IB Physics Topic 1 Measurement and Uncertainties.

IB Physics Topic 1 Measurement and Uncertainties

1.1 Measurements in Physics

A. Quantities vs. Units

Quantities : things that are measureable ( time, length, current, etc.) IB uses italics for quantities: v = Ds/DtUnits: what you measure quantities in: ( seconds, hours, feet, meters, amperes)

IB uses Roman upright : s, hr, ft, m, A


B. Fundamental (or Base) Quantities and

Units in the SI System Quantity Unit Name Symbol

length meter mtime second smass kilogram kg

current ampere Atemperature kelvin K

amount mole mol

luminous intensity candela cd


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


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


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?

Example: A newton is the SI force unit derived from: F = ma 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

kg m s-2


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.)


D. Significant Figures - Examples

Round Each Number to 3 s.f.

a. 2.04b. 1.005c. 0.002455

a. 2.04 b. 1.01 c. .00246


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.)


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


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


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

10m/200mm = (10m)/(200x10-3m) = (10)/(.200) = 50

Example: Divide 10m by 200mm.


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 stickPrecision = ± .05 cm(some may use ± .1cm)

Digital caliperPrecision = ± .01 mm

Example

The precision is ± .05 cm (1/2 of .1) The reading is 3.45cmThe 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 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 UncertaintiesConsider 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

Value = average ± (1/2 of the range) = 12.5 ± .2o

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

Absolute uncertainty = .2oFractional 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)


Example: Combine two masses ifm1 = 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


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 ExampleA 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

ExampleA 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


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 JAnd 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

The independent variable should be varied 5 times – use 5 different weights. This should be done 3 times – do 3 different trials.

How much data should she take?

Data Trial 1 Trial 2 Trial 3

F ± .020Ns ± .01

cms ± .01

cms ± .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 48.00

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.

Average Stretch

s ± 1.25 cm

9.87

19.50

30.00

38.67

47.83

Where did she get ± 1.25cm for the uncertainty in the average value?

Data and GraphsThe 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 ± .020

N

s ± .01 s

cm

s ± .01 cm

s ± .01 cm

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 1.00

0.490 48.00 47.50 48.00 0.25

Average Stretch

s ± 1.25 cm

9.87

19.50

30.00

38.67

47.83

Points to PlotAverage Stretch/ cm Force/N

Ds ± 1.25 cm DF ± .020N

9.87 3.430

19.50 12.250

30.00 9.800

38.67 9.800

47.83 2.450

Horizontal Error Bars are 2Ds = 2.50cm and vertical are 2DF = .040N

Graph with Best Linear Fit

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.000.000

0.100

0.200

0.300

0.400

0.500

0.600

f(x) = 0.0101157198386278 x

Stretch /cm

Force/N

Finding the Uncertainty in the Slope and y-intercept

Find the two extreme lines thatStill pass through the error barsand 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

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.000.000

0.100

0.200

0.300

0.400

0.500

0.600

f(x) = 0.0101157198386278 x

Stretch /cm

Force/N

m = .010 ±.001 N cm-1

b = .006 ± .030 N

IB Physics Topic 1 Measurement and Uncertainties.

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