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

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

Can you complete writing the results of 5 other people into your results from yesterday’s “measuring” experiment?

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.

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.

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

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

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

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 ± 0.01 (NOT or 19.2)

Significant figures Go through the results of our measuring experiment and make sure all results are written to the correct number of significent figures Write the uncertainty to each measurement.

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.

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

Repeated measurements Now can you choose 3 of the measurements we did last lesson (i.e. current in circuit, height of your teacher, length of table) and calculate the average value AND its uncertainty.

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.

Precision and Accuracy The same thing?

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)

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.

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)

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)

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

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.

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 )

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.

Systematic/zero errors Systematic errors are sometimes hard to identify and eradicate. Let’s look at your measurement results – where might there have been systematic uncertainties?

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 = The percentage uncertainty is 5/1563 x 100 = 0.3%

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 = The percentage uncertainty is 0.01/1.23 x 100 = 0.8%

Let’s try some questions. 1.2 Uncertainty questions

Let’s read! Pages 7 to 10 of Hamper/Ord ‘SL Physics’

Homework Complete “1.2 Measuring Practical” Taking one measurement; i.Decide whether it is precise and/or accurate. Explain your answer. ii.Are there liable to be systematic or random uncertainties? (Explain) iii.How could a better measurement be obtained? DUE Friday 11 th September

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.

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.

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

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 cm 3 % 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 cm 3 This means the actual volume could be anywhere between 286 and 314 cm 3

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.

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

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

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

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

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

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)

Let’s try some more questions! 1.2 Propagating uncertainties

HOMEWORK Complete both “Uncertainty” sheets for Wednesday 16 th September.

1.2 Graphing uncertaintities practical

Can you get your homework out?

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

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

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

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

Error bars/line of best fits

Some Maths! B α L

Proportional? If B α L then B = kL

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

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

Inversely proportional?

U α 1/W Uniform conformity/ U Number of weeks of school/W

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

U 1 W 1 = U 2 W 2 UW = k Uniform conformit y/U Number of weeks of school/W U1U1 U2U2 W1W1 W2W2

y = mx + c y x

y x c c Δx Δy m = Δy/Δx

E = ½mv 2

Energy /J v 2 /m 2 /s -2 ½m

R = aT b lnR = lna +blnT

lnR lnT lna b

Gradient to a curve

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.

1.2 Period of a pendulum practical

HOMEWORK Complete “Pendulum investigation (DO what it says on the sheet!) Due NEXT TUESDAY 22 nd September