Year 13 Physics

Uncertainti

es and

Graphing

THE BASICS........

These units are:

Physics involves measuring physical quantities such as the length of a spring the charge of an electron.

Units are used to help us know what we are measuring.

There are 7 quantities that make up the fundamental units which are called the Systeme International d’Unites’or SI Units.

Physical Quantities & their Units

Quantity SI Unit

Name Symbol(s) Name Symbol

LengthMassTimeElectric Current TemperatureLuminous IntensityAmount of substance

l, x, y,d,M, mT, tITIN

MetreKilogramSecondAmpereKelvinCandelaMole

mkgsAKcdmol

• Prefix multipliers can be added to each of the units in both numerically and in the form of names.

• You are expected to be able to convert from name to numbers and back again.

• You already use some of the these prefixes:Kilogram = 1000grams OR 1 x 103g

The standard SI prefixes are as follows:

Standard Prefix Multipliers

Power Prefix Symbol

109

106

103

10-3

10-6

10-9

10-12

GMkmnp

Solution

a. 43 g = 0.043 kg

b. 137 mA = 0.137 A

c. 2.8 km = 2800 m

d. 38 cm2 = 0.0038 m2

e. 6.8 x 104 cm3 = 6.8 x 10-2 m3

a. Convert 43 g to kg.

b. Convert 137mA to A

c. Convert 2.8km to m

d. Convert 38cm² to m²

e. Convert 6.8 x 104 cm3

to m3.

Quick Exercise:

Significant FiguresThese give us an indication as to how accurate a measurement can be. For instance:

23.0cm is 3 sig. figs23.01cm is 4 sig. figs and is more accurate.

Our measurement tool can measure to 0.01cm.

In Physics we use sig. figs to show the level of accuracy we are measuring to. In a lot of cases we measure to 2 significant figures.When converting to specific sig. figs we count from the first digit, not zero.

• Exercise:A metre ruler was used to make distance measurements. The following results were recorded by different students.a. 43cm b. 43.0cm c. 43.03 cmComment on each of these measurements and explain which of the three measurements are valid?Measurement A is given

to the nearest cm. A metre ruler can measure to the nearest mm, and so the measurement should have been recorded as 43.0cm.

Measurement C implies it is possible to accurately read to one tenth of a millimetre. This is not possible using a metre ruler, so the measurement should have been recorded as 43.0cm.

Measurement B correctly expresses the measurement to the nearest mm and so is valid.

• the measurement should be the least number of significant figures of the data values used.Multiplication

or Division

• the measurement should be rounded to the least number of decimal places of the data values used.Addition or

Subtraction

Rules for Processing Data

Example:

SOLUTION

a. Perimeter of the plate:

= 15.4 + 0.94 + 15.4 + 0.94

= 32.68

= 32.7 cm (to 1 decimal point)

b. Area of the plate:

= 15.4 x 0.94

= 14.476

= 14 cm2 (to 2 sf)

A metal plate has the following dimensions: Length 15.4cm. Width 0.94cm.

Calculate :Perimeter of the plateArea of the plate

Uncertainties of Measurement

MeasurementMeasuring necessarily involves some element of inaccuracy.This may be due to such things as:• Limits to the precision which a scale may be read.• Reaction times when using a stopwatch.• An incorrectly calibrated meter.• A zero error on a measuring device.

This means that any measurement taken must be recorded with its absolute uncertainty.

eg. t = 5.3s + 0.1sThe +0.1s is the ‘uncertainty’ or ‘error’.

Systematic UncertaintiesThese are errors arising due to such things as:• An incorrectly zeroed meter.• An inaccurate timing device.• Using an approximate theory. (ie. Ignoring friction in a pulley) etc.

Random UncertaintiesThese errors tend to arise due to limitations on our ability to measure or judge. For example:• Timing with a stopwatch.• Estimating the sharpest image formed by a lens.• Marking the amplitude on a swinging pendulum. etc

Measuring necessarily involves some element of inaccuracy.This may be due to such things as:• Limits to the precision which a scale

may be read.• Reaction times when using a

stopwatch.• An incorrectly calibrated meter.• A zero error on a measuring device.

This means that any measurement taken must be recorded with its absolute uncertainty.

eg. t = 5.3s + 0.1sThe +0.1s is the ‘uncertainty’ or ‘error’.

Systematic UncertaintiesThese are errors arising due to such things as:• An incorrectly zeroed meter.• An inaccurate timing device.• Using an approximate theory. (ie.

Ignoring friction in a pulley) etc.

Random UncertaintiesThese errors tend to arise due to limitations on our ability to measure or judge. For example:• Timing with a stopwatch.• Estimating the sharpest image formed

by a lens.• Marking the amplitude on a swinging

pendulum. etc

Measurement

Origin of Uncertainties

1. Limitations in the accuracy of the measuring instrument.

2. Limitations in the skill of the experimenter.3. Fluctuations in the physical quantity being measured.4. External influences on the measurement procedure.

Uncertainties are not caused by mistakes rather they are due to such things as:

If the measurements are used to draw a graph, there will be uncertainty in the gradient of the graph line. This will cause uncertainty in the value of any physical quantity calculated from the gradient.

An absolute uncertainty is always given to 1 sf only.

A measurement must not be expressed to a greater accuracy that the absolute uncertainty.

A processed measurement must not be expressed to a greater number of sf than the raw data from

which it was found.

Significant figures and uncertainties

• The number of sig. figs (s.f) used when giving a measurement should reflect the amount of uncertainty involved. The following principles should be applied whenever decisions about s.f are made.

Calculating Uncertainty from data: The absolute uncertainty “X” for a number

of data entries can be calculated by finding the range. This is done as follows:

Absolute Uncertainty = Difference between highest and lowest entry

2

Written as (measurement) X ± X (uncertainty)

• The relative uncertainty is a percentage and is used when processing uncertainties. The relative uncertainty can be calculated by:

Relative Uncertainty = X x 100

X

Rules For Manipulating Uncertainties

1. When adding or subtracting quantities – ADD the absolute errors.

2. When multiplying or dividing quantities – ADD the relative errors.

From these rules we can infer that:• When raising to a power, multiply the relative error by that power.

Note: That includes fractional powers.• When multiplying a quantity by a constant, multiply the absolute error by

that constant.

Examples: a = 5.2 + 0.2, b = 7.5 + 0.5Find:

ba

bav

biv

aiii

abii

bai

2

3)(

)(

)(

)(

)(

3

2

Express the errors as percentages. a = 5.2 + 0.2 (3.8%)b = 7.5 + 0.5 (6.7%)

(in case they are needed)

113

7.07.12)(

bai The error is 0.2 + 0.5 Note that the final result

should be given to no more than 2 sig. fig. The error must be given to only 1 significant figure.

439)( abii Adding percentage errors means finding 10.5% of 39

227

06.205.27)( 2

aiii The error is 7.6% (double 3.8%)

1.00.2

04.096.1)( 3

biv The percentage error is 2.2% (one third of 6.7%)

2.2% of 1.96 is 0.04 which is best rounded up tobe on the safe side. The final result can still be only2 sig. fig.

(v) Working should be presented as follows:3a + b = 23.1 + 1.1 (4.8%) (3 x 0.2 +

0.5)2a – b = 2.9 + 0.9 (31%)

3a + b = 7.97 + 2.9 (36%)2a – b

= 8 + 3 NB. Only 1 s.f for the error means that in this case

there will only be 1s.f. in the result.

Complex formulae calculations with uncertainties.

HWK

Find the time period of a mass on a spring under simple harmonic motion including uncertainties using the following formula:

T = 2(m/k)Where m = 0.5 ± 5% Kg k = 30.0 ± 0.2 Nm-1

T = 0.81 ± 0.02 s

SOLUTION:a. Convert k to relative: 0.2 / 30 x 100% = 0.67%b. Divide m by k: 0.5 / 30 = 0.0167c. Add relatives: 5 + 0.67 = 5.667%d. Thus: T = 2(0.0167 ± 5.66%)e. Square root measurement: 0.0167 = 0.129f. Divide relative by 2: 5.667 / 2 = 2.833%g. Thus: T = 2 (0.129 ± 2.833)h. 2 is pure so no uncertainty:6.283± 0%i. Multiply out: (6.283 ± 0%) x (0.129 ±

2.833%)j. Thus T=0.811 ± 2.83%k. Convert to absolute: 0.811 / 100 x 2.83 =

0.023sl. Thus: T = 0.811 ± 0.023sm. Convert to 1 sig fig: T = 0.81 ± 0.02s

Using these skills complete

exercises from page 4-12

Rutter

GRAPHINGSKILLS......

CONVERTING RELATIONSHIPS TO EQUATIONS

In order to produce an equation from data the following steps should be observed:

1. Tabulate your raw data and process data including uncertainties.

2. Sketch measurement data onto a graph.

3. Decide what relationship has been produced i.e. y x; y 1/x; y x2 e.t.c.

4. Do the function to measurement and uncertainty i.e. if y x2 then do x2 to all

the x values and multiply relative uncertainty by two.

5. Plot y versus x2 and this should produce a straight line if you have drawn it

correctly.

6. Error bars need to be drawn on to show uncertainty above & below point.

7. Produce a line of best fit and then worst fit (using error bars)

8. Calculate the gradient of the line of best fit (m) and worst fit (m’) using the rise/run method.

9. Calculate uncertainty for line of best fit (m) by subtracting worst fit (m’) from best fit (m). m = m’ – m

10. Substitute m ± m into the ‘y = mx + c’ equation.

11. Calculate ‘c’, from y intercept of line of best fit and its uncertainty (c) from subtracting y intercept for the line of best fit (c) from y intercept for the line of worst fit (c’). c = c’ – c.

12. Rewrite the equation y = mx +c, with the function of ‘x ± x’, the new gradient (m ± m ) and the y intercept if there is one (c ± c) .

Graphing with Uncertainties

Experimental data will contain uncertainties

These uncertainties must be considered when trying to determine the relationship between variables

Error bars are used to show the uncertainty in individual data points

Lines of best & worst fit are used to find the overall uncertainty in the relationship

Uncertainties on GraphsWhen plotting measured, or calculated points, error bars must be shown. The length of the bar will depend on the size of the uncertainty. In most cases only the dependent variable errors are plotted as the independent only show equipment errors.

mm’

The ‘line of best fit’ is drawn and its gradient m calculated.Another line (either a maximum or a minimum) error line is drawn. This line should still be within the bounds of the error bars. It’s gradient m’ is calculated.

The error in the gradient is |m – m’|. ie. The absolute value of the difference.

Graphing:Your graph must show:

Simple Pendulum (heading)

Period vs Length (a statement of what is being plotted)

Dep

end

ent

vari

able

Independent variable

T(s) (symbols with appropriate SI unit)

l(m)

Suitable scales.

A

m

lθ

Graphing and Data CollectionConsider the experiment to find the relationship between

The period and length of a simple pendulum

Independent variable • Usually the variable that you have control over. eg: The length of

a pendulum.• (You are physically able to adjust and measure that.)

Dependent Variable • This is the quantity that you have to measure and is affected by

how you alter the independent variable.• eg. The period of the pendulum.

Other variables: • If the aim is to discover a relationship between the period and length of a pendulum, then other variables that may affect the

period must be identified and kept constant. eg: the mass of the pendulum bob, the amplitude or the size of the semi-vertical

angle.

Data

Time (s) Speed (ms-1)0.5 2.71.0 4.01.5 5.42.0 6.72.5 8.13.0 9.43.5 10.84.0 12.1

Uncertainty in time values is ±0.2 s and speed values are ±10%

Calculate the value of each

uncertainty

Data

Time (s) Time Speed (ms-1) Speed0.5 0.2 2.7 0.271.0 0.2 4.0 0.41.5 0.2 5.4 0.542.0 0.2 6.7 0.672.5 0.2 8.1 0.813.0 0.2 9.4 0.943.5 0.2 10.8 1.084.0 0.2 12.1 1.21

Uncertainty in time values is ±0.2 s and speed values are ±10%

Graph the data

Add error bars

How to calculate error bars

+ 0.2 s

- 0.2 s

Total length of error bar is 0.4 s

Each bar shows the possible values for data point

Draw line of BEST fit

Draw line of WORST fit

LABEL the lines

Line of worst fit

Line of best fit

Calculate gradients &

record intercepts for

BOTH lines

Line of worst fit

Line of best fit

Line of worst fit:Intercept = 2.0 ms-1

Gradient = 7.6/3.4 = 2.235

Calculate the overall error

Line of best fit:Intercept = 1.4 ms-1

Gradient = 10/3.9 = 2.564

Line of worst fit:Intercept = 2.0 ms-1

Gradient = 7.6/3.4 = 2.235

Final Errors:Intercept = (2.0 – 1.4) = ± 0.6 ms-1

Gradient = (2.235 – 2.564) = ± 0.329 ms-2

State the relationship between the

variables

This statement shows • linear relationship between

speed and time• error in the GRADIENT • error in the INTERCEPT• units

Using these skills complete

exercises from Rutter

Page 13-27

LE 06

To investigate the relationship between applied force and the amount of deflection when a ruler is bent

• Page 29 Rutter.• Follow instructions and draw up a table similar to the one below.• Hwk: complete the write-up.

m m x x F50g

100g150g

500g