1watt=1W=1kg m 2 /s 3

Transcription

1 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory we will mainly use te SI or metric system. Te SI unit of lengt is te meter (m). Te mass unit is te kilogram (kg). Te time unit is te second (s). In nearly all laboratory situations, units of meter, kilograms and seconds sould be used. rom tese base units, we will derive all te oter quantities we will study, e.g. volume, acceleration, energy, etc. or example, a watt can be expressed as: 1watt=1W=1kg m /s 3 Because some of te tings we will use are very big or small, i.e. te mass of an atom or te distance to te sun, we often employ scientific notation. Tis is just expressing te number wit powers of ten. or example te distance to te moon is approximately: distance to moon = 38,000,000 m Tat is a lot of zeros so we will write it as: distance to moon = 3.8 x 10 8 m Proper scientific notation is a number between 1.0 and and a power of ten. Te number of decimal places we give wit a number is called te significant digits. Tese depend on ow well we know te number. In te above distance to te moon, we ave 3 significant digits. In te SI system tere are standard prefixes wic tell us te power of ten to associate wit te unit. or example kilo (k) means 10 3,soakilometeris1000meters.Someofte common SI prefixes are: 77

2 actor Prefix Symbol actor Prefix Symbol 10 1 tera- T 10 1 pico- p 10 9 giga- G 10 9 nano- n 10 6 mega- M 10 6 micro- µ 10 3 kilo- k 10 3 milli- m 10 ecto- 10 centi- c 10 1 deka- da 10 1 deci- d An important issue is ow to convert units. You may be given a time in minutes and want to convert to seconds. Te important concept is to multiply te quantity by 1. or example, 1m=3.81ftso1= 3.81ft = 1.0m.Toconvert57fttometers: 1.0m 3.81ft 57 ft = 57 ft 1 m 3.81 ft =17.37m You sould always put te units in your calculation and sow all conversion factors. If te units come out wrong, e.g. a lengt in kg, you ave done someting wrong. In pysics, dimension refers to te pysical nature of te quantity. Dimensions cancel out just like algebraic quantities, so aving te units work out is te best ceck you can make of your calculation. Some basic conversion factors are given in te following list: Lengt: 1 in =.54 cm Energy 4186 J = 1 kcal 1m=10 cm = 10 3 mm 1 ev = 1.60 x J Mass 1 kg = 1000 grams Angle 1 radian = o 1 AMU = x 10 7 kg 1 o = radian Some important pysical constant are given in te following table: Constant Symbol Value Acceleration of gravity g 9.80 m s Universal gravitational constant G 6.67 x N m kg Speed of ligt in vacuum c.997 x 10 8 m s Avogadro s Number N A 6.0 x mol Electron carge e 1.60 x C Planck s constant 6.66 x J s Permeability of free space µ o 4 x10 7 T m/a Permittivity of free space o x 10 A. Graps and Plotting 1 C N m Graps are often te most important part of your report and an excellent way to display matematical relationsips (functions). Tere are several tings to keep in mind wen you 78

3 make a grap eiter on te computer or drawn by and. It sould fill most of te page if drawn by and. Bot axes sould be labeled wit an axis title and units and ave tick marks. Data points sould ave error bars if appropriate. If you draw a best fit line troug te data, it sould be your estimate of te line tat passes nearest to all te points. It sould not necessarily connect te first and last points or be a connect te dots series of line segments. If you take a slope, you sould indicate on te grap wat points on te line (not single data points) were used for te vertical and orizontal di erence as well as te final value. A grap sould look someting like tis sample grap: My Grap Title 4 Velocity (m/s) slope = x y = (m/s) = (m).86 (s 1 ) Distance (m) A.3 Statistics Wen a measurement is made, it is often useful to compare te measured result to an accepted or standard value. Percent error is used wen comparing a result to an accepted value. %error= (X Xs) X s x100% were X s is te standard or accepted value and X is te experimental value. Percent difference is used wen comparing two results from di erent experimental metods or measurements. Te average of te two measurement is probably closer to te actual value tan eiter measurement. So, te average is used in te denominator. %di erence= X 1 X X avg x100% were X 1 is one experimental value and X is te oter experimental value. Te average, X avg is given by X avg = X 1+X Wat if tere tere are several measured values x 1,x,...,x n?tearitmetic mean is defined as: X = 1 nx X i. n i=1 79

4 were X is te aritmetic mean, n is te number of data points and X i are te values for te individual points. Te mean tells us te average value of all of te measurements. Anoter important quantity tells us te spread or uncertainty in te set of measurement. irst, te deviation is defined by: D i = X i X. were D i is te deviation, X i are te values measured, and X is again, te aritmetic mean. Tere is a Deviation value for eac data point and it is eiter positive or negative in sign. Sometimes te term residual is used instead of deviation. Te standard deviation ( ) is defined by: = 1 nx (D i ) n i=1 were D i is te it deviation and n is te number of measurements. Te standard deviation tells us te widt or uncertainty in te set of n measurements. Te assumption beind tis is tat any data measurement as an equal likeliood of being iger or lower tan te mean. Te percent probability of te absolute value of data points falling witin integral multiples of are as follows: 68% -,95.4%-,99.7%-3. Eac individual measure is not exact. Te normal metod by wic uncertainty or error is assigned for single measurements wit analog devices suc as as voltmeter, meter sticks, etc. is to assume te error in any given measurement is one-alf te smallest scale division of te device. Tus, te ruler tat is graduated in millimeters as an error of ±0.5 mm. or example, you ave measured ow far a glider as traveled before it comes to a complete stop. Tis distance is ten determined wit a ruler and is found to be 1.34 meters and a small amount more or less tan one millimeter. By convention, you are allowed to round up or down to te nearest alf millimeter. So, te measurement appears nearer te middle of te distance between te millimeter marks tan eiter side. You will ten express te number as meters. If te distance is nearer to te millimeter marks te distance is ten 1.34 or 1.35 depending on te position of te glider s edge. So, to completely state your measured quantity, you can write te following: ± m As anoter example, you ave measured te time tat te glider was in motion. You found it to be.5 seconds by means of a stopwatc tat as digits to te undredt of a second. Tis is expressed as:.5 ± 0.01 seconds Note tat it is not possible to determine weter or not te time point is nearer te middle of te undredt or te end marks. Te stopwatc only gives wole digits to te undredt. Tis is an exception to te 1 smallest scale division rule (ssd) and is applicable to digital measurements made wit computers. 80

5 Wen you do calculations wit measured numbers to arrive at anoter number, te error in te calculated number is determined by wat calculations are done. If you add two numbers, te error in te sum is di erent from te error in two numbers you multiply togeter. Te following equations contain formulas for propagating uncertainties troug your calculations. Using tese basic equations, you can calculate te uncertainty in complicated expressions. = x + y, error:( ) =( x) +( y). = x y, error: = q ( x x ) +( y y ). = x/y, error:seeexpressionfor = x y. = x error: =x( x). = x n y m z k,error: = Cx y,error: = q (n x x ) +(m y y ) + k(z z z ). = q (y x x ) +( y ln x). Te above expressions will enable you to express te error equations of all of te calculations you will encounter in tis lab. Normally te values are te standard deviations ( ) from previous calculations or te ssd rule. Tese can be substituted back to arrive at expressions for te more complicated equations. A.4 Trigonometry You must be able to do simple algebra and trigonometry. Trigonometry relates te sides and angle for rigt triangles. or te angle te trig functions are defined as: sin = o cos = a tan = o a 81

6 Note tat te coice of wic side is te adjacent and opposite is determined by te angle. If you look up te value of a trig function, you can determine te angle. Tese are called inverse or arc trig functions: =sin 1 o =cos 1 a =tan 1 o a Note tat sin 1 does not means 1 sin. inally, te trig functions are related by te Pytagorean teorem: = o + a or 1 = cos +sin 8