23-Feb-2015: Finding the relationship between mass and period for an inertial balance

Objective: 

       This lab is conducted to find out an equation to describe the relationship between mass(kg) and period(s). Further, we use this equation to calculate the mass of unknown objects. 

The set-up:

       First, we use a C-clamp to secure the balance at the edge of the table. Then we attach a small piece of tape at the end of the balance toward the photo gate. The tape is long enough in order for the censor get the signal when the tape passes through the beam of the photogate. When the balance is oscillating, the tape passes through the beam, thus we can measure the period. The set-up basically is shown in the figure 1.

Figure 1: The set up of the experiment

Data collecting.

      After setting up the equipment, we start measure the  period with different mass at each time. The first time we don’t put any weight on the tray. We just pull and release the balance, then record the period (Figure 2). We continue the experiment by adding more weights to the tray. The range of weights is from 100g to 800g. We do the same process: add 100g weight, pull and release the balance, then record the period. Continuing to do it until we add up to 800g weight on the tray (Figure 3).    

Figure 2: Pull and release the balance without any weight, then record the period(s)

Figure 3: 800g weight on the balance and ready to be released to measure the period(s)

Lab Notes and Calculation.

      When we are done collecting the data, we get a table with one column describing mass(kg) and one column describing period(s).

      Now to find an equation to describe the relationship between mass(kg) and period(s), we guess that these two variables relate to each other through the equation:

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

Using the data above, we then plotted our result, and get a graph like below.

 Figure 4: The graph of mass(kg) and period(s)

The line in figure 4 is not a straight line which is not our desirable result, thus we take  natural logarithm both side and come up with a new equation

ln T = n*ln(m+Mtray) + ln A

       This new equation is similar to the linear equation y=mx+b, which means that if we can get a straight line, we are able to find the correct value of Mtray, then work backward and find the original equation to describe the relationship between mass(kg) and period(s).

       We create three more columns: lnT, m+Mtray, and ln(m+Mtray). We calculate lnT, m+Mtray, and ln(m+Mtray) then plot a new result. However, to get a straight line, we need to find out the correct value of Mtray which we don’t know. Therefore, we take a guess and continue doing it until we get the correlation coefficient is equal 0.9999 or 1. Here is the result we got (Figure 5). At the end, we find out that the lowest value of the tray to get a straight line is 0.3g and the highest value is 0.349g. 

Figure 5: The graph of lnT and ln(m+Mtray). The correlation coefficient is 0.9999 according to the graph.

       Since we get a straight line, we now can work backward to find the equation. The slope of  equation ln T = n*ln(m+Mtray) + ln A gives us the value of n because this equation is similar to y=mx+b. To get the value of A, we do e^(lnA) which is also known as the y-intercept from a straight line. After doing the calculation, we come up with two equations based on the lowest and highest value of the Mtray we found above (Figure 6).   

Figure 6: The calculation after we figured out the value of A, n, and Mtray, and plugged back into the original equation and solved for T(period).

       After finding the equation, we check whether it is correct or not by using these two equations to calculate the mass of two objects: wallet and calculator. We first measure the “true” mass of wallet and calculator using the balance. Below is the mass of wallet and calculator  (Figure 8 and 9 respectively)

 Figure 7: The mass of the calculator 

                                                                     Figure 8: The mass of the wallet.

          Since we know the “true” mass of calculator and wallet, we use our equation to find the mass and check whether the results are matched. If the mass we find using our equation is close to the “true” mass, then we achieve the goal of the lab: finding an equation to describe the relationship between mass(kg) and period(s). After doing the calculation, our results match the “true” value. Seen in Figure 9.

Figure 9: The calculation to check the mass of wallet found from our equation whether it matches the "real" mass measured by the balance.

Clear summary: 

        What we did in lab that day was to find an equation to describe the relationship between mass(kg) and period(s). Further, we use our equations to check whether the mass we found match the “true” mass measured from the balance. We first started to set up the equipment which is shown in the Figure 1. After finishing set-up, we collected the data using different weights at each time to measure the period(s). We then guess an equation to describe the relationship between mass(kg) and period(s). Because the graph between the mass and period was not a straight line, we took natural logarithm both side of the equation and came up with a new equation. We ended up finding the correct value of the tray and other values such as A and n; we plugged back to the original equation and finished our calculation. Finally, we used our equation to check whether the mass found from our equation was close to the “real” mass of wallet and calculator.