26- Aug- 2014: Deriving a Power law for an Inertial Pendulum

Inertial Balance Lab

Objective: Find a relationship between inertial mass and the resulting period on an inertial balance through deriving a power law and testing it.

The Set-up:
  Using an apparatus similar to the one seen in (fig.1) we clamped it firmly to a desk with a flapping piece of tape sticking out of its edge making an obvious flap. We then align this flap to a sensor that counts and times how many times the piece of tape swing between its sensors, thus, calculating its average period ( T ) onto lab pro on our laptops.

Data Collecting:
  After initial set up and opening the labpro file for the lab we then started with the inertial balance as it was as a control. So we would pull carefully toward a direction perpendicular to the balance and, when ready, released to time the average period of the tape flap to pass the sensor for approximately 5 seconds then recorded the results in our lab packet. After this initial trial, we tape 100g to the end and repeating the action for a consistent time of approximately 5 seconds. From that point on for the next 7 trials we would add another 100 grams each time until we reached 800 grams, seen in (fig.2).  

Figure 1. The apparatus used for today's

lab with exception to the laptop and       labpro connections.                                  

  Figure 2. Resulting periods ( T ) for each trial   from 0g (control) to 800g on the inertial balance.

  We then plotted our results, including the results of two unknown masses having our x- axis being our mass in kg, and our y- axis was our resulting periods shown below in (fig. 3). Plotting our unknown masses early on had proven to be a mistake later on when we attempted to linear- fit our graph.

  Figure 3. The initial scatter plot that had also contained our unknown masses and periods as well as the Data Table results.

  Now at this point, we have to find a relationship between a period of a mass in an inertial balance and the mass itself. So we assume some sort of power law equation a period T being  the total mass to the power of n times a constant:

T= A(m + Mtray)^(n)

  If we log both sides of the equation we can algebraically change the order to a more recognizable form like that of a linear graph:

lnT= lnA + (n) ln (m + Mtray)

lnT =(n) ln(m + Mtray) +lnA

y =(m)         (x)         +b

  From here it is apparent that in order to have a linear- fit of our Power law labeling our n as our slope, our lnA as our y- intercept, and making our Mtray a parameter. So as a result, we labeled three more calculated columns for  lnT, ln(m + Mtray), and (m + Mtray) respectively in our scatter plot graph to plot a linear graph of lnT vs. ln(m + Mtray). Now our  linear function attempts to meet all points within a correlation to as close to 1 as possible. However, there was difficulty getting our linear- fit to reach the correlation we wanted because we had to put in a parameter of the the mass of the apparatus itself which affected the overall linear- fit correlation depending how accurate your guess was to the Mtray as it was labeled. We then found that our mistake of adding in our unknown masses to the plot was a mistake and so  after recording their periods and the masses of the unknown (for reference of later calculations), we came up with a linear- fit to as close as .9998, seen in (fig.4).

  Figure 4. Our resulting Linear- fit of our scatterplot with a correlation of .9998 and the unknown masses and their periods were removed. We also added 3 new "calculated" columns representing: 
ln( T ), ln (m + Mtray), and (m+ Mtray) respectively.

  This was how far our progress had gone as far as the 26th of August. However, we then continued our lab on August 28th to derive an equation showing the relationship masses have to their periods. So by finding the lowest limit of Mtray and upper limit of Mtray that maintains a correlation of .9998, we found their slopes of n and found the value of A and made them constants for all values of m. How we got that was by plugging in the lower and upper limits of the parameter to the linear graph, in fig 4. the lower limit of parameter Mtray came out to be .257kg, and our linear- fit feature in Labpro gave us values of slope n and y- intercept lnA. Since we wanted the value of just A from lnA, which at this point was given, we had to algebraically find that value. Now that we had our values of n, A, and Mtray (within a upper and lower limit) it was not time to find the relationship between inertial mass to its period by finding our 2 unknown masses. We have the period of each unknown mass so with that we have to make a rearranged equation m( T ) to find the masses from their periods shown in fig 5. 

   Figure 5. The resulting periods of the two "unknown" inertial masses labeled as Mu(1) and Mu(2). As reference we recorded the unknown's true mass to compare to our results.                                                                             

Lab Notes and Final Calculations:

  In fig 6., you can see what we calculated to be our upper and lower limits of the parameter Mtray that maintains a correlation of C= .9998 and our average Mtray so that way we could find the unknown masses within a range of uncertainty.

ex):

m( .4)= .220± .004 

  We also labeled all of our variables and showed the values of n and A for each limit to use in our m( T ) once we found what m( T ) was from the Power Law we started with.


































  



Figure 6. My rewritten lab notes for organizational and photo quality purposes, none of the values or results were altered. 

  The calculations resulted to the unknown masses to be Mu(1)= .103± .001 kg and Mu(2)= .252± .004 kg which swayed from their true masses of 100.0kg and .256 kg respectively. However, it appears that our calculations were not too far off to call this a utter failure. Nevertheless, we had reached our objective of finding a relationship between a Period on an inertial balance to the mass that is placed on it through the Power Law:

T= A (m + Mtray)^(n)