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Edexcel International A Level Your notes

Physics
Practical Skills II: Analysis
Contents
Calculations using Experimental Data
Plotting Graphs
Using Units Correctly
Trend & Patterns in Experimental Data
Interpreting Graphs
Precision & Accuracy
Reducing Errors
Suggesting Improvements
Uncertainties
Calculating Percentage Uncertainties

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Calculations using Experimental Data

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Calculations Using Experimental Data
Collecting experimental data is only some of the work required when carrying out a practical
When setting up a data table, space must always be left for some calculations
The most common calculation is the mean for repeat readings

sum of the readings

Mean =
number of readings
The value of the mean is then used in for further calculations
Anomalous readings should be ignored in this calculation
The mean should have the same number of significant figures as the readings used to calculate it
The experiment may require to calculate a variable which you can't directly measure
E.g., the area of a wire
In this case, other measurements are taken, which then by using an equation, the variable that is
required can then be calculated
E.g., area of a wire = πr2, so r is measured for the wire using a micrometer and substituted into this
equation to calculate the corresponding area for each value of r
Another example of this is finding the 'log' of a value
One column in a data table should be for the measurement
A column next to it should be the for the 'log' of that measurement
In the Hooke's law experiment, the 'extension' cannot be measured direction, but can be calculated
from the final and initial length, which can be measured
Hooke's Law Table of Results

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Worked Example
A student wants to find the resistivity of a constantan wire. They set up the experiment by attaching
one end of the wire to a circuit with a 6.0 V battery and the other with a flying lead and measuring the
length with a ruler. Attaching the flying lead onto the wire at different lengths, they obtain the
following table of results.

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Calculate the the missing values from the table.

Answer:
The average current is calculated by
I1 + I2 + I3
3
The resistance is calculated using the equation

All readings are to 3 significant figures, so all values calculated should also be to 3 s.f.

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Your notes

Examiner Tips and Tricks

These calculations show why it's important not to draw your data tables too big, without leaving
space for more columns and rows. Think carefully about what data you need to measure, but also
what you may need to calculate in order to draw graphs in the future. Thinking ahead this way will
reduce the change of drawing messy tables that you'll have to keep redoing in the exam!

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Plotting Graphs
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Plotting Graphs
When plotting graphs, it is important to consider the importance of the following factors:
Selecting appropriate scales
Labelling axes with quantities and units
Carefully plotting the points

Choice of Scale
When choosing a scale, it must be big enough to accommodate all the collected values using as much
of the graph paper as possible
At least half of the graph grid should be occupied in both the x and y directions
Scales should be clearly indicated and have suitable, sensible ranges that are easy to work with
For example, scales with multiples of 3 should be avoided
The scales should increase outwards and upwards from the origin
Each axis should be labelled with the quantity that is being plotted, along with the correct unit

Labelling the Axes

Label each axis with the name of the quantity and its unit
For example, F / N means force measured in Newtons
The convention is that a forward slash ( / ) is used to separate the quantity and the unit
In general:
The independent variable goes on the x-axis
The dependent variable goes on the y-axis

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Your notes

Example of labelled axes with the name of the variable, its symbol and its unit
Plotting the Points
Points should be plotted so that they all fit on the graph grid and not outside it
All values should be plotted, and the points must be precise to within half a small square
Points must be clear, and not obscured by the line of best fit, and they need to be plotted with a sharp
pencil so that they are thin
There should be at least six points plotted on the graph, with any major outliers identified

Line or Curve of Best Fit

There should be equal numbers of points above and below the line of best fit
Using a clear plastic ruler will help with this
Not all lines will pass through the origin and nor should they be forced to
The line (or curve) of best fit should not be too thick or joined dot-to-dot like a frequency polygon
Anomalous values that have not been identified during the implementation stage should be ignored if
they are obviously incorrect
This is because they will have a large effect on the gradient of the line of best fit

Determining the y-intercept

The y-intercept is the y value obtained where the line crosses the y-axis at x = 0
Values should be read accurately from the graph, with the scale on the y-axis being interpreted
correctly

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Worked Example
A student investigates the effect of placing an electric fan in front of a wind turbine. The wind turbine
is connected to a voltmeter. When the wind turbine turns, it generates voltage. The student obtains
the following results:

Plot the student’s results on the grid and draw a curve of best fit on the graph.

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Answer:
Step 1: Identify the independent and dependent variables
Independent variable = blade angle / °
Dependent variable = voltage / V
Step 2: Choose an appropriate scale
The range of the blade angle is 0 – 90°
Ideally, every small square represents 10°
The range of the voltage is 0 – 2.2 V
Ideally, each small square represents 0.5 V
Both axes should occupy at least 50% of the grid
Step 3: Label the axes
The dependent variable (voltage / V) goes on the y-axis
The independent variable (blade angle / °) goes on the x-axis
Both axes should be labelled with a quantity and a unit
Step 4: Plot the points
Each point should be accurate within half a small square

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Step 5: Draw a curve of best fit

The curve should be smooth with a roughly equal distribution of points on either side of the
curve
It must start at (0,0) and peak at (20, 2.2)

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Your notes

Examiner Tips and Tricks

Remember that 'sketching' and 'plotting' a graph are two different command words
'Sketch' means – Produce a freehand drawing. For a graph, this would require a line and labelled
axis with important features indicated, the axes are not scaled.
'Plot' means – Produce a graph by marking points accurately on a grid from data that is provided
and then drawing a line of best fit through these points. A suitable scale and appropriately
labelled axes must be included if these are not provided in the question
The difference between these two command words is the use of scales. A plotted graph has scaled
axes, whilst a sketch doesn't have to be but both times the axes should be clearly labelled

Logarithmic Scales
Graphs can be logarithmic in nature
A logarithmic (log) scale is a non-linear scale often used for analysing a large range of quantities
The log of a number is always greater than 1, so all log values are only positive
Hence, when drawing a log-log graph, the graph will only have a positive quadrant

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Often, in practicals, if the log of a value is required, then a separate column is needed in the data table
to calculate this, for example:
Table of Results Using ln Your notes

A separate column is often needed to calculate ln(V)

In the above case, the potential difference V is determined from a voltmeter, but the ln(V) values are
calculated using a calculator
The most common example of this in A level physics is in:
Radioactive decay
capacitor charge and discharge equations

Using Natural logs (ln)

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Taking natural logs (ln) of an equation with an exponential function means the equation can become
linear i.e. in the form y = mx + c
Your notes
Straight-line graphs tend to be more useful than curves for interpreting data
Gradients and intercepts are useful values that can be seen from a straight-line graph
Nuclei decay exponentially, therefore, to achieve a straight-line plot, logarithms can be used
Take the exponential decay equation for the number of nuclei
N = N0 e–λt
Taking the natural logs of both sides
ln N = ln (N0e–λt) = ln (N0) + ln(e–λt)
ln N = ln (N0) − λt
In this form, this equation can be compared to the equation of a straight line
y = mx + c
ln N = − λt + ln (N0)
Where:
y = ln (N) is plotted on the y-axis
x = t is plotted on the x-axis
gradient, m = −λ
y-intercept = ln (N0) is a constant
The exponential decay version of the equation could produce a curve, whilst the ln(N) equation
produces a straight line

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Your notes

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Linear decay curve vs. a log graph

Examiner Tips and Tricks

Remember that log and ln are subtly different! There are two different functions on your calculator.
By default, log is to the base 10, log10 E.g., log 100 = log10 100 = 2
This is very rarely used, if at all, in A level physics
'ln' is just log to the base e (the exponential function). Therefore, ln = loge E.g., ln(ex) = loge(ex) = x
Therefore, if you ever have an exponential function, e in the equation - use 'ln' and not 'log'
'ln' follows all the same laws of logarithms of addition, subtraction and power. You can find more in
the 'Logarithmic Function' A level Maths notes here on Save My Exams

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Using Units Correctly

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Using Units Correctly
SI Base Units
Every time you measure or calculate a quantity, you need to give its units
All units in Physics can be reduced to six base units from which every other unit can be derived
These other quantities are called derived units
These seven units are referred to as the SI Base Units; this is the only system of measurement that is
officially used in almost every country around the world
SI Base Quantities Table

Derived Units
Derived units are derived from the seven SI Base units mathematically
The base units of physical quantities such as:
Newtons, N
Joules, J
Pascals, Pa, can be deduced

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To deduce the base units, it is necessary to use the definition of the quantity
The Newton (N), the unit of force, is defined by the equation: Your notes
Force = mass × acceleration
N = kg × m s–2 = kg m s–2
Therefore, the Newton (N) in SI base units is kg m s–2
The Joule (J), the unit of energy, is defined by the equation:
Energy = ½ × mass × velocity2
J = kg × (m s–1)2 = kg m2 s–2
Therefore, the Joule (J) in SI base units is kg m2 s–2
The Pascal (Pa), the unit of pressure, is defined by the equation:
Pressure = force ÷ area
Pa = N ÷ m2 = (kg m s–2) ÷ m2 = kg m–1 s–2
Therefore, the Pascal (Pa) in SI base units is kg m–1 s–2
It is essential that the correct scientific measurements are used when discussing experiments in
physics
Ensure that the correct symbols are used in conjunction with the unit of measurement
E.g. m3 for cubic metres
Units of Measurement Table

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Your notes

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Note:
cm3 is the same as millilitre (ml) Your notes
dm3 is the same as litre (l)

Examiner Tips and Tricks

Units are extremely important in physics, and should always be stated when calculating any values if
they are not already given on the paper. Units should always be included on the axes for graphs
(either sketches or plotted) and table headings. Some variables may not have units, such as straight,
refractive index and number of particles.

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Trend & Patterns in Experimental Data

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Trend & Patterns in Experimental Data
Graphs are used to visualise the relationship between two sets of data from two different variables
Trends and patterns can be identified from experimental data
Common trends are:
Linear
Directly proportional
Inversely proportional
Rate of change
A linear graph set of data is any data that creates a straight line

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Your notes

An example of a linear graph

The rate of change of a graph is how quickly a variable is increasing or decreasing with something else
This can be seen from the change in gradient of the graph, an increasing gradient has an increasing
rate of change and a decreasing gradient has a decreasing rate of change
A direct proportionality relationship is where as one amount increases, another amount increases at
the same rate
This is represented by a straight-line graph with a positive gradient
For two variables, y and x this looks like:
y∝x

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An inverse proportionality relationship is where as one amount increases, another amount decreases at
the same rate
Your notes
This is represented by a curved graph with a decreasing gradient
For two variables, y and x this looks like:

1
y∝
x

Sketched graphs show relationships between variables

In the first sketch graph, above you can see that the relationship is a straight line going through the
origin
This means as you double one variable the other variable also doubles so we say the independent
variable is directly proportional to the dependent variable
The second sketched graph shows a shallow curve
This is the characteristic shape when two variables have an inversely proportional relationship
The third sketched graph shows a straight horizontal line,
This means as the independent variable (x-axis) increases the dependent variable does not change
or is constant

Worked Example
Comment on the trend of the graph.

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Answer:
Stress and strain are proportional to each other, but not directly
The graph is linear with a positive gradient up to a strain of 1.0 × 10-3
After this, the rate of change of the strain with stress decreases, as the gradient of the graph
decreases up to the breaking stress at 190 MPa

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Interpreting Graphs
Your notes
Relationships between Variables
Identifying the relationship between two variables shows how one variable changes with another
This is best figured out from an equation that links both variables together
Two variables can be:
Directly proportional
Inversely proportional
For two variables, y and x that are directly proportional, their relationship looks like:
y∝x
This means that as y increases, x also increases at the same rate (and vice versa)
An example of this is
F = ma
Since m is a constant, F and a are directly proportional
If a is doubled, then so is F
For two variables, y and x that are inversely proportional, this looks like:

1
y∝
x
This means that as y increases, x decreases at the same rate (or vice versa)
An example of this is

V
R=
I
If V is constant, then R and I are inversely proportional
If I is doubled, the R is halved
It is important to note that in both of these examples, the remaining variable is constant, this is
important to consider and check before stating the relationship between two variables

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In F = ma, if m changed as well with a then F would not increase by the same amount as a (i.e. it
would no longer be directly proportional)
Therefore, the directly proportional relationship can be turned into an equation by replacing '∝' with an Your notes
'=' sign, and adding a constant k
y = kx
This now means that as y increases, x increases with the amount determined by the constant k
If k is 3 then y = 3x so the both increase by a factor of 3
The same happens for an inversely proportional relationship

k
y=
x
This now means that as y increases, x increases with the amount determined by the constant k

3
If k is 3 then y = so y decreases by a factor of 3
x
Another common relationship is the inverse square law
For two variables, y and x that are related by the inverse square law, this looks like:

1
y∝
x2
This means that if x increases by a factor of 2, then y decreases by a factor of 22 = 4!
An example of this is

L
F=
4πd2
If L is constant, then d and F are inversely proportional
If the distance d is 3 times larger, then the flux intensity, F is 32 = 9 times smaller

Worked Example
A student collects the following data of the count rate on a geiger counter increasing in distance
from a gamma ray source.

Distance d / cm Count rate C / counts min-1

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10 512

20 128 Your notes

30 57

40 32

Show that the relationship between distance and count rate is an inverse square law relationship.
Answer:
Step 1: State the inverse square law relationship
The count rate is inversely proportional to the distance squared
1
C∝
d2
Step 2: Change relationship into an equation

k
C=
d2
Step 3: Rearrange for constant, k
k = Cd2
Step 4: Show all pairs of C and d have the same constant, k
Row 1: k = 512 × (10)2 = 51200
Row 2: k = 128 × (20)2 = 51200
Row 3: k = 57 × (30)2 = 51300
Row 4: k = 32 × (40)2 = 51200
Step 5: Comment on constant and refer back to relationship
Since all values have the same constant k to 2 significant figures (51000), C and d have an
inverse square law relationship

Examiner Tips and Tricks

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When you're comparing two variables you must check whether the other variables are constant
before declaring that they're inversely or directly proportional. Otherwise, one will not increase at
the same rate as the other, which goes against the definition! Your notes

Determining Constants from Graphs

Straight line graphs (linear relationships) are common in A-level physics
These graphs are useful because we can calculate a constant value from them
This is the gradient
The gradient can be calculated by dividing the rise (change in y) by the run (change in x)

The full calculation of the gradient needs to be shown in the working out, including the correct
substitution of identified plotted points from the axes into the equation
The triangle used to calculate the gradient should be drawn on the graph and it needs to be as large as
possible
Small triangles are not acceptable for working out a gradient

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When using the results from a table of values, the triangle that is used to obtain the gradient can utilise
points that lie on the line of best fit but not values that lie away from the line
Your notes
Try to avoid using data points to calculate this where possible
The units of the gradient will be the ratio of the units of the y variable and units of the x variable
E.g., For a graph for extension x (in m) against force F (in N) the units of the gradient would be N m-1

Worked Example
Calculate the gradient of the following graph.

Answer:
Step 1: Draw a large gradient triangle

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Your notes

Step 2: Use the gradient equation

27. 00 − 5 . 00
Gradient = = = 15.7 Ω m-1
1.7 − 0.3

Examiner Tips and Tricks

The general rule is to draw a gradient triangle that takes up more than half of the graph. Drawing
triangles too small, even if you get the correct answer, will not achieve full marks in the practical
paper!
There is normally a range of answers accepted in the mark scheme for gradients, as everyone's line
of best fit may be slightly different, but don't count on this! This range will often be very small so
always use a sharp pencil and ruler to draw lines where possible. Show your working out on the paper
always to help this.

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Remember to always check the units and scale when reading values from a graph! Don't just assume
that all lengths are in m or that forces will be in N. They could be in mm or kN. Also watch out for the
powers of ten e.g., Force F × 103 / N which means a value of '5' on the graph will actually be 5 × 103 N Your notes
(or 5 kN).

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Precision & Accuracy

Your notes
Precision & Accuracy
Precision
Precision is how close the measured values are to each other
If a measurement is repeated several times, then they can be described as precise when the values are
very similar to, or the same as, each other
The precision of a measurement is reflected in the values recorded
Measurements to a greater number of decimal places are said to be more precise than those to a
whole number

Accuracy
Accuracy is how close a measured value is to the true value
Accuracy can be increased by repeating measurements and finding a mean average

The difference between precise and accurate results

Measurements of quantities are made with the aim of finding the true value of that quantity
In reality, it is impossible to obtain the true value of any quantity, there will always be a degree of
uncertainty
The uncertainty is an estimate of the difference between a measurement reading and the true value

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Random and systematic errors are two types of measurement errors which lead to uncertainty

Your notes

Representing precision and accuracy on a graph

Sensitivity
A measuring instrument can have a sensitivity
This is the ratio of the changes in the output of an instrument to the change in value of the quantity
being measured
In other words, this is the smallest change the instrument can detect
This is slightly different to resolution
Resolution is the smallest change the instrument can observe but sensitivity is the smallest change that
the instrument can detect
If what you are measuring (the dependent variable) changes significantly when the independent
variable is changed, then the instrument is deemed to be sensitive
If the value does not change significantly, with big changes to the independent variable, then the
instrument is not sensitive
An instrument with better sensitivity detects changes in a variable much better
For example, a thermometer could have a sensitivity of 1 °C (it only detects change of 1 °C)

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A digital thermometer could have a sensitivity of 0.1 °C (it can detect changes as small as 0.1 °C)
If you are taking measurements in very small intervals of temperature increase, the digital thermometer
will be able to give much more accurate results Your notes

Examiner Tips and Tricks

Try not to confuse precision with accuracy - measurements can be precise but not accurate if each
measurement reading has the same error. Precision refers to the ability to take multiple readings
with an instrument that are close to each other, whereas accuracy is the closeness of those
measurements to the true value.

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Reducing Errors
Your notes
Reducing Errors
Reducing errors in an experiment is vital for obtaining more accurate results
Even if the experimental result is close to the true value, there are always potential limitations of
experimental methods such as the presence of random errors
Random errors cannot be completely removed but their effect can be reduced by taking as many
repeats as possible and using the average of the repeats
There are always opportunities to identify limitations of the procedure, some common examples
include:
Parallax error when reading scales
Not using a fiducial marker (eg. when measuring the time period of a pendulum using a stopwatch)
Not repeating measurements to reduce random errors
Not checking for zero errors to reduce systematic errors
The equipment not working properly or not checking beforehand with small tests
Equipment with poor precision and resolution (eg. using a ruler over a micrometer)
Difficult to control variables (eg. the temperature of the classroom)
Unwanted heating effects eg. in circuits
Parallax error is minimised by reading the value on a scale only when the line of sight is perpendicular to
the scale readings (i.e.. at eye level)
Examples of where parallax error is common are:
Determining the volume of liquid
Making sure two objects are aligned
Reading the temperature from a thermometer
If it makes it easier, use a marker to help where possible

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Your notes

Reading the value of the needle head-on (left image) looks different to reading it from the right (right
image). This is parallax error
A fiducial marker is a useful tool to act as a clear reference point, such as when measuring the time
period of a pendulum using a stopwatch
This improves the accuracy of a measurement of periodic time by:
Making timings by sighting the pendulum as it passes the fiducial marker
Sighting the pendulum as it passes the fiducial marker at its highest speed. The pendulum swings
fastest at its lowest point and slowest at the top of each swing

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A fiducial marker is used to mark the centre of the oscillation of the pendulum
Zero errors must be checked for in both digital and analogue instruments Your notes
E.g., If there is no current through the circuit, an ammeter must read 0 A
The common way to reduce unwanted heating effects in circuits is to turn off the power supply in
between readings
As the temperature of a component increases, so does its resistance (e.g., in wires). This will affect
the experiment and produce an error in your final result

Worked Example
A student wants to determine the radius of a wire for an experiment to calculate its Young Modulus.
They measure the radius using a ruler from one part of the wire.
Discuss ways in which the student can reduce the error in this reading.
Answer:
Step 1: Comment on the instrument used
Since the radius of a wire is on the order of < 1 mm, and has a circular cross section, a micrometer
screw gauge should have been used instead
Step 2: Comment on the method
The student did not take any repeat readings
They should take between 3-5 repeat readings for each value of the radius from the
micrometer
Step 3: Suggest improvements to the method
The experiment assumes the wire is uniform the whole way through (i.e. has the same radius)
This can be checked by measuring the radius at different points on the wire

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Suggesting Improvements
Your notes
Suggesting Improvements
Improvements to an experiment help make it more reliable and reproducible to gain more trust in the
results
A common method is to use digital methods of data collection where possible
These subsequently reduce uncertainties that are a result of human error (e.g., reaction time)

Software & Tools

Graph plotting and data analysis software, such as Microsoft Excel, can be an invaluable tool
Spreadsheets provide a very effective way of processing data, particularly when the amount of
data is large
Cameras can be used to take photos in experiments that happen too quickly to read a scale
A camera can be used to take a photo burst as the experiment happens
The scale can then be read from the photos afterwards
If the time each photo is taken is known, or if the frame rate is known, then properties such as
velocity can be calculated

Data Loggers
Data loggers are a tool that allows for the quick and efficient gathering of data
They are more accurate, quick and reliable than manual logging
The information contained within a data logger can be inputted into a computer and formatted into a
table
After this is done the computer is able to calculate the average and plot graphs using the data and
calculate gradients which quicker and more accurately than humans
They are electronic devices that automatically monitor and record environmental parameters over time
such as temperature, pressure, voltage or current
It contains multiple sensors to receive the information and a computer chip to store it

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Your notes

A data logger measuring and displaying temperature using a probe

The benefits of using data loggers and ICT (information and communication technology) include:
Readings are taken with higher degrees of accuracy
Reduction of human error (e.g. human reaction times, subjectiveness)
Readings can be taken over a long period of time e.g. hourly readings of temperature over many
days
Readings can be taken in a very short period of time, which would be too quick for humans to see a
difference
Reduction in safety risks with extreme conditions such as measuring the temperature of boiling
water

Computer Modelling
Computer modelling is commonly done in conjunction with devices such as a data logger
Modelling is about processing the data collected from a physics experiment into software or a
spreadsheet
Graphs and charts can be generated from a table of values
These can then be exported to a scientific report

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One of the benefits of these computer programs is that time can be sped up to predict the future
outcome of an experiment
Your notes

Computer modelling uses a computer and sensors to analyse and display data
Making Methods Reproducible
To improve upon an experimental method, it could be made more reproducible
This is the ability to be properly reproduced for other scientists to also see if they get the same
results
For example, when measuring the resistivity of a wire, a constantan wire may be used
If the same method is used to measure an accurate value of the resistivity of copper or aluminium,
then this means the method is properly reproducible
A further discussion of similarities and / or differences between the three wire materials can then be
analysed
Another example could be when measuring the count rate of a gamma ray source
By using a more or less active source, more differentiation in their readings can be achieved

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Uncertainties
Your notes
Uncertainties
Uncertainties can be represented in a number of ways:
Absolute Uncertainty: where uncertainty is given as a fixed quantity
Fractional Uncertainty: where uncertainty is given as a fraction of the measurement
Percentage Uncertainty: where uncertainty is given as a percentage of the measurement
Percentage uncertainty is defined by the equation:

uncertainty
Percentage uncertainty = × 100 %
measured value
To find uncertainties in different situations:
The uncertainty in a reading: ± half the smallest division
The uncertainty in a measurement: at least ±1 smallest division
The uncertainty in repeated data: half the range i.e. ± ½ (largest - smallest value)
The uncertainty in digital readings: ± the last significant digit unless otherwise quoted

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Your notes

How to calculate absolute, fractional and percentage uncertainty

Always make sure your absolute or percentage uncertainty is, at a maximum, to the same number of
significant figures as the reading
Absolute uncertainties are compounded when adding or subtracting data

Adding / Subtracting Data

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Add together the absolute uncertainties

Your notes

Examiner Tips and Tricks

Remember:
Absolute uncertainties have the same units as the quantity
The uncertainty in numbers and constants, such as π, is taken to be zero
In Edexcel International A level, the uncertainty should be stated to at least one few significant
figures than the data but no more than the significant figures of the data.
For example, the uncertainty of a value of 12.0 which is calculated to be 1.204 can be stated as 12.0 ±
1.2 or 12.0 ± 1.20.

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Calculating Percentage Uncertainties

Your notes
Percentage Uncertainties
Percentage uncertainties are compounded when multiplying or dividing data

Multiplying / Dividing Data

Add the percentage or fractional uncertainties

Raising to a Power
Multiply the percentage uncertainty by the power

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Your notes

Examiner Tips and Tricks

Remember:
Percentage uncertainties have no units
The uncertainty in numbers and constants, such as π, is taken to be zero
In Edexcel International A level, the uncertainty should be stated to at least one few significant
figures than the data but no more than the significant figures of the data.

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For example, the uncertainty of a value of 12.0 which is calculated to be 1.204 can be stated as 12.0 ±
1.2 or 12.0 ± 1.20.
Your notes
Single & Multiple Readings
Single Reading
Percentage uncertainty for a single reading (measured value) is defined by the equation:

uncertainty
Percentage uncertainty = × 100 %
measured value
The (absolute) uncertainty in a single reading is half the resolution of the instrument

Multiple Readings
The percentage uncertainty in measurements from multiple readings (e.g. repeat readings) use half
the range of the readings
The range of the readings is the difference between the highest and lowest reading

Worked Example
A student achieves the following results in their experiment for the angular frequency, ω.
0.154, 0.153, 0.159, 0.147, 0.152
Calculate the percentage uncertainty in the mean value of ω.
Answer:
Step 1: Calculate the mean value

0 . 154 + 0 . 153 + 0 . 159 + 0 . 147 + 0 . 152

mean ω = = 0.153 rad s–1
5
Step 2: Calculate half the range (this is the uncertainty for multiple readings)

1
× (0.159 – 0.147) = 0.006 rad s–1a
2
Step 3: Calculate percentage uncertainty

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uncertainty ±half the range

× 100 % = × 100 %
measured value mean Your notes
0 . 006
× 100 % = 3.92 %
0 . 153

Examiner Tips and Tricks

Remember that percentage uncertainties have no units, only the % sign! Always make sure your
percentage uncertainty is at least one significant figures smaller than the measurement.

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