Induction Tutorial 8  Orders of Magnitude
Contents 
When people want a job doing on their house, they get a builder in to give them an estimate. The builder looks at the job that needs doing and uses his experience to work out in his mind:
What materials he needs;
What techniques he will need to use;
What, if any, specialist equipment he will need to hire in;
How many people he will need to help him to do the job;
How long the job will take to do.
From those mental calculations, he will tell the owners how much, roughly, the job will cost. He has given them an estimate. (A quotation is a much more detailed affair and gives an exact price that will be charged, which will form the basis of a contract.)
If the builder is not very good at giving estimates, he will find that his price is way out. Either the customer will not be very pleased at having to pay a larger sum than expected, or the job will be done at a loss. Either way the builder won’t be in business for very long…
Physicists need to acquire the same skill. They need to use estimates to:
See if an answer is reasonable;
Make a comparison;
Provide data to ask questions;
To see if a question is worth following.
A good general understanding of physics will allow us to make reasonable estimates. It is a good idea to get used to doing back of the envelope calculations, using the physics principles you know and realistic quantities that you may have to look up, for example, the mass of a car being about 1200 kg.
Image from Wikimedia Commons
Consider this question about a cheetah. The
cheetah is the largest of the purring cats and is the world’s fastest land
animal. It feeds on antelopes in Africa.
Here is the question:
A cheetah is running at 30 m s^{1}. He sees an antelope and accelerates at 4 m s^{2} for 10 seconds. He then maintains this new speed for a further 500 s.
(a) What is his new speed?
(b) How far does he travel while running at that speed?
Show that the answer to (a) is 70 m s^{1} and (b) is 35 000 m 

Do you think that these answers are reasonable? 
The question above appeared in resource materials that were produced commercially and schools paid considerable prices for the photocopy masters. Think about the numbers:
70 m s^{1} is double the motorway speed limit, about 230 km/h. A cheetah can sprint at about 35 m s^{1}.
The cheetah can sprint at that speed for about 5 seconds. Although the cheetah runs fast, it can run for no more than 100 m. Most antelopes (which are no slugs) get away.
After 3 seconds, the oxygen debt in the cheetah is so severe that it collapses in a panting heap, and does not feed for several minutes. In that state, it is actually very vulnerable to passing jackals, hyenas, or leopards that would quite happily help themselves to its hardcaught meal, and, quite often, to the cheetah itself.
A wolf, like all dogs, has very high stamina, and will trot at 8 m s^{1} for hours. It could run 35 km quite easily.
Website: https://www.grc.nasa.gov/www/k12/Numbers/Math/Mathematical_Thinking/fermis_piano_tuner.htm
As a lecturer, Enrico Fermi used to challenge his classes with problems that, at first glance, seemed impossible. One such problem was that of estimating the number of piano tuners in Chicago given only the population of the city.
The population of Chicago was, at the time, 3 × 10^{6}.
There is now a little problem:
Chicago is a very run down area. Not many people own traditional pianos.
If they have a keyboard, it would be electronic. So it doesn't need tuning.
So there...
To make sense of the physical world, we often make comparisons to things we are familiar with. For example we could say that London (a city in the South of England) is ten times bigger than Leeds (a city in Yorkshire). Or that a very large animal is the length of two doubledecker buses.
We saw how a badly written question compared the sprint speed of a cheetah to the speed of a highspeed electric train, or an aeroplane.
By making comparisons, we come on to the important skill of getting a scale of objects.
What is wrong here? How high would you estimate the room to be? 
Using scales, we can represent bigger or smaller objects in a context that is meaningful to us. This is particularly useful when the object is too big for us to handle, for example a fullsized aeroplane, or when the object is far too small for us to see, for example an arrangement of atoms.
Here is a scale drawing of a car.


How wide is this shape?
Does it matter that the view is at an angle? 
Magnitude is a word that means size or value.
Journalists tend to use the term orders of magnitude as a piece of meaningless padding.
It is a scientific term that means this:
If an object is ten times greater (or smaller) than another object, it is an order of magnitude greater (or smaller).
The size changes as a power of 10. So, if an object is 4 orders of magnitude bigger than another object, it is 10^{4} times bigger (10 000 times bigger). Mathematicians call this a logarithmic scale.
Look at this table:
1 m 
Human scale – the average British person is 1.69 m 
10 m 
The height of a house 
100 m 
The diameter of a city square, like George Square 
10^{3} m 
The length of an average street 
10^{4} m 
The diameter of a small city like Perth 
10^{5} m 
Distance between Aberdeen and Aviemore or Stirling and Ayr 
10^{6} m 
Length of Great Britain 
10^{7} m 
Diameter of Earth 
Of course this includes things that are bigger than we are, but there is no reason why we cannot go to much smaller things. Also we are tending to think of distances, but we can apply the same arguments to other quantities, like currents, temperatures, and so on.
The table on the next couple of pages illustrates the orders of magnitude from the very small, to the very large.
Theoretical physics has suggested distances of 10^{38} m, and particle physics experiments have modelled conditions 10^{44} s after the Big Bang.
Size 
Powers of 10 
Examples 


10^{–18} m 
Size of an electron? Size of a quark? 


10^{–17} m 



10^{–16} m 


1 fm (femto) 
10^{–15} m 
Size of a proton 


10^{–14} m 
Atomic nucleus 


10^{–13} m 


1 pm (pico) 
10^{–12} m 



10^{–11} m 


1Å (Angstrom) 
10^{–10} m 
Atom 

1 nm (nano) 
10^{–9} m 
Glucose 


10^{–8} m 
DNA Antibody Haemoglobin 


10^{–7} m 
Wavelength of visible light Virus 

1 μm (micro) 
10^{–6} m 
Lyosome 


10^{–5} m 
Red blood cell 


10^{–4} m 
Width of a human hair Grain of salt 

1 mm (milli) 
10^{–3} m 
Width of a credit card 

1 cm (centi) 
10^{–2} m 
Diameter of a shirt button 


10^{–1} m 
Diameter of a DVD 

1 m 
10^{0} m 
Height of door handle 


10^{1} m 
Width of a classroom 


10^{2} m 
Length of a football pitch 

1 km (kilo) 
10^{3} m 
Central span of the Forth Road Bridge 


10^{4} m 
Typical altitude of an airliner, diameter of Large Hadron Collider, CERN 


10^{5} m 
Height of the atmosphere 

1 Mm (mega) 
10^{6} m 
Length of Great Britain 


10^{7} m 
Diameter of Earth Coastline of Great Britain 


10^{8} m 


1 Gm (giga) 
10^{9} m 
Moon’s orbit around the Earth, The farthest any person has travelled. Diameter of the Sun. 


10^{10} m 



10^{11} m 
Orbit of Venus around the Sun 

1 Tm (tera) 
10^{12} m 
Orbit of Jupiter around the Sun 


10^{13} m 
The heliosphere, edge of our solar system 


10^{14} m 



10^{15} m 



10^{16} m 
Light year Distance to Proxima Centauri, the next closest star 


10^{17} m 



10^{18} m 



10^{19} m 



10^{20} m 



10^{21} m 
Diameter of our galaxy 


10^{22} m 



10^{23} m 
Distance to the Andromeda galaxy 


10^{24} m 



10^{25} m 



10^{26} m 



10^{27} m 
Distance to the next galaxy cluster 


10^{28} m 



10^{29} m 
Distance to the edge of the observable universe 
In the following table the words represented by the letters A, B, C, D, E, F and G are missing. They are at the bottom.
Match each letter with the
correct words from the list below. 

(Challenge problem for Alevel students)
Here is a back of the
envelope calculation.
How far will the car go
before the battery goes flat? 