 10-3
k - kilo 103
M - Mega 106
G - Giga 109

Thus if we have 8900000.0 J we would write it as 8.9 MJ, similary 0.000000555 m would be written as 555 nm

6) Scientific notation

Scientific notation is another way to express our values. In scientific notation we express our numbers as a number (often called a co-efficient) multiplied by a power of ten. The coefficient is between 1 and 10. The table below provides some examples.
Number Scientific Notation
678000 6.78 x 105
2.0 2.0 x 100
589 nm 5.89 x 10-7 nm
0.00003 3.0 x 10-5
840 GJ 8.4 X 1011J

Higher Physics .. Unit 1 Mechanics .. 4.1 Units, prefixes and scientific notation .. BR (2012)