Finding a relationship between mass and period for an inertial balance

1.     Lab conducted by Mohammed Karim (author), Curtis, and Lynel on August 31, 2016.

2.     Objective: Find an equation that can relate mass and period for inertial balance.

3.     Theory/Introduction: Normally, we would use a scale that utilizes spring force to find the mass of an object. However, since mass is independent of weight and independent of gravity, we can use many other methods to find the mass of an object. In this lab, we use inertia as a method of finding an object's mass. In order to find the mass of an object, we first had to find an equation that could relate mass and period. What we did was measure the period of the oscillation at different masses. Once we collected our data, we would derive an equation. With this equation, we could find the mass of an object with an unknown mass.

4.     Apparatus/Experimental Procedure: 

 Lab 1 Apparatus

As shown above, our apparatus consists of an inertial balance clamped to the tabletop with a thin piece of masking tape on the end. A photogate is set up and placed in front of the masking tape. This is so that it can record the number of oscillations. Using LoggerPro, we used an application that measures the period.

After ensuring that the timing of the test run is somewhat accurate, we began to collect data using variable weights, ranging from 0 grams to 800 grams not including the weight of the inertial balance. The data for this portion can be found under the "Data" section (See Figure 1.1). With this data, we decided to find a relationship with a power-law type equation:

T=A*(m+M(tray)) ^n (m+M(tray) as the mass of the tray is always present.)

By taking the logarithm of both sides, we can make the equation resemble the equation of a linear line (y=mx+b). This is done so we could incorporate it into a graph and find the value that would make the line straight. If the line is straight, we could find the value of the tray.

After adjusting many parameters, we found a linear fit with a correlation coefficient of 0.9999. We then found the upper and lower limit of the correlation as there was some uncertainty in conducting the experiment. (This will be further discussed in the "Limitations" portion of the blog.) The graphs of the upper and lower limits can be found under the "Data" section of the blog. (See Figure 1.2/1.3). 

Looking back at the equation: ln(T) = n*ln(m+M(tray)) + lnA, we are finally able to determine the value of the constants A and n. They can be found by looking at the equation of the graph in Figures 1.2/1.3. These equations utilize these values and if we remove the natural log, we can find the value. This gives us a final, working equation:

(T/A) ^(1/n) -M(tray) = m

With this equation, we have to keep in mind that we do not know the exact value of M(tray), and therefore the exact value of A and n. To find an accurate measurement, we take the upper and lower limit and plug them both into the equation to find the range of what the expected mass should be.

Now that we have a working equation, we can put it to the test by finding the mass of two random objects. The objects we picked consisted of a stapler and tape dispenser. First, we put each object on the inertial balance and recorded the period of oscillation. Next, we plugged these values into the equation, twice, bearing in mind the upper and lower limits. The worked out results for both the stapler and tape dispenser can be seen in the “Data” section of the blog. (See Figures 1.4/1.5).

5.     Data

Figure 1.1 (Data Table of Masses)

Figure 1.2 (Graph of Upper Limit)

Figure 1.3 (Graph of Lower Limit)

Figure 1.4 (Worked out Calculations of Stapler)

Figure 1.5 (Worked out Calculations of Tape Dispenser

Figure 1.6 (Period of Stapler)

Figure 1.7 (Period of Tape Dispenser)

• Table of Calculated Results

View figures 1.4/1.5 for calculated results. Figures contain calculations for unknown masses using the derived equation. Each mass required the equation to be used twice, once for the upper limit and once for the lower limit. This was done in order to calculate the range of the object.

• Analysis of Graphs

View figures 1.2/1.3 for graphs. These graphs are the power-law equations, after being manipulated to look similar to y=mx+b form and with a linear fit. As mentioned above, a linear fit means that the parameters are correct and the mass of the tray has been found. Since we cannot find the mass of the tray directly (due to reasons that will be explained in the conclusion), we get a range, hence the need for two different graphs. One graph is for the upper limit, which is the highest amount the tray can be before the line loses its linear fit, while the other is the lower limit, which is the lowest amount the tray can be before the line loses its linear fit.

• Conclusion

Using the equations to find the estimated mass of the stapler and tape dispenser answered some basic level questions and answered our objective in that we've found an equation to relate the two. However, much more work is needed to find a much more accurate answer. While I was able to find a suitable range for the tape dispenser, the stapler's actual mass was outside of my range. However, this could be for many reasons.

First, we do not know how the center of mass of both the object and the inertial balance affects the period. Perhaps if placed differently, the period could change drastically. Another potential foresight would be the propogated uncertainty of the mass of the object and the correlation of the line. We do not exactly know the limits, therefore finding a much closer limit would result in a closer answer. Another reason could be the damping constant of the inertial balance or the amplitude in which the balance is released at. Basically, there were a countless number of factors that could have swayed the results.