How to Get From a bunch of Data to a (Hopefully) Linear Plot.
Physics seeks to reduce the seemingly complex into something more understandable. Intrinsically, a graph is usually more understandable than a data table. Happily for physicists, many
natural phenomena “obey” mathematical formulae such as E=mc2. This means we get “nice” graphs. Why this should be so and what is so special about mathematics is a “hot topic” currently
amongst physicists, philosophers and mathematicians. Like so many before us, let us just accept this useful fact and use it to further our understanding.
Using the E = mc2 example it is fairly clear that if we plot energy against mass (said this way mass is the x value) we will get a straight line, with a gradient of the speed of light squared. In
the old days (Before Computers or BC) fitting a straight line was as good as we hoped to do. By “fitting” we mean using a mathematical system to get the line as close as possible to as many
points as possible. Most people can see where the line of best fit should go remarkably well with a bit of practice. Now, with spreadsheets, it is quick and easy to fit just about any curve to
any data set. Physicists, however, still often manipulate their data into a linear form: y = mx + b. Which of the two variables represents the independent and dependent variable is far less
important than having a meaningful or “nice” gradient. In an Ohm’s law experiment why not have the gradient be the resistance rather than its inverse? This means putting the potential
difference, which was probably the independent variable, on the y-axis. Relax, this is OK, just don’t let your grade seven science teacher find out .
If you are verifying a relationship between the variables that is already known, or using the relationship to find some constant or controlled variable, half the battle is won, all you need do
now is get the formula into a linear form. If not, you have to do as Kepler did, and search for it by wading through the data. Things are easier these days as the magical spreadsheet can fit all
sorts of trend-lines quickly and accurately.
Getting the formula into a linear form seems to worry many students so let’s run through two examples:
1.
What do you think the formula is?
Height and time of fall: s = ut + ½ at2
2.
Get the dependent and dependent
variables on opposite sides. Tidy up
the equation.
Make sure all the constants and
controlled variables are on one side.
u=0
3.
4.
5.
Examine what you have and it will
become clear what needs to be
graphed.
Add the necessary columns to the
spreadsheet (processing data) and
draw the graph.
Photoelectric effect, changing frequency changes the
maximum kinetic energy of the photoelectrons: hf = ϕ +
Emax
hf = ϕ + Emax
s = ½ gt2
All good
All is good:
s = ½ gt2
But who wants a gradient of 1/h?
Looks like y = mx
y is s,
x is t2 not t
We need to add a t squared column to our data.
The gradient is g/2, notice this is constant.
Emax = hf - ϕ
So y = mx + b: y = Emax , x = f and m = h
OMG!! The y intercept isn’t the systematic error!
Now you can try a few. The first one is done for you. (Yay!)
Experiment
Newton’s second
law
Independent
variable
Force
Dependent
variable
Acceleration
Controlled
variables
mass
Formula(e)
F = ma
Manipulated
equation
F = ma
Too easy!
Graph
Gradient
F
m = mass
b=0
a
Impulse
Force
Time to stop
Initial velocity;
mass
Conservation of
energy
Height ping pong
ball rolled from
Final velocity
Mass of ball
Centripetal force
Period of orbit
Orbiting mass;
Radius of orbit
Whirling a stopper
II
Orbiting mass
Period of orbit
Centripetal force
Radius of orbit
Whirling a stopper
III
Radius of orbit
Period of orbit
Centripetal force
Orbiting mass
Initial velocity
Distance to stop
Decelerating force;
Mass of trolley
Whirling a stopper I
Distance to stop a
trolley when it runs
onto a rough carpet.
I = FΔt = m Δv