23-Feb-2015: Deriving a power law for an inertial pendulum

Purpose: To derive a power-law equation that describes the relationship between mass and period of oscilation. This will be based on the data we collect and parameters we will create and adjust using Logger Pro. We will then use our equation and our calculated values of M-tray to find a tight range of mass for two unknown objects.

Procedure: We first set up an inertial balance and a photogate connected to a laptop to collect data. We used a c-clamps to hold the balance against the surface of the table so that all the motion is in horizontal oscilations. To collect data on each oscillation time we placed a piece of tape at the end of the balance and set up a stand to hold the photogate at the end. The piece of tape attached to the balance breaks the led light and records the time for each oscilation. We preformed several different trials using masses starting with zero mass on the tray and adding 100 grams each trial up to 800 grams. 

Below are the data table and logger pro graph from our trials.

The power-law type of equation we used, T=A(m+Mtray)^n, closely resembles the linear equation y=mx+b when we take the Natural Log of both sides.

 Using Logger Pro we entered the data and created additional columns for the Natural Logs of both the periods and masses and plotted out results.

 However, we did not weigh the actual tray, so we estimated and adjusted the Mtray parameter until it gave us the closest correlations possible. 

We chose to use 282 g as our lower and 311 g as our upper limits for our Mtray parameter. This later helped us select a very tight range when calculating the unknown masses of other objects by using the time each took for their oscillations.

 This lab seemed really tough because we started with so many unknown variables. Collecting data on different known masses with the Inertial Balance and photogate made things go smoothly when using Logger Pro. We also found the natural logs we needed, graphed the modified formula and found values for some of our variables. This also involved estimating, using and adjusting values for our Mtray parameter. Using very close values for our lower and upper limits of Mtray we were able to calculate a tight weight range for our unknown masses.