Lab #1: Finding a Relationship between Mass and Period for an Inertial Balance
May Soe Moe
Lab Partners: Ben Chen, Steven Castro
Date: 27 February 2017
Introduction: When trying to find a relationship between mass and period, we want to know the inertial mass--independent of gravity--by using an inertial balance, a device that can be used to measure the inertial mass without depending on gravity. By using an inertial balance, we will be able to figure out the oscillation period of the inertial pendulum for various masses that we will use throughout the experiment. Once we get the measurements of the oscillation periods for different masses, we can develop a mathematical model that represents the relationship between period and added mass (mass of different objects used during the experiment+mass of the tray). After we produce a mathematical model, which is power law in this case, we can use that model to determine the unknown masses of the various objects that are used in this experiment.
Experimental Procedure:
(4) Create a data table as above in a new Logger Pro document. Add three more columns in the data table-- m+Mtray, ln T, and ln (m+Mtray).
May Soe Moe
Lab Partners: Ben Chen, Steven Castro
Date: 27 February 2017
Objective: We will try to find a relationship between mass and period for an inertial balance through this experiment.
Introduction: When trying to find a relationship between mass and period, we want to know the inertial mass--independent of gravity--by using an inertial balance, a device that can be used to measure the inertial mass without depending on gravity. By using an inertial balance, we will be able to figure out the oscillation period of the inertial pendulum for various masses that we will use throughout the experiment. Once we get the measurements of the oscillation periods for different masses, we can develop a mathematical model that represents the relationship between period and added mass (mass of different objects used during the experiment+mass of the tray). After we produce a mathematical model, which is power law in this case, we can use that model to determine the unknown masses of the various objects that are used in this experiment.
Experimental Procedure:
(1) First, we set up the inertial balance by using a C-clamp to secure to the table so that the balance can oscillate horizontally. After that, we put a thin piece of tape on the end of the inertial balance in order to figure out the period of oscillation.
(2)We set up the photogate by giving it enough space in between the tape and the photogate so that the beam of the photogate can capture the oscillation period.
We set up the LabPro by plugging into a power source and to computer, which will allow to record the measurements. We used the Logger Pro application to help with the graphs and getting measurements.
(3) Record oscillation periods by adding 100g mass each time on the balance and pulling back the balance until 800g of mass.
(4) Create a data table as above in a new Logger Pro document. Add three more columns in the data table-- m+Mtray, ln T, and ln (m+Mtray).
(5) In order to develop a mathematical model for mass vs. period, we will use the power law:
T=A [m(added)+M(tray)]^n, and derive it to
ln T=n*ln[m(added)+M(tray)]+ln A, so that the model will be in a linear equation form--y=mx+b.
By deriving it, we can find values of A, Mass of the tray and n.
(6) We can graph a ln T vs. ln[m(added)+M(tray)] and find out the slope( the value of n) and y-intercept (the value of ln A), which will help with finding out the mass of various objects.
(7) Since we want the graph to be in a linear or straight line, we want the correlation coefficient( which you can check from the graph using linear fit) to be as close as 0.9999. In order to do that we will guess the mass of the tray, which will give us the correlation 0.9999.
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| ln T vs. ln(m+Mtray) graphs looks as in above picture by adjusting the Mtray to be 253g, 269g, 280g. |
(8) After trials of adjusting the mass of the tray, we get the minimum, intermediate, and maximum values of the mass of the tray that gives us the correlation closest to 0.999.
M(tray) minimum value=253 g
M(tray) intermediate value=260 g
M(tray) maximum value=280 g
(9) Record the values of the slope and y-intercept you get after plotting the graph and adjusting the mass of the tray to be 253g, 260g, and 280g respectively. We use linear fit to give us those values.
(10) We find oscillation periods of two other different objects-- a cellphone, and tape dispenser by putting it on the inertial balance.
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| Photogate collects the period of oscillation for different objects as in above picture. |
(11) After recording the two new oscillation periods for the cellphone and tape dispenser, we will use the derived power law formula to find the masses of those two objects by plugging in the slope as n, y-intercept as ln A, the two new periods recorded as T, and also the adjusted masses of the mass of the tray. We will solve three different times for the different masses using the minimum, intermediate, and maximum values of the mass of the tray.
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| Calculation of the two masses by using the minimum value of the mass of the tray, slope and y-intercept from the ln T vs. ln (m+Mtray) graph |
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| Calculation of the two masses by using the intermediate value of the mass of the tray, slope and y-intercept from the ln T vs. ln (m+Mtray) graph |
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| Calculation the two masses by using the maximum value of the mass of the tray, slope and y-intercept from the ln T vs. ln (m+Mtray) graph |
(12) We also measure the mass of the cellphone (158g) and tape dispenser (688g) by using electronic balance.
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| The calculated masses of the cellphone and the tape dispenser |
Conclusion: We tried to find the relationship between mass and period by using inertial balance. We recorded the oscillation periods for 100g to 800g. We created a table and plotted it in order to get the values for A and n, to theoretically find out the unknown masses of different objects. The formula used here was T=A*[m(added)+M(tray)]^n, and mass and period are directly proportional to each other by the formula. But we used the derived formula ln T= n*ln[m(added)+M(tray)]+ln A to make it in the straight line formula y=mx+b. When graphing, ln T vs. ln (m+Mtray) came out to be a straight line, which let us know that n was the slope, and y-intercept was the ln A, according to the formula. We collected different periods for two other objects--cellphone and tape dispenser-- and calculate the theoretical masses of the objects. The calculated masses came out to have different values, which are not close to the actual masses of the objects, when compared to the actual masses from the electronic balance.
Errors: There may be errors that led us to get different values for theoretical masses and actual masses. We could have used different forces for each time to pull back the balance when recording oscillation periods for different masses. We assumed that forces to pull back the balance are equal, which could make a difference in comparing masses. The actual measured mass from electronic balance could actually be weight, including the gravitational mass.
In your introductory paragraph, you repeat the end of one sentence to begin the next sentence. It is okay to just use a lot fewer words: "We developed a mathematical model that represents the relationship between period and added mass (mass of different objects used during the experiment+mass of the tray)."
ReplyDeleteDo this all in the past tense. The blog is about what you did already rather than about what you will do.
In your first curve fit graph you didn't include the last point in coming up with your slope.
Here are the steps you did in finding the best set of curve fit equations:
--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
You essentially say all of this. The last part, about guessing the mass of Mtray until you get a range of Mtray values that give you the best correlation, is a little unclear.
Your calculations are very clear. Thank you. The middle value for the unknown mass 2 is very different from the other values. I suspect a calculation error somewhere.
Unlike an essay for an English class, the conclusion here doesn't restate the entire lab procedure. Look at the syllabus for details of what goes into it.
As far as errors go, the actual measured mass from the electronic balance IS the gravitational mass. This isn't an error. They should be the same measurement either way.
When we set up our original equation all of the masses were cylinders centered in the tray.
Our unknown objects had different shapes and perhaps different placement in the tray. We didn't test separately to see if placement or shape made a difference. This is an assumption (that mass is the only variable) that maybe turns out not to be true.