3.7 Square Relations: Side Opening Parabolas

Consider an experiment in which a student investigates the effect of changing the length of a pendulum on the time required for the pendulum to make on swing (or the period).

Notice that the shape of this graph resembles the original Position vs. Time graph on the previous page, and that it could be a parabola which opens to the right

Once again, since the graph is not linear, we can't determine if the equation directly. The trend of the graph shows that as the length of the pendulum increases in constant increments, the period increases in decreasing increments. This is why the graph bends away from the period axis. Since length increases at a greater rate than period, what mathematical manipulation of the data would you perform which might allow you to plot a linear graph? Since squaring the variable, which was increasing at the lower rate worked in the previous example, why not try it again. This time the reasoning might be: If length is increasing at a greater rate than time, isn't it possible that squaring period, which will make it increase at a greater rate, might make it keep up with length? Based on our speculation that this might be a sideways opening parabola, we will make a new table in which we will square every period value. We square period because it is the variable, which is not keeping up in this case. Our test plot will then be Period² vs. Length. How will we know if this mathematical relationship we have predicted is correct? Of course! If the new graph is linear and passes through the origin, we can say that period² is proportional to length.

Since the resulting test plot is linear and passes through the origin, our prediction is confirmed. The original graph was indeed a sideways opening parabola. To determine the mathematical model that describes the relationship between the Period and Length of a pendulum, we will follow the standard procedure as outlined below:

The resulting mathematical model is:

Period² = 0.041 s²/cm · Length

Remember that this equation describes not only the linear test plot but is also the equation of the original parabolic graph. As in the direct proportion, the constant slope of 0.041 s²/cm may be replaced with a variable name that describes the nature of the rate of change represented by the slope of the straight-line graph, Period squared vs. Length. The meaning of this graph’s slope is a future topic of study in physics, so we will not explore the relationship at this time.