27-Feb-2017: Finding a relationship between mass and period for an inertial balance

Lab 1: Finding a relationship between mass and period for an inertial balance
27-Feb-2017

Idea: I have to determine the period of an inertial balance first for various known masses, then the period of various unknown objects, and be able to calculate those masses based on the data obtained from the graph of mass/period of the known masses.
Summary: First, with regards to materials, in this lab I used an inertial pendulum with tape for a marker, and a photogate that measures the period of the oscillation which was connected to the Logger Pro application, which records the period of the oscillations over a length of time. In addition to a stand, various known masses, and two random objects (a Samsung S6 phone and this clamp we found in the room), as shown here:

1. Then, we loaded the inertial pendulum's tray with masses ranging from 0 (nothing on the tray)  to 800 grams in increments of 100 grams, measuring the period over 4-5 seconds with Logger Pro.
2. We then opened up a data set (x and y) and set Mass in grams to be X, and Period in seconds to be Y and created 4 new columns, Mtray, m+Mtray, lnT, and ln(m+Mtray).
3. Based upon the graph that these parameters gave, we adjusted the value of Mtray until the graph gave us a straight line of best fit.
4. Simultaneously, we also experimented with different values of Mtray that would give us the range of uncertainty for Mtray, that is, we found the minimum and maximum values of Mtray that would still give us the best possible straight line. 
5. In addition, we recorded the slope and y-intercepts that resulted from these different values of Mtray, for each different value of Mtray.
6. These values would allow us to find the value of constants A and n.
7. Subsequently, we measured the period of two different objects with unknown masses, and using the values of Mtray and their respective values for A and n in the previous step, determined the range of uncertainty of unknown masses 1 and 2.
8. Finally, I used Excel in order to graph my raw data.

Measured Data

Part 1: Data Table for Period of the various known masses

Part 2: Data Table of the two unknown masses. Shows data from the lowerbound, middle, and upperbound values of Mtray.

Graph/Calculations: These are the graphs and calculations for the known masses based off our data.

Graph of ln(t) and ln(m+Mtray) for known masses

Graph for lower value of Mtray (240 g). This shows the graph with the lowerbound value of Mtray that would give me the best possible correlation closest to 1. For me, the best I could get to was .9996.

 Graph for likely value of Mtray (270g), which was closer to its lowerbound and subsequently shown in the graph as its slope is most similar to the Mtray lowerbound graph.

Graph for upper value of Mtray (300 g), similarly, with 300 grams being the absolute maximum that would allow the graph to keep the best possible correlation of .9996.

Calculations

Values I needed to calculate

Finding the mass of the added object

Resulting data

Graph of Resulting Data

The topmost line is the graph of the second mass (the clamp), and the bottom line is of the phone, the first mass, and both for the middle value of Mtray. As I could see, the bottom-most graph (of the phone) is noticeably far more accurate, as its slope is closer to 1 than mass 2's.

Analysis: 
In the first three graphs, line of best fit allowed us to find the slopes (n), y-intercepts, and best correlation for the three variations of the Mtray value. As the value of Mtray increased, I observed that the slope of the line also increased.
Then, in order to find the unknown masses, I used the A and n constants (for the three values of Mtray) I found based on the previous known masses, which allowed me to later rearrange the equation of period in order to solve for the two unknown masses. I repeated the calculations for both objects, with the various values of Mtray.
Afterwards, in graphing my raw data, I again set my set my parameters and columns exactly the same as before, but instead entered in the calculated masses of my two unknown masses. In order to get a line of best fit I utilized Excel, which allowed the comparison to the other graphs a lot easier. The graph essentially showed me that the experiment had been more accurate with the first unknown object (the phone) rather than the second, as its constants (slope, for example) is far more accurate than those of object 2 with a slope of 1.0086.

Conclusion: I was able to graph the data obtained from the two known masses, and calculate the mass of two unknown objects, as 179 grams and 1008 grams, respectively. Weighing the two objects on a scale gave us their actual mass, 184 grams (with a phone case) and 1300 grams (with the clamp), showing that the margin of error varied greatly between the two throughout my calculations.

• In hindsight, an area of uncertainty in this case was possibly (though not likely) first in the photogate, as the condition of the equipment is something we could not ascertain. 
• Then, in the inertial pendulum itself, as we observed that the metal connecting the tray would twist/bend if the mass in the tray was too heavy, which could certainly affect the oscillation of the pendulum, and subsequently its period. The clamp we used in the second part of the lab was much heavier than the other masses in the lab, and as a result I did observe a slight twist to the pendulum. In the end, this could explain the discrepancy between the constants of my last graph.