29-Aug-2016: Deriving a power law for an inertial pendulum.

Student: Xavier Lomeli
Partners: Matthew Ibarra, Billy Justin
Date of Lab: 29 August 2016 

Deriving a power law for an inertial pendulum

Mission Statement:To measure the inertial mass of an object by comparing the resistances experienced by the object when subject to changes in its motion. 

Theory: Due to mass not being dependent on gravity, gravity is not required within the derivation to find the inertial mass of the object, since mass is ultimately a quantitative measure of an object's inertia, or resistance to motion. So if the force of gravity were unknown, such as on another planet, the mass of any given object could still be identified by utilizing a mechanism that exploits this characteristic of constant inertial mass.
Experimental Procedure: For this experiment, we were supposed to determine the mass of any given object (stapler, pencil, etc.) by clamping an inertial balance to the edge of the worktable, placing a thin piece of masking tape on the end of said balance, and then setting up a photogate so that when the balance oscillated the tape completely passed through the beam of the photogate. Every two times the strip of masking tape blocked the beam, the computer program Logger Pro recorded the period duration via a connected LabPro. We recorded 9 different periods corresponding to 9 different masses atop the inertial balance. With this data we then proceeded to find the relationship between mass and period via some power-law type of equation described below.
Lists/Tables/Graphs of Collected Data with Explanation:
The table below shows the different masses we tested together with their respective periods (T)-

From the data on the table above, we could now use the equation T = A(m+Mtray)^n to determine the mass (m) of any given object. However, first we needed to solve for the unknown variables A and n, which correspond to the y-intercept and slope, respectively, of the above equation when written in the equivalent form Ln(T) = n*Ln(m + Mtray) + Ln(A). Furthermore, we needed to arbitrarily determine the appropriate value for Mtray such that the plot of the equation would be linear with a correlation coefficient of 0.9998. We discovered that there were a range of values for Mtray, which we discovered to be 210 +/- 20 grams. Beyond this range, the plot lost linearity. Below, I took pictures of three graphs, one depicting the lowest possible value, one the highest, and the third depicting the median value for Mtray.
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Lowest value for Mtray (190 g)

    n = slope = 0.5351
   A = e^(-4.083) = 0.0168

Highest value for Mtray (230 g)

    n = slope = 0.5950
   A = e^(-4.505) = 0.0110

Middle value for Mtray (210 g)

    n = slope = 0.5497
   A = e^(-4.186) = 0.0152
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Now that we found the proper values for A, n, and Mtray, we could put our equation into practice and calculate the inertial mass of any given object upon knowing the period of said object. Since we would be inputting three different values of Mtray per object, we expected three different values for the inertial mass per object. We decided that these two objects would be my partner's phone and the classroom stapler. Below is my work.
*Keep in mind Ln(A) = Y-intercept, thus A = e^(Y-intercept)

Next, we went and weighed both the phone and the stapler to record the gravitational mass. Below are pictures of the scale.

Conclusion- Fittingly, these observed gravitational masses corresponded very well with their respective inertial counterparts, falling well within the range of calculated values for the inertial masses. Thus, the end purpose of this lab, to find a mathematical relationship between mass and period for an inertial balance, has been proven to be satisfied by the equation described above.