Question 4: How do we determine how certain we are when taking data? Measurement = Best Estimate +/- Uncertainty Or Examples:
Representing uncertainty: Absolute or Raw uncertainty Percentage Uncertainty time = 1.98 0.01 s time = 1.98 9%
Determining uncertainty with common measuring devises. Measurement = Best Estimate +/- Least Count
Determining Uncertainty with common measuring tools:
Uncertainty isn’t only due to the tool… * Limitations for the data collector * YOU decide the uncertainty, but if it is anything other than the uncertainty of the tool, you must justify your choice Ex. The ability to stop a timer the instant a ball rolls past a given point adds an uncertainty greater than that of the least count. It’s hard to see just when the ball passes the point you want. You make a judgment that this additional uncertainty is perhaps 0.02s. Hense the uncertainty in a single measurement of the time might be +/-0.03s.
Using a Caliper:
Fig 2 What is the reading that should be recorded here? Answer: mm
What is the reading that should be recorded here? Answer: mm
Give a value for the following: State which tool you used and why. State why you chose the uncertainty that you did. Length of the lab table Thickness of a 1 zloty coin The height of your calculator
Question 5: What is the difference between precision and accuracy?
Precision vs. Accuracy Accuracy: an indication of how close a measurement is to the accepted value. Precision: the degree of exactness to which the measurement of a quantity can be reproduced (how close are repeated measurements to each other? If someone else were to make the same measurement how close would they be? Often defined by the tool)
Precision vs. Accuracy Label each of the ducks with the either: High Accuracy, High precision High Accuracy, Low precision Low Accuracy, High precision Low Accuracy, Low precision
Precision vs. Accuracy
Understanding uncertainty and improving accuracy from random error: Repeated Measurements help us quantify the uncertainty due to random error.
Question 7: How do we know what the uncertainty of a calculation is? Ex: You have just measured the radius of a ball to be (6.1 +/- 0.02)mm, but now you want to find the volume.
Repeated measurements - Averaging If repeated trials of the same measurement are made, the greatest deviation from the average of the trials can be taken as the uncertainty. For example, Time Trials 2.01 1.82 1.97 2.16 1.94
Adding and Subtracting An uncertainty range can be found by examining the minimum and maximum values for the calculated value. Ex:T i = 16 2 o C = 14 to 18 o C T f = 34 2 o C = 32 to 36 o C best value for T = ___________ minimum value for T = ____________ maximum value for T=___________ T = T f – T i = ____________
Adding and Subtracting Shortcut: Whenever two quantities with uncertainties are added or subtracted, the uncertainty of the answer is equal to the SUM of the individual ABSOLUTE uncertainties. eg. T = T f – T i = (34 2 o C) – (16 2 o C) = (34 – 16 ) (2 + 2) o C = 18 4 o C
Multiplying and Dividing An uncertainty range can be found by examining the minimum and maximum values for the calculated value. ex.m = 84.2 0.1 g = 84.1 to 84.3 g V = 25 2 mL = 23 to 27 mL Best value for D = m V = Minimum value for D = Maximum value for D = D =
Multiplying and Dividing Shortcut: Whenever two quantities with uncertainties are multiplied or divided, the uncertainty of the answer is equal to the SUM of the individual PERCENT uncertainties. Ex. m = 84.2 0.1 g = 84.1 g 0.12% V = 25 2 mL = 25 mL 8% D = (84.1 g 0.12%) (25 mL 8%) = (84.1 25 g / mL ) ( %) = g / mL 8.12% = g / mL = 3.4 0.3 g / mL
You try: The lengths of the sides of a rectangular plate are measured. The width is (50+/-1)mm and the length is (25 +/- 1)mm. What is area (with uncertainty) of the plate. Find the area of the top of your lab table. Report your answer with uncertainties. Show all measurements and calculations.
Power Functions Power functions are just like multiplication!
Other Functions If the calculation involves mathematical operations other than +, -, or (eg. root functions, trigonomeric functions, logarithmic functions, … ), then the short cuts do not apply and we must find the minimum and maximum values and use the greatest deviation from the best value as the uncertainty. ex. = 33 C 2 C = 31 C to 35 C best value for sin = sin(33) = minimum value for sin = sin(31) = maximum value for sin = sin(35) = Take the maximum: sin =
Volume of a sphere Ex: You have just measured the radius of a ball to be (6.1 +/- 0.02)mm, but now you want to find the volume.
You try: The power dissipated in a resistor of resistance R carrying a current I is equal to I^2R. The value of I has an uncertainty of +/- 2% and the value of R has an uncertainty of +/-10%. The value of the uncertainty in the calculated power dissipation is: