CAPE Physics Unit 1

CAPE Physics Unit 1Lecturer: Henry SmallWhy Study CAPE Physics?My parents want me to do it.It is an interesting subject!

It provides good support for other Sciences (Chemistry, Biology, Environmental science, etc.)

It facilitates a general understanding of many observations in the world which we live.

It is a prerequisite for many careers.

Some Careers in PhysicsPhysics Teacher, University or College Physics Lecturer, Theoretical Chemist,Theoretical Physicist, Physical Chemist,Geologist, Medical PhysicistMeteorologist, Medical Doctor, Nuclear Scientist, Astronaut, Radiographer (X ray Technician), Airline Pilot,Air Traffic Controller,EngineerMiscellaneousPrerequisitesPass in CSEC or O-level Physics or their equivalent.Pass in CSEC or O-level Math or their equivalent.Pass in CSEC or O-level English language or their equivalent.Textbook(s) One of:A level Physics by Roger MuncasterUnderstanding Physics for Advanced Level by Jim BreithauptAdvanced Physics by Tom DuncanOR CAPE Physics Course Notes by me (Which will be ready next week)Monthly Tests will consist of5 to 10 MCQ 2 to 4 Written QuestionsAlways bring a calculator to the testMiscellaneousGradesA : 80 % and aboveB: 70 - 79 %C: 60 - 69%D: 50 - 59%E: 40 - 49%Maintain at least a C and you should do well in CAPE.ScheduleTue 8 10 AM, Thur 12 2 PMUnit 2 will be on: Tuesday 4 6 PM, Friday 8 10 AMPhysical QuantityA Physical Quantity is a physical property that can be measured.A quantity is usually written as the product of a number and a unit.Quantity = Number x UnitExample: mass = 2.5 kg This makes it fairly easy to convert the mass to grams i.e. mass = 2.5 x 103 g = 2500 g

If this mass of 2.5 kg were to be written on a graphical axis which is labeled mass/kg, only the number, 2.5, would be written. This is so because when the quantity is divided by its unit the result is the number part of the quantity. Therefore,

If mass = 2.5 kg, then mass/kg = 2.5

N.B. Some quantities have no unit and are called dimensionless quantities e.g. Refractive index, relative density, magnification.

Basic QuantityPhysical QuantitySymbolName of UnitSymbollengthlmetermmassmkilogramkgtimetsecondstemperatureTKelvinKelectric currentIAmpereAamount of substancenmolemolluminous intensityIvcandelacdThe mole is the amount of substance containing 6.02 x 1023 particles. This number is called Avogadros constant and it is equal to the number of atoms in 0.012 kg of the C-12 isotope.Derived quantities combination of basic quantities via multiplication of divisionUnits standard size of quantity used for comparisonSI Units internationally accepted units (determined in Paris in a meeting in 1960) Le System International dUnites

Scientific NotationWhen working with numbers that are extremely large or extremely small it is necessary to express these numbers in scientific notation, as a number between 1 and 10 multiplied by 10 raised to some exponent.

The exponent in scientific notation is equal to the number of times the decimal point must be moved to produce a number between 1 and 10.

Example: 10,300,000,000,000,000,000,000 = 1.03 x 1022 and 0.000,000,000,000,000,000,000,020 = 2.0 x 10-23

Factors to consider when choosing an instrument for measurementRange the interval between the highest and lowest dimensions.Sensitivity the ability of a measuring apparatus to detect small changes of the physical quantity.Precision the agreement between similar readings. If an instrument is precise there is consistency in the readings taken (i.e. the uncertainties are small).Accuracy the closeness of a measurement to the actual value.

Uncertainty in MeasurementMeasurements are always accompanied by a finite amount of error or uncertainty, which reflects (1) limitations in the sensitivity of the instruments used and (2) imperfections in the techniques used to make them.These errors can be divided into two classes: systematic and random.Systematic errors result from faulty instruments and are not easily allowed for while Random errors result from the Experimenters technique and can be minimized by making a large number of measurements and taking an average.Errors

Absolute Uncertainty.Uncertainties may be expressed as absolute uncertainties, giving the size of a quantity's uncertainty in the same units as the quantity itself.

Example. A piece of metal is weighed a number of times, and the average value obtained is: M = 34.6 g. By analysis of the scatter of the measurements, the uncertainty is determined to be m = 0.07 g. This absolute uncertainty may be included with the measurement in this manner: M = 34.60 0.07 g.

The value 0.07 after the sign in this example is the estimated absolute uncertainty in the value 34.60.

Relative (Fractional) Uncertainty and Percentage Uncertainty.Uncertainties may be expressed as relative uncertainties, giving the ratio of a quantity's uncertainty to the quantity itself. In general:

Example. In the previous example, the uncertainty in M = 34.6 g was m = 0.07 g. The relative uncertainty is therefore:

as a percentage uncertainty this would be 0.2%

Rules for Estimating Uncertainties in Calculated ResultsIf a measured quantity is multiplied or divided by a constant then theabsoluteuncertainty is multiplied or divided by the same constant. (In other words the relative uncertainty stays the same.)If two measured quantities are added or subtracted, their absolute uncertainties add.If two measured quantities are multiplied or divided, their relative uncertainties add.If a measured quantity israised to a powerthen therelativeuncertainty ismultiplied by that power. (If you think about this rule, you will realise that it is just a special case of rule 3.)

www.saburchill.com/physics/chapters/0070.html#1Example problem 1Suppose the thickness of 100 pages of a book was measured as 9.00.1 mm what would be the average thickness of one page of the same book?ANSWERThe average thickness of one page, t = 9.0/100So the result can be given as: t = 9.0/100 0.1/100 mm t = 0.090 0.001 mm Example problem 2The temperature of an object was increased from T1 = 25 1 oC to T2 = 40 1 oC. Calculate the temperature change, T.ANSWERT = T2 T1 = 40 25 = 15 oCUncertainty in T = (1 + 1) oC = 2 oCTherefore T = 15 2 oCExample problem 3Using a rule marked in mm, the length of a rectangular surface was measured as 80 1 mm. If the width of the surface was 50 1 mm, calculate its surface area.ANSWERSurface area = l w = 80 50 = 4000 mm2Relative uncertainty in l = 1/80 = 0.0125Relative uncertainty in w = 1/50 = 0.02Relative uncertainty in area = 0.0125 + 0.02 = 0.0325Absolute uncertainty in area = 0.0325 4000 mm2 = 130 mm2 Surface area = 4000 130 mm2Example problem 4Calculate the volume of a sphere if its diameter is measured and found to be 50.0 0.1 mm.ANSWERRadius, r = 25.0 0.05 mmVolume, V = (4/3) r3 = (4/3) (25.0 mm)3 = 65450 mm3Relative uncertainty in V = 3 relative uncertainty in rSo Relative uncertainty in V = 3 0.05/25.0 = 0.006Absolute uncertainty in V = 65450 0.006 = 393 mm3 V = 65450 393 mm3 Significant FiguresAll non-zero digits are significant

Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures

Zeros that do nothing but set the decimal point (place holders) are not significant. Thus, 470,000 and 0.0036 both have two significant figures

Zeros that are the last digits after a decimal point are significant. For example, 4.00, 20.0 and 3.10 all have three significant figures

Significant figures and decimal places in calculation.As a rule, the last digit of all measurement is uncertain.When combining measurements with different degrees of accuracy and precision, the accuracy/precision of the final answer can be no greater than the least accurate/precise measurement.

When measurements are added or subtracted, the answer can contain no more decimal places than the least accurate/precise measurement.

When measurements are multiplied or divided, the answer can contain no more significant figures than the least accurate/precise measurement.

EquationsCorrect physical equations are always homogenous, i.e. the units on both sides of the equal sign are the same.Prove that the following equations are homogeneous with respect to units:(a)

(b)Kinetic energy = mass (velocity)2

(c)Centripetal acceleration =

Use quantity algebra to obtain the base units equivalent for the unit of power. That is the Watt (W).

1999 Paper 2 Question 5(b)

Linear EquationsThe equation of a straight line isx and y are variablesm is a constant called the gradient or slope of the linec is a constant called the Y-interceptA straight line that passes through the origin may be represented by the expression y x. We say y is directly proportional to x.It may be converted to y = kxk is called the constant of proportionality

Think of some examples of these types of equations in Physics.

Linearizing equationsA plot of experimental data is better than an average since it may indicate systematic errors in the data.An equation for a line of best fit replaces a bunch of data with a few parameters (i.e. the slope, error in slope, Y-intercept, error in Y-intercept).A straight line is easy to spot with the unaided eye.If the data fits an equation of the form y = mx+c, then it is easy to plot a straight line graph and interpret the slope and yintercept, but it is rarelythat simpleIn most cases, the equation must be modified or linearized so that the variables plotted are different than the variables measured but produce a straight line.Linearizing equationsWe can linearize an equation if we can get it in the formvariable1 = constant1 variable2 + constant2Note thatSeveral constants combined together produces another single constant.Powers or functions of constants are also constants.Constants may have special values of 0 or 1 so they appear invisible. For exampley = mx is still the equation of a straight line, where c = 0,y = c is the equation of a line where m = 0, andy = x + c is the equation of a line with m = 1.Variables may be combined together to form new variables.Powers or functions of variables are also variables.Logarithm Suppose 10A = x then A is called the logarithm of x, or the logarithm to the base 10 of x.Therefore 10log x = xAs an example 103 = 1000 so that log(to the base 10) of 1000 is 3. We write this as log10 1000 = 3If the base is not specified we assume it to be 10, so log 5 is interpreted to mean log105.Laws of Logarithms

LogarithmWe will use only two basesbase 10 and base ee is the exponential function and has a value close to 2.718log (to the base e) is usually written as lnloge10 means ln10It should be obvious then that ln obeys all the rules of log

Graph of y = ex (Sometimes written as y = exp x)

Linearizing y = exIf you ln both sides of this equation you getln y = xPlotting ln y on the Y-axis and x on the X-axis gives a straight as shown below:

2003 Paper 2 ques 2 - Assignment

GraphsYou tell me!