Experiment IIb and III-b

A Determination of g, the Acceleration Due to Gravity,

from Newton's Laws of Motion (Week Two).

Objective

To analyze your data from last week by means of a computer and a modern spreadsheet program.

Before you can reach your objective this week, you will need to study the material in the following sections of this write-up.

I. Brief Comments on the Classification of Experimental Errors.

II. Error Analysis in 2048C Laboratory.

III. Computer Analysis of Experiment II - Acceleration Due to Gravity.

Be aware that you will be called upon to use the knowledge you gain this week in future experiments. Each of you, therefore, must gain hands-on experience with the computer.

I. Brief Comments on the Treatment of Experimental Errors

Experimental error or uncertainty is inherent in any experimental result at some level, however small. It is set by a combination of the design of the experiment, the quality of the apparatus and the skill of the experimenter. Generally it is possible to separate the sources of experimental errors into two categories: random and systematic.

A. Random Errors

The existence of random errors in a measurement can be inferred if repetition of the measurement does not give the same result each time. By definition, random errors are those that tend to average out upon repetitions of the measurement. For random errors then, the more repetitions of a measurement the less the uncertainty in the resulting average. From the mathematics of statistics it can be shown that the uncertainty due to random error in the average of N measurements decreases as when N is "large". Thus 400 measurements should give an average with half the uncertainty due to random error as compared to 100 measurements.

The first step in treating the random error in a large number N of repeated measurements is to calculate the average. The average is the desired result and it is the uncertainty of the average due to random errors which must then be determined.

If the N measurements are labeled y₁, y₂, y₃, ... y_N the average as denoted by is defined as:

(The average is also sometimes called the mean in statistics.)

To establish the uncertainty in the average, the usual procedure is to first calculate what is called s (sigma), the standard deviation of measurements. If it is believed that no one measurement is more accurate that any of the others (as we will assume) then s is defined by:

The significance of s is that any one additional y measurement has about a 2 in 3 chance of falling between ± s of the average, . Statistical analysis further shows that the average, has about a 2 in 3 chance of falling within ± s of the true value. Thus, after specifying the average of N repeated measurements, it is common to append ± s as a measure of the uncertainty due to the randomness of the measurements.

Example

Suppose you have the following 5 measurements of the same quantity.

Note: One or two more decimal points must be kept in the calculations than are in the data until the calculations are complete.

The standard deviation of these measurements is:

The uncertainty in the average is = 0.0104. The result is then 1.0434 ± 0.0104. At this point some rounding off is appropriate and the result would be reported as 1.04 ± 0.01 or possibly as 1.04₃ ± 0.01₀.

It should be pointed out that the number 5 is stretching the concept of large N beyond reasonable bounds but that this is common practice when a large number of measurements are not available.

B. Systematic Errors

A systematic error is one which tends to repeat and thus create a shift in the average from the true value. Systematic errors may be provided by the experimenter, the apparatus or by poor design of the experiment. Because they are not revealed by repeated measurements, care must be exercised to investigate and account for all possible sources of systematic errors. This can be very difficult to do. Such errors sometimes remain unknown until other experimenters with other apparatus obtain convincing evidence that a previous result is off by more than the originally reported uncertainty.

II. Error Analysis in 2048C Laboratory Experiments

A. Treatment of Random Errors

Generally in your lab experiments in 2048C you will not measure the same quantity over and over. Often, however, you will have a moderate number of data points which are used to determine a single experimental quantity. Under such circumstances the data can be treated in two ways. The first way is the graphical method such as you used in the g experiment where the data are used to create a straight line plot the slope of which is the desired experimental quantity. The second way is to convert the data to a form equivalent to a series of repeated measurements. In the case of the g experiment, this would correspond to determining the accelerator between every pair of data points on the tape and creating a whole set of values for a which you would then average. In general the graphical method is preferable and it is the only one you will use. It has the advantage that it gives a useful visual picture of the data. It allows you to see whether, in fact, the data exhibit the expected linear behavior and it also makes it easy to see the scatter or random errors in the data.

To treat the random errors in your data using the second (non-graphical) approach you would use the results presented earlier for repeated measurements to calculate the uncertainty in the average. In the case of the graphical approach, the random errors in the data points produce a statistical error or uncertainty in the slope. The statistical treatment of random data errors in the graphical approach involves the same principles, but somewhat different mathematics, as the treatment for the effect of random errors in repeated measurements. These mathematics will not be discussed in this course in any detail but have been embodied in a computer program which you will use inside Excel. This program will analyze and plot your data and give you the uncertainty in the slope produced by random errors in your data.

Drawing a best fit straight-line graph by hand through a number of data points with scatter requires some thought and it also takes some effort to determine what range of slopes is consistent with the data. The advantage of using the computer program is that it does all of this for you in a defined and consistent manner.

B. Treatment of Systematic Errors

You will be expected to discuss for each experiment possible sources of systematic errors. In general, this means giving plausible reasons as to why your results might differ from the expected result by more than the uncertainty due to random error.

III. Computer Analysis of Your Data for Acceleration Due to Gravity

Run Excel and enter your data from last week in the following way. Using the data from one of your tapes, in cell A1:A3 type m1, m2 and m, and in cells B1:B3 type in the values of m1, m2, and m, respectively. In cells A4:C4 type in Time, delta D, and v, respectively. Now supposing you have 14 values of time t, in cell A5 type in 0.0333, in cell A6 type in 0.0667, in cell A7 type in 0.1000, etc down to cell A18 which will contain 0.4667. In cells B5:B18 enter the corresponding values of ΔD that you measured for each value of time t. In cell C5 type in the formula =B5/0.0333 (or whatever is your value of Δt) and hit enter. To easily copy this formula downwards, just select the range C5:C18 and type control-d and you should see the computed values of v filled in down the entire column.

Now we have to make a least-squares fit to v versus t. Click a blank cell on the worksheet, then click Tools, Data Analysis, and select Regression. In the box that appears, for input y range enter C5:c18, for input x range enter A5:A18, click the button for New Worksheet Ply, and click the boxes for Residuals, Residual Plots, and Line Fit Plots. Now click OK, and you should see a new worksheet appear with plots and analysis of your data.

A. Interpreting the Results

On the analysis screen you will see a column called "Predicted Y". This is the value your measured y would need to be if it were to fall exactly on the computer fitted line. Hence the difference between measured y and best fit y is how far the data point is above or below the line. Also on the screen is a column labeled "Residuals". This represents the difference between the best fit and your measured values. Looking at this column is a good way to quickly see if you have an input error or a bad data point. If you have entered a bad data point the difference will be much larger than the adjacent values. Alternatively, you data points should scatter equally above and below the best fit straight line in your graph without and "stray" individual points. There is a plot on your screen on the residuals versus “X Variable 1”, which is t in this application. Find the plot and verify that the residuals look reasonable, i.e. there are no obvious outliers.

In computing the best fit straight line, the computer program chooses a line such that the following sum is a minimum:

[y₁(measured) - y₁(best fit)]² + [y₂(measured) - y₂(best fit)]² +

[y₃(measured) - y₃(best fit)]² + ... + [y_N(measured) - y_N(best fit)]²

This method is called, for evident reasons, "The Method of Least Squares Fitting." Your data is fitted by a best fit straight line of the form y = mx + c where m is the slope and c is the intercept on the y-axis (i.e., when x = 0). Along with these important values their associated uncertainties (or error) are given. On the screen find the values of m (labeled “X Variable 1”) and c (labeled “intercept”) and their statistical uncertainties, or ‘errors’, labeled “Standard Error”.

Now find the second plot on your screen, which is labeled “X Variable 1 Line Fit Plot”. We are going to change that plot to see things better. Select the plot by clicking on it, then right-click the plot, select Location, and select “As New Sheet”, and select OK (or hit Enter). Go to the new sheet that has appeared (probably tabbed as Chart1). Right-click one of the Predicted Y Values, select Chart Type, and select the type that has the data points connected by a line. The resulting graph will now look similar to the graph you drew last week by hand, i.e. a straight line going between your measured data points.

Now repeat all of the above for each of your remaining data tapes.

B. Lab Report and Discussion

This section of your report should contain:

1. Presentation of your results. This includes printouts of your data tables and graphs from Excel. Make sure you complete a summary sheet which neatly summaries all your results from your own hand graph analysis and those from this weeks computer analysis. Also, prepare a single chart which shows (a) your three values of g determined last week by hand, (b) the three values of g you got today from the computer fits, with their estimated uncertainties shown as vertical error bars, and (c) the real value of g = 9.81 m/s^2.

2. Answers to the questions posed below (a to d).

Questions:

a. By how much do each of your values for g differ from 980 cm/s²? How does each difference compare with the corresponding uncertainty Dg due to the random errors in the data? Does it appear that there is a systematic error in either or both of your results?

b. If you worked carefully and your equipment worked properly, you may see the effect of a small systematic error inherent in the design of this experiment. In order to get the cart wheels to spin faster and faster as the cart accelerates, the table must exert a static friction force on the wheel rim in a direction opposite to the cart motion. This is not the friction force you measured when the cart was moving at constant velocity and so it has not been accounted for. A calculation of this retarding force on the cart based on the mass and geometry of the wheels and the physics of circular motion shows that it should cause the measured value of g to come out about 2% low for the values of m₁, and m₂ used here.

What result do you obtain if your g values are each increased by 2% (about 20 cm/s²) to account for the systematic errors above? Are your data accurate enough to see this effect? Do you still have systematic errors larger than the above?

c. List a few other possible sources of systematic errors that should be checked if time allowed. Hint. Think about some of the assumptions you likely have made about the equipment that may or may not be valid.

Because each run was made under different conditions, i.e., with different masses, they do not represent an exact repetition of a single measurement. Each run may be subject therefore, to different systematic errors. Until such errors are accounted for, averaging of your results is not strictly valid.

d. Compare the slopes you obtained last week with those generated by the computer analysis. Did you do a good job by hand?