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 bestfit 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 bestfit 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, N_{y} between y_{1}
and y_{2} (see diagram), and work out
the change that this figure represents.
vertical
change = N_{y} × (vertical scale) 
4. 
Count
the number of small squares, N_{x} between x_{1}
and x_{2} (see diagram), and work out
the change that this figure represents.
horizontal
change = N_{x} × (horizontal scale) 
5. 
Now,
the slope of the graph is defined as
and must, of
course, be stated in the appropriate units 


(x_{1}, y_{1}) and (x_{2} , y_{2}) can be
the coordinates 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 bestfit 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 1^{x} = 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. 



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 



See also Radioactive Decay for more about
decreasing exponential variations. 



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 = x^{2} then we could also write
x = √y.


We say that "squaring" and "square rooting"
are inverse operations. 

Similarly, if y = a^{x} 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 



For more information about logs; read a mathematics book... 



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
= a^{x}, we can plot a graph of log_{a}y
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 
log_{10}y 
(log_{10}y)/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 log_{10}y α
x, in other words,
a graph of log_{10}y 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 log_{10}
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 =
ax^{n} 

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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