# How to analyse this data obtained from a simple physics experiment on attractive forces?

I did a simple physics experiment that measures the attractive force a plate experiences towards the other plate as a function of the applied voltage and distance between the plates. Now I have to know whether the gathered data confirms the relation $F \propto V^2/d^2$. Suppose the following is what I gathered.

```voltage\distance 8.000 10.000 12.000 14.000 16.000
             4.0   3.3    2.0    1.7    1.2    0.8
             6.0   8.1    4.8    3.8    2.5    1.9
             8.0  13.4    9.2    6.0    4.4    3.5
            10.0  22.9   14.2    9.4    7.0    5.6
            12.0  32.7   20.1   13.5   10.5    7.9
```

The uncertainties are $\pm 0.05$ for the voltage, $\pm 0.005$ for the distance, and $\pm 0.1$ for the force.

As far as I know, there are a few ways to analyze the data with R.

1. `lm(log(force) ~ log(voltage) + log(distance))`
2. `lm(I(force * distance^2) ~ -1 + I(voltage^2))`
3. `aov(I(force * voltage^-2 * distance^2) ~ voltage * distance)`

Which one should I use?

-

## 1 Answer

If you think that the errors of measurement (that you referred to as "uncertainties") are on the relative scale, then analysis in logs (model 1 on your list) will be the only one that will likely handle heteroskedasticity (non-constant variance, one of the assumptions of the textbook linear regression and ANOVA models). Other than that, the most important issue is to perform the analysis of residuals. That is, you would want to check for any remaining patterns that may give you an indication that your $V^2/d^2$ model is not performing well. With model 1, you can also test for the coefficients to be equal to 2 and -2, respectively, as another indication of the model performance. A test of the functional form should also be easily available from model 3: any deviation from the null intercept-only model is bad. On these accounts, model 2 is probably the weakest one, in the sense of allowing you rather little diagnostic capabilities per se. I would personally run several different forms, to see if they all agree that your data support your physical model.

-
default
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9492358565330505, "perplexity_flag": "head"}