Geometry
In this lab, we will study the area of various two-dimensional shapes, comparing direct measurements with standard formulas for the result.
- Assorted cardboard shapes:
- Six rectangles
- One triangle
- One circle
- One trapezoid
- Five parabola segments
- 6" (15cm) rulers
- Scale (.01g precision)
- Parabola and circle printout
- Record data in this Google Sheets data table
Area Formulas
In this lab, we will be computing areas, using formulas you should have learned in high school. You will need the following formulas:
- The area of a rectangle of length \(l\) and width \(w\): $$A=lw$$
- The area of a circle of radius \(r\): $$A=\pi r^2$$
- The area of a trapezoid of bases \(b_1\) and \(b_2\) and height \(h\): $$A=\frac{b_1+b_2}{2}h$$
You will also need to know how to propagate uncertainties through these formulas. You can find these cases as worked examples in section 2.4 of our Error Analysis Guide: section 2.4.1 for the rectangle, 2.4.2 for the circle, and 2.4.3 for the trapezoid.
We will also experimentally determine the area underneath a parabola \(y=ax^2\), proving that the area underneath the curve between \(0\) and a point \(x_\text{max}\) is:
$$A=\frac{a}{3}x_\text{max}^3\label{parabeq}$$This result requires either calculus or some convoluted trickery to show mathematically. However, we can observe it experimentally without any serious mathematics.
Concept of Area Density
In order to show these formulas, we're going to need a way to measure area directly. We will do this by cutting our shapes out of cardboard and weighing them.
Using the mass, we can compute the area if we know the area mass density, \(D\), defined as mass per unit area:12
$$D=\frac{M}{A}$$This formula applies for any part of the material - that is to say, the (average) area density is always the mass of the object divided by the area of that object. (Note: area, not "surface area"; we are assuming our object is approximately two-dimensional.)
In general, this density can vary from point to point. We are going to assume that our cardboard is uniform, so the area density is constant everywhere.
In physics, this idea of area density is often used to characterize phenomena that happen on a surface, such as a soap bubble. You may encounter this quantity later this semester when discussing sound produced by the a drum (as the vibrations are partially characterized by the area density of the drum head).
Note: unlike most of our labs, this one is designed to be done individually (not in pairs). There should sufficiently many shapes for each person to have a set, although sharing is fine (just do your own data collection & calculations).
Part I: Computing Area Mass Density of Cardboard
Take five of your six rectangles. For each rectangle:
- Measure it's length and width.1 In this part, you do not need to estimate uncertainties in these quantities.
- Take it to the scale and record it's mass. You do not need to estimate uncertainty in this quantity.
With this done (and for each part you do), you have a choice: either analyze the data you have (see the "Analysis" section below), or move on to the next part (to get all the measurements done), and do the analysis at the end.2
Part II: Single Measurement of Area Mass Density
Now, we are going to do the same set of measurements on another rectangle, but use them a little differently.
Take the last rectangle (i.e., a different one than the ones you already used). Again, measure the length and width. This time, however, you should also estimate an uncertainty in those quantities.
At a minimum, this will be the precision of your ruler. However, if your cut edges are not straight (or not at perfect right angles), such that the length and width vary somewhat over the course of the "rectangle," then your uncertainty may be larger than that of the ruler alone. Your estimate is your own, although your TA may give you further guidance.1
Then, measure mass. Take your uncertainty in this mass measurement (and all mass measurements in this lab) as the uncertainty of the scale, which should be the last decimal point to which said scale reads (namely, .01g).
Part III: Area of a Circle
Next, take your circle. Measure the diameter of the circle, and estimate uncertainty (again, considering both the ruler and variations in your cutting precision).
Then, weigh your circle (again, just the cardboard part), and record alongside the scale uncertainty.
Part IV: Area of a Trapezoid
Next, take your trapezoid.3 Measure both bases of the trapezoid as well as the height. Estimate uncertainties in all three of these measurements.
Weight the trapezoid and take the uncertainty as you did in the previous part.
Part V: Area under a Parabola
Now, take your parabola segments. You should have five pieces that assemble into (approximately) a parabola, like so:
Start by taking just the first segment. Measure its width (\(x\)-distance) and maximum height (\(y\)-distance on the right-hand edge), then weigh it to get the mass. Take appropriate uncertainties in each of these quantities as usual.
Next, take slices 1 and 2, and combine them into a larger portion of a parabola. Measure the total width (of both pieces together), the maximum height (the right-hand side of slice 2), and the total mass (of both pieces together).
Repeat that set of measurements for the combination of slices 1-3, etc., until eventually you are measuring the width of all five pieces together (and weighing them all together).
Part I: Computing Area Mass Density of Cardboard
For each rectangle, compute the area. Then, divide the mass by that area to compute the area density for that rectangle of cardboard.3 Since you're doing a bunch of identical calculations, take advantage of Google Sheets' (or Excel's) "click-and-drag" abilities to make these easier.
We now have five measurements of area density. Take their average, and compute the uncertainty both in their average and in a single measurement, according to the formulas in section 2.1 of the Error Analysis Guide.
Part II: Single Measurement of Area Mass Density.
First, compute the area of the rectangle. Compute also the uncertainty in the area, following section 2.4.1 of the Error Analysis Guide.
Now, divide the mass by this area to compute the area density of this rectangle. Following similar logic to how you computed the uncertainty in area, compute the uncertainty in this area density. (This propagation is deliberately left up to you to figure out - which formula from section 2.2 can we apply here, and how do we do it?)
Compare this measurement of area mass density (with uncertainty) to your measurement of "average mass density" in part I. The part I measurement has an uncertainty as well, the uncertainty in that mean. Using the rules for comparison given in section 1.3 of the Error Analysis Guide, determine: do these two measurements agree, to within uncertanty?
Part III: Area of a Circle
For this part, follow section 2.4.2 of the Error Analysis Guide.
Compute the radius from diameter, then the circle area from radius, and propagate uncertainties.
The spreadsheet should autofill the area density based on your measurements in part I. If it doesn't, use your area density measurement from part I, along with the uncertainty in the mean.4
Using that and your measured mass, determine the area of the circle as measured directly. Compare these two areas, with their corresponding uncertainties: do these areas agree?
Part IV: Area of a Trapezoid
First, compute the area of the trapezoid from your length and height measurements. Following section 2.4.3 of the Error Analysis Guide, propagate uncertainty into the area.
Then, use the mass and area density (which should autofill, just like Part III) to compute the area of the trapezoid directly, as you did for a circle. Propagate uncertainties.4
Part V: Area under a Parabola
For each value of \(x\), compute \(x^2\) and \(x^3\), and propagate uncertainties with the appropriate formulas.
Make a plot of \(y\) vs. \(x^2\) using the PHY121/122 Plotting Tool, with error bars in both quantities. Include a fit, and ensure it passes through the origin. Ensure your plot has all the appropriate accoutrements: title, axis labels, etc. (see the Guide to Making and Using Plots).
Note the slope (which is called \(A\) by the plotting tool) has nothing to do with the area; the plotting tool just gives the slope a generic letter \(A\), which happens to conflict with the way we use \(A\) in this lab.
In any case: since our curve is \(y=ax^2\), the plot we make of \(y\) vs. \(x^2\) should be linear, and the slope of this plot will be the quadratic coefficient \(a\).
Next, compute the area \(A\) between \(0\) and \(x_\text{max}=x\) (for each value of \(x\)) using mass and area density, as you did in the previous two parts. Propagate uncertainty as usual.
Now, make a plot of area \(A\) vs. \(x^3\) (again, with a fit through the origin). Using equation \eqref{parabeq} above, how does the slope of this plot relate to the coefficient \(a\) of the parabola? (You may find the last section of our Guide to Making and Using Plots to be useful to understand this.) Compute the coefficient \(a\). Does it agree with your previous measurement, to within uncertainty?
Click here for more information about error in the mean vs. in a measurement in this experiment.
In parts III-V, it's not immediately clear whether we should use uncertainty in the mean or uncertainty in a single measurement. On the one hand, since we're using the average in our calculations, it seems we should use that uncertainty in the average. On the other hand, since the later measurements are all single measurements, it seems that we should use uncertainty in a single measurement. So which is it?
Unfortunately, the correct answer is: it depends on the source of the uncertainty. And determining that (in this experiment) could be an entire experiment in itself.
There are two (main) potential sources of uncertainty for our area mass density: it could be measurement imprecision, or it could be that the cardboard actually varies somewhat in density from place to place. Let's consider each one in turn.
Suppose first that the cardboard was perfectly uniform, and the uncertainty is entirely measurement-based. In this case, we are using the mean as an approximation of the "true value" of the area density, and we should just use uncertainty in the mean, since that is the uncertainty in this "true value."
On the other hand, suppose that our measurements were perfect, and the cardboard actually varied in area mass density from point to point. Then, we actually have two consecutive approximations: first, our mean is an approximation of the true average (differing, typically, by an amount of the "uncertainty in the mean"); then, this true average is an approximation of the mass of the single "new" shape (typically differing by "the uncertainty in a single measurement").
In this case, to do our "uncertainty in the new shape's area density," we would have to combine these two "steps" of uncertainty (which, in this case, consists of adding absolute uncertainties in quadrature). If we pretend we took "very many measurements" in part I (in practice, much more than 5), such that the uncertainty in the mean is very small compared to the uncertainty in a single measurement, then we can neglect that uncertainty, and our uncertainty would just be the uncertainty in a single measurement.
In reality, we aren't exactly sure which error dominates; we haven't determined that with our experiment.5 In fact, if both are of a relevant size, we have the additional complication of separating them in part I, then re-combining them in part III...
So, we make a reasonable guess: we assume that the cardboard is highly consistent in area density, and the only variation is our measurement rather than the cardboard itself. With this assumption, we use uncertainty in the mean in parts III-V.
(Also, even if we did all of that correctly, we would be ignoring the fact that uncertainty is likely to be larger with a bigger shape, which we should also account for, in principle, nor have we even started talking about systemic errors... and of course, in a real experiment [which might involve some unknown physics], all of these problems just get all the more difficult!)
Your TA will ask you to answer some of the following questions (they will tell you which ones to answer):
Experimental questions:
- Suppose you have a stone statue of mass \(M\), and you want to measure its volume. Suppose you have a small cube of the same material, which has mass \(m\) and side length \(s\). How could you compute the volume \(V\) of the statue, just with those quantities?
- Suppose we wanted to compare our uncertainty estimate (in area density) from Part II to our calculations in Part I. We have two choices: compare to the uncertainty of the mean, or compare to the uncertainty of a single measurement. Which number do we expect to be the same (plausibly speaking)? [Hint: which of those numbers "should" change with more measurements, and which "should" remain the same? If you're not sure, review section 2.1 of the Error Analysis Guide.]
Theoretical questions:
- Suppose we had calculated the circumference \(C\) of the circle from our diameter \(d\), using the formula \(C=\pi d\). What would be the (absolute) uncertainty \(\sigma_C\) in the circumference, in terms of any of the following quantities: \(\sigma_d\), \(d\), and \(C\)?
- Suppose we had cut out a triangle, of base \(b\) (uncertainty \(\sigma_b\)) and height \(h\) (uncertainty \(\sigma_h\)). Recalling that the area of a triangle is \(A=\frac{1}{2}bh\), what would the (absolute) uncertainty of the area of the triangle be (computed according to that formula), in terms of any of the following quantities: \(b\), \(h\), \(\sigma_b\), \(\sigma_h\), and \(A\)?
- Suppose we had cut out an equilateral triangle of side length \(s\), which thus has formula \(A=\frac{\sqrt{3}}{4}s^2\). What would be the uncertainty in the area of this shape, in terms of any of the following quantities: \(s\), \(\sigma_s\), and \(A\)?
For further thought:
- Suppose that you had a velocity vs. time plot that looked like \(v=at^2\), for some number \(a\). Based on the results of this lab, what would the position look like, as a function of time?
- Compare your uncertainty in area density from Part I and Part II. (Choose the uncertainty from Part I which is appropriate to compare, in the sense that the value does not [approximately] change with N.) If your Part I value were appreciably bigger, what might you deduce from that? What if your Part II value were bigger? [Answer those questions regardless of whether your uncertainties actually differ.]
- Consider the fact that, with many measurements, your uncertainty in the mean is proportional to \(\frac{1}{\sqrt{N}}\). Suppose the cardboard varied in thickness such that each square centimeter has some variation \(\sigma_0\) of its mass. How would the area density of a large shape (neglecting measurement uncertainties, just considering the variation of the cardboard itself) increase as the area increased? I.e., if you had a large shape of area \(A\), what would be the expected deviation from the average value of mass for that shape, \(\sigma_M\), in terms of \(\sigma_0\) and the area \(A\)?
Hovering over these bubbles will make a footnote pop up. Gray footnotes are citations and links to outside references.
Blue footnotes are discussions of general physics material that would break up the flow of explanation to include directly. These can be important subtleties, advanced material, historical asides, hints for questions, etc.
Yellow footnotes are details about experimental procedure or analysis. These can be reminders about how to use equipment, explanations of how to get good results, or clarifications on details of frequent confusion.
The standard symbol for area density is actually \(\sigma\). However, since we call our uncertainties \(\sigma\), to avoid confusion, we instead just call our area density \(D\). (Also, it would be somewhat silly to write \(\sigma_\sigma\) for the uncertainty in area density.)
One can also work with densities that are not based around mass. E.g., in thermodynamics, you may work with "number density" - the number of particles per unit volume (or area, or length).
There's actually an industrial term for this quantity; it's called the grammage of the paper or cardboard.
Whether to use "uncertainty in measurement" or "uncertainty in mean" for this measurement is a fairly subtle issue. (Unfortunately, it's a bit long for a footnote, so look to the end of the analysis section for more information.)
In principle, our measurement in part II should give a sense: the "uncertainty" determined in part II, as stated, only considers the "measurement uncertainties," not the cardboard uncertainty itself. Thus (assuming our uncertainty estimates were accurate, which - for technical reasons [see appendix B.1 of the Error Analysis Guide] - they probably aren't), from our independent estimate of uncertainty in our measurements, we can estimate variation in the cardboard thickness. But that's pretty deep rabbit hole to go down, beyond the skills we're aiming for in this course.
You can choose what units to measure in - cm, mm, m, or even inches if you really want to (although we encourage using metric units, since that's what we'll be doing for the rest of the course). In this lab, there's no need to put everything in SI units (i.e., convert from cm to m), so you may as well record in whatever units are most convenient.
This is actually a decision you will end up making in all labs. In general, for all introductory labs, either method will work if you're efficient. However, if you choose to analyze data before completing all measurements, make sure to keep track of time - analysis can be done at home if need be, but measurements rarely can.
Remember that not all trapezoids are the prototypical isosceles trapezoid; a trapezoid is any quadrilateral in which two of the edges are parallel.
Hint: this is exactly the same computation that you did in the previous part. Google Sheets (or Excel) can therefore make this part easy: what happens if you directly copy-paste the cells from the previous part?
For more guidelines on how to estimate basic uncertainties, see section 2.1 of the Error Analysis Guide.