PocketLab is a perfect device for determining the quantitative relationship between the length of a pendulum and its period of oscillation. Pendulums of known lengths were made from balsa wood strips such as those available from Michaels and other hobby stores. The photo below shows six such pendulums of lengths 15, 30, 45, 60, 75, and 90 cm alongside a meter stick. The picture shows that PocketLab was taped with double-stick mounting tape to the pendulum whose length is 45 cm.
The photo below shows the apparatus setup. The balsa wood pendulum with PocketLab attached is hung with masking tape from a ring stand supported by the weight of several books to keep it stable. The orientation of PocketLab shows that it will be swinging in the XZ plane. Therefore, the Y angular velocity data will provide information necessary to compute the period of the pendulum.
The video below shows the PocketLab graphs superimposed on the actual moving pendulums. The Y angular velocity, shown in blue, contains the data of interest in the analysis.
The Excel graph below shows the Y angular velocity in deg/s obtained from the gyroscope data file for the pendulum of length 15 cm. (Note that half the length (3.2 cm) of PocketLab is subtracted from 15 cm, giving a length of 11.8 cm to the approximate center of mass of PocketLab.) As shown in red in the graph, the period is calculated by averaging the time for ten complete oscillations. This process is repeated for all six pendulum lengths. All gyroscope data for these six pendulums can be found in the attached gyroscope Excel file.
Below is shown an Excel chart the summarizes the results of the experiment. When a power regression type is applied to the data, it is seen that the power turned out to be 0.4773, very close to the theoretical 0.5 expected for such a pendulum. It can be concluded that the period of a simple pendulum is proportional to the square root of the length of the pendulum.