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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
1.1 Measurements in Physics
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
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?
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
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.
a. 2.04b. 1.005c. 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
10m/200mm = (10m)/(200x10-3m) = (10)/(.200) = 50
Example: Divide 10m by 200mm.
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 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)
1.2 Propagation of Uncertainties
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
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 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
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 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