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Topic 1Physics and Physical MeasurementsContents:1.1 The realm of physics1.2 Measurement and uncertainties1.3 Mathematical and graphical techniques1.4 Vectors and scalars
Introduction WHAT 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 = vtspace velocity timePrediction from theoryObservation from experiments
MEASUREMENTSallow us to make quantitative comparisonsbetween the laws of physics and the naturalworldCommon 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 dUnités (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 worlds increasingly demanding requirements for measurement * Also mks
SI FUNDAMENTAL UNITS There are seven well-defined units which by convention are regarded as dimensionally independentPhysical quantity unit symbolLENGTH metre mMASS kilogram kgTIME second sELECTRIC CURRENT ampere ATHERMODYNAMIC TEMPERATURE kelvin KAMOUNT OF SUBSTANCE mole molLUMINOUS INTENSITY candela cd
SI FUNDAMENTAL UNIT OF LENGTH Previously: 1 metre (from the Greek metron=measure)=one ten-millionth of the distance from the North Pole tothe equator; standard metre (platinum-iridium alloy rodwith two marks one metre apart) produced in 1799 The metre is the length of the path travelled by light invacuum during a time interval of 1/299,792,458 of asecond
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 FUNDAMENTAL UNIT OF MASSThe kilogram is equal to the mass of the international prototype of the kilogram. Cylinder of platinum and iridium 0.039 m in height and diameterThe 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 FUNDAMENTAL 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 sToday 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 ofthe radiation corresponding to the transition betweenthe two hyperfine levels of the ground stateof 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 FUNDAMENTAL UNIT OF TEMPERATUREThe kelvin, unit of thermodynamic temperature, is the fraction1/273.16 of the thermodynamic temperature of the triple point ofwater.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 fundamental units according to the algebraic relations linking the corresponding quantitiesPhysical quantity unit equivalentFREQUENCY Hertz Hz = 1/s=s-1FORCE Newton N = kg.m.s-2PRESSURE Pascal Pa = N.m-2 = kg. m-1 s-2ENERGY, WORK Joule J = N.m = kg.m2.s-2POWER Watt W = J.s-1 = kg.m2.s-3
COMMON SI PREFIXESPower Prefix Abbreviation1015 peta P1012 tera T109 giga G106 mega M103 kilo k102 hecto h101 deka da10–1 deci d10–2 centi c10–3 milli m10–6 micro μ10–9 nano n10–12 pico p10–15 femto f
TYPICAL DISTANCESThe metric system makes writing very large and verysmall numbers very easy.But what changes when you increase or decrease themeasurement by a Power of Ten?
CGS SYSTEM• centimetre cm 1 cm= 10-2 m• gram g 1 g = 10-3 kg• second sDerived unitsEnergy: erg 1 erg = g.cm2.s-2= 10-3kg.10-4m2.s-2 =10-7kg.m2.s-2= 10-7 JForce: dyne 1dyn = 1 erg.cm-1 = 10-7 J/ 10-2 m =10-5 N
DIMENSIONAL ANALYSISdimension = 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 ANALYSIS Notation: L length; M mass; T timeQUANTITY DIMENSION Distance [L] Area [L2] Volume [L3] Velocity [L] . [T-1] Acceleration [L] . [T-2] Energy [M] [L2] . [T-2]
SIGNIFICANT DIGITS The result of a measurement is known only within a certain degree of accuracy• significant digits are the number of digits reliably known (excluding digits that indicate the decimal place)• 3.72 and 0.0000372 have both 3 significant digits
SIGNIFICANT DIGITS Scientific notation power of ten 3.50 x 10-3number ofSig digits
SIGNIFICANT DIGITS d=21.2 m t=8.5 s v=? v=d/t=2.4941176 m.s-1 ?TOK - What is a more accurate answer; 2/3 vs 0.667?Rule of thumb (multiplication and division): The number ofsignificant digits after multiplication or division is equal to thenumber of significant digits in the least accurate known quantity v=d/t=2.5 m.s-1
SIGNIFICANT DIGITS t1=16.74s t2=5.1 s t1+t2=? t1+t2=21.84 s?Rule of thumb (addition and subtraction): The number of decimalplaces after addition or subtraction is equal to the smallest numberof decimal places of any of the individual terms. t1+t2=21.8 s
SIGNIFICANT DIGITSHow many significant digits are in 35.00 4 35 2 3.5x10-2 2 3.50x10-3 3 ?
CONVERTING UNITSYou will need to be able to convert from one unit toanother for the same quantity. Example: Convert 72 km.h-1 to m.s-1 1 km 1000m 1h 72km .h 72 h 1km 3600s 72 1 1 m .s 20m .s 3.6
ConversionsYou will need to be able to convert from one unit toanother 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 eV1 kWh = 1kW x 1 h = 1000W x 60 x 60 s = 1000 Js-1 x 3600 s = 3600000 J = 3.6 x 106 J1 eV = 1.6 x 10-19 J
SI FormatThe 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 MAGNITUDESAn order of magnitude calculation is a rough estimatedesigned to be accurate to within a factor of about 10To get ideas and feeling for what size of numbers areinvolved in situation where a precise count is notpossible 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 soccer pitch 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 MagnitudesYou will need to be able to state (express) quantities to the nearest order ofmagnitude, that is to say to the nearest 10x Range of magnitudes of quantities in our universeSizes 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)
RatiosYou will also be required to state (express) ratios of quantities asdifferences 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
Now try these ratiosMass of electron to your massRadius of atom to your heightMass of electron to mass of uranium atomRadius of Earth to size of universe
Solutions Mass of electron to your mass10-30 / 102 = 10-32 so 32 orders of mag. Radius of atom to your height10-10 / 100 = 10-10 so 10 orders of mag. Mass of electron to mass of uranium atom10-30 / 10-25 = 10-5 so 5 orders of mag. Radius of Earth to size of universe107 / 1026 = 10-19 so 19 orders of mag.
Errors and UncertaintiesErrorsErrors can be divided into 2 main classes Random errors Systematic errors TOK What are the assumptions in measurement?
MistakesMistakes 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 equationsThese are a source of error but are not considered asan experimental error
Systematic ErrorsCause a random set of measurements to be spreadabout a value rather than being spread about theaccepted valueIt is a system or instrument value
Systematic Errors result fromBadly made instrumentsPoorly calibrated instrumentsAn instrument having a zero error (off-set error), aform of calibrationPoorly timed actionsInstrument parallax errorNote that systematic errors are not reduced bymultiple readings
Random ErrorsAre due to variations in performance of theinstrument and the operator.Even when systematic errors have been allowed for,there exists error.
Random Errors result fromVibrations and air convectionMisreadingVariation in thickness of surface being measuredUsing less sensitive instrument when a more sensitiveinstrument is availableHuman parallax error
Reducing Random ErrorsRandom errors can be reduced bytaking multiple readings, and eliminating obviouslyerroneous resultor by averaging the range of results.
AccuracyAccuracy is an indication of how close a measurementis to the accepted value indicated by the relative orpercentage error in the measurementAn accurate experiment has a low systematic errorTOKHow do we know we are accurate?Does more data mean we are more accurate?
PrecisionPrecision is an indication of the agreement among anumber of measurements made in the same wayindicated by the absolute errorA precise experiment has a low random error
Diagramming Accuracy and PrecisionAccurate •Accurate and preciseprecise
UncertaintiesIn any experimental measurement there is alwaysan estimated last digit for the measured quantity.You are not certain about the last digit.The last digit varies between two extremesexpressed as AExample: a length on a 20cm ruler is expressed as 3.25 0.05cm
Expression of physical measurements and uncertaintiesAny experimental measure is expressed in the form A Ao A Real value or Approximate value or Uncertainty final value measured value
Types of uncertainties.1. Absolute uncertainty A written as A2. Relative uncertainty A A or Ao A3. :Percentage uncertainty A % ARemark: the absolute uncertainty is always positive
Working with uncertainties. Uncertainty on a sum or difference.Rule: in addition or subtraction uncertainties just add S A B S A B D A B D A B Uncertainty on a product or a quotient.Rule: in a product or a quotient relative or percentage uncertainties add. P A B P A B P A B A Q A B Q B Q A B
Working with uncertainties cont. P A B Or P A B % % % P A B A Q A B Q % % % B Q A BAlso for n P A P A P A n or % n % P A P A An Q A B Q n m Bm Q A B Q A B or % n % m % Q 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 UncertaintiesTake multiple readingsWhen a series of readings are taken for ameasurement, then the arithmetic mean of thereading is taken as the most probable answerThe greatest deviation or residual from the mean istaken as the absolute error
Accuracy and Precision inrelation to error and uncertainty
Accuracy & Errors(208 ± 1) mm is a fairly accurate measurement(2 ± 1) mm is highly inaccurate
Graphical TechniquesGraphs are very useful for analysing the data that iscollected during investigationsGraphing is one of the most valuable tools usedbecause
Why Graphit gives a visual display of the relationship between twoor more variablesshows which data points do not obey the relationshipgives an indication at which point a relationship ceasesto be trueused to determine the constants in an equation relatingtwo variables
You need to be able to give a qualitative physicalinterpretation of a particular graphe.g. as the potential difference increases, theionization current also increases until it reaches amaximum at…..
Plotting GraphsIndependent variables are plotted on the x-axisDependent variables are plotted on the y-axisMost graphs occur in the 1st quadrant however somemay appear in all 4
Plotting Graphs - Choice of Axis When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis
Plotting Graphs - ScalesSize of graph should be large, to fill as much space aspossiblechoose a convenient scale that is easily subdivided
Plotting Graphs - LabelsEach axis is labeled with the name and symbol, as wellas the relevant unit usedThe graph should also be given a descriptive title
Plotting Graphs - Line of Best Fit When choosing the line or curve it is best to use a transparent ruler Position the ruler until it lies along an ideal line The line or curve does not have to pass through every point Do not assume that all lines should pass through the origin Do not join the dots!
Analysing the GraphOften a relationship between variables will firstproduce a parabola, hyperbole or an exponentialgrowth or decay. These can be transformed to astraight line relationshipGeneral equation for a straight line isy = mx + c y is the dependent variable, x is the independent variable, m is the gradient and c is the y-intercept
The parameters of a function can also be obtainedfrom the slope (m) and the intercept (c) of a straightline graph
GradientsGradient = vertical run / horizontal runor gradient = y / xuphill slope is positive and downhill slope is negativeDon´t forget to give the units of the gradient
Areas under GraphsThe area under a graph is a useful tool e.g. on a force displacement graph the area is work (N x m = J) e.g. on a speed time graph the area is distance (ms-1 x s = m)Again, don´t forget the units of the area
Standard Graphs - linear graphs A straight line passing through the origin shows proportionalityy y x k = ryse/run y=kx Where k is the constant of proportionality x
Standard Graphs - parabola A parabola shows that y is directly proportional to x2y y x x2i.e. y x2 or y = kx2 where k is the constant of proportionality
Standard Graphs - hyperbolaA hyperbola shows that y is inversely proportional to xy y x 1/x i.e. y 1/x or y = k/x where k is the constant of proportionality
Standard Graphs - hyperbola again An inverse square law graph is also a hyperbolay y x 1/x2 i.e. y 1/x2 or y = k/x2 where k is the constant of proportionality
Non-Standard Graphs You need to make a connection between graphs and equationsy If this is a graph of r against t2 plotted from data having an expected relationship r = at2/2 +r0 where a is a constant x Then the gradient is a/2 and the y-intercept is r0 - it is not the case that r t2, it is a linear relationship The intercept is therefore important too
Plotting Uncertainties on GraphsPoints are plotted with a fine pencil crossUncertainty or error bars are requiredThey are short lines drawn from the plotted pointsparallel to the axes indicating the absolute error ofmeasurement
Scalars QuantitiesScalars can be completely described by magnitude(size)Scalars can be added algebraicallyThey are expressed as positive or negative numbersand a unitexamples include :- mass, electric charge, distance,speed and energy
Vector QuantitiesVectors need both a magnitude and a directionto describe them (also a point of application)When expressing vectors as a symbol, you needto adopt a recognized notatione.g. draw an arrow across top of the letterThey need to be added, subtracted and multipliedin a special wayExamples :- velocity, weight, acceleration,displacement, momentum, force
Addition and SubtractionThe Resultant (Net) is the vector that comes fromadding or subtracting a number of vectorsIf vectors have the same or opposite directions theaddition can be done simply same direction : add opposite direction : subtract
Co-planar vectorsThe addition of co-planar vectors that do nothave the same or opposite direction can besolved by using scale drawings to get an accurate resultant Or if an estimation is required, they can be drawn roughly or by Pythagoras’ theorem and trigonometryVectors can be represented by a straight linesegment with an arrow at the end
Triangle of VectorsTwo vectors are added by drawing to scale and withthe correct direction the two vectors with the tail ofone at the tip of the other.The resultant vector is the third side of the triangleand the arrow head points in the direction from the‘free’ tail to the ‘free’ tip
Parallelogram of VectorsPlace the two vectors tail to tail, to scale and with thecorrect directionsThen complete the parallelogramThe diagonal starting where the two tails meet andfinishing where the two arrows meet becomes theresultant vector
More than 2If there are more than 2 co-planar vectors to beadded, place them all head to tail to form polygonwhen the resultant is drawn from the ‘free’ tail to the‘free’ tip.Notice that the order doesn’t matter!
Subtraction of VectorsTo subtract a vector, you reverse the direction of thatvector to get the negative of itThen you simply add that vector
Multiplying ScalarsScalars are multiplied and divided in the normalalgebraic mannerDo not forget units!5m / 2s = 2.5 ms-12kW x 3h = 6 kWh (kilowatt-hours)
Multiplying VectorsA vector multiplied by a scalar gives a vector withthe same direction as the vector and magnitudeequal to the product of the scalar and a vectormagnitudeA vector divided by a scalar gives a vector withsame direction as the vector and magnitude equalto the vector magnitude divided by the scalarYou don’t need to be able to multiply a vector byanother vector
Resolving VectorsThe process of finding the Components of vectors iscalled Resolving vectorsJust as 2 vectors can be added to give a resultant, asingle vector can be split into 2 components or parts
The RuleA vector can be split into two perpendicularcomponentsThese could be the vertical and horizontalcomponents Vertical component Horizontal component
Or parallel to and perpendicular to an inclined plane
These vertical and horizontal components could bethe vertical and horizontal components of velocity forprojectile motionOr the forces perpendicular to and along an inclinedplane
Doing the TrigonometryV Sin = opp/hyp = y/ V y Therefore y = V sin In this case this is the vertical component x Cos = adj/hyp = x/ VV sin Therefore x = V cos In this case this is the horizontal component V cos
Quick WayIf you resolve through the angle it is cosIf you resolve ‘not’ through the angle it is sin
Adding 2 or More Vectors by ComponentsFirst resolve into components (making sure that allare in the same 2 directions)Then add the components in each of the 2 directionsRecombine them into a resultant vectorThis will involve using Pythagoras´ theorem
QuestionThree strings are attached to a small metal ring. 2 ofthe strings make an angle of 70o and each is pulledwith a force of 7N.What force must be applied to the 3rd string to keepthe ring stationary?
AnswerDraw a diagram 7 cos 35o + 7 cos 35o 7N 7N 70o 7 sin 35o 7 sin 35o F
Horizontally7 sin 35o - 7 sin 35o = 0Vertically7 cos 35o + 7 cos 35o = FF = 11.5NAnd at what angle?145o to one of the strings.