Physics lab worksheet

Lab03-MGO-03.pdf

BCIT Department of Physics

Version: Jan-23 - 1 - © Dr. M. Harder

MGO-03: Graphical Analysis

Learning Objectives

In this lab you will improve your ability to:

Follow a definite set of instructions for the analysis of data.

Plot data using graphical analysis software (GraphIt).

Choose a model equation when analyzing experimental data graphically.

Determine a fit equation using graphical analysis software (GraphIt).

Extract information from fit coefficients and perform uncertainty calculations.

Apparatus

• Computer with Excel

Theory

The ability to plot data and analyze/interpret graphs is important for any technological field. By plotting measured data we can determine relationships between variables and also identify data points that look out of the ordinary.

The standard way to refer to a plot is to say “the 𝑦 data as a function of the 𝑥 data” or “the 𝑦 data versus the 𝑥 data”. For example, if you are asked to plot the mass of a cylinder as a function of its diameter, then you would plot mass on the 𝑦-axis and diameter on the 𝑥-axis. By contrast, a plot of the diameter of a cylinder versus its mass would have diameter on the 𝑦-axis and mass on the 𝑥-axis. By convention, we plot the independent variable (the one we control in an experiment) on the 𝑥-axis and the dependent variable (the one we measure) on the 𝑦-axis.

It is important that:

1. All graphs are numbered and include a descriptive title, e.g., Graph 1: Voltage Across a Resistor as a Function of Current.

2. All graphs have axes labels that include the name and unit of the quantity, e.g., Voltage (V).

3. All graphs follow the formatting requirements discussed in Sec. 2 of the Introduction.

MGO-03 BCIT Department of Physics

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To practice graphical analysis, two data sets are provided below and in the MGO-03.xlsx spreadsheet. For each data set a hypothesis, i.e., the function expected to describe the data, and models are given.

When selecting a model to analyze your data:

1. The model should have the same functional form as the hypothesis, e.g., if our hypothesis is 𝐹 = 𝑚𝑎, we should choose a model that is linear, like 𝑦 = 𝐵𝑥, 𝑛𝑜𝑡 𝑎 quadratic function, like 𝑦 = 𝐵𝑥!.

2. The model should have the least number of parameters possible, while maximizing the CD value. e.g., if our hypothesis is 𝐹 = 𝑚𝑎, we should choose a model like 𝑦 = 𝐵𝑥, not like 𝑦 = 𝐴 + 𝐵𝑥, unless the additional parameters are physically meaningful. This also means that if one parameter is consistent with zero, we should refit the data and exclude that parameter.

3. Choose a model that allows for the easiest interpretation and analysis of your data.

Analyzing Data Set 1

Experimental Description: Different currents are applied to a resistor and the voltage drop across the resistor is measured. The data is given in Table 1.

Table 1: Data Set 1

Hypothesis: V = V" + RI Model 1.1: 𝑦 = 𝐴 + 𝐵𝑥 Model 1.2: 𝑦 = 𝐵𝑥

1. By comparing the hypothesis to Model 1.1, identify the independent and dependent variables as well as the relationship between the model and hypothesis fit coefficients.

2. Plot the data in Table 1, and using Model 1.1 determine the fit equation in GraphIt.

3. Plot the data in Table 1, and using Model 2.1 determine the fit equation in GraphIt.

Voltage, V (V) 1.00 2.00 3.00 4.00 5.00 6.00

Current, I (mA) 0.581 1.163 1.745 2.326 2.908 3.489

BCIT Department of Physics MGO-03

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Analyzing Data Set 2

Experimental Description: The mass of different-sized ball bearings is measured. The mass 𝑚 is related to the density ρ and the volume 𝑉 according to 𝑚 = ρ𝑉 and the volume of a sphere is related to the diameter 𝑑 by 𝑉 = #

$ 𝑑%.

Table 2: Data Set 2

Hypothesis: 𝑚 = 5 π 6 ρ8 𝑑

% Model 2.1: 𝑦 = 𝐷𝑥% Model 2.2: 𝑦 = 𝐵𝑥

1. By comparing the hypothesis to Model 2.1, identify the independent and dependent variables, as well as the model and hypothesis fit coefficients.

2. Plot the data in Table 2, and using Model 2.1 determine the fit equation in GraphIt.

3. Now linearize the data and enter your values into Table 3.

4. By comparing the hypothesis to Model 2.2, identify the independent and dependent variables in your linearized data, as well as the model and hypothesis fit coefficients.

5. Plot the linearized data in Table 3, and using Model 2.2 determine the fit equation in GraphIt.

In our comparison in Step 4, we find that

B = π 6 ρ (1)

and therefore

ρ = 6 πB (2)

where 𝐵 is the slope of the fitted line and 𝜌 is the density. Following the rules given in Sec. 4 of the introduction, the uncertainty ∆𝜌 is:

Δ𝜌 = 6 𝜋 Δ𝐵.

(3)

6. Use your fit results from Step 5 to calculate the density, 𝜌, and the uncertainty, ∆𝜌, for the ball bearing.

Mass, m (g) 0.034 0.259 0.882 2.11 4.09 7.07

Diameter, d (cm) 0.200 0.400 0.600 0.800 1.00 1.20

MGO-03 BCIT Department of Physics

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Worksheet

Complete the MGO-03 worksheet on separate sheets of paper and attach a PDF of your data and analysis tables and all plots. See Sec. 9 of the lab manual introduction. Submit your PDF to the appropriate Learning Hub Assignment folder.

Lab03-MGO-03-Worksheet.pdf

BCIT Department of Physics MGO-03 - Worksheet

Version: Jan-23 - 1 - © Dr. M. Harder

MGO-03: Graphical Analysis – Worksheet

Maximum Score: 18 1. [4] Include the following four GraphIt outputs (plot and fit results) with your submission:

(1) Data Set 1 fit with Model 1.1 (2) Data Set 1 fit with Model 1.2 (3) Data Set 2 fit with Model 2.1 (4) Data Set 2 fit with Model 2.2

Ensure that: (i) Your axes are properly labelled, including units; (ii) Your graph has an appropriate title; (iii) You follow the formatting requirements of Sec. 2 in the introduction. From now on all of these conditions must be met as a default when submitting any plot.

2. [1] For Data Set 1, write out the relationship between Model 1.1 and the hypothesis. The answer for this question is:

𝒚 = 𝑽; 𝒙 = 𝑰; 𝑨 = 𝑽𝟎; 𝑩 = 𝑹 You can simply write out this answer on your worksheet. The reason I am writing it here is so you know what I mean when I ask for “the relationship between the model and hypothesis equations, and you can now perform this same analysis for the other hypotheses and models in this experiment.

3. [1] Write out your fit equation for Data Set 1, using Model 1.1. Ensure: (i) That you use the appropriate variables and that all fit coefficients: (ii) Include uncertainties; (iii) Are properly rounded; and (iv) Include appropriate units.

From now on all of these conditions must be met as a default when writing out any fit equation, in any experiment we do.

4. [1] Based on your fit, is 𝐴 consistent with zero? In other words, does 𝐴 overlap with zero

within experimental uncertainty?

5. [1] For Data Set 1, write out the relationship between Model 1.2 and the hypothesis. 6. [1] Write out your fit equation for Data Set 1, using Model 1.2.

MGO-03 - Worksheet BCIT Physics Department

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7. [1] Based on your answer to question 3 and the conditions for selecting an appropriate model equation (see p. 2 of the MGO-03 lab description), should you use Model 1.1 or Model 1.2 to fit Data Set 1?

8. [1] For Data Set 2, write out the relationship between Model 2.1 and the hypothesis. 9. [1] Write out your fit equation for Data Set 2, using Model 2.1.

10. [1] For Data Set 2, write out the relationship between Model 2.2 and the hypothesis.

11. [1] Write out your fit equation for Data Set 2, using Model 2.2.

12. [2] Show your calculation of ρ and Δρ using Data Set 2 and your Model 2.2 fit. 13. [1] State your final result for ρ ± Δρ. Make sure to properly round your result, and

include your uncertainty and units. From now on, all of these conditions should be met by default when stating an experimental result.

14. [1] Would you choose Model 2.1 or 2.2 to fit Data Set 2? Explain your reasoning.