labreport

PHYS 2010 Lecture 02 (Laboratory) Measurement and Uncertainty We’ve introduced the concept of units as it applies to measurements. While that’s a good start, there’s a bit more in play. Specifically, we also have to worry about how well we can measure something, and how multiple measurements combine. The purpose of this activity is to familiarize yourself with these deeper concepts.

Materials (see Appendix):

 Mainstays Measuring Cup 4 cups, single

 Diamond Toothpick Dispenser Packs, 3 Count

 Westcott Stainless Steel Ruler, 18"

Lab Preparation (do this before anything else!) Pull the plastic wrap off of the Diamond Toothpick Dispenser Packs, and remove all the toothpicks from

each box by gently popping the tops off. Fill the plastic boxes with some pocket change or other weights

to ensure it will sink (since this will eventually be going in water). Put the tops back on, and using a thin

(preferably electrical or other waterproof) tape, seal all three boxes back together arranged as they

were originally packaged (winding up with effectively a single large rectangular shape). Look carefully,

and make sure you’ve sealed every opening.

What you start with How it should look when prepped

Congratulations! You’ve made the reference block we all will be using to make measurements.

Proceed to the next page and start the lab.

Precision vs Accuracy Experiments often require a number of different measurements (with the appropriate units). The reliability of any measurement has two components: precision and accuracy. Consider the results from aiming at a target.

“Accurate” would be an indication of how well your average value agrees with the desired location (aka, the measurement). Using that definition, it’s easy to see that the right column of attempts is more accurate than the left. Estimating the average of the four shots (aka, data points) of each of the two attempts in the right column, you get a spot that is very close to the center of the target. Not so much for the attempts in the left column… “Precise” has more to do with the spread of the data, how close they are among themselves. Using that definition, it’s also easy to see that the bottom row of attempts are more precise than the upper row. Note that it’s possible to be precise but not accurate, and accurate but not precise. A lot of people use accurate and precise pretty interchangeable, but not scientists like us. You’ll find that a lot of what messes people up is not paying close attention to the words in play. Physics terminology is incredibly specific, and helps to define the rules of the game we’re playing. We mean exactly what we mean, and we have to be nitpicky in our phrasing so others understand the limits of how we’re applying something. For example, you’ll come to see that there is a difference between speed and velocity (but that’s a conversation for another day).

Uncertainty Uncertainty is a concept that is closely related to precision. It’s a statement of confidence in the measurement you are making. No measurement is exact (infinitely precise), there is always some uncertainty. Consider the following line, and the three measuring rulers (one in decimeters (every ten centimeters), one in centimeters, and one in millimeters:

The line that is trying to be measured is the same in each, but how well could you express that measurement in the different units? What if your rent depended on an accurate (correct) value?

In decimeters, we could say it’s a bit more than 0.5dm. But that’s pretty vague; scientists like specific numbers, and so instead we define a range of a reasonable answer: somewhere between 0.5dm and 0.6dm. If the rent was on the line if we got it wrong, however, I think we can all agree it wouldn’t be wrong to hedge our bets a bit and assert that the line is somewhere between 0.35dm and 0.75dm. You could also go further and go overboard, claiming it’s between 0dm and 1dm (but would you hire the guy giving that answer instead of the previous one?)

That range is an estimate of your uncertainty. You’re sure that the length of the line is between certain identified values, and that range depends on your confidence (which can be influenced by quite a lot). In “physics-speak”, we’d state those ranges as:

0.55 ± 0.05dm <- covers all possibilities between 0.5dm and 0.6dm 0.55 ± 0.20dm <- covers all possibilities between 0.35dm and 0.75dm, a larger uncertainty!

It’s important to note that we’re being a bit simple here. We took an average of our upper and lower values, which placed the measurement nicely centered between the two. It won’t always be that way. It’s possible to have a larger or smaller uncertainty above your stated value than below – isn’t math fun?!

Now, if I use a more precise scale, the picture changes

Now we’re talking centimeters, with ten times the resolution we had before.

With the increased resolution, our confidence increases, and we feel pretty good with a measurement of 5.5 ± 0.20cm (or 0.55 ± 0.02dm). Note that this choice of ruler reduces our uncertainty. What if we go to an even more precise scale of mm?

Here we get 56.25 ± 0.25mm (or 0.5625 ± 0.0025dm). This drastically has improved our precision and uncertainty from the first attempt, and in fact, changes our reported value for the measurement itself (0.5625dm from 0.55dm). Clearly, then, what you use to measure an object, and how you make your measurement affects the precision of the measurement and the associated uncertainty. In general, the guiding rules are:

 For Analog Devices (such as a ruler), the uncertainty can be broadly estimated as 1/4th or 1/5th the smallest division. It’s usually pretty easy to eyeball halfway, or three- quarters between 2 ticks, and this level of uncertainty covers you pretty well.

 For Digital Devices (with an LED display or some such), the uncertainty can be broadly estimated as half the smallest displayed digit. After all, if the readout was 1.5cm, clearly it must have been closer to 1.5cm than 1.4cm or 1.6cm, so the measurement would read 1.5 ± 0.05cm.

It’s also worth noting that these guidelines are for the smallest possible reasonable uncertainty, and as you become more skilled, there are more formal rules that will guide you. Regardless, there’s nothing stopping you from making it larger if that makes you more confident. Your uncertainty is an ever-changing balance between wanting to be as precise as possible, and yet ensuring that your range covers the correct value. So, when measuring, you need to be aware of what’s in play. Common sources of uncertainty include:

• the effects of environmental conditions on the measurement (heat and cold can appreciably change the dimensions of an object in some cases, so can dampness, etc);

• how well you can in reading analog instruments (i.e., rulers); • the sensitivity of your instruments (e.g. using a cheaper or more expensive digital scale); • the rating or stated calibration of the instrument (how long ago did somebody actually

confirm it was giving good values?); • approximations and assumptions that you make while doing the experiment; • fluctuations in a given measurement (say, the value keeps changing a little bit as you’re

trying to read it); • variations in repeated readings made under apparently identical conditions; • …and many others not listed here.

As a final word on this, a measurement uncertainty is not meant to be an indication of “mistakes” that you might make in an experiment. Despite its common appearance in labs you may have had before this class, “human error” is not a valid source of uncertainty. If you know that you did something “wrong”, then why didn’t you go back and do it correctly?

Activity 1: Measuring an Object So, now that we have the basics of a measurement, let’s apply that to an object. In this case we will be using the reference block you prepared for lab.

1.1) Using a ruler, measure the dimensions of the block (in both inches and centimeters). Consider what we covered regarding uncertainty earlier, and include that in your measurement.

Using Inches (in) Using Centimeters (cm)

Length ± in ± cm

Width ± in ± cm

Height ± in ± cm

1.2) Which measurement unit (inches or centimeters) has the smallest uncertainty? You can’t just make a direct comparison, since inches and centimeters are different units. Refer back to Lecture 1, and convert the uncertainty in inches to cm. Once you’ve done that, you can then directly compare the two values (cm to cm)

Propagated Uncertainties (multiplication) Measurements are almost always part of a larger picture, so we need to figure out a way for uncertainties to combine. Fortunately, we can work a lot of that out using standard algebra. Let’s consider volumes for a moment, which is purely a multiplication problem. Volume = Length x Width x Height

V = (L ± L) (W ± W) (H ± H)

The basic definition for volume

Replacing the words of the definition with letters, a standard equation trick. Note that we’ve also included our uncertainties, which can be indicated

with that “” symbol (read “delta”)

V = LWH ± L(WH) ± W(LH) ± H(LW) ± (LW)H ±LH)W ± (WH)L ± LWH We’re going to pause here for a moment and let the hyperventilating get out of your system. All we did was expand the multiplication, and we’re about to toss a lot of this away. Don’t panic, and read on.

Yeah, that looks pretty intense, but there are some patterns in the madness…

V + V = LWH ± L(WH) ± W(LH) ± H(LW) ± (LW)H ±LH)W ± (WH)L ± LWH

V 1 ’s 2 ’s 3 ’s

Why is that important? Because uncertainties are usually small compared to the measurement. As an example, let’s say each of those measurements (L/W/H) were 10 ± 0.1cm. The expanded volume would work out to be

V + V = 1000 ±  ±  ±  ± 0.1 ± ± 0.1 ±  cm3

V 1 ’s 2 ’s 3 ’s

If you look closely, you’ll see that the “0 ’s” part is the mathematical outcome of evaluating

the volume, the “1 ’s”, “2 ’s”, and “3 ’s” parts are the propagated uncertainty. Looking

even more closely, that propagated uncertainty is mostly contained in just the “1 ’s” part

(30 vs 30.301 if I include it all). So a lot of the time, we’ll just drop anything with more than 1 

(and call it "good to first order” since we’re only keeping terms with 1 ). Thus, to first order, the volume of our example rectangular solid is given by:

V + V = 1000 ± 30cm3

It’s a rough approximation, but is good enough for our needs in this class. Should you go further, you’ll discover more formal and precise approaches.

Activity 2: Applying your Measurement 2.1) Using your measured dimensions from Activity 1, determine the total volume (including

uncertainty) for your block

Using Cubic Inches (in3) Using Cubic Centimeters (cm3)

Volume ± in3 ± cm3

Validating Approaches You just completed a basic exercise in physics, taking data and working with uncertainties to get a result. However, the question may remain in your head – “Did I do it right?” After all, you’re applying unfamiliar concepts, we don’t have an answer per se to check against, and you don’t quite know just yet. Guess what, that happens more often than you think with the professionals, too. The only difference is that they’ve been in this position many times before, and know “The Secret” - Do it again (and if possible, use a different methodology) and see if you get the same outcome.

So, let’s work the same problem from a different angle. The block is effectively a rectangular solid, and you took measurements of the dimensions to work out the volume. This time, let’s just go for the volume directly, using displaced water. Displaced is a fancy phrase (as are so many things in physics) describing a simple act. We’re going to take the block, dunk it into water, and use the change in volume to assess the volume of the block. That’s an easy process to describe, but we’ll need to go back and see how our approaches to uncertainty work

Propagated Uncertainties (Addition/Subtraction) We’ll be taking an initial, measurable volume of water (with a certain amount of uncertainty) and adding a block of unknown volume to it. The combination will then yield a new, measurable, total volume, the excess over our initial measurement should be due to the added block. Total Volume = Initial Water Volume + Object Volume This is an addition problem and, just like multiplication, there are different ways you can approach the uncertainty depending on how rigorous you want to be. At its simplest, you could simply add the uncertainties together. However, If I do that, I’m allowing for the possibility that any uncertainties in the data are in the same direction (either only + or only -); that approach actually overestimates the likelier possibility that sometimes I’d be high and others low. Instead, a good general approach is to use quadratures (which wind up looking a bit like the Pythagorean Theorem for reasons beyond this lab). Suffice to say, that if I add or subtract two numbers, the propagated uncertainty of the combination is:

(A ± A) + (B ± B) = (A + B) ± √∆A2 + ∆B2 This combination will always be a bit bigger than the larger uncertainty alone, but smaller than simply adding the uncertainties directly. So, let’s apply this to our approach, putting all the things were we know the uncertainty on one side, and leaving the unknowns on the other Total Volume = Initial Water Volume + Object Volume

VT = VW + VO

The basic definition

Replacing concepts with variables. Note that since all three of these are volumes, I used the same symbol, V, to describe them mathematically. To help keep them separate, I added subscript labels to act as “nametags for the identical triplets”

VT - VW = VO

(VT ± VT) - (VW ± VW) = (VO ± VO)

Rearranging things so that the measurements I can make are on the left, and the value I’m trying to determine is on the right

Adding in uncertainties

So, looking at quantities and uncertainties separately, we can say:

VO = VT - VW

VO = √∆𝑉𝑇 2 + ∆𝑉𝑊

2

Activity 3: Volume Measurement via Water Displacement 3.1) Take your measuring cup, and partially fill it with water. You’re going to want to have

enough water in the cup so that when you add your block, it will be fully submerged. However, you also don’t want so much that the raised water level exceeds the measurements on the side of the cup.

Note the initial volume of the water. Depending on your cup, it might be in cups, oz, or mL, regardless, we’re going to want it in cubic inches and cubic centimeters. We covered units conversions in Lecture 01 which tells you how to make the conversions, but I’ll give you some relevant conversion factors that will help you get to where you need to be: 1 mL = 1 cm3

1 cup = 240 cm3 = 8.11537 oz Note the exposed lie that 8 oz is 1 cup. It’s close to 1 cup, but not precisely; however, most of the time it doesn’t matter – that time, however, is not in a physics class  Also, remember that you’ll need to convert your estimated uncertainties as well…

Using Cubic Inches (in3) Using Cubic Centimeters (cm3)

Initial Volume

± in3 ± cm3

3.2) Add your block to the measuring cup with water, and as before, record the final volume of

water in the measuring cup (making the necessary conversions).

Using Cubic Inches (in3) Using Cubic Centimeters (cm3)

Final Volume

± in3 ± cm3

Activity 3: Volume Measurement via Water Displacement (continued)

3.3) Knowing that the volume of the block is the difference between your Final and Initial Volumes, determine the volume of the block, including the propagated uncertainty.

Using Cubic Inches (in3) Using Cubic Centimeters (cm3)

Volume ± in3 ± cm3

You have made volume determinations of the block using two methods. The first was by taking

dimensional measurements, and combining them to get the volume. The second was by indirectly

measuring the volume using displaced water. Both these results have some uncertainty associated with

them.

If you compare the two, the whole range defined by the value and uncertainty for each method should

have some overlap. If they do, you can feel fairly confident you have a good result. If not, it’s time to

start looking for what went wrong, fixing it, and doing it again. If one particular method’s results lie

entirely within the other method’s results, it’s a good indication that the smaller range is the better

approach (because you are confident in the results, with a smaller uncertainty).

Activity 4: Comparing Results

4.1) Make a reasonably-“to-scale” sketch of the results of the two methods with each result side by side

for easy comparison (one pair for in3, and a second pair for cm3). Using a few sentences, interpret

what you are seeing, and explain why you believe what you do and what, if anything, should be

done the next time this experiment is conducted to improve the outcomes.