PHYS 4A/ Fall 2014: Finding a Relationship Between Mass and Period of an Inertial Balance

Purpose: By determining the mass of the balance, we will be able to find the relationship between the total mass of the balance and its period using the equation T=A(m+Mtray)^n.

An inertial balance was mounted to the table with a photo gate mounted upon a metal rod. With a piece of masking tape stuck to the front of the balance, the photo gate was able to record the oscillations of the balance.

In order to determine the mass of the balance, we first had to set up a way to obtain useful data. In order to do this, we mounted the balance as seen in the picture above and aligned the photo gate in order to record the period of oscillations. Using LoggerPro, we were able to record and analyze the oscillations.

 In order to analyze the obtained data, we needed an equation that related mass to period. Therefore, T=A(m+Mtray)^n was used. However, this equation contains three unknowns which had to be found.

This was done by taking the natural log of the entire equation and comparing it to the equation of a line, y=mx+b. By making this comparison, we were able to translate the data into values that fit into the equation. We created columns that made calculations based on the data in order to get our  lnA and nln(m+Mtray) values that are shown in the table above. 

Taking the slope and y-intercept of the graph gave us our average lnA and nln(m+Mtray) values. By going into user parameters and setting a value for Mtray, we were able to adjust the correlation value of the graph to 0.9999. However, this value was achieved from a range of Mtray values, thus we were able to use that to determine a range of error in our results. 

Combining this information with a reorganized equation, we were able to determine the accuracy of our results.

The new equation solves for a mass value (m). This is used by plugging in the range of values we had found and the recorded period for a specific mass. We took two more mass measurements of random objects in order to see how accurate we were. These objects' masses and periods were recorded as follows.

Using all of this data, we were able to calculate the range of masses and see if the equation T=A(m+Mtray)^n is a valid relationship.

After calculating the range, we saw that the range was about 0.004kg for both masses. The first mass had a range of 0.141kg-0.144kg with an actual mass of 0.142kg. The second mass had a range of 0.306kg-0.310kg with an actual mass of 0.280kg. The first mass fell within the range thus proving the equation valid. However, the second mass was well below the range. This may have been due to the object being a bottle of water. As the bottle was half full, the water sloshed around as the balance oscillated which may have altered the period of oscillations. There would have been an extra force as the water collided with the walls of the bottle thus extending the period. Therefore, the second mass logically makes sense even if the data shows otherwise due to error.

Summary: 

Overall the experiment was a success as the relationship was supported. The mass of the balance was determined using the recorded period, the theoretical equation, and LoggerPro. The analysis of the graph gave a range of mass values which was used to determine the mass of the object within a range. The actual mass for the first object fell within the calculated mass range therefore supporting the theoretical formula of T=A(m+Mtray)^n. The period and mass had a direct relationship, as the mass increased, the period also increased. The experiment method also proved to be accurate as there was an error of only 4 grams.