high school physics textbook principles and problems | download free pdf
The Free High School Science Texts: A Textbook for High
School Students Studying Physics.
December 9, 2005
See http://savannah.nongnu.org/projects/fhsstChapter 1
1.1 PGCE Comments
² Explain what is meant by `physical quantity'.
² Chapter is too full of tables and words; need ¯gures to make it more interesting.
² Make researching history of SI units a small project.
² Multiply by one technique: not positive Suggest using exponents instead (i.e. use the
table of pre¯xes). This also works better for changing complicated units (km=h to
m:s etc....). Opinion that this technique is limited in its application.
² Edit NASA story.
² The Temperature section should be cut-down. SW: I have edited the original section but
perhaps a more aggressive edit is justi¯ed with the details de®ered until the section on
1.2 `TO DO' LIST
² Write section on scienti¯c notation, signi¯cant ¯gures and rounding.
² Add to sanity test table of sensible values for things.
² Graph Celsius/Kelvin ladder.
² Address PGCE comments above.
Imagine you had to make curtains and needed to buy material. The shop assistant would need
to know how much material was required. Telling her you need material 2 wide and 6 long would
be insu±cient you have to specify the unit (i.e. 2 metres wide and 6 metres long). Without
the unit the information is incomplete and the shop assistant would have to guess. If you were
making curtains for a doll's house the dimensions might be 2 centimetres wide and 6 centimetres
3Base quantity Name Symbol
length metre m
mass kilogram kg
time second s
electric current ampere A
thermodynamic temperature kelvin K
amount of substance mole mol
luminous intensity candela cd
Table 1.1: SI Base Units
It is not just lengths that have units, all physical quantities have units (e.g. time and tem-
1.4 Unit Systems
There are many unit systems in use today. Physicists, for example, use 4 main sets of units: SI
units, c.g.s units, imperial units and natural units.
Depending on where you are in the world or what area of physics you work in, the units will
be di®erent. For example, in South Africa road distances are measured in kilometres (SI units),
while in England they are measured in miles (imperial units). You could even make up your own
system of units if you wished, but you would then have to teach people how to use it
1.4.1 SI Units (Systµ eme International d'Unit¶ es)
These are the internationally agreed upon units and the ones we will use. Historically these
units are based on the metric system which was developed in France at the time of the French
All physical quantities have units which can be built from the 7 base units listed in Table 1.1
(incidentally the choice of these seven was arbitrary). They are called base units because none of
them can be expressed as combinations of the other six. This is similar to breaking a language
down into a set of sounds from which all words are made. Another way of viewing the base units
is like the three primary colours. All other colours can be made from the primary colours but
no primary colour can be made by combining the other two primaries.
Unit names are always written with lowercase initials (e.g. the metre). The symbols (or
abbreviations) of units are also written with lowercase initials except if they are named after
scientists (e.g. the kelvin (K) and the ampere (A)).
To make life convenient, particular combinations of the base units are given special names.
This makes working with them easier, but it is always correct to reduce everything to the base
units. Table 1.2 lists some examples of combinations of SI base units assigned special names. Do
not be concerned if the formulae look unfamiliar at this stage we will deal with each in detail
in the chapters ahead (as well as many others)
It is very important that you are able to say the units correctly. For instance, the newton is
another name for the kilogram metre per second squared (kg:m:s ), while the kilogram
metre squared per second squared (kg:m :s ) is called the joule.
Another important aspect of dealing with units is the pre¯xes that they sometimes have
(pre¯xes are words or letters written in front that change the meaning). The kilogram (kg) is a
simple example. 1kg is 1000g or 1£ 10 g. Grouping the 10 and the g together we can replace
4Quantity Formula Unit Expressed in Name of
Base Units Combination
Force ma kg:m:s N (newton)
Frequency s Hz (hertz)
Work & Energy F:s kg:m :s J (joule)
Table 1.2: Some Examples of Combinations of SI Base Units Assigned Special Names
the 10 with the pre¯x k (kilo). Therefore the k takes the place of the 10 . Incidentally the
kilogram is unique in that it is the only SI base unit containing a pre¯x
There are pre¯xes for many powers of 10 (Table 1.3 lists a large set of these pre¯xes). This
is a larger set than you will need but it serves as a good reference. The case of the pre¯x symbol
is very important. Where a letter features twice in the table, it is written in uppercase for
exponents bigger than one and in lowercase for exponents less than one. Those pre¯xes listed
in boldface should be learnt.
Pre¯x Symbol Exponent Pre¯x Symbol Exponent
yotta Y 10 yocto y 10
zetta Z 10 zepto z 10
exa E 10 atto a 10
peta P 10 femto f 10
tera T 10 pico p 10
giga G 10 nano n 10
mega M 10 micro ¹ 10
kilo k 10 milli m 10
hecto h 10 centi c 10
deca da 10 deci d 10
Table 1.3: Unit Pre¯xes
As another example of the use of pre¯xes, 1£ 10 g can be written as 1mg (1 milligram).
1.4.2 The Other Systems of Units
The remaining sets of units, although not used by us, are also internationally recognised and still
in use by others. We will mention them brie°y for interest only.
In this system the metre is replaced by the centimetre and the kilogram is replaced by the
gram. This is a simple change but it means that all units derived from these two are changed.
For example, the units of force and work are di®erent. These units are used most often in
astrophysics and atomic physics.
These units (as their name suggests) stem from the days when monarchs decided measures. Here
all the base units are di®erent, except the measure of time. This is the unit system you are
most likely to encounter if SI units are not used. These units are used by the Americans and
5British. As you can imagine, having di®erent units in use from place to place makes scienti¯c
communication very di±cult. This was the motivation for adopting a set of internationally agreed
This is the most sophisticated choice of units. Here the most fundamental discovered quantities
(such as the speed of light) are set equal to 1. The argument for this choice is that all other
quantities should be built from these fundamental units. This system of units is used in high
energy physics and quantum mechanics.
1.5 The Importance of Units
Without units much of our work as scientists would be meaningless. We need to express our
thoughts clearly and units give meaning to the numbers we calculate. Depending on which units
we use, the numbers are di®erent (e.g. 3.8 m and 3800 mm actually represent the same length).
Units are an essential part of the language we use. Units must be speci¯ed when expressing
physical quantities. In the case of the curtain example at the beginning of the chapter, the result
of a misunderstanding would simply have been an incorrect amount of material cut. However,
sometimes such misunderstandings have catastrophic results. Here is an extract from a story on
(NOTE TO SELF: This quote may need to be removed as the licence we are using allows for
all parts of the document to be copied and I am not sure if this being copied is legit in all ways?)
NASA: Human error caused loss of Mars orbiter November 10, 1999
WASHINGTON (AP) Failure to convert English measures to metric values caused
the loss of the Mars Climate Orbiter, a spacecraft that smashed into the planet instead
of reaching a safe orbit, a NASA investigation concluded Wednesday.
The Mars Climate Orbiter, a key craft in the space agency's exploration of the red
planet, vanished after a rocket ¯ring September 23 that was supposed to put the
spacecraft on orbit around Mars.
An investigation board concluded that NASA engineers failed to convert English
measures of rocket thrusts to newton, a metric system measuring rocket force. One
English pound of force equals 4.45 newtons. A small di®erence between the two
values caused the spacecraft to approach Mars at too low an altitude and the craft
is thought to have smashed into the planet's atmosphere and was destroyed.
The spacecraft was to be a key part of the exploration of the planet. From its station
about the red planet, the Mars Climate Orbiter was to relay signals from the Mars
Polar Lander, which is scheduled to touch down on Mars next month.
\The root cause of the loss of the spacecraft was a failed translation of English
units into metric units and a segment of ground-based, navigation-related mission
software," said Arthus Stephenson, chairman of the investigation board.
This story illustrates the importance of being aware that di®erent systems of units exist.
Furthermore, we must be able to convert between systems of units
61.6 Choice of Units
There are no wrong units to use, but a clever choice of units can make a problem look simpler.
The vast range of problems makes it impossible to use a single set of units for everything without
making some problems look much more complicated than they should. We can't easily compare
the mass of the sun and the mass of an electron, for instance. This is why astrophysicists and
atomic physicists use di®erent systems of units.
We won't ask you to choose between di®erent unit systems. For your present purposes the SI
system is perfectly su±cient. In some cases you may come across quantities expressed in units
other than the standard SI units. You will then need to convert these quantities into the correct
SI units. This is explained in the next section.
1.7 How to Change Units the \Multiply by 1" Technique
Firstly you obviously need some relationship between the two units that you wish to convert
between. Let us demonstrate with a simple example. We will consider the case of converting
millimetres (mm) to metres (m) the SI unit of length. We know that there are 1000mm in
1m which we can write as
1000mm = 1m:
Now multiplying both sides by we get
1000mm = 1m;
which simply gives us
1 = :
This is the conversion ratio from millimetres to metres. You can derive any conversion ratio in
this way from a known relationship between two units. Let's use the conversion ratio we have
just derived in an example:
Question: Express 3800mm in metres.
3800mm = 3800mm£ 1
Note that we wrote every unit in each step of the calculation. By writing them in and
cancelling them properly, we can check that we have the right units when we are ¯nished. We
started with `mm' and multiplied by ` '. This cancelled the `mm' leaving us with just `m'
the SI unit we wanted to end up with If we wished to do the reverse and convert metres to
millimetres, then we would need a conversion ratio with millimetres on the top and metres on
71.8 How Units Can Help You
We conclude each section of this book with a discussion of the units most relevant to that
particular section. It is important to try to understand what the units mean. That is why
thinking about the examples and explanations of the units is essential.
If we are careful with our units then the numbers we get in our calculations can be checked
in a `sanity test'.
1.8.1 What is a `sanity test'?
This isn't a special or secret test. All we do is stop, take a deep breath, and look at our answer.
Sure we always look at our answers or do we? This time we mean stop and really look does
our answer make sense?
Imagine you were calculating the number of people in a classroom. If the answer you got was
1 000 000 people you would know it was wrong that's just an insane number of people to have
in a classroom. That's all a sanity check is is your answer insane or not? But what units were
we using? We were using people as our unit. This helped us to make sense of the answer. If we
had used some other unit (or no unit) the number would have lacked meaning and a sanity test
would have been much harder (or even impossible).
It is useful to have an idea of some numbers before we start. For example, let's consider
masses. An average person has mass 70kg, while the heaviest person in medical history had a
mass of 635kg. If you ever have to calculate a person's mass and you get 7000kg, this should
fail your sanity check your answer is insane and you must have made a mistake somewhere.
In the same way an answer of 0:00001kg should fail your sanity test.
The only problem with a sanity check is that you must know what typical values for things
are. In the example of people in a classroom you need to know that there are usually 2050
people in a classroom. Only then do you know that your answer of 1 000 000 must be wrong.
Here is a table of typical values of various things (big and small, fast and slow, light and heavy
you get the idea):
Category Quantity Minimum Maximum
Table 1.4: Everyday examples to help with sanity checks
(NOTE TO SELF: Add to this table as we go along with examples from each section.)
Now you don't have to memorise this table but you should read it. The best thing to do is
to refer to it every time you do a calculation.
We need to make a special mention of the units used to describe temperature. The unit of
temperature listed in Table 1.1 is not the everyday unit we see and use.
Normally the Celsius scale is used to describe temperature. As we all know, Celsius temper-
atures can be negative. This might suggest that any number is a valid temperature. In fact, the
temperature of a gas is a measure of the average kinetic energy of the particles that make up the
gas. As we lower the temperature so the motion of the particles is reduced until a point is reached
8where all motion ceases. The temperature at which this occurs is called absolute zero. There is
no physically possible temperature colder than this. In Celsius, absolute zero is at¡273 C.
Physicists have de¯ned a new temperature scale called the Kelvin scale. According to this
scale absolute zero is at 0K and negative temperatures are not allowed. The size of one unit
kelvin is exactly the same as that of one unit Celsius. This means that a change in temperature
of 1 degree kelvin is equal to a change in temperature of 1 degree Celsius the scales just start
in di®erent places. Think of two ladders with steps that are the same size but the bottom most
step on the Celsius ladder is labelled -273, while the ¯rst step on the Kelvin ladder is labelled 0.
There are still 100 steps between the points where water freezes and boils.
102 Celsius 375 Kelvin
101 Celsius 374 Kelvin
water boils - 100 Celsius 373 Kelvin
99 Celsius 372 Kelvin
98 Celsius 371 Kelvin
2 Celsius 275 Kelvin
1 Celsius 274 Kelvin
ice melts - 0 Celsius 273 Kelvin
-1 Celsius 272 Kelvin
-2 Celsius 271 Kelvin
-269 Celsius 4 Kelvin
-270 Celsius 3 Kelvin
-271 Celsius 2 Kelvin
-272 Celsius 1 Kelvin
absolute zero - -273 Celsius 0 Kelvin
(NOTE TO SELF: Come up with a decent picture of two ladders with the labels water
boiling and freezingin the same place but with di®erent labelling on the steps)
This makes the conversion from kelvin to Celsius and back very easy. To convert from Cel-
sius to kelvin add 273. To convert from kelvin to Celsius subtract 273. Representing the Kelvin
temperature by T and the Celsius temperature by T ,
T = To + 273: (1.1)
It is because this conversion is additive that a di®erence in temperature of 1 degree Celsius
is equal to a di®erence of 1 kelvin. The majority of conversions between units are multiplicative.
For example, to convert from metres to millimetres we multiply by 1000. Therefore a change
of 1m is equal to a change of 1000mm.
1.10 Scienti¯c Notation, Signi¯cant Figures and Rounding
(NOTE TO SELF: still to be written)
In this chapter we have discussed the importance of units. We have discovered that there are
many di®erent units to describe the same thing, although you should stick to SI units in your
calculations. We have also discussed how to convert between di®erent units. This is a skill you
Waves and Wavelike Motion
Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in
the ocean, or in a bucket. We are most interested in the properties that waves have. All waves
have the same properties so if we study waves in water then we can transfer our knowledge to
predict how other examples of waves will behave.
2.1 What are waves?
Waves are disturbances which propagate (move) through a medium . Waves can be viewed as
a transfer energy rather than the movement of a particle. Particles form the medium through
which waves propagate but they are not the wave. This will become clearer later.
Lets consider one case of waves: water waves. Waves in water consist of moving peaks and
troughs. A peak is a place where the water rises higher than when the water is still and a trough
is a place where the water sinks lower than when the water is still. A single peak or trough we
call a pulse. A wave consists of a train of pulses.
So waves have peaks and troughs. This could be our ¯rst property for waves. The following
diagram shows the peaks and troughs on a wave.
In physics we try to be as quantitative as possible. If we look very carefully we notice that
the height of the peaks above the level of the still water is the same as the depth of the troughs
below the level of the still water. The size of the peaks and troughs is the same.
2.1.1 Characteristics of Waves : Amplitude
The characteristic height of a peak and depth of a trough is called the amplitude of the wave.
The vertical distance between the bottom of the trough and the top of the peak is twice the
amplitude. We use symbols agreed upon by convention to label the characteristic quantities of
Light is a special case, it exhibits wave-like properties but does not require a medium through which to
11the waves. Normally the letter A is used for the amplitude of a wave. The units of amplitude
are metres (m).
2 x Amplitude
Worked Example 1
Question: (NOTE TO SELF: Make this a more exciting question) If the peak of a
wave measures 2m above the still water mark in the harbour what is the amplitude
of the wave?
Answer: The de¯nition of the amplitude is the height that the water rises to above
when it is still. This is exactly what we were told, so the answer is that the amplitude
2.1.2 Characteristics of Waves : Wavelength
Look a little closer at the peaks and the troughs. The distance between two adjacent (next to
each other) peaks is the same no matter which two adjacent peaks you choose. So there is a
¯xed distance between the peaks.
Looking closer you'll notice that the distance between two adjacent troughs is the same no
matter which two troughs you look at. But, more importantly, its is the same as the distance
between the peaks. This distance which is a characteristic of the wave is called the wavelength.
Waves have a characteristic wavelength. The symbol for the wavelength is ¸. The units are
The wavelength is the distance between any two adjacent points which are in phase. Two
points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They
don't have to be peaks or trough but they must be separated by a complete number of waves.
2.1.3 Characteristics of Waves : Period
Now imagine you are sitting next to a pond and you watch the waves going past you. First one
peak, then a trough and then another peak. If you measure the time between two adjacent peaks
you'll ¯nd that it is the same. Now if you measure the time between two adjacent troughs you'll
12¯nd that its always the same, no matter which two adjacent troughs you pick. The time you
have been measuring is the time for one wavelength to pass by. We call this time the period and
it is a characteristic of the wave.
Waves have a characteristic time interval which we call the period of the wave and denote
with the symbol T. It is the time it takes for any two adjacent points which are in phase to pass
a ¯xed point. The units are seconds (s).
2.1.4 Characteristics of Waves : Frequency
There is another way of characterising the time interval of a wave. We timed how long it takes
for one wavelength to pass a ¯xed point to get the period. We could also turn this around and
say how many waves go by in 1 second.
We can easily determine this number, which we call the frequency and denote f. To determine
the frequency, how many waves go by in 1s, we work out what fraction of a waves goes by in 1
second by dividing 1 second by the time it takes T. If a wave takes a second to go by then in
1 second two waves must go by. = 2. The unit of frequency is the Hz or s .
Waves have a characteristic frequency.
f : frequency (Hz or s )
T : period (s)
2.1.5 Characteristics of Waves : Speed
Now if you are watching a wave go by you will notice that they move at a constant velocity. The
speed is the distance you travel divided by the time you take to travel that distance. This is
excellent because we know that the waves travel a distance ¸ in a time T. This means that we
can determine the speed.
v : speed (m:s )
¸ : wavelength (m)
T : period (s)
There are a number of relationships involving the various characteristic quantities of waves.
A simple example of how this would be useful is how to determine the velocity when you have the
frequency and the wavelength. We can take the above equation and substitute the relationship
between frequency and period to produce an equation for speed of the form
v = f¸
v : speed (m:s )
¸ : wavelength (m)
f : frequency (Hz or s )
Is this correct? Remember a simple ¯rst check is to check the units On the right hand
side we have velocity which has units ms . On the left hand side we have frequency which is
measured in s multiplied by wavelength which is measure in m. On the left hand side we have
ms which is exactly what we want.
2.2 Two Types of Waves
We agreed that a wave was a moving set of peaks and troughs and we used water as an example.
Moving peaks and troughs, with all the characteristics we described, in any medium constitute a
wave. It is possible to have waves where the peaks and troughs are perpendicular to the direction
of motion, like in the case of water waves. These waves are called transverse waves.
There is another type of wave. Called a longitudinal wave and it has the peaks and troughs
in the same direction as the wave is moving. The question is how do we construct such a wave?
An example of a longitudinal wave is a pressure wave moving through a gas. The peaks in
this wave are places where the pressure reaches a peak and the troughs are places where the
pressure is a minimum.
In the picture below we show the random placement of the gas molecules in a tube. The
piston at the end moves into the tube with a repetitive motion. Before the ¯rst piston stroke
the pressure is the same throughout the tube.