SI Units and Uncertainties
-
Upload
neve-hoover -
Category
Documents
-
view
90 -
download
4
description
Transcript of SI Units and Uncertainties
![Page 1: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/1.jpg)
SI Units and Uncertainties
Unit 1: Measurements
![Page 2: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/2.jpg)
SI Units and Uncertainties
SI Unit (Le Système International d’Unités)
Fundamental units meter (m) kilogram (kg) second (s) ampere (A) Kelvin (K) mole (mol) candela (cd)
![Page 3: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/3.jpg)
SI Units and Uncertainties
Derived Units Any unit made of 2 or more
fundamental units m s-1
m s-2
Newton (N) = kg m s-2
Joule (J) = kg m2 s-2
Watt (W) = kg m2 s-3
Coulomb (C) = A s
![Page 4: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/4.jpg)
Estimation with SI Units
Fundamental Units Mass: 1 kg – 2.2lbs / 1 L of H2O /
An avg. person is 50 kg Length: 1 m - Distance between one’s
hands with outstretched arms Time: 1 s - Duration of resting heartbeat
Derived Units Force: 1 N- weight of an apple Energy: 1 J- Work lifting an apple off of
the ground
![Page 5: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/5.jpg)
Scientific Notation and Prefixes
SI prefixes Table
1 Gm = 1,000,000,000 m = 1,000,000 km1 GM = 1 x 109 m = 1 x 106 km
0.0000000001 s = 1 ?s = ? ms
![Page 6: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/6.jpg)
Uncertainties & Errors
A. Random Errors1. Readability of an instrument2. A less than perfect observer3. Effects of a change in the
surroundings
Can be reduced by repeated readings
B. Systematic Errors1. A wrongly calibrated instrument2. An observer is less than perfect
for every measurement in the same way
Cannot be reduced by repeated readings
![Page 7: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/7.jpg)
Uncertainties & Errors (cont.)
•An experiment is accurate if……it has a small systematic error
it has a small random error
x
x
x
x
Systematic error
Random errors
Perfect
•An experiment is precise if……
![Page 8: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/8.jpg)
Uncertainties & Errors (cont.)
Accuracy and Precision:
Precise but not accurate
Accurate but not precise
Precise and accurate!
Precision– uniformityAccuracy- conformity
to a standard
![Page 9: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/9.jpg)
Determining the Range of Uncertainty
1) Analogue scales (rulers,thermometers meters with needles)
±half of the smallest division
2) Digital scales
±the smallest division on the readout
If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g
10
40
30
20
50
Since the smallest division on the cylinder is 10 ml, the reading would be 32 ± 5 ml
Absolute Uncertainty- has units of the measurement
![Page 10: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/10.jpg)
Range of Uncertainty (cont.)
3. Significant Figures
•The measurement is 14.742 g, the uncertainty of the measurement is 14.742 ± .001 g•The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml
If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure
Examples:
![Page 11: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/11.jpg)
Range of Uncertainty (cont.)
4. From repeated measurements (an average)
Find the deviations between the average value and the largest and smallest values.
Example: A student times a cart going down a ramp 5 times, and gets these numbers: 2.03 s, 1.89 s, 1.92 s, 2.09 s, 1.96 s Average: 1.98 s
The average is the best value and the largest deviation is taken as the uncertainty range:
Largest: 2.09 - 1.98 = 0.11 sSmallest: 1.98 - 1.89 = 0.09 s
1.98 ± 0.11 s
![Page 12: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/12.jpg)
Mathematical Representation of Uncertainty
Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3
= g5.0 cm3
= 3.0 g cm-3
For calculations, compare the calculated value without uncertainties (the best value) with the max and min values with uncertainties in the calculation.
Example 1:
Best value
mv
Density =
![Page 13: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/13.jpg)
Mathematical Representation of Uncertainty
Find the density of a block of wood if its mass is 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3
= g4.7 cm3
= 3.40 g cm-3
Example 1 (cont.):
Maximum value:
mv
Density =
Minimum value:
mv
Density = = g5.3 cm3
= 2.64 g cm-3
![Page 14: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/14.jpg)
Mathematical Representation of Uncertainty (cont.)
•The uncertainty in the previous problem could have been written as a percentage
In this case, the density is 3.0 g cm-3 ± 13%
yy
= 3
X 100% = 13%
•The uncertainty range of our calculated value is the largest difference from the best value..
In this case, the density is 3.0 ± 0.4 g cm-3
![Page 15: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/15.jpg)
Mathematical Representation of Uncertainty (cont.)
Example #2: What is the uncertainty of cos if = 60o ±5o?
•Best value of cos = cos 60o = 0.50•Max value of cos = cos 55o = 0.57•Min value of cos = cos 65o = 0.42
The largest deviation is taken as the uncertainty range:
In this case, it is 0.50 ± .08 OR 0.50 ± 16%
Deviates 0.07
Deviates 0.08
![Page 16: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/16.jpg)
Mathematical Representation of Uncertainty: Shortcuts!
When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties.
Addition and Subtraction:
y = a + b Uncertainty of 2nd quantity
Uncertainty of 1st quantity
Total uncertainty
![Page 17: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/17.jpg)
Mathematical Representation of Uncertainty: Shortcuts! (cont.)
•Determine the thickness of a pipe wall if the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm
Example for Addition and Subtraction:
Internal radius = 3.6 ± 0.1 cm
External radius = 4.0 ± 0.1 cm
Thickness of pipe: 4.0 cm – 3.6 cm = 0.4 cm
Uncertainty = 0.1 cm + 0.1 cm = 0.2 cm
Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 cm ± 50%
![Page 18: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/18.jpg)
Mathematical Representation of Uncertainty: Shortcuts! (cont.)
The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity.
Multiplication and Division:
y = a + b + cy a b c Denominators
represent best values
Total percentage/ fractional uncertainty
Fractional Uncertainties of each quantity
![Page 19: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/19.jpg)
Mathematical Representation of Uncertainty: Shortcuts! (cont.)
Using the density example from before (where the mass was 15 g ± 1 g and its volume is 5.0 ± 0.3 cm3)
Example for Multiplication and Division:
y = a + by a b
= 1 + 0.3
15 5= 0.07 + 0.06 = .13 ( this means 13%)
13% of 3 g cm-3 is 0.4 g cm-3
3.0 ± 0.4 g cm-3 or 3.0 g cm-3 ± 13%
The result of this calculation with uncertainty is:
![Page 20: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/20.jpg)
Mathematical Representation of Uncertainty: Shortcuts! (cont.)
Just multiply the exponent by the percentage (or fractional) uncertainty of the number.
For exponential calculations (x2, x3):
Cube- each side is 6.0 ± 0.1 cm Example:
Percent uncertainty
= 1.7%0.16
x 100 %=
Volume = (6 cm)3 = 216 cm3
Uncertainty in value = 3 (1.7%) = ± 5.1% (or 11 cm3)
Therefore the volume is 216 ± 11 cm3
![Page 21: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/21.jpg)
Problems:
1. If a cube is measured to be 4.0+_ 0.1 cm in length along each side.
Calculate the uncertainty in volume.
Answer: 64+_5 Cm
![Page 22: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/22.jpg)
Problem ( IB 2010)
The length of each side of a sugar cube is measured as 10 mm with an uncertainty of +_2mm. Which of the following is the absolute uncertainty in the volume of the sugar cube?
a.+_6 mm c. +_400 mmb. +_8 mm d. +_600 mm
![Page 23: SI Units and Uncertainties](https://reader035.fdocuments.net/reader035/viewer/2022081419/56813347550346895d9a4074/html5/thumbnails/23.jpg)
Problem:
3. The lengths and width of a rectangular plates are 50+_0.5 mm and 25+_0.5 mm. Calculate the best estimate of the percentage uncertainty in the calculated area.
a. +_0.02% c. +_3%b. +_1 % d. +_5%