IB Physics Topic 1 Measurement and Uncertainties

1.1 Measurements in Physics Quantities vs. Units Quantities : things that are measureable ( time, length, current, etc.) IB uses italics for quantities: v = Ds/Dt Units: what you measure quantities in: ( seconds, hours, feet, meters, amperes) IB uses Roman upright : s, hr, ft, m, A

1.1 Measurements in Physics Fundamental (or Base) Quantities and Units in the SI System Quantity Unit Name Symbol length meter m time second s mass kilogram kg current ampere A temperature kelvin K amount mole mol luminous intensity candela cd

1.1 Measurements in Physics Note: units written out in full always use the lower case: meter, newton, pascal, ampere, second When the symbol for the unit is written, it may be upper case if named for a person: m, N, P, A, s

1.1 Measurements in Physics C. Derived Quantities and Units in the SI System - made of Base Units Example: Volume comes in cubic meters. The volume of a cube, 2.0m on each edge is 8.0 m3 Example: Density comes in kg per cubic meters. The density of a 16.0 kg cube, 2.0m on each edge is : r = m/V = 16.0 kg/ 8.0 m3 = 2.0 kg m-3 Example: Velocity comes meters per second. The velocity of a car that moves 10.0m in 2.00 seconds is: v = Ds/Dt = 10.0m / 2.00 s = 5.00 m s-1

1.1 Measurements in Physics Note: instead of m/s, IB uses : m s-1 instead of kg/ m3 IB uses: kg m-3 Example: Acceleration is derived from: a = Dv/Dt What are the derived units? Units of velocity are m s-1 and of time are s. Therefore the units of acceleration are : m s-1 / s = m s-2 Example: A newton is the SI force unit derived from: F = ma What are the derived units? kg m s-2

1.1 Measurements in Physics D. Significant Figures Counting Significant Figures For a decimal number: Find the 1st non-zero digit and count all the way to the right. Trailing zeros are significant. 002.00 ( 3 s.f.) 0.02050 ( 4 s.f.)

1.1 Measurements in Physics D. Significant Figures - Examples Round Each Number to 3 s.f. 2.04 1.005 0.002455 a. 2.04 b. 1.01 c. .00246

1.1 Measurements in Physics D. Significant Figures Counting Significant Figures For a non – decimal number, trailing zeros are not significant: 2450 ( 3 s.f.) 10205 ( 5 s.f.) 1000 (1 s.f.)

1.1 Measurements in Physics E. Scientific Notation The speed of light is 3.00 x 108 m s-1 This number is in Scientific Notion: The Coefficient is 3.00 The Base is 10 The Exponent is 8

1.1 Measurements in Physics E. Exponent Rules Division: xa / xb = xa-b Example: 12 x 105 / 4 x 103 = (12/4) x 105-3 = 3 x 102 Multiplication: xa . xb = xa+b Example: (12 x 105 )( 4 x 103) = (12.4) x 105+3 = 48 x 108

1.1 Measurements in Physics E. Exponent Rules Addition or Subtraction: ( Must have same exponent! ) Axa ± Bxa = (A ± B)xa Example: (12 x 105 ) + (4 x 106 ) = (1.2 x 106 ) + (4 x 106 ) = 5.2 x 106

SI Prefixes

SI Prefixes Example: Put into scientific notation: a. 300mm b. 368mm c. 200MV a. 300mm = 300 x 10-3m = 3.00 x 10-1 m b. 368mm = 368 x 10-6m = 3.68 x 10-4 m c. 200MV = 200 x 106V = 2.00 x 108 V

SI Prefixes Example: Change .000000056m into mm. .000000056m = .056 x 10-6m = .056mm Example: Divide 10m by 200mm. 10m/200mm = (10m)/(200x10-3m) = (10)/(.200) = 50

1.1 Measurements in Physics F. Order of Magnitude A number rounded to the nearest power of 10 is called an “order of magnitude”. Example: What’s the order of magnitude for a 3.8 gram sheet of paper? 3.8 grams = 3.8 x 10-3 kg rounds to 10-3 kg ( 3.8 is closer to 1 than 10), so : The order of magnitude is 10-3 kg.

IB Physics Topic 1.2 Uncertainties and errors Systematic Error - Due to the system used to make the measurements - examples: Improperly calibrated instrument, zero error, damaged instrument - Cannot be corrected by repeat measurements. - Causes poor accuracy

IB Physics Topic 1.2 Uncertainties and errors Random Error - Due to the estimating a scale reading- The precision of an analog scale is usually ± ½ of the smallest division but this may not always be true. You look at each situation and make a reasonable estimate. The precision of a digital readout is ± one of the last digit place shown

Example Meter stick Precision = ± .05 cm (some may use ± .1cm) Digital caliper Precision = ± .01 mm

Example The precision is ± .05 cm (1/2 of .1) The reading is 3.45cm The measurement is expressed as: L = 3.45 ± .05cm

1.2 Precision from several measurements To reduce the random error - take several measurements. - find the average and round off to the same decimal place as the precision of the instrument - find ½ of the range Report your measurement M as: M = average ± (1/2 of the range)

Example The reported measurement is: 12.33 ± .10cm The following lengths were measured with a meter stick: 12.30cm, 12.40cm, 12.20cm, 12.35cm, 12.40cm 1st: Get the average (12.33 cm) 2nd: Get ½ of the range .5( 12.40-12.20) = .10cm The reported measurement is: 12.33 ± .10cm

Reporting a Measurement Summary The precision of a scale is ½ of the smallest division. For a single reading: Value = quantity ± precision For several readings ( best method) : Value = average ± (1/2 of the range) Each part of the measured Value must go to the same number of decimal places.

1.2 Uncertainties Consider this reported length: Length = 12.5 ± .2 cm We can generalize this to: Measured Value = Quantity ± Uncertainty The Absolute Uncertainty is the absolute value of ±.2cm = | ±.2cm | = .2cm The Fractional Uncertainty = Uncertainty/Quanatity = .2/12.5 = .008 The Percent Uncertainty = 100 x Fractional Uncertainty = .008 x 100 = .8%

Example For the following temperature readings find the absolute, fractional, and percent uncertainties. 12.5o, 12.7o, 12.4o Average = (12.5 + 12.7 + 12.4)/3 = 12.5 (Round to 10ths ) Uncertainty = ½ of range = .5( 12.7-12.4) = .15 , rounds to .2 Value = average ± (1/2 of the range) = 12.5 ± .2o Absolute uncertainty = .2o Fractional uncertainty = .2 / 12.5 = .016 Percent uncertainty = 100 x .016 = 1.6%

1.2 Propagation of Uncertainties Value = Quantity ± Uncertainty = a ± Da Addition or Subtraction of 2 Values: If V1 = a ± Da and V2 = b ± Db then V1 + V2 = (a + b) ± (Da +Db)

1.2 Propagation of Uncertainties Example: Combine two masses if m1 = 120 ± 5g and m1 = 150 ± 5g mtotal = (120+150) ± (5+5)g = 270 ± 10g Example: Subtract 35 ± .5mm from 55 ± .5mm (55-35) ± (.5+.5)mm = 20 ± 1mm

1.2 Propagation of Uncertainties Multiplication or Division of 2 Values: If V1 = a ± Da and V2 = b ± Db then V1 x V2 = (ab) ± (bDa +aDb) OR……………………. %Uncertainty of the product = %Uncertainty of a + %Uncertainty of b Dp/p = Da/a +Db/b

Area Example Take a = 10 and Da = 2, b = 12 and Db = 2, then A rectangle’s sides are measured as L1 = 10 ± 2cm and L2 = 12 ± 2cm. Find the area and the uncertainty in the area. Take a = 10 and Da = 2, b = 12 and Db = 2, then Area = (ab) ± (bDa +aDb) = 120 ± (12(2) + 10(2)) cm = 120 ± 44cm Alternately : DA/A = Da/a +Db/b = 2/10 + 2/12 = 11/30 A = 10(12) = 120cm and DA = A (11/30) = 44cm Area = 120 ± 44cm

Example A solid cylinder has a measured mass of 2.00 kg with a 2.5% uncertainty and a measured volume of .00100 m3 with a 5% uncertainty. Find the density and the uncertainty of the density. %uncertainty in the density = % uncertainty of the mass + % uncertainty of the volume = 7.5% = .075 Density = m/v = 2.00kg/ .00100 m3 = 2.00 x 103 kg/m3 Density Value = 2.00 x 103 ± .075(2.00 x 103) kg/m3 = 2.00 x 103 ± .15 x 103 kg m-3

1.2 Propagation of Uncertainties Quantity Raised to a Power: If a = bn Da /a = n |Db /b| %Uncertainty of the product = %Uncertainty of the base x the power

1.2 Exponent Example The potential energy, U, of a compressed spring is given by: U=1/2kx 2 where k = 500 ± 10 N m-1 And x = .25 ± .02 m. Find the quantity and uncertainty of U. DU/U = Dk/k + 2Dx/x = 10/500 + 2(.02/.25) = .02 + .16 = .18 or 18% Then U=1/2kx 2 = .5(500)(.252) = 6.25 J And DU = .18U = .18(6.25) = 1.13 J U = 6.25 ± 1.13 J

Data and Graphing A student wants to determine the relationship between the distance a spring stretches and the force used to do the stretching. She does this by hanging known weights on the spring and measuring the stretch. What is the independent variable? That’s what she controls – the weight F / N How much data should she take? The independent variable should be varied 5 times – use 5 different weights. This should be done 3 times – do 3 different trials.

Data Note: 5 variations of the independent variable ( F) and 3 trials.   Trial 1 Trial 2 Trial 3 F ± .020N s ± .01 cm 0.098 9.90 9.50 10.20 0.196 21.00 19.00 18.50 0.294 31.00 29.00 30.00 0.392 39.50 37.50 39.00 0.490 48.00 47.50 Average Stretch s ± 1.25 cm 9.87 19.50 30.00 38.67 47.83 Note: 5 variations of the independent variable ( F) and 3 trials. We want to plot F on the y- axis and the average value of s on the x-axis. Where did she get ± 1.25cm for the uncertainty in the average value?

Data and Graphs The uncertainty in the average value is ½ of the greatest range of the dependent variable.   Trial 1 Trial 2 Trial 3 1/2 Range F ± .020N s ± .01 s cm s ± .01 cm 0.098 9.90 9.50 10.20 0.35 0.196 21.00 19.00 18.50 1.25 0.294 31.00 29.00 30.00 1.00 0.392 39.50 37.50 39.00 0.490 48.00 47.50 0.25 Average Stretch s ± 1.25 cm 9.87 19.50 30.00 38.67 47.83

Points to Plot Horizontal Error Bars are 2Ds = 2.50cm Average Stretch/ cm Force/N Ds ± 1.25 cm DF ± .020N 9.87 3.430 19.50 12.250 30.00 9.800 38.67 47.83 2.450 Horizontal Error Bars are 2Ds = 2.50cm and vertical are 2DF = .040N

Graph with Best Linear Fit

Finding the Uncertainty in the Slope and y-intercept Find the two extreme lines that Still pass through the error bars and find their equations. The uncertainty in the slope is ½ of the range: Dm = (.011-.009)/2 =.001 The uncertainty in the intercept is ½ the range: Db =(.03-(-.03))/2 = .03

Final Results