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Graphs
The results of an experiment are often used to plot a graph.
A graph can be used to verify the relation between two variables and, at the same time, give an immediate impression of the precision of the results.
When we plot a graph, the independent variable is plotted on the horizontal axis.
Think of the independent variable as the cause and the dependent variable the effect.

If one variable is directly proportional to another variable, then a graph of these two variables will be a straight line passing through the origin of the axes.
So, for example, Ohm's Law has been verified if a graph of voltage against current (for a metal conductor at constant temperature) is a straight line passing through (0,0).
Similarly, when current flows through a given resistor, the power dissipated is directly proportional to the current squared.
If we wanted to verify this fact we could plot a graph of power (vertical) against current squared (horizontal).
This graph should also be a straight line passing through (0,0).

The "Best Fit" Line

The best-fit line is the (in this case, straight) line which passes as near to as many of the points as possible.

By drawing such a line, we are attempting to minimize the effects of random errors in the measurements.

So, if the points look like these...

then the best fit line should be something like this...

Notice that the best-fit line does not necessarily pass through any of the data points.

In this case I have drawn the line through the origin but even this is not necessarily the case: a best fit line that does not pass through the point (0,0) might be suggesting that there was a systematic error in the experiment.

If systematic error has been ruled out one can perhaps consider the origin to be an especially accurate point.

To Measure the Slope of a Graph
The slope of a graph tells us how a change in one variable affects the value of another variable.
To measure the slope, proceed as follows.

 1 Find out what one small square represents vertically. For example, one small square might represent 0.01volts or 10metres etc. This is the vertical scale. 2 Find out what one small square represents horizontally. For example, one small square might represent 0.002Amps or 0.5seconds etc. This is the horizontal scale. 3 Count the number of small squares, Ny between y1 and y2 (see diagram), and work out the change that this figure represents. vertical change = Ny × (vertical scale) 4 Count the number of small squares, Nx between x1 and x2 (see diagram), and work out the change that this figure represents. horizontal change = Nx × (horizontal scale) 5 Now, the slope of the graph is defined as and must, of course, be stated in the appropriate units

(x1, y1) and (x2 , y2) can be the co-ordinates of any two points on the line but for best precision, they should be as far apart as possible.

In this graph, it is clear that y decreases as x increases so in this case, the slope is negative.

Error Bars

Instead of plotting points on a graph we sometimes draw lines representing the uncertainty in the measurements.

These lines are called error bars and if we plot both vertical and horizontal bars we have what might be called "error rectangles", as shown here.

In the graph shown here:
x was measured to ±0.5s and
y was measured to ±0.3m

Once we have represented the uncertainties on the graph (by drawing the error bars/rectangles) we can see that there must exist a range of possible values for the slope of the graph.

The best-fit line could be any line which passes through all of the rectangles.

If we assume that the line passes through zero, the maximum and minimum slopes of lines which are consistent with these data are about 1.06ms-1 maximum and about 0.92ms-1 minimum.

However, notice that if we have no reason to believe that the line must pass through the origin, the range of possible slopes of best fit line increases greatly.

Measuring the Slope at a Point on a Curved Graph
Often we will plot results which we expect to give us a straight line.
If we plot a graph which we expect to give us a smooth curve, we might want to find the slope of the curve at a given point; for example, the slope of a displacement against time graph tells us the (instantaneous) velocity of the object.

To find the slope at a given point, draw a tangent to the curve at that point and then find the slope of the tangent in the usual way.

The slope of the graph at this point is given by
Δy/Δx = (approximately)6ms-1

Exponential Graphs
Many situations in physics can be described by equations of the form

where a is a constant.
Equations of this form are called exponential equations.
In a case like this, how does a change in x affect the value of y?
The answer to this question obviously depends on the value of the constant.
Suppose a = 1.
Now 1x = 1 whatever the value of x so this is not a very interesting example.
However, if we now put a = 2 the situation is very different, as shown in the table below.
 x y 0 1 1 2 2 4 3 8 4 16

Notice that each time we add one to x we multiply y by the same number, the constant a (so in this case the number is 2).

More generally, if a relationship is exponential, for equal intervals of the x variable, the y variable changes by a constant factor.
It is clear that a graph of x against y will not be a straight line.
If we plotted a graph from the results in the table we would have a curve something like this

If the exponent in the equation is negative then the value of y decreases (exponentially) with increasing x.
So, if the relation between x and y is

we have a graph something like this

It is usually easier to work with straight line graphs.

There are two possible ways to obtain a straight line graph from a set of results based on an exponential relationship.
i) using a calculator/computer
ii) using special graph paper.

Using a Calculator/Computer
Remember that any mathematical relation can be written in different forms.
For example, if y = x2 then we could also write x = √y.

We say that "squaring" and "square rooting" are inverse operations.
Similarly, if y = ax then, reading this “in reverse” we could say, “x is the power to which a must be raised to give y”.
This is more conveniently stated as

So, if we have a set of results from an experiment, and we suspect that they are related by an equation of the form y = ax, we can plot a graph of logay against x and this should give us a straight line.
To illustrate this we will consider again the previous set of “results” but now we add two more columns to the table.
 x y log10y (log10y)/x 0 1 0 indeterminate 1 2 0.3 0.3 2 4 0.6 0.3 3 8 0.9 0.3 4 16 1.2 0.3

The fourth column shows that log10y α x, in other words, a graph of log10y against x will be a straight line.
In this table log “to base 10” has been used.
We  could also use natural logs (written ln), which have base e = 2.71828…
We would, of course, have a different constant in the fourth column.
In theory, logs to any base could be used but log10 and ln are the two which your calculator will give you easily.

Using Log Scale Graph Paper
Special “log scale” graph paper can be obtained on which the spacing between the lines varies in a logarithmic way.
Using this graph paper, if variables related by an exponential relation are plotted directly (that is, x against y; no other calculations needed) a straight line will result.
Imagine plotting y against x using the special “log/linear” graph paper below.

You will find that the points form a straight line.

Log/Log Graph Paper
If both the scales are log scales we have “log/log” graph paper.
This type of paper is useful if we suspect that the relation between two variables is of the form

where a and n are constants.
Taking logs of this equation gives

so, if we plot a graph of logy against logx, we should have a straight line.
Alternatively, we could plot y against x on log/log paper.

Example
 x y 2 22.6 3 41.6 4 64.0 5 89.6

The numbers in the table above are related by an equation of the form y = axn
Plot a graph of y against x on log/log graph paper and find the values of a and n.
In order to calculate the slope of a log/log graph, simply measure the Δx and Δy in millimetres, don’t look at the numbers on the axes.

You should find n = 1.5 and a = 8

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