Finding a relationship between mass and period for an inertial balance.
Daniel Guzman
James Okamura
Rodrigo
February 27, 2017
What this experiment is trying to accomplish is to understand the mathematical relationship between mass and the oscillation period of an inertial balance, to then test, the relationship that exist by measuring the period of unknown masses, so the mass could be calculated for the objects.
In this lab we were trying to measure the different unknown masses of three different objects: an inertial balance, a calculator, and a tape dispenser. In order to calculate the masses for these objects, we measured the period of oscillations of the analytical balance using different cylindrical wights, which ranged from 0 to 800 grams. The period of oscillations was measured for each weight individually.
The apparatus consisted of a photo gate a clamp and an inertial balance. The inertial balance was clamped into the table and a piece of tape was put on the inertial balance, so the tape will break the light and the photo gate could record the period of oscillations.
The first part for this experiment was to set up the apparatus correctly and measured the period of oscillations for the cylindrical weights. Once the period of oscillations was measured we had to use the power law equation T=A(m+mtray)^n, which is going to model the data; nevertheless, before modeling the data the ln of the equation has to be taken, so the data can be modeled as a straight, with an equation in the form y=mx+b, which is a straight line. When the ln of the equation is taken the equation would be lnT=nln(m+mtray)+lnA. After obtaining the equation in this form one can proceed and modeled data as a straight line, on the y axis we used lnT and on the x axis we put (m+Mtray), which gave us a straight line from where we extracted important information, such as the y intercept of the graph and the slope of it. The y intercept of the graph would give us the amplitude of the oscillation and the slope of the graph would give us the exponent of the equation, which are crucial in order to find the unknown masses. However, to find the unknown masses we had first to determine or find the value for the tray. This particular values was found using different numbers when grpahing, until we found the number that will yield the better correlation of the line, which gives us a hint of how the values plotted tend to vary together. Having this Valuable infromation from the graphs we proceed and found the unknown masses of the objects listed previously.
The first part for this experiment was to set up the apparatus correctly and measured the period of oscillations for the cylindrical weights. Once the period of oscillations was measured we had to use the power law equation T=A(m+mtray)^n, which is going to model the data; nevertheless, before modeling the data the ln of the equation has to be taken, so the data can be modeled as a straight, with an equation in the form y=mx+b, which is a straight line. When the ln of the equation is taken the equation would be lnT=nln(m+mtray)+lnA. After obtaining the equation in this form one can proceed and modeled data as a straight line, on the y axis we used lnT and on the x axis we put (m+Mtray), which gave us a straight line from where we extracted important information, such as the y intercept of the graph and the slope of it. The y intercept of the graph would give us the amplitude of the oscillation and the slope of the graph would give us the exponent of the equation, which are crucial in order to find the unknown masses. However, to find the unknown masses we had first to determine or find the value for the tray. This particular values was found using different numbers when grpahing, until we found the number that will yield the better correlation of the line, which gives us a hint of how the values plotted tend to vary together. Having this Valuable infromation from the graphs we proceed and found the unknown masses of the objects listed previously.
Calculation to see if the period of oscillations is relatively close to the period that one measured using a stopwatch and data table
Graphs of the data
Graphs Analysis (Graphing process and interpretation of the graphs)
How it was mentioned before the data was modeled using the equation T=A(m+Mtray)^n. However in order to obtain a straight line the log of this graph had to be taken, which will yield the equation of a straight line that looks like this : lnT=n ln(m+Mtray)+lnA, where n represents the slope of the line and lnA represent the y intercept of it (the slope of the line is the exponent for the power law equation and the y intercept represents the amplitude of the oscillation). The key point when analyzing the graph is to obtain almost a perfect correlation, to obtain a good or almost perfect correlation we had to modify the mass of the tray until we found a good correlation, and once this is done we had to come up with a range of values that yielded a very good correlation; these sets of values are going to have an upper bound and a lower bound which will have specific values for the y-intercept and the slope of the line. These specific values found in the upper and lower bounds were used to find the unknown masses of : the calculator, the tape dispenser and the tray.
How it was mentioned before the data was modeled using the equation T=A(m+Mtray)^n. However in order to obtain a straight line the log of this graph had to be taken, which will yield the equation of a straight line that looks like this : lnT=n ln(m+Mtray)+lnA, where n represents the slope of the line and lnA represent the y intercept of it (the slope of the line is the exponent for the power law equation and the y intercept represents the amplitude of the oscillation). The key point when analyzing the graph is to obtain almost a perfect correlation, to obtain a good or almost perfect correlation we had to modify the mass of the tray until we found a good correlation, and once this is done we had to come up with a range of values that yielded a very good correlation; these sets of values are going to have an upper bound and a lower bound which will have specific values for the y-intercept and the slope of the line. These specific values found in the upper and lower bounds were used to find the unknown masses of : the calculator, the tape dispenser and the tray.
Calculations using lower and upper bounds to find the unknown masses : The values obtained in the graphs are used here to calculate the unknown values for the objects. A set of different values is used for the upper and lower bound.
Table of results for the unknowns
The sources of error in this experiment could possible come from the distribution of the mass that was put on the tray;for instance, putting the calculator vertically might yield a different period of oscillation than putting it horizontally. Another source of error is the amplitude that the inertial balance had when recording the period of oscillations for the unknown masses, due that it might record a slightly longer or shorter period of oscillations.
Conclusion
After having completed the lab one was able to calculate the unknown masses of three different objects using the relationship that exists between the period of oscillations and the mass of any body or object. The calculated unknown masses were relatively close to the actual masses of the different bodies, and the percent errors calculated when using the upper and lower bounds were lower than 10%, which let one infer that the results were relatively accurate.
Most of what is needed is here. Some could be made clearer.
ReplyDeleteThe purpose was to find the mathematical relationship between mass and oscillation period for an inertial pendulum, and to test that relationship by measuring the period for some "unknown" masses. I would put the picture of the apparatus up near the top so that the "4A student" reader knows what the experiment is about. Otherwise you are talking about how you are going to model its behavior but the reader still doesn't know what the apparatus is that you are modeling.
If we take the natural log to model the equation as a straight line, it would be best to show what the straight line equation is. Again, you have most of these things in your description, but not quite all:
-- Power law equation
-- ln form
-- what will be plotted on the y axis and on the x-axis
-- what the slope and y-intercept of that graph will tell you
--how you are going to find the mass of the tray
Then a check that the photogate measures what it should.
Then data.
Then your fit graphs.
Then your derived upper and lower bounds for you fit equation.
Then your measured periods of the unknowns.
Then your calculation of their masses (upper and lower bounds)
Then a table with your results for the unknowns, comparing them with the electronic balance results.
Then your sources of uncertainty.
Again, you have most of this stuff here.
The lab handout suggests a format for reporting your results.
It is fine to say "we did" or I did" in your lab report, rather than "one did".
Your conclusions are fine.