23-Feb-2015: Using an inertial balance to find a relationship between mass and period by deriving a power-law equation.

     The purpose of this experiment was to determine if there is a relationship between mass and period for an inertial balance. To determine this, an inertial balance was used, as well as masses ranging from 0 to 800 grams and a photogate to measure the balance's period. We hypothesized that adding more mass to the inertial balance will cause the period to increase. 

     The inertial balance was attached to the edge of a table, with a piece of tape extended out from the main tray. A photogate, connected to a computer using the Logger Pro system, was set up in front of the main tray so that the piece of tape was able to fit between the photogate sensors. The purpose of this was so the tray can oscillate left to right allowing the piece of tape to move through the photogate, which would measure the period of the oscillation.

     As a control for the period, the inertial balance with no added mass other than the tray itself, was pulled to the side just past the photogate sensor and released. The period was recorded under a mass of zero (Table 1). To determine if the period would change with different masses, masses in increments of 100 grams were centered on the inertial balance tray. 

The inertial balance with mass added to the tray.

     Each time 100 grams of mass was added, the balance tray was pulled to the side just past the photogate sensor and released, exactly as performed in the control. The period for each increment was recorded under its respective mass total in a table (Table 1). 

Data on the Logger Pro system shows a period of 0.604 seconds for a mass of 600 grams.

     After all data was recorded and graphed, a mathematical model was developed to describe the mass to period relationship: T=A(m+Mtray)^n.

     The mass of the tray was a constant that needed to be accounted for when using the mathematical model. To find it, the equation needed to be converted to a y=mx+b form by taking the natural log of it. After taking the natural log of both sides of the equation, a graph of ln(T) versus ln(m+Mtray) was formed (Figure 1.) 

     A range of masses was guessed for the unknown mass of the tray to attempt to get the graph to a correlation as close to 1.0 as possible. A correlation being the computer's report out on how well it can minimize the sum of deviations. A lower bound of mass 280g (Figure 2) and a higher bound of mass 362g (Figure 3) were found and graphed. The other two unknowns of A and n were found but were relative to the estimated mass of the inertial balance tray. They can be found by looking at the graph's y-intercept for unknown A and the slope for unknown n.

Figure 1. Graph of ln(T) versus ln(m+Mtray)

Figure 2. Data set for the lowest value of the mass of the inertial balance tray with the correlation as close to 1.0 as possible.

Figure 3. Data set for the highest value of mass of the inertial balance tray with the correlation as close to 1.0 as possible.

     To verify the new mathematical model, two objects of unknown mass were placed individually in the inertial balance tray and their periods recorded. Using the low and high boundaries of the mathematical model and the periods recorded from the new objects, simple algebra solved for an estimated range of masses for the objects. The range of mass was compared to the gravitational mass recorded by a balance in order to determine the accuracy of the experiment and the derived mathematical model (Table 2). 

     The results were promising. Table 1 and its graph show the heavier the mass the longer it takes to oscillate. Thus confirming the hypothesis. Table 2 shows the results from the calculations using the derived mathematical model and its accuracy. 

Graph of data in Table 1.

Table 1. Mass versus Period

Mass in balance, g	Period, sec	Mass in balance, g	Period, sec
0	0.288	500	0.556
100	0.354	600	0.604
200	0.409	700	0.650
300	0.460	800	0.704
400	0.509

Table 2. Calculations using the new mathematical model and measured masses of new objects.

Objects of unknown mass	Water Bottle (full)	Tape Roll
Lowest Mass Value, g	512	132
Highest Mass Value, g	515	135
Gravitational Mass. g	496	125
% error	3.4%	5.6%

     After all calculations, the results seen above strongly support the hypothesis. Although the percent error from the mathematical model is not as close to zero as would have hoped to be, it could have been cause by several uncertainties in the lab.  In future experiments, the masses might be shifted to one end of the inertial balance tray to see if there is a different effect than centering them. The piece of tape that triggers the photogate sensor might be narrowed or widened. Also, the gravitational mass of the tape used to hold each object in place might be measured. Since there was a small gap between calculated and actual values, the mass of the tape might not be so negligible.