Topic 1Physics and Physical

Measurements

Contents:1.1 The realm of physics 1.2 Measurement and uncertainties 1.3 Mathematical and graphical techniques 1.4 Vectors and scalars

IntroductionWHAT IS PHYSICS?

• Physics (from a Greek term meaning nature)

is historically the term to designate the study of natural phenomena (also natural philosophy till early in the 19th century)

• Goal of physics: to understand and predict how nature works

• Everything in nature obeys the laws of physics

• Everything we build also obeys the laws of physics

PHYSICS & MATHS

• The laws of physics can be expressed in terms of mathematical equations

MOTION WITH CONSTANT VELOCITY

x = vt space velocity time

Prediction from theory Observation from experiments

MEASUREMENTS

• allow us to make quantitative comparisons

between the laws of physics and the natural

world• Common measured quantities: length, mass,

time, temperature…• A measurement requires a system of units

Measurement = number x unit

THE INTERNATIONAL SYSTEM OF UNITS (SI)*

• The 11th Conférence Générale des Poids et Mesures (1960) adopted the name Système International d'Unités (International System of Units, SI), for the recommended practical system of units of measurement.

• The 11th CGPM laid down rules for the base units, the derived units, prefixes and other matters.

• The SI is not static but evolves to match the world's increasingly demanding requirements for measurement

* Also mks

SI BASE UNITS

• a choice of seven well-defined units which by convention are regarded as dimensionally independent

Physical quantity unit symbol

LENGTH meter m

MASS kilogram kg

TIME second s

ELECTRIC CURRENT ampere A

THERMODYNAMIC TEMPERATURE kelvin K

AMOUNT OF SUBSTANCE mole mol

LUMINOUS INTENSITY candela cd

SI BASE UNIT OF LENGTH

• Previously: 1 meter (from the Greek metron=measure)=

one ten-millionth of the distance from the North Pole to

the equator; standard meter (platinum-iridium alloy rod

with two marks one meter apart) produced in 1799

• The meter is the length of the path traveled by light in

vacuum during a time interval of 1/299,792,458 of a

second

TYPICAL DISTANCES• Diameter of the Milky Way 2x1020 m• One light year 4x1016 m• Distance from Earth to Sun 1.5x1011 m• Radius of Earth 6.37x106 m• Length of a football field 102 m• Height of a person 2x100 m• Diameter of a CD 1.2x10-1 m• Diameter of the aorta 1.8x10-2 m• Diameter of a red blood cell 8x10-6 m• Diameter of the hydrogen atom 10-10 m• Diameter of the proton 2x10-15m

SI BASE UNIT OF MASS

The kilogram is equal to the mass of the international prototype of the kilogram.

Cylinder of platinum andiridium 0.039 m in heightand diameter

The mass is not the weight (=measure of the gravitational force)

TYPICAL MASSES

• Galaxy (Milky Way) 4x1041 kg• Sun 2x1030 kg• Earth 5.97x1024 kg• Elephant 5400 kg• Automobile 1200 kg• Human 70 kg• Honeybee 1.5x10-4 kg• Red blood cell 10-13 kg• Bacterium 10-15 kg• Hydrogen atom 1.67x10-27 kg• Electron 9.11x10-31 kg

SI BASE UNIT OF TIME

• Previously: the revolving Earth was considered a fairly accurate timekeeper.

Mean solar day = 24 h = 24 x 60 min = 24x60x60 s = 84,400 s

Today the most accurate timekeeper are atomic clock

(accuracy 1 second in 300,000 years)

• The second is the duration of 9,192,631,770 periods of

the radiation corresponding to the transition between

the two hyperfine levels of the ground state of the

caesium 133 atom.

TYPICAL TIMES• Age of the universe 5 x 1017 s• Age of the Earth 1.3 x 1017 s• Existence of human species 6 x 1013 s• Human lifetime 2 x 109 s• One year 3 x 107 s• One day 8.6 x 104 s• Time between heartbeat 0.8 s• Human reaction time 0.1 s• One cycle of a high-pitched sound wave 5 x 10-5 s• One cycle of an AM radio wave 10-6 s• One cycle of a visible light wave 2 x 10-15 s

SI BASE UNIT OF TEMPERATURE

• The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

The triple point of any substance is that temperature and pressure at which the material can coexist in all three phases (solid, liquid and gas) at equilibrium.

SI DERIVED UNITSformed by combining base units according to theAlgebraic relations linking the correspondingquantities

Physical quantity unit equivalentFREQUENCY Hertz Hz = 1/s=s-1

FORCE Newton N = kg.m.s-2

PRESSURE Pascal Pa = N.m-2 = kg. m-1 s-2

ENERGY, WORK Joule J = N.m = kg.m2.s-2

POWER Watt W = J.s-1 =

kg.m2.s-3

COMMON SI PREFIXES

Power Prefix Abbreviation1015 peta P 1012 tera T 109 giga G106 mega M103 kilo k102 hecto h 101 deka da 10–1 deci d10–2 centi c 10–3 milli m 10–6 micro μ 10–9 nano n 10–12 pico p10–15 femto f

CGS SYSTEM

• centimeter cm 1 cm= 10-2 m

• gram g 1 g = 10-3 kg

• second s

Derived unitsEnergy: erg 1 erg = g.cm2.s-2= 10-3kg.10-4m2.s-2

=10-7kg.m2.s-2= 10-7 J

Force: dyne 1dyn = 1 erg.cm-1 = 10-7 J/ 10-2 m =10-5 N

DIMENSIONAL ANALYSIS

dimension = type of quantity independent from units

1 foot≠ 1.1 mile ≠ 5 km ≠ 2.5 m ≠ 1 light-year

but

they have all the same dimension = length

Any valid formula in physics must be

dimensionally consistent

DIMENSIONAL ANALYSISNotation: L length; M mass; T time

QUANTITY DIMENSION

Distance [L]

Area [L2]

Volume [L3]

Velocity [L] . [T-1]

Acceleration [L] . [T-2]

Energy [M][L2] . [T-2]

DIMENSIONAL ANALYSISDimensional consistency

distance velocity time distance

x = vt + x0

[ ][ ] [ ] [ ] [ ] [ ] [ ]

[ ]

LL T L L L L

T

SIGNIFICANT FIGURES• The result of a measurement is known only

within a certain accuracy

• Significant figures are the number of digits reliably known (excluding digits that indicate the decimal place)

• 3.72 and 0.0000372 have both 3 significant figures

SIGNIFICANT FIGURES

Scientific notation

3.50 x 10-3

number oforder unity

power of ten

SIGNIFICANT FIGURES

d=21.2 m

t=8.5 s

v=?

v=d/t=2.4941176 m.s-1?

• Rule of thumb (multiplication and division): The number of significant figures after multiplication or division is equal to the number of significant figures in the least accurate known quantity

v=d/t=2.5 m.s-1

SIGNIFICANT FIGURESt1=16.74s

t2=5.1 s

t1+t2=?

t1+t2=21.84 s?

• Rule of thumb (addition and subtraction): The number of decimal places after addition or subtraction is equal to the smallest number of decimal places ofany of the individual terms.

t1+t2=21.8 s

SIGNIFICANT FIGURES

How many significant figures are in

• 35.00

• 35 • 3.5x10-2

• 3.50x10-3

?

4

3

2

2

CONVERTING UNITS

• You will need to be able to convert from one unit to another for the same quantity.

Example:

Convert 72 km.h-1 to m.s-1

1

1 1

1000 172 . 72

1 360072

. 20 .3.6

km m hkm h

h km s

m s m s

Conversions

• You will need to be able to convert from one unit to another for the same quantity– J to kWh– J to eV– Years to seconds– And between other systems and SI

KWh to J and J to eV

• 1 kWh = 1kW x 1 h

= 1000W x 60 x 60 s

= 1000 Js-1 x 3600 s

= 3600000 J

= 3.6 x 106 J

• 1 eV = 1.6 x 10-19 J

SI Format

The accepted SI format is– m.s-1 not m/s– m.s-2 not m/s/s

• i.e. we use the suffix not dashes

ORDER OF MAGNITUDES

• An order of magnitude calculation is a rough estimate designed to be accurate to within a factor of about 10

• To get ideas and feeling for what size of numbers are involved in situation where a precise count is not possible or important

ORDER OF MAGNITUDE TYPICAL DISTANCES

• Diameter of the Milky Way 2x1020 m• One light year 4x1016 m• Distance from Earth to Sun 1.5x1011m• Radius of Earth 6.37x106m• Length of a football field 102m• Height of a person 2x100 m• Diameter of a CD 1.2x10-1m• Diameter of the aorta 1.8x10-2 m• Diameter of a red blood cell 8x10-6 m• Diameter of the hydrogen atom 10-10 m• Diameter of the proton 2x10-15 m

ORDER OF MAGNITUDE

EXAMPLEEstimate the number of seconds in a human"lifetime."You can choose the definition of "lifetime."Do all reasonable choices of "lifetime" give answersthat have the same order of magnitude?

The order of magnitude estimate: 109 seconds• 70 yr = 2.2 x 109 s• 100 yr = 3.1 x 109 s• 50 yr = 1.6 x 109 s

Summary for Range of Magnitudes

• You will need to be able to state (express) quantities to the nearest order of magnitude, that is to say to the nearest 10x

Range of magnitudes of quantities in our universe • Sizes

– From 10-15 m (subnuclear particles)– To 10+25 m (extent of the visible universe)

• masses– From 10-30 kg (electron mass)– To 10+50 kg (mass of the universe)

• Times– From 10-23 s (passage of light across a nucleus)– To 10+18 s (age of the universe)

• You will also be required to state (express) ratios of quantities as differences of order of magnitude.Example:– the hydrogen atom has a diameter of 10-10 m– whereas the nucleus is 10-15 m– The difference is 105

– A difference of 5 orders of magnitude

Errors and Uncertainties

Errors

Errors can be divided into 2 main classes

• Random errors

• Systematic errors

Mistakes

• Mistakes on the part of an individual such as– misreading scales– poor arithmetic and computational skills– wrongly transferring raw data to the final

report– using the wrong theory and equations

• These are a source of error but are not considered as an experimental error

Systematic Errors

• Cause a random set of measurements to be spread about a value rather than being spread about the accepted value

• It is a system or instrument value

Systematic Errors result from

• Badly made instruments

• Poorly calibrated instruments

• An instrument having a zero error (off-set error), a form of calibration

• Poorly timed actions

• Instrument parallax error

• Note that systematic errors are not reduced by multiple readings

Random Errors

• Are due to variations in performance of the instrument and the operator.

• Even when systematic errors have been allowed for, there exists error.

Random Errors result from

• Vibrations and air convection

• Misreading

• Variation in thickness of surface being measured

• Using less sensitive instrument when a more sensitive instrument is available

• Human parallax error

Reducing Random Errors

• Random errors can be reduced by

• taking multiple readings, and eliminating obviously erroneous result

• or by averaging the range of results.

Accuracy

• Accuracy is an indication of how close a measurement is to the accepted value indicated by the relative or percentage error in the measurement

• An accurate experiment has a low systematic error

Precision

• Precision is an indication of the agreement among a number of measurements made in the same way indicated by the absolute error

• A precise experiment has a low random error

uncertainties

• In any experimental measurement there is always an estimated last digit for the measured quantity.

• You are not certain about the last digit.• The last digit varies between two extremes

expressed as• Example: a length on a 20cm ruler is expressed

as

A

3.25 0.05cm

Expression of physical measurements and uncertainties

Any experimental measure is expressed in the form

oA A A

Real value or final value

Approximate value or measured value

Uncertainty

Types of uncertainties.

1. Absolute uncertainty written as2. Relative uncertainty

3. :Percentage uncertainty

Remark: the absolute uncertainty is always positive

A A

o

A Aor

A A

%A

A

Working with uncertainties.

• Uncertainty on a sum or difference.

Rule: in addition or subtraction uncertainties just add

• Uncertainty on a product or a quotient.

Rule: in a product or a quotient relative or percentage uncertainties add.

S A B S A B

D A B D A B

P A BP A B

P A BA Q A B

QB Q A B

Working with uncertainties cont.

% % %

% % %

P A BP A B

P A BA Q A B

QB Q A B

Or

Also for % %

% % %

n

n

m

P A P AP A n or n

P A P A

A Q A BQ n m

Q A BBQ A B

or n mQ A B

Limit of Reading and Uncertainty

• The Limit of Reading of a measurement is equal to the smallest graduation of the scale of an instrument

• The Degree of Uncertainty of a measurement is equal to half the limit of reading

• e.g. If the limit of reading is 0.1cm then the absolute uncertainty range is 0.05cm

Reducing the Effects of Random Uncertainties

• Take multiple readings

• When a series of readings are taken for a measurement, then the arithmetic mean of the reading is taken as the most probable answer

• The greatest deviation or residual from the mean is taken as the absolute error

Diagramming Accuracy and Precision

•

precise

•Accurate and precise

Accurate

Diagramming Accuracy and Precision in relation to error and uncertainty

figure 1

Figure 2

Figure 3

Plotting Uncertainties on Graphs

• Points are plotted with a fine pencil cross

• Uncertainty or error bars are required

• These are short lines drawn from the plotted points parallel to the axes indicating the absolute error of measurement

Uncertainties on a Graph