2012-15
CSEC PHYSICS - MANUAL
JERRORD THOMAS
SECTION A: PHYSICAL MEASUREMENTS AND UNITS
GENERAL OBJECTIVES
1. Recognize the need for an internationally agreed system of units;
2. Appreciate that every measurement of a quantity is an attempt to determine its true value and that failure to
obtain this arises from limitations on the part of the experimenter and the apparatus used;
3. Recognize the importance of a graphical presentation of data in Physics
FUNDAMENTAL QUANTITIES AND UNITS
 Recall some of the fundamental quantities of the International System ( S.I. system) and their related
symbols
 Recall that a physical quantity is usually expressed as the product of a number and a unit;
 Recall the base units for some fundamental quantities in the S.I. system and their related symbols;
 Explain the need for, and importance of, standard units in measurements;
What is a quantity?

A quantity is anything we can measure, that is: obtain a numerical value.
What is a physical quantity?

A physical quantity is a physical property of a phenomenon, body or substance that can be quantified by
measurement.

The value of a quantity is expressed as the product of a number and a unit.

A symbol is a letter we use to represent a particular quantity or a unit.
Example:
F = 100 N
Symbol of quantity
number
symbol of unit
The meaning of a physical quantity is usually presented in two ways:
 A word definition. For example: „The density of a substance is the mass per unit volume of the
substance.‟
 A defining equation:
 An equation is a statement which shows the connection between two or more quantities. It may be
expressed in words or symbols.
Page | 1
Example:
Equation in words
Equation in symbols
where P = Pressure F = Force A = Area
FUNDAMENTAL QUANTITIES AND UNITS

Measurements involve comparing unknown quantities with known standard units.

Since 1960 the system of units used in the scientific community is called “Systeme Internacionale” or the
S.I. system.
This system of units is a development of the metric system of measurements and has been adopted in many
countries to help standardize all measurements internationally.
Notes:
It is important to use the correct upper or lower case letters for the symbols. E.g. K (upper case) is Kelvin
but k (lower case) is kilo.
Roman (upright) letters are used in printing for unit symbols while italic (sloping) letters are used for
quantities.
Avoid writing plurals of unit symbols by adding an „s‟ because the symbol „s‟ stands for seconds.
To measure a quantity, such as temperature, requires us first to choose a suitable unit to act as a standard. It





should be easily copied and unchanging.

The word fundamental means basic. This means that these quantities cannot be defined in any other
physical quantity.

The S.I. system of units is based the seven (7) fundamental or basic quantities. How for physics at your level
(CSEC) we concentrate on five (5) of the basic/fundamental quantities.

The fundamental quantities are those physical quantities from which all other quantities are derived.
Fundamental Quantity
Base Unit
Name
Symbol
Name
m
kilogram
Mass
l
metre
Length
t
second
Time
I
Ampere
Electric Current
T
Kelvin
Temperature
Luminous intensity
Candela
Amount of substance
n
mole
Table 1: Fundamental Quantities and Units
Symbol
kg
m
s
A
K
cd
mol
The mnemonic MELTT can be used to memorize the first five (5) fundamental quantities.
Page | 2
What is a unit?

A unit is a specified measure of a physical quantity.

When recording measurements you should give both the numerical value and the unit.

Example:
Numerical value
55.2 metres (m)
5.6 seconds ( 5.6 s)
Unit
Why are units important?
1.
It gives meaning to the expressed values of physical quantities.
2. It is useful when comparing physical quantities.
3. It is used as a means of standardization.
Non-Standard Units
Quantity
Unit/Instrument
Mass
Grains of corn
Time
Position of the Sun, a person‟s shadow
length
The foot of a „King‟ or hand span
Temperature
The human skin
A few physical quantities have no units and are called dimensionless quantities. These are quantities which
are ratios of like quantities. Some examples are: relative density, refractive index, magnification (lens) and
relative atomic mass.
A quantity divided by its unit gives a pure number. Example: F = 100N F/N = 100
Tables and graphs are therefore headed with Quantity/Unit.
Force F/N
Distance x/cm
100
100
150
150
Page | 3
PREFIXES
 Express standard units using prefixes
In the S.I. System the root is the unit that we multiply and subdivide by powers of ten (10).
Examples of roots: gram, metre, second, ampere
Prefixes are names or symbols that precede a unit of measure to represent very big or very small numerical
values in S.I. units. What we add in front of the root is called the prefix.
Multiples and Submultiple Units
The SI system is a decimal one, that is, units are multiplied or divided by ten (10) to give larger (multiple) or
smaller (submultiple) units. In making measurements in the laboratory, the experimenter may need to measure
very large or very small quantities. One method of indicating very large or very small amounts is to use the
standard metric prefixes given in the table below. Again it is important to use the correct upper or lower case
letters. E.g. M (upper case) means „multiply by a million‟ while m (lower case) means „divide by a thousand‟!
Prefixes in frequent use with multiple and submultiple units are:
Prefix
Name




o
o
o
o
o
o

Symbol
T
Tera
G
Giga
M
Mega
k
kilo
centi
c
milli
m
micro
μ
nano
n
pico
p
Submultiples
Multiples
12
Multiple
Submultiple
10
109
106
103
10-1
10-2
10-3
10-6
10-9
10-12
Meaning
Multiples
Number
Standard Form
1012
109
106
103
10-2
10-3
10-6
10-9
10-12
1 000 000 000 000
1 000 000 000
1 000 000
1 000
0.01
0.001
0.000001
0.000000001
0.000000000001
1 x 1012
1 x 109
1 x 106
1 x 103
1 x 10-2
1 x10-3
1 x 10-6
1 x10-9
1 x10-12
Prefix
Tera
Giga
Mega
kilo
deci
centi
milli
micro
nano
pico
symbol
T
G
M
k
d
c
m
μ
n
p
Example
Tm
GW
MJ
kg
dB
cm
mA
μC
ns
pF
Page | 4
Example:
3 km = 3 x 103m = 3000 m
5 MW = 5 x 106W = 5 000 000 W
10 cm = 10 x 10-2m = 10-1 m = 0.1 m
2 mm = 2 x 10-3 m = 0.002 m
6μs = 6 x 10-6 s = 0.000 006 s
CONVERSION FACTOR:
A conversion factor is the factor by which a quantity expressed in one set of numbers must be multiplied or
divided in order to express that quantity in different units.
To obtain the conversion factor simply divide the multiple for the larger prefix by the multiple for the smaller
prefix.
Note the conversion factor MUST always have a positive index.
RULES:
 When converting from large to small prefixes you multiply the conversion factor
 When converting from small to large you divide by the conversion factor
Now recall the laws of indices:
an x am = a(m +n) , an /am = a(n-m), a0 =1
For example: m4 x m2 = m4+2 = m6
m4/m2 = m(4-2)= m2
m2/m-5 = m(2--5)= m7
kilo to milli
= 10(3 – (-3) = 106
conversion factor
Mega to nano
= 10(6 – (-9) = 1015
conversion factor
Page | 5
Micro to milli
= 10(3 – (-3) = 106
conversion factor
Giga to kilo
= 10(9 – (3) = 106
conversion factor
kmh-1 = 1000 m x ( 3600 s)-1
1 kmh-1 = 5/18 ms-1
Example:
Convert 100 kmh-1 to ms-1
= 100 x (5/18) ms-1
= 27.77 ms-1
Convert 50 ms-1 to kmh-1
= 180 kmh-1
Page | 6
1. Convert the unit of length, area and volume
below to the units given.
2cm = ________mm
a) 7.2 m = ____________cm
b) 0.32 m2 = ____________cm2
c) 0.0012 m3 = ____________cm3
d) 5.6 cm = ____________m
e) 350 cm2 = ____________m2
f) 45000 cm3 = ____________m3
2. Complete the following unit conversion
a) 12 kmh-1 = __________ ms-1
-1
b) 12 ms = __________ kmh
3. Prefix to base and reverse
1km = _______m
6 km = _______m
200m = ________km
100kg = _________g
ACTIVITY
4. Prefix to prefix and reverse
-1
25 cm = _________mm
3 mm = _________cm
20 m = ____________mm
200KHz = ______________MHz
5. Multiple prefixes to sub-prefixes and reverse
400 km = __________________mm
95 mm = ___________________km
15 cm2 = ___________________m2
10 m2 = ____________________cm2
40 cm3 = ___________________m3
7 m3 = _____________________cm3
30 g cm-3 = ________________kg m-3
100 kg m-3 = _______________g cm-3
6. What do the following symbols stand for?
a) km
b) mg
c) μs
d) MW
e) kJ
0.156 cm = __________mm
7. Convert the following:
a) 19 cm to m b) 3 kJ to J c) 72 MW to W d) 8 km to m e) 7500 kg to g f) 12 km to m
g) 78 cm to mm h) 36 mm to cm i) 97 m to mm j) 9 cm to mm
8. Fill out the following table of multiple and sub-multiple units
Unit
Name
Value
μs
microsecond
1 x 10-6 s
MV
mg
KJ
mPa
cm
MW
ns
μA
Page | 7
SCIENTIFIC NOTATION (STANDARD FORM)
 Use numbers expressed in standard form
In scientific notation you express a number in standard form- that is, as a product of a power of 10- and a
(mixed number consisting of a) whole number part together with a decimal part with the appropriate number of
decimal places. Example 734.6 will be 7.346 x 102 and 0.007346 will be 7.346 x 10-3 in standard form.
CONVERTING TO STANDARD FORM

Place the point in the number given so that it will make the number between one and ten (including one
but not ten). If the number already has a point then ignore it.

To get the power of ten, we look at the point in the number between one and ten and establish how many
places it was moved from where it was in the original number.
o Number of places gives the power of ten value
o Point moves to the left – index is positive
o Point moves to the right – index is negative
Examples:
1. 496 =
5. 0.745
9. 60000
2. 0.0089 =
6. 903
10. 4600
3. 10 000 =
7. 25.4
11. 0.00944
4. 0.032 =
8. 0.000459
12. 3
REVERSING STANDARD FORM

Look at the power of ten and determine how to move the point. The number value gives the number of
places to move the point.
o Positive index – move to the right
o Negative index - move to the left
Examples:
1. 4.31 x 104
3. 4.0 x 101
5. 6.8 x 104
2. 7.63 x 102
4. 6.34 x 10-3
6. 7.8 x 10-1
Page | 8
SIGNIFICANT FIGURES:
 Express the result of a measurement or calculation to an appropriate number of significant figures or
decimal places
The significant figures in a number give an indication of the precision of the number. In physics, the number is
usually obtained from a measurement. The more significant figures we have, the more precise is our
measurement. Generally, to find the number of significant figures, count how many numbers are in the number
but bearing in mind the following rules:

Zeroes in front of a non-zero digit are not significant e.g. 0.05 ( 1 s.f.)

Zeroes between two non-zero digits are significant e.g. 505 ( 3 s.f.)

Zeroes behind a non-zero digit without a point is not significant if they are put in to maintain place value
only e.g. 5500 (2 s.f.)
Example:
State the number of significant figures in each of the following:
1. 9.7000
5. 0.502
9. 0.1400
2. 0.87000
6. 0.109
10. 147
3. 4.30000
7. 1260
11. 0.0025
4. 301
8. 8100
12. 0.00608
Page | 9
DERIVED UNITS
 Recall that derived quantities and their related units are produced by multiplying and dividing
fundamental quantities and their units;
 recall special names given to the units for some derived quantities;
 express derived units using index notation.

Derived quantities are those quantities that are obtained from one or more of the fundamental quantities.
All derived quantities are really different combinations of the five fundamental quantities.

A derived quantity is a Physics quantity that is not a base quantity. It is the quantities which derived
from the base quantities through multiplying and/or dividing them.
Example
(Speed is derived from dividing distance by time.)
The units of the derived quantities can also be derived from the base units. If we multiply two quantities then to
get the unit of the resulting quantity we must multiply the units of the two quantities. If we divide quantities
then we also divide the units. To simplify our units we give special names to some combinations of the base
units.

Example: kg m/s2 is a Newton and has the symbol N
Page | 10
Derived quantities with units and symbols
Derived Quantity
Defining Equation
Unit derived for
Special equivalent SI name
base SI
Speed(s) or Velocity (v)
-
Acceleration (a)
-
Force (F)
kg m/s2
Newton(N)
Work (w)
kg ms-2 x m
Joule (J) = 1Nm
momentum
kg ms-1
Pressure
N/m2 or
Pascal
-1 -2
kg m s
Derived Quantity
Unit
Name
Symbol
acceleration
a
metre per second squared
m s-2
Area
A
square metre
m2
density
ρ
kilogram per cubic metre
kg m-3
Electric charge
Q
Coulomb
Electromotive force
E
Volt
V

Energy
E
Joule
J

Force
F
Newton
N

Frequency
f
Hertz
Hz

momentum
ρ
kilogram metre per second
kg ms-1
Potential Difference
V
Volt
V

Power
P
Watt
W

Pressure
P
Pascal
Pa

Resistance
R
Ohm
Ω

Velocity
v
metre per second
ms-1
volume
V
cubic metre
m-3
Symbol
Special Name

Page | 11
INDEX NOTATION
The index notation is a convenient and elegant means of expressing derived units. This is used to express units
solely as products, with the sign of the index indicating whether the term in the expression is a numerator or a
denominator.

A positive index means multiplication

Negative index means division
1. m/s is written as ms-1
5. J/K is written as JK-1
2. m/s2 is written as ms-2
6. kg m/s is written as kgms-1,
3. kg m/s2 is written as kg ms-2
7. J/kgK is written as Jkg-1K-1,
4. kg/m3 is written as kg m-3
8. kgm2/s2 is written as kgm2s-2
Determine the derived unit for the following derived quantities.
Page | 12
MEASUREMENT
SCALES:
 Differentiate between linear and non-linear scales
 Differentiate between analogue and digital scales;
SCALES
A scale is a set of marks (graduations) at intervals on a measuring instrument. The smaller the value of the
subdivisions on a scale, the greater the precision of the instrument.
The scale of any instrument is meaningless unless that instrument or its scale has been calibrated. Calibration
essentially involves the checking of the scale of an instrument against some accepted standard to ensure
accuracy of its readings.

A linear scale is one in which equal changes in the value of the physical quantity being measured are
indicated by equal changes on the measuring instrument (e.g. metre rule).
Notice the equal intervals between the marks on the scale

A non-linear scale is one in which equal changes in the value of the physical quantity being measured
are indicated by unequal changes on the scale of the measuring instrument (e.g. a conical flask).
Notice the unequal intervals between the marks on the scale
Page | 13
Types of Scales:

An analogue scale is one in which a pointer continuously deflects over a calibrated scale.
Notice the pointer which deflects across the scale

A digital scale is one which has a digital decimal light emitting diode (LED) or liquid crystal display
(LCD). The readings displayed on a digital scale cannot change continuously from one to the next but
does so by making small “jumps” from one to the next.
Notice the readings make small “jumps” from one to the next
In assessing the suitability of instruments the following criteria should be considered: accuracy, sensitivity,
precision, and range.
ERROR
Error is the difference between the actual value of a quantity and the value obtained in measurement.
Any measurement of a physical quantity has errors or uncertainty. There are two types of error


Systematic Error
Random Error
Page | 14
Systematic Error
Systematic errors are errors in the measurement of a physical quantity due to instruments, the effects of
surrounding conditions and physical constraints of the observer.
The main characteristic of systematic error is that its magnitude is almost constant or appears in one direction
only. The value of the measurement is always greater or is always less than the actual value.
Systematic errors are errors which tend to shift all measurements in a systematic way so their mean value is
displaced. Systematic errors can be compensated if the errors are known.
Examples of systematic errors are
1. zero error or end errors, which cause by an incorrect position of the zero point
Zero errors occur when the instrument gives a non- zero reading when in fact the actual reading is zero.
2. Personal error of the observer.
Physical constraints or limitations of the observer can cause systematic errors.
An example is the reaction time.
3. an incorrect calibration of the measuring instrument.
The examples are;
A stopwatch which is faster than normal would give readings which are always larger than the actual
time.
A thermometer which is used under different conditions from which it was calibrated.
A voltmeter manufactured in Germany had been calibrated under different temperature and earth‟s
magnetic field from Malaysia where the voltmeter is used.
4. consistently improper use of equipment.
5. Errors due to wrong assumption.
For example, we assumed that the value of the acceleration due to gravity g is 9.81 ms-2, but the actual
value may 9.79 ms-2. Hence there is a positive error of 0.02 ms-2.
Systematic errors cannot be reduced or eliminated by taking repeated readings using the same method, the same
instrument or by the same observer.
Systematic error can be reduced by
1. Improving the procedure of taking the measurements
2. Conducting the experiment with care.
3. Repeating the experiment by using different instruments.
4. Getting somebody else to make the measurements
Page | 15
Zero error
1. A zero error arises when the measuring instrument does not start from exactly zero.
2. Zero errors are consistently present in every reading of a measurement.
3. The zero error can be positive or negative.
(NO ZERO ERROR: The pointer of the ammeter place on
zero when no current flow through it.)
(NEGATIVE ZERO ERROR: The pointer of the ammeter does not place on zero
but a negative value when no current flow through it.)
(POSITIVE ZERO ERROR: The pointer of the ammeter does not place on zero
but a negative value when no current flow through it.)
Page | 16
Random errors
1. Random errors arise from unknown and unpredictable variations in condition.
2. It fluctuates from one measurement to the next.
3. Random errors are caused by factors that are beyond the control of the observers.
4. Random error can cause by

personal errors such as human limitations of sight and touch.

lack of sensitivity of the instrument: the instrument fail to respond to the small change.

natural errors such as changes in temperature or wind, while the experiment is in progress.

wrong technique of measurement.
Examples of random errors are:
(i)
Parallax errors – occur when the position of the eye is not perpendicular to the scale.
(ii)
Different pressures are applied when closing the gap of the micrometer screw gauge when it is
used to measure the diameter of a wire.
(iii)
Changes in the temperature during an experiment.
(iv)
Recording the wrong reading.
(v)
Mistake in counting
Random error can be reduced by

taking repeat readings

find the average value of the reading.
Page | 17
Parallax error
A parallax error is an error in reading an instrument due to the eye of the observer and pointer are not in a line
perpendicular to the plane of the scale.
Precision
Precision is the degree of uniformity or reproducibility of the measurements.
OR
Precision is the degree of a measuring instrument to record consistent reading for each measurement by the
same way.
OR
The precision of a measurement is the degree of exactness (sometimes the number of significant figures) to
which the measurement of a quantity or value can be obtained and reproduced consistently. It does not
require us to know the correct or true value.
When we say the measurements are consistent, we mean that all the values of the measurements are close
together.
The precision of an instrument is the smallest non-zero reading that can be measured using the instrument.
1. Precision is the ability of an instrument in measuring a quantity in a consistent manner with only a small
relative deviation between readings.
2. The precision of a reading can be indicated by its relative deviation.
3. The relative deviation is the percentage of mean deviation for a set of measurements and it is defined by the
following formula:
Page | 18
Accuracy
Accuracy is the degree of closeness of the measurements to the actual or accepted value.
1. The accuracy of a measurement is the approximation of the measurement to the actual value for a certain
quantity of Physics.
2. The measurement is more accurate if its number of significant figures increases.
3. Table above shows that the micrometer screw gauge is more accurate than the other measuring instruments.
4. The accuracy of a measurement can be increased by

taking a number of repeat readings to calculate the mean value of the reading.

avoiding the end errors or zero errors.

taking into account the zero and parallax errors.

using more sensitive equipment such as a vernier caliper to replace a ruler.

use magnifying glass to aid in reading.
5. The difference between precision and accuracy can be shown by the spread of shooting of a tar-get (as
shown in Diagram below).
When we say the measurements are accurate, we are actually saying that the values of the measurements are
close to the true or accepted value. The diagram shows the result for four shooters A, B, C and D in a
tournament. Every shooter shot five times.
The table shows the conclusion.
Shooter
Precision
A
High
B
Low
C
High
D
Low
Accuracy
Low
High
High
Low
Page | 19
Sensitivity
Sensitivity of a measuring instrument is defined as the capability of that instrument to respond to physical
stimuli or to register small physical amount or differences.
OR
Sensitivity is the degree of a measuring instrument to record small change in its reading.
OR
Sensitivity is a measure of how well an instrument responds (the magnitude or time of response) to an input
signal.
1. The sensitivity of an instrument is its ability to detect small changes in the quantity that is being measured.
2. Thus, a sensitive instrument can quickly detect a small change in measurement.
3. Measuring instruments that have smaller scale parts are more sensitive.
4. Sensitive instruments need not necessarily be accurate.
The smallest scale division on the measuring instruments shows the sensitivity of the instruments. Thus the
more sensitive the measuring instruments the finer the scale divisions.
Vernier calipers are more sensitive than a ruler or a miliammeter is more sensitive than an ammeter.
A sensitive instrument is not always an accurate instrument.
Range
Range is the size of the interval between the maximum value and the minimum value that an instrument can
measure, that is the range of a set of readings or calculated values or measuring instrument is the difference
between the smallest and the largest values.
Example:
If you time the fall of a paper parachute over a fixed distance, the times may vary slightly.
Taking measurements: Range
The results were:
10.1 s, 10.2 s, 9.9 s, 10.0 s, 10.3 s
Find the minimum value and the maximum value
Range = from min to max = 9.9 to 10.3
Range = max value – min value
= 10.3 s – 9.9 s
= 0.4 s
Page | 20
VERNIER CALIPERS
The vernier caliper is used in length measurements to gain an additional digit of accuracy compared to a simple
ruler. The vernier calipers have two scales … the main scale and the vernier scale. The vernier is a convenient
tool to use when measuring the length of an object, the outer diameter (OD) of a round or cylindrical object, the
inner diameter (ID) of a pipe, and the depth of a hole. The inside jaws are used to measure internal diameters
and the outside jaws are used to measure external diameters and thickness. The tail is used to measure depths.
The vernier consists of a main scale engraved on a fixed ruler and an auxiliary vernier scale engraved on a
movable jaw). The movable auxiliary scale is free to slide along the length of the fixed ruler. This vernier's
main scale is calibrated in centimeters with the smallest division in millimeters. The auxiliary scale has 10
divisions that cover the same distance as 9 divisions on the main scale. Therefore, the length of the auxiliary
scale is 9.0 mm.
When the vernier is closed and properly zeroed, the first mark (zero) on
the main scale is aligned with the first mark on the auxiliary scale. The
last mark on the auxiliary scale will then coincide with the 9 mm-mark
on the main scale. This is read 0.00 cm.
Once the vernier is positioned to make a reading, make a
note of where the first mark on the auxiliary scale falls on
the main scale. In Figure, we see that the object's length is
between 3.4 cm and 3.5 cm because the first auxiliary
mark is between these two values on the main scale. The
last digit (tenths of a millimeter) is found by noting which
line on the auxiliary scale coincides with a mark on the
main scale. In our example, the last digit is 7 because the
seventh auxiliary mark lines up with a mark on the main scale. Therefore, the length of the object is 3.47 cm.
Page | 21
An improperly zeroed vernier caliper
Care must be taken to insure that the vernier caliper is properly zeroed. (With misuse, it is possible that the
vernier will not read zero when the jaws are closed, thus leading to systematic errors.) The vernier in Figure 5 is
improperly zeroed. To correct this, a zero correction must be made. A correction may be either positive or
negative. If the first mark on the auxiliary scale lies to the right of the main scale, then the reading is too large
and the error is positive. The zero reading in Figure 5 is +0.05 cm and should be subtracted from any
measurement reading. Similarly, if the first mark on the auxiliary scale lies to the left of the main scale zeromark, then the error is negative and the correction should be added from the measurement reading.
MICROMETER SCREWGAUGE
Range and Accuracy
The range of a micrometer is 0-25mm.
The accuracy of a micrometer is up to 0.01mm.
How to Use a Micrometer?
1. Turn the thimble until the object is gripped gently between the anvil and spindle.
2. Turn the ratchet knob until a "click" sound is heard. This is to prevent exerting too much pressure on the
object measured.
3. Take the reading.
Page | 22
Micrometer Screw Gauge
To measure diameter of fine wires, thickness of paper and small lengths, a micrometer screw gauge is used
The micrometer has two scales:
Main scale on the sleeve
Circular scale on the thimble
There are 50 divisions on the thimble. One complete turn of the thimble moves the spindle by 0.50 mm
Two scales: main scale and circular scale. One complete turn moves the spindle by 0.50 mm. Each division
on the circular scale = 0.01 mm
Reading
Reading
of
of
main
scale
=
5.5mm
thimble
scale
=
0.27mm
Actual Reading = 5.5mm + 0.27mm = 5.77mm
Precautions when using a micrometer
1. Never tighten thimble too much
–
Modern micrometers have a ratchet to avoid this
2. Clean the ends of the anvil and spindle before making a measurement
–
Any dirt on either of surfaces could affect the reading
3. Check for zero error by closing the micrometer when there is nothing between the anvil and spindle
–
The reading should be zero, but it is common to find a small zero error
–
Correct zero error by adjusting the final measurement
Page | 23
GRAPH
How to draw a graph
a) Choosing the correct axis:
The first variable mentioned goes on the “vertical” axis ( y-axis).
i.
Example – if you are told to plot displacement against time, then displacement is plotted on the y-axis.
ii.
The independent variable (the one which we set) is plotted on the x-axis and the dependent
variable (the one which we measure) is plotted on the y-axis
b) Choosing a suitable scale for each axis:
i.
Scales are chosen to ensure that the graph occupies as much of the graph as it can without
making the scale to use. The graph should occupy at least 75% (3/4) of the page, both „vertically
and horizontally‟. (The more expanded the scale the greater the precision of the results)
ii.
A scale which is NOT awkward is easy to read and makes points easy to plot and read-off.
Example a 1:5 or 1: 10 scale. Note that scales like 1:3 or 1:4 etc are awkward.
iii.
Broken scales may be used unless an intercept has to be found
c) Labelling:
i.
The title of the graph must be given
ii.
Each axis must be labelled with quantity/unit (given in words or symbols) Example I/A or
time/second
iii.
Axes should have scale markings every 2 cm (or less)
d) Plotting:
i.
ii.
All recorded readings must be plotted
Points can be FINE circled dots
or SHARP upright crosses x whose intersection is exactly
the intersection of the two values producing the point.
iii.
Plotted reading must NOT be rounded off.
e) Drawing the ‘Best fit’ line:
i.
To obtain the best curve or straight line from a scatter of points the LEAST SQUARES
principle may be used. Using a transparent ruler, the line is drawn so that the
PERPENDICULAR DISTANCE of points on one side of the line when added (by inspection)
give the same total as those taken from the other side.
ii.
The best fit line need not pass through any of the plotted points.
iii.
Straight line graphs do not have to pass through the origin.
Page | 24
iv.
A straight line must not be forced through points which show a curved trend.
v.
A THIN line must be drawn.
Why do we use a ‘best fit’ line?
– Drawing a „best fit‟ line is a process which finds a good average from a set of values which have
some random variations.
– It is better than an arithmetic mean because it can ignore results which are too far off the straight
line and give greater weighing to those which fit it well.
– It can identify results which are obviously incorrect and allow the readings to be repeated or
checked.
– It provides an accurate method of extracting numerical results from a set of data.
f) Finding the gradient of a graph:
i.
A large triangle must be drawn ( whose hypotenuse must be greater than half the length of the
line). This increases accuracy
The gradient is calculated from: ( change in vertical/change in horizontal) i.e y/x = (y2 - y1 )/
ii.
(x2 - x1 )
iii.
GRADIENTS HAVE UNITS, which come from (vertical units/horizontal units). If both axes
have the same units, the gradient is a pure number.
iv.
The read-off must be from the graph and NOT from the table.
v.
The read-off must be to the same number of significant figures as the plotted readings.
Why do we often find the gradient?
– The gradient often gives the numerical value of a quantity which is required, e.g. If we plot
displacement (y-axis) against time (x-axis), the gradient gives the velocity
h) Interpretation of graphs:
Its shape can give the relationship between the measured quantities
i.
The measured quantities are DIRECTLY PROPORTIONAL: a straight line passing through
the origin (y = mx)
ii.
There is a linear relationship, but not proportionality: a straight line which cuts an axis ( y =
mx + c)
iii.
There is a non-linear relationship: a curve
i) Intercept:
Page | 25
i.
The intercept is read directly off the graph . It is the point on an axis that the line passes through.
In physical terms the intercept is the value of one of the quantities/variables plotted when the
other is zero.
ii.
Do not use a broken scale if the intercept has to be found.
iv.
INTERCEPTS have UNITS which are the same as those for the axis cut.
v.
The read-off must be to the same number of significant figures at the plotted readings.
vi.
The intercept must NOT be calculated from the equation ( y = mx + c)
j) Rationale for use of a graph to represent measurements:
i.
It gives a total picture
ii.
It shows trends in measurement e.g. When there is a change from a linear to a curved trend
iii.
Its shape can give the relationship between measured variables
iv.
It can be extrapolated in any direction to predict results.
v.
It shows up systematic errors which are way off. They can then be checked.
vi.
The „best fit‟ reduces the random errors which show up a scatter of points about the line.
Page | 26