Tutorial 5 - Basic Graphical Skills

 Contents

Recording Results

The chances are that you were told how to present data in tables when you were in Year 7 (1st Year).  It is clear that many students weren’t listening, because the presentation of tables of data causes many problems and lost marks in the practical assessments.  It shouldn’t; it is dead easy.  Even if you know no physics, you can still pick up several marks for making sure that your data are presented well.

• Make sure you make a table.

• It should be boxed in with ruled lines, please.

• There should be headings for each column.

• With units.

• Data should be to no more than three significant figures.

In an experiment you should get into the habit of taking two or three repeat readings.  This helps to reduce anomalous results (those that don’t fit into the pattern).  Show these in your table and do an average.

In an experiment all students are expected to have their own copy of the results

It is depressing how often the excuse is made that “Zack’s got my results.”

Dealing with Uncertainty

We dealt with this in Tutorial 4.  In experimental there is always a certain amount of uncertainty.  Some books call it error, but error implies operator carelessness, which is not always the case.  Uncertainty can be:

• Random, where there is no pattern.  For example a digital meter takes readings every 0.2 s.  Was the result caught exactly as the stopwatch read 10 s?

• Systematic, where there is uncertainty in the calibration of an instrument.  A school voltmeter may read 3.45 V, but the real voltage could be 3.41 V.

In general, a school physics experiment will produce at best accuracy of one part in 100.  Therefore it makes no sense mindlessly to reproduce all ten digits from a calculator.

The writing up of practical reports is an essential skill in physics.  You might have got brilliant results in a required practical, but that's no good if you don't tell anyone about them.  I have written at length about practical reports in the induction notes in Tutorial 7 There is more stuff about graphs in the induction notes Tutorial 5 and Tutorial 6.

A sure-fire way of getting data that are unreliable is to jot them down on a scrap piece of paper with no semblance of order.  While you may know what the data items refer to when you are taking them, it is very easy to forget.  Or the bit of paper gets lost.  It is depressing to hear the excuse for an experiment not being handed in, "Sean's got my results".

Data are recorded on a table of results, and each student must have its own set of data.   This should be drawn up before you start your experimental work.  The picture below shows a neat table of results.   While it may not be of an experiment at AS level, it still shows the principle:

Your data must be recorded to the precision of the instruments.  The averages from repeat readings must be recorded to the same number of significant figures as the data.

If you process data, the number of significant figures must be no more than the minimum number of significant figures.  Let us look at the resistance of a length of resistance wire:

 Length / m Current / A Potential difference (V) Average / V Resistance  / W Reading 1 Reading 2 Reading 3 0.100 0.25 2.30 2.25 2.32 2.29 9.0 0.200 0.25 0.300 0.25 0.400 0.25 0.500 0.25 0.600 0.25 0.700 0.25 Etc…

• The p.d. readings are consistent with the readout of a digital voltmeter.

• The current was kept constant at 0.25 A.  The number of significant figures is consistent with the precision of an analogue ammeter.

• The average is consistent with the number of significant figures of the p.d. readings.

• The resistance is calculated to two significant figures.  It calculated to 9.04 W, but is written as 9.0 W to make it consistent with the current which is recorded to two significant figures.

Graphical Skills

On their own, numbers do not mean a lot.  A table of numbers can be confusing.  A graph allows us to see a picture of how the numbers relate to each other.

You have harvested data, done repeats, etc.  But what then?  A table of numbers is not very helpful.  So we need to represent the data as a picture which shows us a great deal of information that we could not pick up from the data in the table.

• Whether one quantity is proportional to another;

• What the proportionality is;

• How a derived quantity is obtained from the gradient.

All graphs in physics are line graphs.  This is because quantities in Physics tend to be continuous.  A set of results is not very easy to use to judge the way quantities are related to each other.  The graph shows it instantly.

#### Size

Graphs should always:

• Be large – they should occupy the whole of a side of A4 paper.

• Have a title to tell the reader what the graph is about;

• Have sensible calibration of axes, with simple scales;

• Have both axes labelled with the quantity and units.

Note that sometimes the points are compressed to one side of the graph.

You really need to consider the scale of your graph to stretch out the points.  As such the graph would not get any marks for its size.

You can draw a graph in portrait (long side vertical), or landscape (long side horizontal).   Sometimes it’s quite clear which way it should go.  Other times it’s a matter of what seems best.

Just a reminder of the rules for drawing a graph:

1. Always use a sharp pencil and a ruler.

2. Draw the axes

3. Label the axes with the quantity and the units

4. When you plot Quantity 1 against Quantity 2, you put Quantity 2 on the horizontal axis.

5. Look for the highest value in each range.  You calibrate (put numbers on) your axes to the nearest convenient step above your highest value.

6. Use a sensible scale.

7. Plot your points with crosses (+ or ×).  Points get lost.

8. Join your points with a line, but not dot-to-dot!

The table below shows some data to plot:

 Voltage (V) Current (mA) 0 0 1 20 2 30 3 65 4 98 5 174 6 280

The graph below is nonsense.  Can you see why?  Although graphs drawn like this are quite useless, they are depressingly common.

 Why is this graph so pitifully useless?

The correct graph is shown below:

Notice:

• Axes labelled with quantities and units;

• Scales are sensible;

• Line of best fit drawn through the points.  No dot-to-dot, please.

In the coursework, you will get marks for a graph that is big enough (more than 1/2 the size of A4).  You will get marks for labelled axes and sensible scales.

Axes and Scales

In general, the independent variable goes on the horizontal axis (the x-axis, or ordinate), and the dependent variable on the vertical axis (y-axis or abscissa).  When you are told to plot quantity 1 against quantity 2, quantity 2 goes on the horizontal axis.

Sometimes the scales go the other way round, for example voltage (independent variable) against current (dependent variable).  This enables us to work out the resistance from the gradient.

Your scales should go up in multiples of 1, 2, 5, or 10.  Do not use multiples of 3.

The quantity should be separated from its units by a solidus (/), for example you would label the velocity axis velocity /m s-1.  In effect you are dividing the quantity by its unit to give a pure number. It is “less scientific” to put the units in brackets, but you would not lose marks in the exam.

Plotting and Lines of Best Fit

It goes without saying that data need to be plotted accurately.  Your scales should allow you to do this.  In AQA examinations, the tolerance is within 1 mm of the true position.

A line of best fit needs to be drawn, because the data points will always be slightly out.

Consider whether the origin is a valid data point.  For example, at zero volts, we get zero amps.  So in a voltage-current graph, the origin is a data point in its own right.

It is important to draw a line of best fit so that goes through the middle of a scatter of points.  Roughly half should be above, and half below, unless the outlying points are clearly anomalous.

Sometimes the data form a curve.  Draw a smooth curve, not doing dot-to-dot.

Do not force a straight line through the curved progression.

Decide whether the origin is a valid data point.  If it is, include it.

Sometimes it is not at all easy to decide whether the graph is a curve or a straight line.  In this case, you should take more data points.  If it’s clearly a straight line, draw your line of best fit with a ruler.  If the graph is a curve, then try to make a smooth curve.  A flexi-curve can help you with this.

You may well get a data point that does not fit in with the rest of the data.  This is called an anomalous result.

If possible, check out anomalous results by doing a repeat.  Your new data will most likely fit in with the other points, and the anomalous data can be discarded.

Not all graphs are a straight line.  When the progression is clearly curved, do not force a straight line through the points.

Sometimes it can be quite hard to decide.  Look at these data on the graph:

You can see how hard it is to decide whether these data form a straight line or curve.  It might be worthwhile starting the scale for Quantity A at a non-zero value.  Suppose your minimum value is 700 units and your maximum value is 1000 units.  The data now form the graph like this:

The progression is clearly a curve.  Make sure that it's clear where you are starting the scale.

 It is clear that in this case the data do not pass through the origin.  Therefore do no include the origin as a data point.

 A student draws a graph like as below and joins the points with a “line of best fit” as shown.   (a)  The title of the graph is ____________________ against ________________. (b)  What is wrong with this line of best fit? (c)  Draw the correct line of best fit. (d) In the exam, he writes that Quantity A is directly proportional to Quantity B. Discuss how you would mark his answer.

### Direct Proportionality

A graph that is straight-line of positive gradient and goes through the origin shows that the two quantities are directly proportional.

In this case:

• If the force is doubled, the extension is doubled.

• If there is zero force, there is zero extension.

The term m is a constant.

Descriptions like “…if weight increases, the extension increases…” are too vague and will not gain any credit.  You need to say, “The extension is directly proportional to the weight (1 mark), as the line is straight (1 mark) with a positive gradient (1 mark) and passes through the origin (1 mark).”

### Other Proportionalities

It is possible for a relationship to show inverse proportionality:

The graph is a hyperbola and looks like this:

Note that the line is asymptotic for both axes, which means that it never touches the x or the y axis.

If we plot acceleration against 1/mass, we get a straight line.

If we plot 1/acceleration against mass, the two would also be proportional.

 Make sure that, when you invert a quantity, that you invert the units.   You will lose marks if you write 1/mass / kg.  You must write it as 1/mass / kg-1.    For the 1/acceleration, the units would be s2 m-1, NOT m-1 s2.  Note the change in order.

 Explain how this graph shows that acceleration is directly proportional to 1/mass.

Suppose we have a relationship like:

We can say that:

The graph of power against current looks like this and is called a parabola

We could plot a graph of power against current2.

 What would the graph look like? What units would you put in for current2?

### Linear Graphs that are not proportional

This graph does NOT go through the origin.

# This is a linear progression (i.e. it’s a straight line), but is NOT proportional.  If the x-value is doubled, the y-value is not doubled.

The general equation for this graph is:

The term c is the value of the intercept, which is the point at which the graph crosses the y-axis.

You can have an x-axis intercept, of course.

Use y = mx + c to give a general relationship for the x-axis intercept.

Reactance is a kind of resistance found in electrical circuits that use alternating currents. Use the data below to plot a graph of Reactance against Frequency. Remember the rules!

 Frequency (Hz) Reactance (W) 500 100 1000 200 1500 310 2000 396 2500 420 3000 600 4000 820 5000 1040

What is the value of the reactance at a frequency of 4200 Hz?

What frequency gives a value of reactance of 700 ohms?

Reading values off the graph is called interpolating, which you did in Question 6.

The graph above is a straight line. The reactance is directly proportional to the frequency, as the line goes straight through the origin. No frequency, no reactance. In a directly proportional relationship, if Quantity A doubles, Quantity B doubles as well.

Remember that straight-line graph has the general equation where m is the gradient and c is the intercept, the point at which the line cuts the vertical axis.  A function with a graph of this type is NOT directly proportional.  If Quantity A doubles, Quantity B does not double

When we extend the graph to read a value outside the range of plotted data, we are extrapolating.

To determine the gradient (or slope) of your graph, you work out the rise and the run.

 Don't just use a pair of values to calculate the gradient.  In most cases the answer will be wrong, especially if the graph does not go through the origin.

When you work out the gradient in the exam, you must have a large triangle.  The minimum length must be 8 cm

Rise is worked out by:

Dy = highest y – lowest y

Run is worked out by:

Dx = highest x – lowest x

The gradient will give a reading that could be used directly, for example resistance is the gradient of the graph of voltage against current.

In a directly proportional relationship, if Quantity A doubles, Quantity B doubles as well.  The graph of this function goes through the origin.

A straight-line graph has the general equation y = mx + c where m is the gradient and c is the intercept, the point at which the line cuts the vertical axis.

A function with a graph of this type is NOT directly proportional.  If Quantity A doubles, Quantity B does not double.

 Work out the gradient of your graph in question 6.  Show on your graph how you got the gradient. Write down the units you think the value of gradient should have.

For other relationships, you may need to do some further processing to get the value that you want.

Consider this graph that shows the relationship between resistance and length of a resistance wire:

The formula for resistivity is:

R - resistance (W);

r - resistivity (W m);

l - length (m);

A - area (m2).

# To get the area, we would need to measure the wire with a micrometer which gives us the diameter.  To work out area, we then need to use the formula:

 How would you work out the resistivity from the graph of resistance against length?

When you do electricity, you will learn how to find the internal resistance of a cell (battery).  This is done by measuring the voltage of the battery in open circuit (the EMF), when no current is drawn.  Then a variable resistor is connected and a range of readings is taken at different resistances.

Voltages and currents are noted and (what a surprise) a graph is plotted.  It looks like this:

The equation for this graph is:

where:

Vvoltage (V);

ɛ - EMF (V);

Icurrent (A);

r internal resistance (W).

 (a) Match up the symbols with y = mx +c. (b) What does the y-axis intercept represent? (c) What does the x-axis intercept represent? (d) How would you find the internal resistance?