1. Ranges of Magnitudes & Quantities
Order of Magnitude- Power of ten
Comparison:
Diameter of an atom: m
Diameter of a proton: m
Atom is 5 order of magnitudes larger
Length of a railway station: 102 m
Diameter of the Earth: 107 m
Railway is 5 order of magnitudes smaller

2. Ranges of Quantities Mass (kg) 1052 –total mass of universe
10-30 –mass of an electron
Length (m)
1026 –radius of universe
10-15 –diameter of a proton
Time (sec)
1019 –age of universe
10-23 –passage of light across a nucleus
Energy (J)
1044 –released during a supernova
10-18 –needed to remove an electron from the surface of a metal

3. Types of Units Fundamental Derived
Base or foundational units for everything we measure (they cannot be broken down into other units)
A unit which is a combination of 2 or more fundamental units which then is given a new name for simplicity.

4. Ex. of Fundamental and Derived Units

5. Estimation with SI Units
Fundamental Units
Mass: 1 kg- box of sugar/ 1 L of H2O/ An avg. person is 50 kg
Length: 1m- Distance between one’s hands with outstretched arms
Time: 1 sec.- Duration of resting heartbeat
Derived Units
Force: 1 N- weight of an apple
Energy: 1 J- Work lifting an apple off of the ground

6. Graphing Graphs are titled and axes labeled with units
Normally include the origin (0,0) in the plot
Trend (or best fit) line (or curve) is drawn
If it is a straight line, use a straight edge
If it is a curve, make it a smooth curve
Generally, there should be the same # of points above the line as there are below the line (be mindful of curves)
Identify points that don’t agree with the best fit line (outlier)

7. Graphing Examples: Data set #1 Data set #2 Measurement in error
The graph to the left has the same # of points above and below the line. Although there is one point out of sync, the pattern of the data for set 1 is best represented by a line
Data set #2
Although the graph to the left has the same # of points above and below the line, the pattern of the data for set 2 is best represented by a curve as shown to the right

8. Uncertainties & Errors
A. Random Errors
Readability of an instrument
A less than perfect observer
Effects of a change in the surroundings
Can be reduced by repeated readings
B. Systematic Errors
Cannot be reduced by repeated readings
A wrongly calibrated instrument
An observer is less than perfect for every measurement

9. Uncertainties & Errors (cont.)
An experiment is accurate if……
it has a small systematic error
An experiment is precise if……
it has a small random error
Systematic error x
Perfect
Random errors

10. Uncertainties & Errors (cont.)
Accuracy and Precision:
Precise but not accurate
Accurate but not precise
Precise and accurate!
Precision– uniformity, # of decimal places
Accuracy- conformity to a standard

11. Determining the Range of Uncertainty
1) Analogue scales (rulers, thermometers, meters with needles) 1 4 3 2 5
± half of the smallest division
Since the smallest division on the cylinder is 1 ml, the reading would be 3.5 ± 0.5 ml
2) Digital scales
± the smallest division on the readout
If the digital scale reads 5.052g, then the uncertainty would be ± 0.001g
Absolute Uncertainty- has units of the measurement

12. Range of Uncertainty (cont.)
3. Significant Figures
If you are given a value without an uncertainty, assume its uncertainty is ±1 of the last significant figure
Examples:
The measurement is g, the uncertainty of the measurement is ± .001 g
The measurement is 50ml, the uncertainty of the measurement is 50 ± 1 ml

13. Range of Uncertainty (cont.)
4. From repeated measurements (an average)
Example: A student times a cart going down a ramp 5 times, and gets these numbers:
2.03s, 1.89s, 1.92s, 2.09s, 1.96s Average: 1.98s
Find the deviations between the average value and the largest and smallest values.
Largest: = 0.11s
Smallest: = 0.09s
The largest deviation is taken as the uncertainty range:
1.98 ± 0.11 s

14. Mathematical Representation of Uncertainty
For calculations, we compare the value of our calculated value without uncertainties (the best value) with the max and min values we would get with uncertainties in the calculation.
Example 1:
Find the density of a block of wood if its mass is 15g ± 1g and its volume is 5.0 ± 0.3cm3
Best value m v
Density = =
15 g
5.0cm3
= 3.0 g/cm3

15. Mathematical Representation of Uncertainty
Example 1 (cont.):
Find the density of a block of wood if its mass is 15g ± 1g and its volume is 5.0 ± 0.3cm3
Maximum value: m v
Density = =
16 g
4.7 cm3
= 3.40 g/cm3
Minimum value: m v
Density = =
14 g
5.3 cm3
= 2.64 g/cm3

16. Mathematical Representation of Uncertainty (cont.)
The uncertainty range of our calculated value is the largest difference from the best value. In this case, the difference between the max and best value is larger- 0.4.
In this case, the density is 3.0 ± 0.4 g/cm3
The uncertainty in the previous problem could have been written as a percentage, which is found by dividing the uncertainty (Dy) by the best value (y) Dy y =
0.4 3
X 100%
= 13%
In this case, the density is 3.0 ± 13%

17. Mathematical Representation of Uncertainty (cont.)
Example #2:
What is the uncertainty of cos q if q = 60o ± 5o?
Best value of cos q = cos 60o = 0.50
Max value of cos q = cos 55o = 0.57
Min value of cos q = cos 65o = 0.42
Deviates 0.7
Deviates 0.8
The largest deviation is taken as the uncertainty range:
In this case, it is 0.50 ± .08

18. Mathematical Representation of Uncertainty: Shortcuts!
The techniques given in the past few pages can sometimes be a bit time consuming. The following are shortcuts for the given mathematical operations:
Addition and Subtraction:
When 2 or more quantities are added or subtracted, the overall uncertainty is equal to the sum of the individual uncertainties.
Uncertainty of 2nd quantity
Uncertainty of 1st quantity
Total uncertainty
Dy = Da + Db

19. Mathematical Representation of Uncertainty: Shortcuts! (cont.)
Example for Addition and Subtraction:
Determine the thickness of a pipe wall it the external radius is 4.0 ± 0.1 cm and the internal radius is 3.6 ± 0.1 cm
Internal radius = 3.6 ± 0.1 cm
External radius = 4.0 ± 0.1 cm
Thickness of pipe: 4.0 – 3.6 = 0.4cm
Uncertainty = 0.1cm cm = 0.2 cm
Thickness with uncertainty: 0.4 ± 0.2 cm OR 0.4 ± 50%

20. Mathematical Representation of Uncertainty: Shortcuts! (cont.)
Multiplication and Division:
The overall uncertainty is approximately equal to the sum of the percentage (or fractional) uncertainties of each quantity.
Dy = Da + Db + Dc
y a b c
Denominators represent best values
Total percentage/
fractional uncertainty
Fractional Uncertainties of each quantity

21. Mathematical Representation of Uncertainty: Shortcuts! (cont.)
Example for Multiplication and Division:
Using the density example from before (where the mass was 15g ± 1g and its volume is 5.0 ± 0.3cm3)
Dy = Da + Db
y a b =
= = .13
( this means 13%)
13% of 3g/cm3 is 0.39
The uncertainty is 3.0 ± 0.39 g/cm3 or 3.0 g/cm3 ±13%

22. Mathematical Representation of Uncertainty: Shortcuts! (cont.)
For exponential calculations (x2, x3):
Just multiply the exponent by the percentage (or fractional) uncertainty of the number.
Example:
Cube- each side is 6.0 ± 0.1cm
Volume = (6cm)3 = 216 cm3
Percent
uncertainty
0.1 6
x 100 % =
= 1.7%
Uncertainty in value
= 3 (1.7%) = ± 5.1% (or 11cm3)
Therefore the volume is 216 ± 11cm3

23. Uncertainties in Graphs
When graphing a quantity, error bars are drawn to represent the absolute uncertainties in the variables.
Uncertainty in t- ± 0.2s
Uncertainty in d- ± 0.5 m

24. Uncertainties in Graphs (cont.)
The best fit line is drawn so that it falls within the error bars around each point.
Speed = slope of graph
= Dd / Dt
= 6-3 / 4-2
=1.5 m/s

25. Uncertainties in Graphs (cont.)
The absolute uncertainty of the slope is found by finding the average of the max and min slopes of the data.
Maximum slope
*Notice that the lines of max and min slope still fall within the uncertainties of every data point.
Minimum slope

26. Uncertainties in Graphs (cont.)
Max slope= 7.9-0/ =1.76 m/s
Min slope= / =1.26 m/s

27. Uncertainties in Graphs (cont.)
The uncertainty in the slope is found by the largest difference from the best value (slope of trendline. In this case, the difference between the maximum slope and bet value was 0.26, while the difference from the minimum slope was 0.24, so the uncertainty of the speed was ± 0.25 m/s, and you would write 1.5± 0.25 m/s
Notice the y intercept (starting distance) varied between 0.6m in front and 0.6m behind the “starting point”

28. Transforming Graphs
In Physics, the slope of a graph that represents 2 quantities that are directly proportional means something. In the previous case, the slope of a distance-time graph is equal to the speed of the object. What about these ones:
time
velocity
acceleration
Force
time
Work
Slope = acceleration
Slope = mass
Slope = Power

29. Transforming Graphs (cont.)
This technique can only be used for directly proportional relationships, but we can transform graphs of quantities that are inversely related, square related, or logarithmically related by doing the following:
1 / mass
acceleration
mass
acceleration
Since a a 1/m, if we graph
a vs. 1/m, we will get a straight line
Since Fnet=ma or a= Fnet/m
Slope =
1 / Force

30. Transforming Graphs (cont.)
For an object accelerating uniformly from rest:
Since d= ½ at2 (from rest), d a t2. So if we graph d vs. t2, we will get a straight line.
time
distance t2
distance
The slope of this line is equal to Dd/ Dt2. This slope is equal to ½ a (since d= ½ at2 -> d/t2 = ½ a)

31. Transforming Graphs (cont.)
The gravitational force between 2 objects and the distance between them:
Since F= Gm1m2 / r2, F a 1/r2. So if we graph F vs. 1/r2, we will get a straight line.
distance
Force
1/r2
Force
The slope of this line is equal to DF/ D(1/r2). This slope is equal to Gm1m2
(since F= Gm1m2/r2 -> F/r2 = Gm1m2)

32. Proportional vs Directly Proportional
A linear or straight line relationship between two variables
Directly Proportional
A linear relationship between 2 variables which passes through the origin