1. 1.2 Uncertainties and errors
Random/systematic uncertainties
Absolute/fractional uncertainties
Propagating uncertainties
Uncertainty in gradients and intercepts

2. Let’s do some measuring!
1.2 Measuring practical
Do the measurements yourselves, but leave space in your table of results to record the measurements of 4 other people from the group

3. Errors/Uncertainties

4. Errors/Uncertainties
In EVERY measurement (as opposed to simply counting) there is an uncertainty in the measurement.
This is sometimes determined by the apparatus you're using, sometimes by the nature of the measurement itself.

5. Estimating uncertainty
As Physicists we need to have an idea of the size of the uncertainty in each measurement
The intelligent ones are always the cutest.

6. Individual measurements
When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)
4.20 ± 0.05 cm

7. Individual measurements
When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)
22.0 ± 0.5 V

8. Individual measurements
When using a digital scale, the uncertainty is plus or minus the smallest unit shown.
19.16 ± 0.01 V

9. Significant figures
Note that the uncertainty is given to one significant figure (after all it is itself an estimate) and it agrees with the number of decimal places given in the measurement.
19.16 ± 0.01
(NOT or 19.2)

10. Repeated measurements
When we take repeated measurements and find an average, we can find the uncertainty by finding the difference between the highest and lowest measurement and divide by two.

11. Repeated measurements - Example
Pascal measured the length of 5 supposedly identical tables. He got the following results; 1560 mm, 1565 mm, 1558 mm, 1567 mm , 1558 mm
Average value = 1563 mm
Uncertainty = (1567 – 1558)/2 = 4.5 mm
Length of table = 1563 ± 5 mm
This means the actual length is anywhere between 1558 and 1568 mm

12. Average of the differences
We can do a slightly more sophisticated estimate of the uncertainty by finding the average of the differences between the average and each individual measurement. Imagine you got the following results for resistance (in Ohms)
13.2, 14.2, 12.3, 15.2, 13.1, 12.2.

13. Precision and Accuracy
The same thing?

14. Precision
A man’s height was measured several times using a laser device. All the measurements were very similar and the height was found to be
± 0.01 cm
This is a precise result (high number of significant figures, small range of measurements)

15. Accuracy
Height of man = ± 0.01cm
This is a precise result, but not accurate (near the “real value”) because the man still had his shoes on.

16. Accuracy
The man then took his shoes off and his height was measured using a ruler to the nearest centimetre.
Height = 182 ± 1 cm
This is accurate (near the real value) but not precise (only 3 significant figures)

17. Precise and accurate
The man’s height was then measured without his socks on using the laser device.
Height = ± 0.01 cm
This is precise (high number of significant figures) AND accurate (near the real value)

18. Precision and Accuracy
Precise – High number of significent figures. Repeated measurements are similar
Accurate – Near to the “real” value

19. Random errors/uncertainties
Some measurements do vary randomly. Some are bigger than the actual/real value, some are smaller. This is called a random uncertainty. Finding an average can produce a more reliable result in this case.

20. Systematic/zero errors
Sometimes all measurements are bigger or smaller than they should be by the same amount. This is called a systematic error/uncertainty.
(An error which is identical for each reading )

21. Systematic/zero errors
This is normally caused by not measuring from zero. For example when you all measured Mr Porter’s height without taking his shoes off!
For this reason they are also known as zero errors/uncertainties. Finding an average doesn’t help.

22. Systematic/zero errors
Systematic errors are sometimes hard to identify and eradicate.

23. Uncertainties
In the example with the table, we found the length of the table to be 1563 ± 5 mm
We say the absolute uncertainty is 5 mm
The fractional uncertainty is 5/1563 = 0.003
The percentage uncertainty is 5/1563 x 100 = 0.3%

24. Uncertainties
If the average height of students at BSW is 1.23 ± 0.01 m
We say the absolute uncertainty is 0.01 m
The fractional uncertainty is 0.01/1.23 = 0.008
The percentage uncertainty is 0.01/1.23 x 100 = 0.8%

25. Let’s try some questions.
1.2 Uncertainty questions

26. Pages 7 to 10 of Hamper/Ord ‘SL Physics’
Let’s read!
Pages 7 to 10 of Hamper/Ord ‘SL Physics’
These pages are on propagating uncertainties and gradients etc. It is important for students to get used to reading their textbooks to learn!

27. Homework Complete “1.2 Measuring Practical” Taking one measurement;
Decide whether it is precise and/or accurate. Explain your answer.
Are there liable to be systematic or random uncertainties? (Explain)
How could a better measurement be obtained?
DUE Friday 12th September

28. Homework due today
On your tables can you compare your answers to the questions
Did you all agree?!

29. Propagating uncertainties
When we find the volume of a block, we have to multiply the length by the width by the height.
Because each measurement has an uncertainty, the uncertainty increases when we multiply the measurements together.

30. Propagating uncertainties
When multiplying (or dividing) quantities, to find the resultant uncertainty we have to add the percentage (or fractional) uncertainties of the quantities we are multiplying.

31. Propagating uncertainties
Data book reference
If y = ab/c
Δy/y = Δa/a + Δb/b + Δc/c
If y = an
Δy/y = nΔa/a

32. Propagating uncertainties
Example: A block has a length of 10.0 ± 0.1 cm, width 5.0 ± 0.1 cm and height 6.0 ± 0.1 cm.
Volume = 10.0 x 5.0 x 6.0 = 300 cm3
% uncertainty in length = 0.1/10 x 100 = 1%
% uncertainty in width = 0.1/5 x 100 = 2 %
% uncertainty in height = 0.1/6 x 100 = 1.7 %
Uncertainty in volume = 1% + 2% + 1.7% = 4.7%
(4.7% of 300 = 14)
Volume = 300 ± 10 cm3
This means the actual volume could be anywhere between 286 and 314 cm3

33. Propagating uncertainties
When adding (or subtracting) quantities, to find the resultant uncertainty we have to add the absolute uncertainties of the quantities we are multiplying.

34. Propagating uncertainties
Data book reference
If y = a ± b
Δy = Δa + Δb

35. Propagating uncertainties
One basketball player has a height of 196 ± 1 cm and the other has a height of 152 ± 1 cm. What is the difference in their heights?
Difference = 44 ± 2 cm

36. Who’s going to win? New York Times Bush 48% Gore 52% Gore will win!
Latest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%

37. Who’s going to win? New York Times Bush 48% Gore 52% Gore will win!
Latest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%

38. Who’s going to win? New York Times Bush 48% Gore 52% Gore will win!
Latest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%
Uncertainty = ± 5%

39. (If the uncertainty is greater than the difference)
Who’s going to win
Bush = 48 ± 5 % = between 43 and 53 %
Gore = 52 ± 5 % = between 47 and 57 %
We can’t say!
(If the uncertainty is greater than the difference)

40. Let’s try some more questions!
1.2 Propagating uncertainties

41. 1.2 Graphing uncertaintities practical
Students do 1.2 Graphing uncertainties practical

42. Error bars/lines of best fit
Mass of dog/kg
Time it takes dog to burn/s

43. Minimum gradient
Mass of dog/kg
Time it takes dog to burn/s

44. Minimum gradient
Mass of dog/kg
Time it takes dog to burn/s

45. Maximum gradient
Mass of dog/kg
Time it takes dog to burn/s

46. Error bars/line of best fits

47. Error bars/line of best fits

48. Some Maths!
B α L

49. Proportional?
If B α L then
B = kL

50. Proportional = straight line through origin
B = kL
Boredom/B
Length of time in class/s

51. k = ΔB/ΔL
B = kL
Boredom/B ΔB ΔL
Length of time in class/s

52. Inversely proportional?

53. Inversely proportional?
U α 1/W
Uniform conformity/U
Number of weeks of school/W

54. Inversely proportional?
U = k/W UW = k
Uniform conformity/U
Number of weeks of school/W

55. UW = k U1W1 = U2W2 Uniform conformity/U U1 U2
Number of weeks of school/W W1 W2

56. y = mx + c y x

57. y = mx + c y
m = Δy/Δx Δy Δx c c x

58. E = ½mv2

59. E = ½mv2
Energy/J ½m
v2/m2/s-2

60. R = aTb
R = aTb lnR = lna +blnT

61. lnR = lna + blnT
lnR b
lna
lnT

62. Gradient to a curve

63. Gradient to a curve

64. Let’s try an IB question!
Paper 3 – Question 1 is always a ‘data response’ question to do with error bars, lines of best fit, gradients etc.
At this point introduce them for the first time to a data response question (or two!)

65. 1.2 Period of a pendulum practical
Now do a simple practical that enables them to put some of topic 1 to test. Period of a pendulum to find g is ideal.

66. HOMEWORK
Complete “Pendulum investigation (DO what it says on the sheet!)
Due NEXT FRIDAY 19th September