PHYS 2016 Lab 2 Testing Mathematical Models I

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University of Minnesota, Duluth *

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2016

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Apr 3, 2024

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Lab Notes PHYS 2016 Lab 2: Testing Mathematical Models I After one team member opens this google doc and before proceeding: File - Make a copy then SHARE your copy with team members with Edit privileges: Everyone needs to be logged in to their UMN Google accounts. Lab section: Tuesday at 10am Team Members (Full Name), roles : Notetaker (Opens, shares the Google Doc): Amanda Brown Critic/Skeptic (Leads comparison of team’s data sets): Anna Serstock , Skylar Schraut Spokesperson - (will speak for the group in class discussions): Mackenzie Morgan These roles are not intended to be restrictive - everyone needs to contribute to data collection in sims, model testing, questioning, and note-taking. Activity I: A. Analyzing, linearizing data: Initial exploration of mathematical models: [Summarize reasons for making different plots for the two mathematical models, expected graph shapes, relation of model parameters to plot characteristics. Sample graphs of test test data for all three graphs (linear; semi-log, and log-log). Resulting values for A and B for the model that best describes the data. 1. POWER LAW: ln(f(x)) = ln(A) + Bln(x) EXPONENTIAL: ln(f(x)) = ln(A) + Bx 2. Power law and exponential graphs are inverses of each other so they are very easily differentiated. Therefore, we would expect the graph to curve upward and follow an exponential trend or follow the natural log trend which curves concave down while increasing upward. 3. In the power law, A is a scalar that will increase the size of the graph and B would increase the slope of the graph. This is a similar case for the exponential law but A moves the graph over to the left. Our Graphs:
Amanda: (x,y) (x,lny) (lnx,lny)
Anna Serstock: (x,y): (x, ln(y)):
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(ln(x),ln(y)): Mackenzie Morgan:
Skylar Schraut: (x,y)
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(x, ln(y)) (ln(x), ln(y))
★ The line of best fit for the graphs is the power law, y = A x B . B. Test the Coulomb’s law model: (Coulomb’s Law; Data, plots, linear fits and interpretation for each team member’s parameters in the Coulomb’s law sim; confirm/reconcile fits’ slopes and intercepts with each member’s choice of simulation parameters.) Coulomb's Law: Like charges repel, opposite charges attract. Coulomb's constant= 8.99x10 9 kgm/s 2 C 2 Experiment Design:[Between force and distance] What variables will you measure? We will measure the force and distance Which variables would you consider to be the independent or dependent variables? Independent variable: Distance Dependent variable: Force What parameters in your experiment will you need to control? Charge on the objects and the size of the objects How many different values of the independent variable will you use? why? 10 different distances because it's enough to make a plot but isn’t unrealistic. -8q 1 to 8q 2 MACRO -8q 1 to 8q 2 ATOMIC Force (N) Distance Apart (cm) Force (N) Distance Apart (m) 57.520 10 2.7e-6 1e-10 71.013 9 3.72e-6 9e-11 89.876 8 5.26e-6 8e-11 117.388 7 7.99e-6 7e-11 159.779 6 1.36e-5 6e-11 230.081 5 2.79e-5 5e-11 359.502 4 8.74e-5 4e-11 639.115 3 1.64e-4 3e-11
1438.008 2 3.69e-3 2e-11 -4q 1 to +4q 2 MACRO -4q 1 to +4q 2 ATOMIC Force (N) Distance (cm) Force (N) Distance (m) 14.380 10 3.69e-7 1e-10 17.753 9 4.56e-7 9e-11 22.469 8 5.77e-7 8e-11 29.347 7 7.53e-7 7e-11 39.945 6 1.02e-6 6e-11 57.520 5 1.48e-6 5e-11 89.876 4 2.31e-6 4e-11 159.779 3 4.10e-6 3e-11 359.502 2 9.23e-6 2e-11 -4q 1 to +4q 2 ATOMIC
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-4q 1 to +4q 2 MACRO
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-8q 1 to 8q 2 MACRO:
-8q 1 to 8q 2 ATOMIC:
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I. Best A and B value: A. Overall, force is dependent on distance because all graphs support that as the distance apart decreases the force increases. B. We chose this graph because it accurately represents the relationship between force and distance. As the distance on the y axis decreases, the force on the x axis increases. C. Because this graph best supports this relationship, our A value is -2 and our B value is 1.319. D. We can approximate the A and B values by using the ln(A)+Bln(x)=ln(y) equation. E. STOP FOR ACTIVITY 1
Activity II: Amanda : a. Made measurements along the axis(to the right of the stack of charges.) q=+1nC and -1nC Electric Field(V/m) Distance to sensor(m) degree(deg) 13.7 .485 81 1.71 1.00 86.7 0.21 2.015 86.4 0.06 3.022 88.9 0.03 3.814 88.4 b.
My data fit power law in relation to distance better than exponential. We can approximate the A and B values by using the ln(A)+Bln(x)=ln(y) equation. By using this equation and using the power-law plot, the A value is approximately -67.74, and the B value is approximately 68.27. c. Measured from equidistance between charges(q=+1nC and -1nC). Made measurements along the axis(to the right of the stack of charges.) Prediction: I think that separating the charges will overall increase the electric field at each distance measured. Electric Field(V/m) Distance to sensor(m) 5.92 1.192 1.03 2.092 0.33 3.053 .14 4.022 .07 4.975
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My data fits power law in relation to distance better than exponential. We can approximate the A and B values by using the ln(A)+Bln(x)=ln(y) equation. By using this equation and using the power-law plot, A value is approximately 44.03, and B value is approximately -12.40. Skylar: a. q=+2nC and -1nC Electric Field (V/m) Distance to Sensor (m) Degree (deg) 4.39 1.725 0.4 1.67 2.619 3.1
0.90 3.481 6.3 0.55 4.372 17.9 0.37 5.252 26.7 b.
My data fits the power law because the correlation is at its highest with the data compared to all the other graphs. Using the equation: ln(y)=A*exp(-C*x)+B, we can calculate the A, B and C values which will approximately be 25.02, -22.23, and 0.09871 respectively. c. d. Changed the charges to be: q=+1nC and -1nC Prediction: I think that lowering the charge will overall increase the electric field at each distance. Electric Field (V/m) Distance to Sensor (m) Degree (deg) 10.3 0.724 23.5 4.0 1.021 18.1 0.51 2.018 8.9 0.12 3.246 5.3 0.05 4.257 4.5
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Looking at my data and my graphs, the power law fits the linear trend best compared to the other two graphs since the correlation was 1.000. Using the sine function equation: ln(y)=A*sin(Bx+C)+D, A, B, and C can be calculated to be approximately 9.768, 0.3214, and 2.717 respectively. Anna Serstock: q= -2nC and +2nc Electric Field (V/m) Distance to Sensor (m) Degree 26.9 .594 -100.7 3.54 1.089 -100.4 1.05 1.585 -101.2 .44 2.088 -100.9 .23 2.591 -99.6
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My data best fits the Base 10 exponent curve fit. We know this by it having a correlation of 1. We can approximate the values of A and B by using the equation, ln(A)+Bln(x)=ln(y). The best fit line has values of approximately 38.91 for A and -0.03678 for B. The units for A and B will correlate to the distance (m) and the Electric Field (V/m) however because we are taking the natural log of these values we don’t need units for A or B.
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Kept the +2nC and -2nC charges in the same spot but moved the sensor and the distance. Prediction : That with the charges being further apart the electric field will be stronger and have a larger magnitude than when they were close together. Electric Field (V/m) Distance (m) Degree 34.6 1.014 -89.8 25.3 1.501 -90.5 12.7 1.996 -90.1 6.17 2.499 -90.2 3.24 3.003 -91.5
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My data fits the power law better than exponential because of the correlation being close to 1. Activity II: D. [As distance increases, the electric field is getting smaller. Follow Coulomb's law. Should see 1/R^2 relationship via coulomb's law. q1Q2/R^2(r=distance) As distance increases, voltage should be 1/R^2
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Some constant/r^3. If distance vs electric field(voltage). Without taking ln of anything it goes down sharper. Follows r^3 vs r^2] Dipole Charge (q) Dipole Charge Separation (d) Dipole Moment p=qd Better Model? Power Law or Exponential Parameter A Parameter B Original +1nC -1nC 0.58 0 Power 43.67 -13.70 +2nC -2nC 0.594 0 Exponential 38.91 -0.03678 +1nC -1nC 0.724 0 Exponential 9.768 0.3214 +1nC -1nC 0.485 0 Exponential -67.74 68.27 Change +1nC -1nC .194 0 Exponential 0.205 4.47 +2nC -2nC 1.014 0 Exponential .5293 3.605 +2nC -1nC 1.725 1.725 Exponential 25.02 -22.23 +1nC -1nC 1.192 0 Exponential 44.03 -12.40 ***This data table shows one point from each dipole experiment from each person.***
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