Tutorial 6 - More Graphical Skills
 Contents Curved Graphs Finding the area under the graph Uncertainty in a Graph Using Absolute Errors Finding the uncertainty

Curved Graphs
Not all graphs are a straight line, as in the following example. When the progression is clearly curved, do not force a straight line through the points.

The data show the power dissipated by a resistor as voltage increases:

 Voltage (V) Power (W) 0 0 2.5 2.0 5.0 4.1 7.5 18.4 10.0 31.8 12.5 52.1 15.0 72.6 17.5 100 20.0 128

(a)  Plot these data and join the points with a line of best fit. Note that there is an anomalous result.

(b)  Which is the anomalous result? What would you do to avoid anomalous results?

The graph below is a curve:

The gradient is worked out by taking a tangent to the curve.  You need to make a line perpendicular to the curve at the point you are interested in.  Then you draw a line at 90 degrees to that line, which will give you the tangent.  You work out the gradient of the tangent using the rise and run in the normal way.  Your triangle is at least 8 cm in its shortest dimension (of course).

The tangent gives us the instantaneous rate of change at that point.

The tangent gives us the rate of change at that point. Consider a speed-time graph of a car accelerating:

In this graph you can see that two tangents have been drawn to work out the accelerations at times t1 and t2.

 Use your graph to state what is happening to the acceleration of the car at times t1 and t2.

 The gradient of a curved graph is changing all the time. So do not do this:   When calculating quantities that are represented by the gradient, it is important to use the graph, not just the values at a given point. These will give you the average gradient, which is not the same as the instantaneous value of the gradient.

The graph below shows the difference between the instantaneous gradient and the average gradient.

 Compare the average accelerations shown in the graph above with the true accelerations shown by the tangents.

Finding the area under the graph

At GCSE, you will have been asked to work out the distance travelled from the speed time graph. You did this by finding the area under the graph.  To get the units for the area, you multiply the units for Quantity 1 by the units for Quantity 2.

If the graph is a simple shape like a straight line, finding the area is really simple; you find the area of the triangle:

An example of this is the energy stored by a spring:

EEnergy (J)
FForce (N)
eextension (m)

You may have come across this while doing Hooke’s Law.

 (a) What are the axes that would be plotted on the graph? (b) What would the units be?

If the graph consists of simple shapes, you can find the area by working out the area of each shape and adding them.

To find the distance travelled, you find the area of the rectangle and the area of the triangle and add them together.

Sometimes the graph is a more complex shape.  If the graph has a known function for example y = x2, then the calculus procedure of integration can be used.  This is widely used by mathematicians, but is not expected for Physics A-level.  You will need to be able to do calculus at university level.

If the graph is a weird shape, then the area under the graph is worked out by counting the squares of the graph paper.  This can only give an approximation.

But there are a couple of rules to make sure that the approximation is not too far out:

• If the area fills more than half the square, count it in;

• If the area fills less than half the square, ignore it.

## Representing Uncertainty in a Graph

Uncertainty can be represented graphically using error bars.  They help you to decide how your line of best fit will go and reduce the guess-work that can happen when you are trying to decide a line of best fit.  This is especially true if the data are scattered about.

If your line of best fit goes within the error bars, it means that the answer lies within experimental uncertainty, and can be considered to be reliable.

Let’s suppose that we are doing a voltage-current measurement.  The voltage range is from 0.5 V to 6.0 V.  The precision of the voltmeter is ± 0.1 V.  The current range is 0.10 A to 1.2 A with a precision in the ammeter of  ± 0.05 A.

 Calculate the percentage uncertainty for the voltage and the current, using the smallest value for each. Comment on the quality of the instruments that have provided these data.

Unless the uncertainty is so small that error bars cannot be seen, we should always use error bars for both axes.  This graph shows the error bars for both axes:

The vertical bars are smaller. Since the horizontal scale has larger steps, the error bars are longer.  In Physics at A-level, absolute uncertainties are used when error bars are shown. More sophisticated statistical methods are used with research.

The lines of worst fit are like this:

 If we did percentage error bars, we would get a pattern like this: You can see the way that the uncertainty for each value gets bigger. Therefore we can see the different slopes it is possible to get. The line of best fit gives a gradient of 5.0 V A-1 (or W). The maximum value is 10 V A-1. The minimum value is 3.3 V A-1. The difference is 6.7 V A-1. This range is very large. Therefore the uncertainty in the answer is so large that it is almost meaningless.

Using Absolute Errors to Measure Uncertainty

Consider an object that is moving at constant speed.  The uncertainty in the timing is 0.3 s and the uncertainty in the measurement of distance is 0.1 m.  Here are the data.

Since the speed is constant, it doesn’t take a genius to see that the speed is 0.4 m s-1.

 Plot these data on a graph.  Time on the horizontal axis, distance on the vertical axis. Draw the line of best fit. Include the error bars on both axes.

Note that the origin does not have any uncertainty.  So the lines of best fit and worst fit still pass through the origin.

 Now add the lines of worst fit. Work out the gradients of the lines of worst fit.

Notice that we have used more significant figures than is appropriate.

In Physics at A-level, absolute uncertainties are used when error bars are shown.  More sophisticated statistical methods are used with research.

In your practical work, you need to get into the habit of drawing error bars as a matter of routine.

## Finding the uncertainty from a gradient

We have now worked out the gradient from our line of best fit and got an answer.  But what is the uncertainty?  A natural way would be to add the percentage uncertainty from the terms that make up the gradient.  However it is not done like this.

Instead we draw a line of worst fit.

This could be the steepest gradient, or the shallowest gradient.   The percentage uncertainty is given by the relationship:

If we have an intercept, its uncertainty can be found in a similar way:

Uncertainty = best intercept – worst intercept × 100 %

best intercept

 The uncertainty of the gradient is not worked out by adding the percentage uncertainty in the rise and the run. Remember that the origin has no uncertainty.