Induction Tutorial 8 - Orders of Magnitude

Estimating quantities in Physics

When some 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:

 

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:

 

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?

 

Question 1

Show that the answer to (a) is 70 m s-1 and (b) is 35 000 m

Answer

Question 2

Do you think that these answers are reasonable?

Answer

 

The question above appeared in resource materials that were produced commercially and schools paid considerable prices for the photocopy masters.  Think about the numbers:

 

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 hard-caught 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.

 

 

Fermi’s piano tuner problem

Website: https://www.grc.nasa.gov/www/k-12/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 × 106.

 

 

There is now a little problem:

 

Making Comparisons

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 double-decker buses.

 

We saw how a badly written question compared the sprint speed of a cheetah to the speed of a high-speed electric train, or an aeroplane.

 

By making comparisons, we come on to the important skill of getting a scale of objects. 

 

 

Question 3

What is wrong here? How high would you estimate the room to be?

Answer

 

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 full-sized aeroplane, or when the object is far too small for us to see, for example an arrangement of atoms.

 

Question 4

Here is a scale drawing of a car.


What is the scale?

Answer

Question 5

How wide is this shape?

 

Does it matter that the view is at an angle?

Answer

 

 

Orders of magnitude

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:

 

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

103 m

The length of an average street

104 m

The diameter of a small city like Perth

105 m

Distance between Aberdeen and Aviemore or Stirling and Ayr

106 m

Length of Great Britain

107 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

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

100 m

Height of door handle  

 

101 m  

Width of a classroom  

 

102 m

Length of a football pitch

1 km (kilo)

103 m

Central span of the Forth Road Bridge

 

104 m

Typical altitude of an airliner,

diameter of Large Hadron Collider, CERN

 

105 m

Height of the atmosphere

1 Mm (mega)

106 m

Length of Great Britain

 

107 m

Diameter of Earth

Coastline of Great Britain

 

108 m

 

1 Gm (giga)

109 m

Moon’s orbit around the Earth,

The farthest any person has travelled.

Diameter of the Sun.

 

1010 m

 

 

1011 m

Orbit of Venus around the Sun

1 Tm (tera)

1012 m

Orbit of Jupiter around the Sun

 

1013 m

The heliosphere, edge of our solar system

 

1014 m

 

 

1015 m

 

 

1016 m

Light year

Distance to Proxima Centauri, the next closest star

 

1017 m

 

 

1018 m

 

 

1019 m

 

 

1020 m

 

 

1021 m

Diameter of our galaxy

 

1022 m

 

 

1023 m

Distance to the Andromeda galaxy

 

1024 m

 

 

1025 m

 

 

1026 m

 

 

1027 m

Distance to the next galaxy cluster

 

1028 m

 

 

1029 m

Distance to the edge of the observable universe

 

Question 6

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.

 

Order of magnitude/m

Object

10-15

A

10-14

B

10-10

Diameter of hydrogen atom

10-4

C

100

D

103

E

107

Diameter of Earth

109

F

1013

Diameter of solar system

1021

G

 

Match each letter with the correct words from the list below.

diameter of nucleus;  diameter of proton;  diameter of Sun; distance to nearest galaxy; height of Ben Nevis;  size of dust particle; your height.
 

Answer

Question 7

(Challenge problem for A-level students)

 

Here is a back of the envelope calculation.

When a car doesn’t start, it is possible to move it a few metres by putting it in gear and moving it with the starter motor.

Assuming:

  • you have a 32 amp hour battery;

  • the battery has a voltage of 12 V;

  • the car moves at 1 m s-1.

     

How far will the car go before the battery goes flat?
 

Answer