1.2 uncertainties and errors random/systematic uncertainties absolute/fractional uncertainties...
TRANSCRIPT
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 practicalDo the
measurements yourselves, but leave space in your table of results to record
the measurements of 4 other people from
the group
Errors/Uncertainties
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.
19.16 ± 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.
• 19.16 ± 0.01
• (NOT 19.160 or 19.2)
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
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
184.34 ± 0.01 cm
This is a precise result (high number of significant figures, small range of measurements)
AccuracyHeight of man = 184.34 ± 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 = 182.23 ± 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.
UncertaintiesIn 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%
UncertaintiesIf 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%
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 12th September
Homework due today
• On your tables can you compare your answers to the questions
• Did you all agree?!
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 = an
• 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 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
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 TimesLatest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%
Who’s going to win?
New York TimesLatest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%
Who’s going to win?
New York TimesLatest opinion poll
Bush 48%
Gore 52%
Gore will win!
Uncertainty = ± 5%
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
1.2 Graphing uncertaintities practical
Error bars/lines of best fitMass of dog/kg
Time it takes dog to burn/s
Minimum gradientMass of dog/kg
Time it takes dog to burn/s
Minimum gradientMass of dog/kg
Time it takes dog to burn/s
Maximum gradientMass of dog/kg
Time it takes dog to burn/s
Error bars/line of best fits
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 = kLBoredom/B
Length of time in class/s
k = ΔB/ΔL
B = kLBoredom/B
Length of time in class/s
ΔL
ΔB
Inversely proportional?
Inversely proportional?
U α 1/WUniform conformity/U
Number of weeks of school/W
Inversely proportional?
U = k/W
UW = kUniform conformity/U
Number of weeks of school/W
U1W1 = U2W2
UW = kUniform conformity/U
Number of weeks of school/W
U1
U2
W1 W2
y = mx + c
y
x
y = mx + c
y
x
c
c Δx
Δy m = Δy/Δx
E = ½mv2
E = ½mv2
Energy/J
v2/m2/s-2
½m
R = aTb
R = aTb
lnR = lna +blnT
lnR = lna + blnT
lnR
lnT
lna
b
Gradient to a curve
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 FRIDAY 19th September