27-Feb-2017: Lab 1: Finding a Relationship Between Mass and Period for an Inertial Balance

Lab 1:

Finding a Relationship Between Mass and Period

for an Intertial Balance

Joel Cook

Nina Song

Cristian Garcia

Lab performed February 22, 2017

In this lab, we attempted to approximate the masses of two unknown objects by using an inertial balance and the relationship between mass and period with an inertial balance. By observing the period of oscillation of objects of known mass, we are able to create a model used to predict the mass of an unknown object.

Introduction: When the tray of the inertial balance is displaced, it oscillates back and forth. The more mass that is on the tray, the slower it oscillates back and forth. Period and mass are related through this equation, T=A(m+mtray)^n, where T is period, A is a constant, m is mass of the object, and mtray is the mass of the tray of the inertial balance. By taking the logarithm of each side and rearranging, this equation becomes a linear equation (y=mx+b). 

Procedure: An inertial balance (shown below) is a metal tray riveted to two flat, springy pieces of metal and clamped to a table at the other end.

A piece of tape is attached to the end of the tray and positioned in such a way as to pass through the beam of a photogate attached to a ring stand. The photogate is connected to a computer in order to collect data for period of oscillation, in seconds. A mass is placed on the tray, the tray is displaced, and the computer records information about the length of the period of oscillation. The period was recorded for masses from 100 grams to 800 grams, in increments of 100 grams.

Data: The time for ten oscillations of the inertial balance, with an empty tray, was recorded with a stopwatch and compared to the time the computer recorded for the same set of oscillations. The period calculated from the stopwatch was 0.574 seconds and the period recorded by the computer was 0.545 seconds. This data shows that the period recorded by the computer for the inertial balance appears to be accurate. The following table shows the data recorded for the period of various masses:

Mass in Balance (grams)	Period (seconds)
0	0.282
100	0.346
200	0.401
300	0.452
400	0.499
500	0.547
600	0.597

700	0.637
800	0.684

The graphs, and associated slopes, for three different lines are shown below:

The first graph is for a mtray value of 325 grams:

The second graph is for a mtray value of 280 grams:

The third graph is for a mtray value of 300 grams:

Analysis: After the data has been gathered, the data is plotted and the slope of the line through those points is calculated. In order to graph the data, a range of values for the mass of the tray of the inertial balance, mtray must be determined. By finding the best range of values for mass of the tray and graphing ln T vs. ln (m+mtray), we are able to determine the slope. The slope of this line is used in the mathematical model used to predict the unknown mass of objects by measuring the period. The linear equation used is: lnT=n*ln(m+mtray) + lnA. 

For each graph, we were able to find a value for mtray that yielded a correlation of 0.9999. The slope shown is the value for n and the y-intercept is ln A. The equation, lnT=n*ln(m+mtray) + lnA, is then rearranged to be able to calculate mass.

This yielded three equations for determining the mass of an object based on the period observed:

m =  (      T      )   ^ (1/0.6989)   - 325

        (e^-5.294)

m =  (      T      )   ^ (1/0.6421)   - 280

        (e^-4.876)

m =  (      T      )   ^ (1/0.6675)   - 300

        (e^-5.062)

Example of a calculation to determine mass based on measured period:

m =  (      0.379    )   ^ (1/0.6989)   - 325 = 161 grams

           (e^-5.294)

	Calculated mass of unknown 1	Calculated mass of unknown 2
Using mtray value of 280	158 grams	685 grams
Using mtray value of 300	159 grams	681 grams
Using mtray value of 325	161 grams	679 grams
	Mass from electronic balance: 163 grams	Mass from electronic balance: 619 grams

Conclusion: The model we created for measuring the mass of an object appeared to be accurate. For the first unknown mass (cell phone), the mass was estimated between 158-161 grams and end up being 163 grams, as measured by a scale. This estimate was off by approximately 1%. The second unknown object (tape dispenser) was estimated to be between 679-685 grams and end up being 619 grams, as measured by a scale. The second estimate is off by approximately 9%. 

There are sources of uncertainty in this lab that may have affected the outcome. First, it is not possible to displace the inertial balance in the same manner each time and record the period at exactly the same moment each time. This may have affected the accuracy of the periods recorded. Performing more samples, with a greater range of known masses, may have helped to improve the accuracy. Finally, the known masses may not have been placed in the exact same location in the tray each time and, in addition, the unknown masses did not fit in the tray exactly the same manner as the known masses. This may have affected the measured period of each object and impacted the model created to estimate unknown masses.