Lab 1 - February 23, 2015: Finding a relationship between mass and period for an inertial balance.

Finding a Relationship Between Mass and Period for an Inertial Balance

Physics 4A, Lab 1:

Author: Brandon Elder

Date of Lab: Feb 23 & 25, 2015

Purpose Statement:   To determine the inertial mass of an unknown object by using the relationship between mass and period on an inertial balance.

Procedure:

Picture 1: C-Clamp is holding the inertial
 balance in place on the table. The piece of tape 
attached to the end of the balance is shown. 
The photogate is set up accordingly as per 
Step 1 in above procedure.

Step 1: The first step in this lab involves the set up of all equipment that will be used for the purpose of eventually measuring an object with an unknown mass. First, assemble the inertial balance to the edge of the table with a C-Clamp, then place a piece of tape on the end of the balance. Additionally, a photogate will be assembled at the same height as the balance so that when the balance is oscillating the piece of tape placed on the balance will pass through the beam of the photogate (Picture 1).

Step 2: Next, connect the computer with LabPro installed. Open the Pendulum Timer.cmbl file and connect the photogate sensor. When all equipment is connected properly the lab experiment will be ready for collection. The green button in the LabPro software is the collect data button.

Picture 2: LoggerPro recording a series of periods from the oscillating balance. 

Step 3: Data collection starts next. Start with a base line test of the equipment and software. The first test will have no mass. Hit the collect button and pull the balance to set it oscillating. The software begins to record the time for each period and plots the data on the given graph (Picture 2).

Picture 3: A total of 800 g added to the
inertial balance.

Picture 4: Recorded results from the various
mass used with the inertial balance.

Next, we added mass in increments of 100g (Picture 3) and collected the results. All results were recorded in the handout provided (Picture 4). After all the required mass measurements were recorded, we added our unknown object which was a cell phone, with 300g on top of it. The measured period for our unknown object was: .531381 seconds.

 

Picture 5: Steps from the handout with information on
adding the three new columns.

Step 4: Once all data was collected with the various mass sizes we added three new columns into LoggerPro. One column was for the mass plus the mass of the tray. One column was for the ln of T. The final column was for the ln of the mass plus the mass of the tray. Picture 5 details out the instruction for adding these rows and the purpose.

Step 5: Analyze the graph of ln T vs. ln (m + Mtray). Adjust the value of Mtray, through the "User Parameters", until the "Correlation (below) is close to 1.000. This step makes it evident to us that a range of mass values inputted into the settings will result in a correlation close to 1.000. In fact, there will be a minimum value and a maximum value that will yield the correct results. Picture 6 shows the values of the max Mtray range.

Picture 6: The graph of  ln T vs. ln (m + Mtray). The value of Mtray was at it's maximum here. 

Picture 7: The graph of  ln T vs. ln (m + Mtray). 
The value of Mtray was at it's minimum here. 

 



In our experiment, we came up with the results in Picture 7 for the minimum Mtray values.

Picture 8: Min and Max values for Mtray. The power-law
equation used to determine the range of mass for the
unknown object that was measured.

Step 6: It's clear at this point that the mass is related to the period by some power-law type of equation. Picture 8 below has the base equation, under the min and max values for Mtray. A and n are constants in this equation. After taking the natural log of both sides and rearranging you have the equation that will allow us to solve for the mass. T represents the period of the unknown object. A represents the value of e to the y-intercept. n is the slope of the line. Both values, A and n, can be taken from the graphs in Pictures 6 and 7 for the min and max values, respectively.

Step 7: With this information, the minimum and maximum range values for mass can be calculated. The math results in a minimum value of .498kg and a maximum value of .498kg. The actual mass of the phone was 197g plus the 300g weight for a total of .497kg. Our estimated weight was within .001 of the actual mass.



Summary: In conclusion, we were able to relate the period of on an inertial balance to the mass that was set upon it. Before we even performed any calculations, we made an assumption that the phone was very close to 200g. We had measured the period of the phone plus the 300g and noticed that the period recorded was very close to the period observed for the 500g mass. Upon further investigation, by performing the math, we concluded that we were correct.