28-February-2015: Finding the Relationship Between Mass and Period for an Inertial Pendulum

The purpose of this lab was to come up with an equation to describe the period of an inertial balance when different masses were set on it.

Materials:
C-Clamp
Inertial Balance
Masking Tape
LabPro- power adapter, USB cable, plug adapter, photogate
Computer with LoggerPro Software
Multiple Different Masses (.1kg - .8kg)

Setting Up the Experiment:

Fig. 1

The C-clamp secured the 

inertial balance to the table, ensuring 

it wouldn't move during oscillation.

To begin, my lab partners and I needed to set up the inertial balance in such away that we could measure its period of oscillation. To do this, we used the C-clamp to secure the inertial balance to the edge of our table (See Fig. 1). Once secured, we needed to set up the LabPro equipment at the other end of the inertial balance. Using a metal pole stand, we placed the photogate to the opposite end of the inertial balance and made sure it was also level with the balance. We also placed a piece of masking tape at the end of the inertial balance. (See Fig. 2). The photogate was then hooked up to the LabPro, along with the power adapter and plug adapter. We then finally hooked up the LabPro to the computer and opened up the LoggerPro Software. The file that we used was the Pendulum Timer.cmbl. (We searched this via the magnifying glass at the top right corner of the macbook). We were now ready to begin collecting Data.

Fig. 2

The photogate produces a light 

beam when turned on. The masking tape

will be used to break that beam during 

oscillation.

      

This is what the final setup looked like. 

Collecting Data and Making Graphs:

  

Fig. 3

We continuously added different masses,

securing them with tape each time. 

To begin collecting data, we intended to measure the oscillation of the inertial balance under a multitude of different circumstances. First, we began with no mass on the balance. We pulled the balance back slightly, released, and collected data of its oscillation. We then added 100g to the balance and repeated. We continuously added 100g after each trial, all the way until we reached 800g (See Fig. 3).  We then measured the period of oscillation of the balance when we added two objects whose masses were unknown. For this experiment, we used a tape dispenser and car keys. Once we had all our data, we recorded it into the data table provided in the lab handout. (See Fig. 4).                                            

Fig. 4

Fig. 5

We created the data table to mimic the 

y=mx+b form. TM is short for "Total

Mass" which is what (m+Mtray) is.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          Once we recorded all the data into our data table, it was time to make a graph of period vs time. From the handout, we were given the initial equation:T=A(m+Mtray)^n. With this equation, we had three unknowns: A, Mtray, and n. To make a more reasonable looking equation, we took the natural logarithm of each side, which gave us: lnT=nln(m+Mtray)+lnA. This equation mimicked that of y=mx+b, in that lnT is y, n is m, ln(m+Mtray) is x, and lnA is b. To begin graphing our data, we first opened a blank LoggerPro document, which allowed us to plot data in an "x" and "y" table. In this table, we labeled the "x" column "Mass (kg)" and the "y" column "Period (sec)." We then created three new columns for (m+Mtray), lnT, and ln(m+Mtray) by first clicking the "Data" tab, and then "New Calculated Column" (See Fig. 5) Finally, we had to create a parameter Mtray, which would be our initial guess for the mass of the inertial balance by itself. To do this, we clicked the "Data" tab, and then "User Parameters."  We first guessed 0.3 kg.

When making the graph, we set the vertical axis as lnT and the horizontal axis as ln(m+Mtray). This gave us a linear graph with a slope "m" (which is n) and a y-intercept "b" (which is lnA). We then had to adjust the parameter Mtray several times to get a correlation coefficient that was as close to 1 as possible. We found that our best range for Mtray was from 0.295kg to 0.299kg (See Fig. 6).

Fig. 6

0.9997 was the closest correlation coefficient we managed to obtain. The fit parameter for Mtray was essentially our best guess for what the mass of the inertial balance on its own would be, since we couldn't actually measure the mass of the part that was oscillating by itself. 

Deriving The Final Equation

Fig. 7

Once we had our originally unknown values for n, lnA, and Mtray, we were now able to complete our equation of T=A(m+Mtray)^n using algebra (See Fig. 7). When testing the equations, we found that we were roughly 0.005 seconds off from the actual period for most of the masses. When evaluating the mass of the tape dispenser (T=0.611s), we found it to be 0.606kg with the equation using Mtray as 0.295kg, and 0.612 with the equation using Mtray as 0.299kg. The actual mass of the tape dispenser was 0.605 kg. As for the keys (T=0.358s), we found its mass to be 0.104kg when using both equations. The actual mass of the keys was 0.105kg. The equations we derived gave us very accurate measurements of our unknown masses. The relationships we found between mass and the period for our inertial balance from this lab are as follows:
T=0.6517(m+0.295)^0.6518 and T=0.65005(m+0.299)^0.6559