# Measurement and Uncertainty

(last updated September 5, 2012)

Dr. Larry Bortner

## Learning Objectives

Understand the concept of uncertainty in measurement and calculations.

Calculate basic laboratory statistics and explain their physical interpretations.

Perform calculations using Excel.

Propagate the uncertainty of a quantity calculated with uncertain values.

Determine if two uncertain values are equivalent.

## Purpose

To use a pendulum to demonstrate the random nature of a physical measurement. To examine experimental methods that can decrease the uncertainty.

## Background

### The Pendulum

The period of a simple pendulum is the time it takes to make one complete cycle, or swing. Galileo noticed that this physical quantity was independent of both the amplitude (size) of the swing and of the mass that was swinging. Please reread that last sentence; the claim may be counterintuitive, but it is true. For small swings, the period T of a pendulum of length  is given by

(1)

where g = 9.80 m/sec2 = 980 cm/sec2 is the acceleration of gravity. This expression is exact to 3 s.f., if the angle of the swing is < 5°. Note that there are no variables in this equation for the mass or the amplitude.

When the pendulum swings, there is one true value of the period. Again (broken record), the time for one complete swing remains the same even as the size of the swing changes. Any attempted measurement of the period is an approximation of the true value, no matter how precise the instrumentation, no matter the method of measurement. If we measure the period two different ways, those two results probably will not equal the true value but they should be close. Plus, they should be close to each other.

Rearranging Eq. 2, we can express the length of a pendulum as

(2)

In this experiment we assume that Eq. 2 is true and that g = 980 cm/sec2. Using a measured period T, the calculated length should be the same as the measured length.

### Measurement and Uncertainty

What follows is a brief summary of pertinent material in Appendices 1 and 2. Refer to them for more details.

In these physics labs we use u{x} to indicate the numerical uncertainty of a physical quantity x, which can be either a measurement or a calculation. E.g., u{x} = 0.2 cm means that the “fuzziness” of the way we are measuring x is 0.2 cm. If we measure x to be 34.7 cm, we’re saying it’s probably in the range of 34.5 cm to 34.9 cm.

No measurement of a continuous physical variable is exact. Some degree of uncertainty or error is associated with the measured data of all experiments. It follows that any calculation involving one or more measurements also has some degree of uncertainty. The process of deriving an expression to calculate this doubt is called the propagation of uncertainty. The details of this are given below.

Outside of blatant mistakes, we classify errors that occur in the data-collection process in two ways, either as random or systematic. Random uncertainties occur due to slightly different measuring conditions and to disparate observer interpretations. Systematic errors occur when there is a disconnect between the theory and experiment. They exist because of conditions not taken into account, are reproducible, and can sometimes be compensated for.

One way to find the random error in a particular measurement is to measure the same quantity many times in the same manner. The higher the number of trials, the closer the average gets to the true value (assuming no systematic errors and a normal distribution of errors about this true value).

The standard deviation σx of a set of values x is a measure of the average variation of the individual measurements from the true value.

The physical interpretation of the standard deviation is that it is the error associated with any single measurement using a specific technique.

The standard error  of a set of values with an average  is calculated by dividing the standard deviation by the square root of the number of trials.

Physically it is the error associated with the average measurement.

A histogram is a plot of the number of measurements that occur in a certain interval versus the average value of the intervals. An example histogram would be to record the height to the nearest inch of everyone in a large class, count the number of people who were at each height, then plot that count versus the height.

The functional relationship between the number and the average of the interval is called the distribution. If the distribution is normal, the peak of the resulting bell-shaped curve occurs at the average measurement value. The width of a normal curve at half its maximum is twice the standard deviation.

We can use these concepts to improve the results of an experiment. Consider what we’ll measure today: the period of a pendulum. (Note that except for the two extremes, the pendulum bob passes through each position in the path twice in a complete swing.) The first measurement method is the obvious: measure the time it takes for the pendulum bob to go through one complete swing. Call this experimental technique the single swing timing (SST) technique.

We make one measurement. Are we through? How certain are we of that number we see on the stopwatch? That is, what is the uncertainty in this measurement? Certainly the error is no smaller than 0.01 seconds, the resolution of the instrument. Could it possibly be larger?

The hallmark of experimentation is repeatability. Say we measure the single swing time a total of ten times and find the average and standard deviation of those ten numbers. As long as we didn’t change the length or go wild with the swing size, we don’t expect the true period to change. According to the laws of statistics, if we time another single swing, there’s about a 68% chance that eleventh time will fall within the standard deviation of the average of our ten measurements. Here’s the cool thing: If we were to take 100 or 1000 similar times, the standard deviation would not appreciably change. The average would certainly get more precise (the standard error would get smaller), but the inherent accuracy of the SST method in a single measurement does not change just because you take more measurements.

As an aside, note that accuracy is how close a number is to the actual value, while precision is how many significant figures you can confidently state.

Making these ten measurements of the pendulum’s period gives us a precision to associate with the SST technique. Assume that the uncertainty comes in the starting and stopping of the watch. This means that the uncertainty is independent of the time interval we are measuring. If we were doing an experiment where we found the period for many lengths using the SST, we would only have to make single measurements for the remaining lengths since the uncertainty established for the period for one length is the same as for the period of any other length.

What if the precision isn’t good enough? Suppose the standard deviation is a larger error than we want to settle for. Can we make it smaller? Making more measurements would decrease the standard error but not the standard deviation, so we could just measure the period for each length ten times and forget about the single measurement. Let’s go with this idea but tweak it a bit.

Assuming that the period is independent of the amplitude, change the measurement to nine swing timing (NST). The following defines this technique:

Make one measurement of the time for nine swings instead of a single swing.

This elapsed time that you record is not the period. It is time that should be nine times larger than the period. To get the time for one swing, divide the measured time by nine. This calculation (as opposed to a straight measurement) is the period.

To find the uncertainty of the NST method, repeat the timing nine more times (for a total of ten) at the same length. In your analysis, find the average and standard deviation of the ten calculated periods.

So we’ve got ten numbers but we had to measure ninety complete swings. Is this really an improvement?

It would be if we had to measure the periods of ten or twenty different lengths. The ninety swings (ten measurements) are for one length, which, after calculation, gives us the uncertainty in the nine swing method of obtaining the period. For each of the other lengths, we make only one measurement of nine swings, since we already have the error in the single measurement (the standard deviation, calculated from the periods of the first length). The assumption, as stated above, is that the error in either method is independent of the time interval.

The NST is an example of how to improve the precision of an apparatus with statistical methods and is one that will be used throughout all three quarters of physics labs.

Where possible in this course we will use statistics to approximate the uncertainty of a method of measurement (which includes the physical instrument along with how it is used). We take a number of readings to establish a standard deviation and use that as the uncertainty in subsequent single readings.

### Estimated Uncertainty

A lot of the measurements you make in the physics lab are single readings we don't have the time or the patience to find standard deviation for each type of measurement. How do you assign an uncertainty without doing the statistics? As a start, your observation of a continuous physical parameter is limited by the resolution of your measuring device. This is known as a scale limited error. In these cases, start your estimate of the error at either ± division or ±1 division of the device. This gives you a minimum value of the uncertainty; that is, it will be at least the scale limited error, and probably more. Among other conditions, parallax, placement of the instrument, and irregular shapes will increase the estimated uncertainty.

It is your task as an experimenter to come up with a reasonable uncertainty estimate.

### Propagation of Uncertainty

As stated above, any calculation that uses one or more of our measurements (which have uncertainties) will itself have an uncertainty. How do we come up with a value for this error?

Starting with the algebraic expression for the calculated quantity and logically deriving the expression that is used to calculate the uncertainty is called the propagation of uncertainty. Rules and justifications for this process are contained in Appendix 2 and are summarized in the Lab References. Familiarize yourself with these rules because you are required to find your own expressions for the errors in calculated values for all experiments in the Physics Labs, starting with this one.

Suppose that an uncertain quantity Z depends on A, B, and/or C, where A and B are uncertain quantities and C is a constant. It is important to note that A or B may be calculations and not measurements. The first four rules, which you should know by heart, are as follows:

1.

2.

3.

4.

You may have to apply these rules several times to get to an expression that you can calculate. As an example of propagating the uncertainty, let’s derive the one expression that you need for today’s lab.

You have to calculate the pendulum length from Eq. 2. What would be the uncertainty of this value? We start with Eq. 2 and assume that g is exact and thus a constant. Since 2 and π are also constants, the value g/(2π)2 is then a constant. Using Rule 1 and basic algebra, we find that

(3)

We know the values of g and π but we don’t know u{T2}. We know that T2 is a calculation. From Rule 4,

(4)

We combine Eqs. 3 and 4 to get the expression for the uncertainty in the calculated length can be written as

(5)

### Equivalence of Two Values

Theories often make exact predictions. How do we determine if our experimental measurements agree with theory? Or, as alluded to above, how can we say that the values from two different techniques agree with each other? At this point in today’s experiment you have two numbers, the SST and NST averages. How do you compare these two numbers to each other?

The error we associate with a measurement means that about 68% of the measurements are within one standard deviation of the average. 95% of the measurements will be within two standard deviations. This 95% level is where we draw the line for equivalency:

In this course, we use the 95% confidence level (agreement within 2σ [two sigma]) as the breakpoint for agreement.

This means that if you are comparing a measurement (e.g., 4.08 ± 0.03 J/cal) with an exact prediction (4.186 J/cal) and find that the prediction is more than two standard deviations away, you say, “My measurement of 4.08 ± 0.03 J/cal is inconsistent with the standard value of 4.186 J/cal. The discrepancy is 3%.”

If the prediction is within two standard deviations, you say, “My measurement of 4.21 ± 0.02 J/cal agrees with the predicted value of 4.186 J/cal.”

When comparing two uncertain numbers A and B to see if they agree or are consistent or are equivalent, you are really asking whether the difference of the two numbers is zero within experimental error. Using Rule 2 for propagating the error of the difference of two quantities leads to the Standard Equivalency Test (SET):

(6)

Eq. 6 is the mathematical test you use to compare two numbers that are supposed to be equal. If the inequality is true, values A and B are equivalent. In practice, we input this as a logical function in Excel, explained below.

A and B are stand-ins for physical quantities. When you show an example of a SET in the Sample Calculations, do not use A and B. Use the variable names of the quantities you are comparing.

## Procedure

You need the following items:

one pendulum for the whole class

stopwatch

The TA will give you an official data sheet. This is where you record your data and observations along with an annotated sketch of the apparatus and definitions and descriptions of what it is you are measuring, how you are measuring it, and estimated uncertainties of each type of measurement. Refer to the Lab Report Format for more details.

Experimentally, you have to do these things:

Record the three lengths of the TA's pendulum, based on the three hypotheses of a real world pendulum’s length.

Measure the period of oscillation (the time of one complete swing) of this pendulum two ways:

10 timings of a single swing (SST).

10 timings of nine swings (NST).

Although most of the experiments in this course are designed for you to work with a partner, for this experiment you are required to take data and do the analysis by yourself.

Figure 1 Three possible definitions of the pendulum length.

1. The TA sets the length of the pendulum. There may be some confusion as to the meaning of the length used in Eqs. 1 and 2. (See Figure 1.) Allow for three different hypotheses: the length of the pendulum is the distance from the point of support to...

a. the top of the bob,

b. the middle line around the bob, and

c. the bottom of the bob.

The TA makes these three measurements along with an uncertainty estimate u{} and announces them for you to record on your data sheet.

2. The TA pulls the bob to the side so that the string makes an angle of no more than 10° from the vertical (a horizontal distance from the rest position that is no more than 1/6 the length of the pendulum), then lets it go.

a. Each student measures one period with the stopwatch (the time it takes to complete a cycle back and forth). To do this:

i. Start timing right when the pendulum changes direction (at the “top” of the swing).

ii. Stop timing after it goes to the other side and comes back.

iii. Be sure to measure the time for a full swing. If your times are about half of everybody else’s, you’re only measuring a half period.

b. Record your times on your data sheet in a table that allows at least ten trials.

3. As the pendulum swings, make nine more measurements of the time for a complete swing, for a total of ten times. The numbers you record will differ. This is due to random errors. No one measurement is any more right or more wrong than another. Each is a statistically independent measurement of the period with an experimental uncertainty associated with it.

4. Repeat Steps 2 & 3, timing nine complete swings of the pendulum instead of a single swing. The TA keeps the pendulum swinging until everybody is finished.

a. At the top of the swing start timing and start counting with a silent remark to yourself of, “Start!” After the bob goes to the other side and comes back, the count changes to one. Continue until you get to a count of nine.

b. Record the time it took for the pendulum to make nine complete swings. Miscounting is a common mistake to watch out for.

## Analysis

Excel has been chosen as the mandatory platform for calculations and graphs because it is ubiquitous, handles large amounts of data easily, and is adaptable to the analytical criteria of the elementary physics labs. The first two experiments are designed to introduce you to the various processes and techniques used in Excel.

Each of you will have a lab computer to work on. Some of you may have to go to an adjoining room.

You must submit the spreadsheet file you work on in class through the In Class Analysis feature in the Meta course for your mu lab on Blackboard. You have access to this file from any computer that can get online, but it is highly recommended that you do not use Blackboard as your only form of file storage.

Your primary storage medium should be a flash drive.

You can store your work on the hard drive of the lab computer you are working on as long as you don’t close Excel. Be aware that files that aren’t open are deleted from all computers every half hour. It is better to work from a flash drive.

The mandatory use of Excel includes creating the layout and formatting of the spreadsheet, as well as the entering of formulas. There are times, such as with the experiment today, when much of the formatting, and occasionally the plotting, is done for you, in an Excel template. This will not be the case for every experiment, so be prepared to work with a blank spreadsheet.

Whenever you have a number on the spreadsheet or your raw data sheet, it should be clear to most people what physical quantity this number is a measure of, with the proper units. Label everything.

Figure 2 Some of the items in a standard Excel 2010 screen.

### Filling the Data Pool

1. In the lower left corner on the lab computer you are using, click on Start> Favorites> Pendulum Time Measurement Form. (Start is the Windows logo.)

i. You may have to wait until other people finish because the server cannot handle the traffic.

b. Type in the first five single swing times, numbers only.

c. Type in the first five nine swing times, numbers only.

d. Click on the Send Data button.

### Starting Excel

2. Click Excel Templates shortcut on the Desktop then open the First Quarter folder. Double click on Measurement and Uncertainty. This starts the Excel template, which has labeled columns and cells (for data and calculations) and a chart that plots your calculated results.

There are two separate worksheets, indicated by tabs at the bottom of the screen: Class Times and Statistics & Comparisons. You should be starting out in cell A1 of the Class Times worksheet. Figure 2 shows an annotated blank Excel 2010 worksheet with standard features. If you have used previous versions of Excel, Excel 2010 has changed a bit in the way all the features are accessed, but the features themselves are the same. You may want to go through some of the online training for Excel 2010 provided by Microsoft or access one of the many tutorials available from a search.

### Transferring Data

3. Both the class data and your non-shared data need to be entered into the spreadsheet. Make sure that everybody has completed Step 1 above before you continue from here.

a. Restore the Pendulum Time Measurement Form.

b. Click on the Display All Data button.

c. Position the cursor just to the left of the Student Name label and click and drag all the way to the last Nine Swing time to select the class times. Copy (Ctrl C) this selection.

d. Switch back to Excel, make sure A1 is the active cell, and Paste the data (Ctrl V).

e. Scroll down to the last data entry. In the next line in the B column, type in your last five single swing times. Do the same for column C and the nine swing times.

i. If you make a mistake in typing, you can select the cell and retype the entire entry which automatically deletes the contents, or you can edit it in the formula bar.

### Formulas and Calculations: Finding the NST Periods

4. Go to cell D1. As the label T9 (s) indicates, this column is for the NST periods, the time for nine swings divided by nine. We could use a calculator to find each of these and enter in the numbers into the appropriate cells, but it would start getting tedious after five or ten entries. Excel offers a better way.

a. Note that in the Formula Bar, the unformatted cell content T9 (s) is displayed and the cell reference D1 appears in the Name Box. Move one cell down to D2. Since there is nothing in the cell, the Formula Bar goes blank.

b. Type in the equal sign from the keyboard. An equal sign appears in the Formula Bar and the last used Excel function appears in the Name Box.

c. Click on the NST time in that row (You can also just type in C2 or c2.), type in /9 to divide by nine then hit Enter. Surprise, surprise, the Formula Bar can act like a calculator! The nine swing period is now in the cell D2

d. The selected cell is now D3. You could continue doing this calculation in each row, just like you would with a calculator, but, as with a calculator, it would get tedious after a few lines. The better way that Excel offers is this:

i. Select the cell D2. A thicker line surrounds the cell and there is a tiny dark square in the bottom right corner.

ii. Position the cursor over the tiny square. The cursor should change from a fat plus to a thin crosshair. Click and drag down column D until you have reached the same row as your last entered NST time. Release the mouse button. Yee-ha! All of your nine swing periods are now calculated.

### Basic Statistics

5. Go to cell H2. As indicated, you need to calculate the average, the standard deviation, and the standard error of columns B, C, and D.

a. Click on the equal sign by the Formula Bar. As before, the last used Excel function appears in the Name Box.

i. If the name is AVERAGE, click on it. If it is not, click on the small triangle to the right that activates a drop-down list of the most recently used functions.

ii. Click on AVERAGE if it is on the list. If it is not, click on More Functions... at the bottom. This activates the Function Wizard.

iii. A Paste Function box appears. Choose the Statistical Function category: and click on AVERAGE in the Function name: list.

iv. A box with two line windows appears; this is the formula palette for AVERAGE. These windows need to be filled out for the function to proceed. Note that Excel anticipates the numbers you want to average, suggesting the cell range B2 to G2. This isn’t what we want; we want to average the SST measurements. Two basic ways you can do this: click and drag to select all the numbers or type b2:b150 into the Number1 box. (Cell references are case-independent; Excel functions ignore blank cells.)

v. Using the Excel function STDEV in cell H3, calculate the SST standard deviation . You want the same cell range as in the previous step.

vi. In cell H4, calculate the standard error  from its definition, the standard deviation divided by the square root of the number of measurements. Use the functions SQRT (finds the square root) and COUNT (totals up all the non-blank cells in a cell range). You’ll have something like =cell reference/SQRT(COUNT(cell range)).

vii. Formulas for the minimum and maximum have already been entered in H5 and H6. These numbers aren’t all that important, they’re just nice to know.

b. Select cells H2:H4. As with Step 4.d.ii, go to the lower right corner of the selection, click and drag across through column J. You now have the statistics for both the NST times and NST periods.

c. Click on the Statistics & Comparisons tab to go to that worksheet. Note that the results of your calculations also appear here.

d. For ID purposes, please fill in your name and date in the appropriate cells.

### Plotting a Histogram

6. We want to get a mathematical picture of the spread, or distribution, of the periods that you and everybody else in the class have measured. This picture is called a histogram. In addition to the numbers that the histogram represents, you need a list of bins, i.e., regular intervals into which the numbers fall. These bin boundaries have been calculated for you, a total of 33 slots covering your SST average ±4σ. This covers 99.997% of random data, but there may be people in your class who have serious systematic errors in their contributions.

a. Click on Data tab in the menu and choose Data Analysis in the Analysis group of the ribbon.

b. Choose the Histogram Analysis Tool.

c. Indicate the cell addresses of

i. your Input Range (all of the SST periods; you have to switch back to the Class Times worksheet to select these),

ii. the Bin Range (cells A12:A44.),

iii. and the Output Range (The cell where you want the upper left corner to be of a table that contains the bins and the frequencies. For the single swing, this is C11, for nine swings, E11.).

iv. Click on OK.

d. The Bin-Frequency table appears, highlighted. Bars should appear on the histogram chart.

e. Repeat Steps a-c for the NST periods, T9. (That’s periods, not times.)

### Saving the File

7. At some point you need to save the blank template with a unique name on your flash drive or on the C: drive in the My Documents folder. Now is a good time.

a. Click on the floppy disk icon at the top, or the File tab and choose Save As.

b. The file type should be an Excel Workbook (file extension xlsx) or an Excel 97-2003 Workbook (xls). If you do not have access to a computer with Excel 2010, you must save the spreadsheet as the earlier file type.

i. The first four characters are mua_ to designate the Measurement and Uncertainty experimental analysis.

ii. The next letters are your Blackboard username. (Obviously, you must be registered on Blackboard. Go to http://www.blackboard.uc.edu or click on the icon on the computer desktop to do this.)

iii. Excel will add the extension .xlsx or .xls.

d. You should manually Save your work frequently and never close your file until you have saved it on an external storage device. Again, closed files stored on lab computers are deleted every half hour and are not recoverable.

### Comparing Periods

8. Compare your two average periods using Eq.(6). (Perform the SET.)

a. Choose the cell G6 to the right of the SET label and type in =abs(b4−d4)<=2*sqrt(sumsq(b6,d6)). A TRUE or FALSE will appear indicating success or failure of the test. This is the Excel formula for Eq. 6, where A=b4, B=d4, u{A}=b6, and u{B}=d6.

b. Another way is to do the following:

i. Click on the fx in the formula bar.

ii. Find the absolute value function ABS and select it.

iii. Click on the cell that contains one of the average periods, type in a minus sign, then click on the cell that has the other average.

iv. Click in the formula bar at the end of the existing formula and type in <=. This makes it a logical formula.

v. Now finish typing in the right side of the inequality in Eq. 6: 2*sqrt(sumsq(b6,d6)). (You can type in the cell address or click on the appropriate cell as you type.)

### Calculating the Pendulum Length from Period

9. Enter in the 3 lengths that were measured in Step 1 of the procedure into cells N5, N6, and N7, along with the estimated length error into N3 and the book value for g in cm/s2 into N4.

When you work with physics lab spreadsheets, follow this convention: experimental data go in shaded cells. If there is a number in a non-shaded cell, it means that it is calculated and that there is a formula in that cell.

a. Select cell N9. Using Eq. (2), calculate the length of the pendulum using the average NST period, which should have the smallest error. (What is the uncertainty of an average?) Refer to the cell where you have entered the value for g and use the Excel function, PI() for π.

b. In cell N10, calculate the uncertainty in this calculated length from Eq. (5).

### Testing Hypotheses

10. Use the SET to compare the calculated length in N9 to each of the three measured lengths in N5, N6, and N7. (Put comparison formulas in the designated cells.) Which of these three lengths, if any, is equivalent to the calculated one? If we had assumed that the length was either of the other two measurements, we would have had a systematic error.

a. Select cell O5 and type in the appropriate formula that compares the top measured length with the calculated length.

b. With appropriate use of absolute cell references, you can enter this formula once and copy it to the other two cells to do all three SET's. The default cell reference in Excel formulas is relative. If you refer to a number in a cell two rows up and one column to the left, when you copy the formula to another cell, that's how Excel looks for the number from that cell location. If the cell reference is absolute, no matter where you copy the formula, it always goes back to the number in that absolutely referenced cell.

i. The only number that is different in the other two SET’s is the measured length, so that cell reference has to be relative while the other three are absolute.

ii. With cell O5 selected after you have typed in the formula, double click on the calculated length cell reference in the formula bar. Hit the F4 key once to make this an absolute cell reference.

iii. Make the cell references to the two uncertainties absolute.

iv. Note that you can make the cell references absolute either as you enter in the formula or after you have entered it

c. With O5 still selected, position the cursor over the lower right corner, double click or click and drag down two rows, and release. You've just done the other two comparisons.

### Submit Preliminary Analysis

You may not be able to finish all of the analysis during the allotted time. You are not expected to do so. But before the end of class you need to submit the work that you have done through the In Class Analysis page on Blackboard. An exclamation point for mua should appear in your grades to acknowledge this. (It is good policy to be sure this exclamation point appears before you leave class because you can’t submit a lab report unless your analysis has that exclamation point.) Your finished analysis spreadsheet must be included in your report, even though you may have finished it here.

a. Click on the Courses tab at the top.

b. From the listed Courses in which you are participating..., select the Physics 1 Lab Meta course in which you are enrolled.

c. Click on In Class Analysis> View/Complete Assignment: mua> Browse (next to Attach local file)> select the spreadsheet, Open> Submit.

d. You should soon get a message from Blackboard that the file has been successfully uploaded.

e. Save the file to your chosen form of external storage then delete any and all working files on the lab computer.

f. If there is no submitted file in your mua channel, you cannot submit a lab report.

## Questions

1. Using algebra and the rules in the Lab References, propagate the error in the quantity (T/2π)2 , where the uncertainty in T is u{T}. That is, derive the expression in the same manner as shown above for u{}, justifying each step. (In all Elementary Physics Labs, this is what "propagate the error" means: to explicitly show the derivation of the expression for the uncertainty in a calculated value.)

2. Three different techniques are used to measure the diameter and circumference of a circular object in order to find pi experimentally. Method A resulted in 3.133 ± 0.007, Method B gave 3.1609 ± 0.0002, and the result of Method C was 3.14 ± 0.03. Which of these methods was the most accurate and which was the most precise. Why?

You may have to finish or rework your analysis outside of class. That’s OK. But you’re by no means finished with your lab class responsibilities. In addition to doing all the spreadsheet stuff, you have to come up with a lab report that follows the Lab Report Format, in less than six days (144 hours counting from the scheduled beginning of class).

The Physics Department is introducing a new type of lab report. It is a multiple-choice assessment called a centort that is administered through WebAssign. You must complete one for this experiment. (The only other experiment that uses the centort is Momentum and Energy in 1D Collisions. The remaining ten experiments require a full written Lab Report.) You are given six possibilities for each section of the Lab Report except for the Data Analysis and you choose what you think is the best one.

Assuming that you have done the experiment, have done most of the analysis, and are still in class, you have to complete the following tasks:

1.     Submit the completed Data Sheet to your TA at the end of class. (You keep the colored copy.)

2.     Submit an Analysis spreadsheet in the mua channel by the end of class. (This is not graded.)

3.     Complete the Analysis and embed the Excel file in the Lab Report Template Word document. (Downloaded from the Lab Manual page on Bb.)

4.     Fill out the ID particulars in this document.

5.     Submit this document in the mu channel on the Submit Lab Report page of the Bb Meta course. (This version of the spreadsheet is the one that is graded.)

6.     Do not write a full lab report! The only section that is graded is the Data Analysis.

7.     Complete the Physics 1 Centort- Measurement and Uncertainty on WebAssign.

## Writing the mu Lab Report

Listed below are the steps you would have to take if you were to write the report. Because the centort is based on the report, you need to be familiar with the process. Plus, you have to write reports for other experiments. Please refer to the Lab Report Format and How to Write a Lab Report for more details.

1.     Finish the Data Analysis.

3.     Embed in the appropriate place in this lab report document the final version of your analysis spreadsheet.

4.     Write the Sample Calculations. The ones to include are easy to figure out for this experiment; they are T9,  l, u{l}, and one SET (not both).

a. You have to insert them as equations, which may take some time to figure out starting from scratch. Word 2007, 2010, and 2011 are set up well for this and most computers on campus have Office 2010.

b. If you are using your own computer, please install either Office 2010 or Office 2011 for the Mac. You can purchase this at a nominal fee from the UC Bookstore.

c. There is a document in the Lab Manual (Equations Physics 1) that contains the starting lines of most equations you need.

d. Copy one of the equations and paste it into your report document, making sure it is left-justified as is the following:

e. Copy this equation and paste it two more times. If you copy it right, both copies will also be left-justified.

f. In the second equation, substitute numbers (with units) where applicable, then in the third equation substitute the final value with units.

g. Repeat until you have samples of all five equations.

5.     Do the Error Analysis.

a. Measurement Uncertainties: This is pretty straightforward, basically indicating what quantities were measured in the experiment and the uncertainties associated with them. What was measured in this experiment? Length and time. You have to...

i. ... describe the quantity and how it was measured.

ii. ... give the numerical value of the uncertainty. Keep it to 1 significant figure! (2 s.f. if the mantissa of the number expressed in scientific notation is between 1 and 2. E.g., u{x}= 0.0346 m should be stated as 0.03 m; u{m}= 1.5739 g should be stated as 1.6 g.)

iii. ... state how the number was found.

Note that this uncertainty generally is the same number that you recorded on the data sheet. The exception is if the uncertainty were found statistically, as is the case in this experiment with the time t. In this case, u{t} is the standard deviation of one of the time measurements, not the scale-limited error.

b. Systematic Errors: This confuses everybody. A systematic error is what happens when the theory does not take everything into account that may have affected the measurements. It can be from an oversimplified theory or from an incorrect measurement done consistently. Sometimes these errors are obvious, sometimes not. If you cannot identify any systematic errors, you state, “No systematic errors were identified.” You might be wrong, but that’s what you say. Anytime you state a systematic error, you must also state how it affected your results. (Results are the results of the analysis, not the measured data.)

c. Propagated Uncertainties: These are errors in calculated values, a number usually calculated from an expression found through the propagation of uncertainty process. You are just stating values here (to 1 or 2 s.f.), you are not showing derivations or calculations. If this number was found from an Excel function like STDEV, you state that fact. For this experiment, you state two values,  and u{l} (this is for the calculated length, not the measurement).

5.     Write the Theory and Concepts. Be careful. The premise in this experiment is not the same as the purpose. Independent of the new techniques and procedures being introduced (which can be overwhelming), you are testing two physical ideas:

a.     ... that measurements of a physical quantity should give the same result regardless of the technique.

b.     ... that the length of a pendulum calculated using the experimental period should tell us how to measure the length.

6.     Write your Experimental Results and Conclusions.

a.     How did the two averages compare?

b.     Which of the three hypotheses is correct?

7.     Answer the Questions. Note that for Question 1, the steps you take to derive u{(T/2π)2} are similar to that for u{l}, but they are not the same.

8.     Write the Abstract. This is short and sweet. Again you are reporting on the experiment, not the exercise.

9.     Save your file as a Word document. It should have a docx or doc extension.

10.  Upload the file to Blackboard (Save) in the mu channel on the Submit Lab Report page from the Meta course menu. If this channel doesn’t appear, you haven’t passed the Academic Misconduct Test at 100% or you haven’t submitted a spreadsheet in the mua channel.

11.  Double check that the file that is on Blackboard is the one you want graded, (It may be different than the file on your computer.) then Submit.

After you submit any lab report, always open the uploaded file that is on Bb using a different computer than the one on which you created the report, specifically a PC running Office 2010. Both the Word file and the embedded Excel spreadsheet should open without any problems, be readable, and be what you want graded.