3. Uncertainty2 Types of Uncertainties Random Uncertainties: result from the randomness of measuring instruments. They can be dealt with by making repeated measurements and averaging. One can calculate the standard deviation of the data to estimate the uncertainty. Systematic Uncertainties: result from a flaw or limitation in the instrument or measurement technique. Systematic uncertainties will always have the same sign. For example, if a meter stick is too short, it will always produce results that are too long.

4. Uncertainty When we measure some physical quantity with an instrument and obtain a numerical value, we want to know how close this value is to the true value. The difference between the true value and the measured value is the error. The term uncertainty is used to refer to “a possible value that an error may have”. Uncertainty can be expressed in either absolute terms (i.e., 5 Volts ±0.5 Volts) or, in percentage terms, 5 Volts ±10% {(0.5/ 5) x 100 = 10% }

5. Step 1 - Estimating Uncertainty Interval Use ± ½ smallest scale division as an estimate. What do you do if you DON’T have a statistically significant sample?

6. Uncertainty5 Accuracy vs. Precision Accurate: means correct. An accurate measurement correctly reflects the size of the thing being measured. Precise: repeatable, reliable, getting the same measurement each time. A measurement can be precise but not accurate.

7. Systematic errors are due to:  faults in the apparatus  wrong experimental technique. Systematic Errors These errors, cause readings to be consistently higher or lower than the true value really is. Your results are systematically wrong.

8. Experimental Technique Consider the error you may make in a reading a scale as shown.

9. Should the gap be wide or narrow for best results? Measurement 2 is best. 1 and 3 give the wrong readings. This is called a parallax error. It is due to the gap here, between the pointer and the scale.

10. Systematic Errors Consider if you make an error consistently on reading on a scale. Your results will always be higher than they should be.

11. Zero Errors A particular type of systematic error is the zero error. Apparatus that does not read zero even when it should. Over a period of time, the spring may weaken, and so the pointer does not point to zero. What affect does this have on ALL the readings?

13. Reading Uncertainties A reading uncertainty is how accurately an instruments scale can be read. Analogue Scales Where the divisions are fairly large, the uncertainty is taken as: half the smallest scale division

14. Where the divisions are small, the uncertainty is taken as: the smallest scale division

15. Digital Scale For a digital scale, the uncertainty is taken as: the smallest scale reading e.g. voltage = 29.7 mV ± 0.1 mV This means the actual reading could be anywhere from

16. For a digital meter the uncertainty is taken as the smallest scale reading. e.g. Voltage = 29.7 mV ± 1 mV

17. The ± 1 second is called the absolute uncertainty Every measurement has an uncertainty or error. e.g. time = 5 seconds ± 1 second There are two main types of uncertainty.  Random Uncertainties (Reading uncertainty)  Systematic Errors Uncertainties

18. Repeated measurements of the same quantity, gives a range of readings. The random uncertainty is found using: Taking more measurements will help eliminate (or reduce) random uncertainties. The mean is the best estimate of the true value. Random Uncertainties

19. give the mean to same number of significant figures as measurements (a) Example 1 Five measurements are taken to determine the length of a card. 209mm, 210mm, 209mm, 210mm, 200mm (a) Calculate the mean length of card. (b) Find the random uncertainty in the measurements. (c) Express mean length including the absolute uncertainty.

20. (b) (c) The “ ± 2mm ” is the absolute uncertainty.

21. Question Repeated measurements of speed give the following results: 9.87 ms -1, 9.80 ms -1, 9.81 ms -1, 9.85 ms -1 (a)Calculate the mean speed. (b)Find the random uncertainty. (c)Express mean speed including the absolute uncertainty. 9.83 ms -1 0.02 ms -1 9.83 ms -1 ± 0.02 ms -1

22. Percentage Uncertainty The percentage uncertainty is calculated as follows:

23. MultiPlicatiOn and DiVisiOn involving errOrs When multiplying and dividing, add the relative or percentage errors of the measurements being multiplied/ divided. The absolute error is then the fraction or percentage of the most probable answer.

24. What is the product of 2.6 ± 0.5 cm and 2.8 ± 0.5 cm? First, we determine the product: 2.6 cm × 2.8 cm = 7.28 cm^2 Relative error 1 = 0.5 ⁄ 2.6 = 0.192 Relative error 2 = 0.5 ⁄ 2.8 = 0.179 Sum of the relative errors = 0.371 or 37.1% Absolute error = 0.371 x 7.28 cm^2 or 37.1% x 7.28 cm^2 = 2.70 cm^2 Errors are expressed to one signiicant igure = 3 cm^2 The product is equal to 7.3 ± 3 cm^2

25. Uncertainty Range for Trigonometric Functions The mean, maximum and minimum values can be calculated to suggest an appropriate uncertainty range for trigonometric functions. For example, if an angle is measured as 30 ±2 0, then the mean value of sin 30 = 0.5, the maximum value is, sin 32 = 0.53 and the minimum value is, sin 28 = 0.47. The answer with correct uncertainty range is (max –min /2) [0.53-0.47 = 0.06/2 = 0.03] 0.5 ± 0.03 (one sig fig)

26. Addition and Subtraction involving errors When adding measurements, the error in the sum is the sum of the absolute error in each measurement taken. For example, the sum of 2.6 ± 0.5 cm and 2.8 ± 0.5 cm is 5.4 ± 1 cm If you place two meter rulers on top of each other to measure your height, remember that the total error is the sum of the uncertainty of each meter rule. (0.05 cm + 0.05 cm).

27. Uncertainties and Powers When raising to the n th power, multiply the percentage uncertainty by n, and when extracting the n th root, divide the percentage uncertainty by n. For example, if the length x of a cube is 2.5 ±0.1cm, then the volume will be given by x^3 = 15.625 cm^3. = 16 cm^3 The percentage uncertainty in the volume = 3(0.1⁄2.5 x 100) = 12%.

28. Uncertainties and Powers Therefore, 12% of 15.625 = 1.875 = 2 Volume of the cube = 16 ± 2 cm^3 If x = 9.0 ± 0.3 m, then √x = x1⁄2 = 3.0 ± 0.15 m = 3.0 ± 0.0.2 m.

29. Example 1 Calculate the percentage uncertainty of the measurement: d = 8cm ± 0.5cm (d = 8cm ± 6.25%)

30. Question 1 Calculate the % uncertainty of the following: a) I = 5A ± 0.5A b) t = 20s ± 1s c) m = 1000g ± 1g d) E = 500J ± 25J e) F = 6N ± 0.5N 10 % 5 % 0.1 % 5 % 8.3 %

31. Consider a simple measurement of the width of a board to be = 23.2 cm. However, measurement is only accurate to 0.05 cm (estimated).  Write width as (23.2  0.05) cm  0.05 cm  Experimental uncertainty Percent Uncertainty:  (0.05/23.2)  100   0.2%

32. Uncertainty31 Absolute and Percent Uncertainties If x = 99 m ± 5 m then the 5 m is referred to as an absolute uncertainty. You may also need to calculate a percent uncertainty ( % error): Please do not write a percent uncertainty as a decimal ( 0.05) because, it is difficult to distinguish it from an absolute uncertainty.

34. Combining Uncertainties Use the following data to calculate the speed, and the uncertainty in speed, of a moving object. Calculation of Speed

35. Calculation of Uncertainty

36. Uncertainty in Speed The biggest uncertainty is used, so get: The absolute uncertainty in the speed: Answer OR

38. Order of Mgnitude Order of magnitude, reflects our emphasis on approximation. Sometimes, we are interested in only an approximate value for a quantity. We are interested in obtaining rough or order of magnitude estimates. Order of magnitude, for all its uncertainty, is a good indicator of size. Order of magnitude estimates: Made by rounding off all numbers in a calculation to 1 sig fig, along with power of 10. –Can be accurate to within a factor of 10

39. Orders of magnitude are generally used to make very approximate comparisons and reflect very large differences. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100.

40. Task Example: It is said that the average person blinks about 1000 times an hour, 1 x 10 3 per hour. This is an order-of-magnitude estimate, that is, it is an estimate given as a power of ten. Consider: If 100 blinks per hour, which is about two blinks per minute is expressed as: 2 x 10 0 10,000 blinks per hour, which is about three blinks per second is expressed as: 3 x 10 0 Neither of these are reasonable estimates for the number of blinks a person makes in an hour.

41. Example Give an order-of-magnitude estimate for the time in seconds for one year: An order of magnitude, usually referred to the nearest power of 10. There are 60 seconds in a minutes, 6 x10 1 60 minutes in an hour, 6 x10 1 24 hours in a day, round it down to 2 x 10 1 365 days in a year, round it up to 4 x 10 2 That is, 6 x10 1 x 6 x10 1 x 2 x 10 1 x 4 x 10 2 = 288 x 10 5 3 x10^2 x 10^5 = 3 x 10^7 seconds

42. Example Estimate the magnitude of your age, considering your age to be 17 years. Assuming the years to be regular year and not counting the leap year.  17 years, round it up to 20, 2 x 10 ^1  365 days, round it up to 400, 4 x 10^2  24 hours in a day, round it down to 20, 2 x10^1  3600 s in one hour rounded up to 4000, 4 x 10^3 Your age in seconds: 2 x 10^1 x 4 x 10^2 x 2 x 10^1 x 4 x 10^3 = 64 x 10^7 seconds rounded up to 1 x 10^9 Order of magnitude = 1 x 10^9 seconds

43. Task Make order-of-magnitude estimates for each of the following: Your age in hours. The number of breaths you take in a year. The number of heart beats in a lifetime. The number of basketballs that would fill your classroom.

44. (a) Find the order of magnitude of your age in seconds. (1)10^3 s (2)10^6 s (3)10^9 s (4)10^12 s (b) Estimate the number of times your heart beats in a month. (1)10^2 (2)10^4 (3)10^6 (4)10^8 (c) Estimate the number of human heart beats in an average lifetime (70 yrs). (1)10^5 (2)10^7 (3)10^9 (4)10^11

45. Example: V = πr 2 d Example: Estimate!

46. Order of Magnitude Suppose that a newspaper article says that the annual cost of health care in the United States will soon surpass $1 trillion. Whenever you read any such claim, you should automatically think: Does this number seem reasonable? Is it far too small, or far too large? You need methods for such estimations, methods that we develop in several examples.