Finding the Appropriate Graph
A substantial part of analysing experimental data involves drawing an appropriate graph, then making conclusions based on this.
This is a skill that has been part of Science since S1, however up until Advanced Higher, the experiments were always set up in such a way that the required axes for a straight line graph through the origin were given.
At Advanced Higher level, this is not the case, with only the statement "Plot an appropriate graph" used.
WHat is an appropriate GrAPH?
In Physics, we are usually trying to prove a relationship between two variables. To do this, we look to find the variables, or a mathematical manipulation of the variables to give a directly proportional graph. This means that the graph will follow the "y = mx + c" straight line graph format.
This has been touched upon in N5 Physics through the Gas Law Graphs relating to temperature.
When the Volume of an ideal gas is varied with temperature, there is a proportional relationship, but not a directly proportional relationship. This means that although when one goes up the other does too, it is not a direct relationship, as when one doubles the other does not.
However, by converting the data to Kelvin by adding 273 to the temperature data, it can be seen that the data does then follow a directly proportional relationship, and therefore passes through the origin.
Why convert to the Appropriate Graph?
When creating graphs from data, the aim is always to find either a "constant of proportionality" or some other constant. This is why the straight line through the origin the aim. By gaining a straight line, the gradient of the graph is constant, and therefore so is any constant of proportionality based upon it.
How to convert to the appropriate graph
To convert a formula to allow the appropriate graph to be plotted from a set of data, a "y = mx + c" format must be used.
In this format, y and x represent the variables in the experiment, with all variables that were kept constant grouped together in m. If this is done perfectly then c will be equal to 0, however in practice the straight line will be close to but not through the origin.
Note - although y and x represent the variables from the experiment, they may not be in the same form they were measured in. See below for examples of this.
Example 1
Use the formula below to investigate how the Voltage across a set of parallel plates affects the Velocity of a charged particle.
In this experiment, the two variables are the Velocity (v) and the Voltage (V). The Charge (q) and the Mass (m) will be kept constant.
If this formula is correct, by rearranging this formula to the y = mx + c format will give a straight line through the origin.
Therefore, if a graph of the Velocity squared (y) against Voltage (x) is plotted and the formula is true, it will give a straight line through the origin, with the gradient (m) equal to 2(q/m).
Example 2
Use the formula below to investigate how the distance between two objects affects the Gravitational Force between them.
In this experiment, the two variables are the radius (r) and the Gravitational Force (F). The Gravitational constant (G), Mass 1 and Mass 2 will be kept constant.
If this formula is correct, by rearranging this formula to the y = mx + c format will give a straight line through the origin.
Therefore, if a graph of the Force (y) against 1 over the radius squared (x) is plotted and the formula is true, it will give a straight line through the origin, with the gradient (m) equal to GM₁M₂.
Example 3
Use the formula below to investigate how the semi-major axis of an orbit affects the Period of that orbit.
In this experiment, the two variables are the semi-major axis (r) and the Period (T). The Gravitational constant (G) and the Central Mass (M) will be kept constant.
If this formula is correct, by rearranging this formula to the y = mx + c format will give a straight line through the origin.
