AC130Nav said:
Wouldn't it be better to plot the displacement versus the cumulative time (x axis)? Gives a more useful sine wave.
A sine wave is pretty much useless. You cannot take the slope of a sine wave, all you can do is say, well, the motion is periodic, and that's nothing you didn't already know.
Other than getting ideas for the form of a function, you should always plot graphs which you think will be linear.
For instance, let's say we have a dependent variable, y, and an independent variable, x. We don't have a theory telling us what y as a function of x is.
What we should do is first look at the graph of y as a function of x, to get an idea of what the functional form is. Suppose we see that the graph reminds us of a high degree polynomial.
Let's assume then that y=kx^n and try and get the full functional form from a graph, given some values (x,y)
How would we go about this? We would linearize the graph, since without great computing power, we don't really have a way of distinguishing one polynomial plot from another.
What we can do, however, is look at the equation y=kx^n and -linearize- it, so that we have a graph we can extract data from (Plotting a trend line and computing the slope and intercept, which we can then relate to our unknown constants, which is something we are fully capable of).
One way of determining k and n, would be to look at the following relation:
y=kx^n
which implies:
\ln{y}=\ln{kx^n}
\ln{y}=\ln{k}+n\ln{x}
When plotting \ln{y} as a function of \ln{x} we would expect a linear graph whose slope is n and whose intercept is \ln{k}
From the y as a function of x graph, we found the form of the function (Which you would usually know from a separate source anyway) and from the linear graph, we managed to extract numerical information from the data.
Since for the most part, slope data is more accurate than intercept data, knowing n from the logarithmic graph, we can then construct a graph of y as a function of x^n, and this graph too, we expect to be linear, with a slope equal to k and an intercept of 0.
Which one goes on the horizontal axis, and which one on the vertical axis is simply a matter of convention where the independent variable (What you change, knowing exactly the change you've made) goes on the horizontal axis and the dependent variable (What you measure after making the change to the system) goes on the vertical axis. In your experiment you, the one conducting the experiment, changed the length of the wire of the pendulum, and what you had measured was the period.
So if you wish to follow the convention, you should plot the period squared on the vertical axis, and the length of the wire on the horizontal axis.
