Even #1
Physics 23 Laboratory Report

Experiment E1:

Measurement and Errors

Prepared By: ______Date:

Partner: ______

Lab Instructor: Lab Section:
Recitation Instructor: Recitation Section
Remarks by Grader:
Grade:

Lab E1 Measurement and Errors

This laboratory exercise differs from the ones to follow. This exercise reviews tools we will need for the rest of the semester.

The Goals for this laboratory are:

Learn to use Vernier Calipers

Learn the precision of measuring instrument

Review Standard Deviation

Learn Least squares Fitting using EXCEL

Learn how to calculate the error in calculated quantities

Review use of Excel

1. Using your Vernier calipers, measure the three dimensions of your aluminum block (record with units)

L=______

W=______

H=______

2. a) What is the smallest difference in length you could measure with a meter stick?

∆ℓ=______

b)  What is the smallest difference in length you could measure with your Vernier calipers? ∆ℓ=______

3. Often we make only a few or even just one measurement of a quantity. In this case, we shall use the smallest difference we can measure as an estimate of our measurement error, ∆x in the quantity x. We will write x = 1.03 + .02 cm, for example, when we measure x with a device which has 0.02 as its smallest measurable difference.

If we measure a reasonable number of samples, x1, x2,…xn, with say n > 10, then we can use the standard deviation, s, as the measurement error. The standard deviation is defined by

sx =

Where is the average of the n measurements. Excel uses the function STDEV to calculate sx. We shall use sx as estimate of our measurement error: x +sx .

4.  How do measurement errors affect quantities calculated using the measurements? For example, we measure two sides of a metal plate and calculate the area. What is the estimated error in the area?

We have measured the length of the two sides of a rectangular plate and wish to calculate its area. One side was found to be length a and the smallest division on our ruler was Da, so we claim that the length of this side is a ±Da. Similarly, for the horizontal side we find a length b±Db. We suspect that our best estimate for the area of the plate is A = ab, but what error bars should we attach to A? At look at the diagram indicates that the smallest area would be AMIN = (a-Da)*(b-Db)=ab - aDb - bDa + DaDb @ A- aDb - bDa  Similarly, the maximum expected value for A is AMAX A + (aDb + bDa). We have assumed that Da < a and Db < b and hence we have neglected the product DaDb. If this is not the case, we do not really need to worry about errors in A. They are large. We can summarize with

It is convenient to use the fractional or relative error in quantities, such as the fractional error in a: ra = (Da/a). Dividing eq (2) by A = ab

Hence, we do a flying generalization and conclude that the relative errors add for multiplication. It can be shown the same is true for division. Convince yourself that for addition, the errors, NOT the relative errors, add.

For your aluminum block measurements, what is ∆ℓ? Δℓ = ______

Hence, the measurements with your error estimates can be written in the form x±∆ℓ and are:

L=______

W=______

H=______

Now calculate the relative errors:

ρL = ∆ℓ/L = ______

ρW = ______

ρH = ______

Calculate the volume of your aluminum block: V = ______

Calculate the relative error in your calculate volumes: ρV = ______

Calculate the error in volume: ∆V = ______

Hence you would report the volume you determined to be V±∆V = ______

5. Now we measure a quantity y(x) for a set of different. Our text book suggests that y(x)=ax+b. How do we determine if our measurements are described well by a straight line, and if so, how do we determine the parameters a and b and their estimated errors? An example: [This example follows the treatment in a recent article in Math Horizons (November 2002) entitled “Is the SAT Irrelevant?” by Mark Schilling.]

A student measured the deflection of a beam (in units of millimeters) as a function of the load in Newtons. A plot of his measurements:

Having read her textbook (and being a physics student looking for the simplest possible description of her measurements) she thinks deflection (D) should be a linear function of load (L). Does the plot suggest this is a reasonable assumption?

What should be the intercept of the straight line which best fits the measurement? The simplest choice would be a line of zero slope (horizontal) with D=average value of the deflection. This is plotted as the coarse dashed line in the figure. Better than nothing perhaps, but not a very good description of the measurements. So what slope should be chose? We need a measurement of what a good fit is.

Here is a plot of the deflection projected onto the y axis so we can look at the variation (the range of values) of the deflection around the average. We cannot use Di - DAVERAGE) as a measure of this variation because it would sum to zero if we added the variations for all the points. WHY? So it is customary to square the variations and then add them to obtain a measure of the amount of variation. This is defined by:

where N is the number of measurements (data points in our jargon) and DAVERAGE has been replaced by the common notation . For our case, (You may recognize this as the square of the standard deviation from the average.)

Now turn to a straight line fit that has a non-zero slope. Think of this straight line as a prediction of the values we should measure. A standard measure of the quality of the fit is called the correlation coefficient, R2. R2 represents the proportion of the variation in our measurements that is eliminated by the fit (or regression.) [EXCEL calls the process of fitting a straight line to a set of measurement “regression analysis.”] This is what I used to plot the solid straight line in the first figure and the equation of the straight line is shown on the figure, where I have switched to the generic names for the vertical and horizontal axes instead of using Deflection and Load. In other words, the slope of this line is -1.52 and the y-intercept is 20.33. R2=0.97. What does this mean? Take the point where the load x=4 and the measured deflection y=15.4. The equation predicts y(4) = 14.25. So the discrepancy between our prediction and our measurement is 14.25-15.4 = 1.15. These discrepancies are call residuals. The variance of the residuals = 0.65, much smaller that the variance from the average (which was 23.75). R2 = 1-(0.65/23.75 = 0.97 which agrees with the value calculated by EXCEL and is shown on the plot. Another way of stating the meaning of this is to say that 97% of the variation in the measurements is explained by the model that predicted a straight line for deflection versus load. In physics lab, we are often interested in the estimated error in the slope. EXCEL’s regression analysis provides this also.

SUMMARY OUTPUT
Regression Statistics
Multiple R / 0.986246
R Square / 0.972682
Adjusted R Square / 0.969646
Standard Error / 0.890557
Observations / 11
ANOVA
df / SS
Regression / 1 / 254.144
Residual / 9 / 7.137818
Total / 10 / 261.2818
Coefficients / Standard Error
Intercept / 20.32727 / 0.502342
X Variable 1 / -1.52 / 0.084911

This is part of the table put out by EXCEL when I used regression analysis to fit a straight line to our measurements. Note R2 in line 5 of the table. Look at the bottom two lines.

”Intercept” is the y intercept. Its value is in the column “Coefficients” and an estimate of the error in the intercept is in the column “Standard Error”.

“X Variable 1” contains the slope under “Coefficients” and the estimated error in the slope in the column “Standard Error”. Thus, in our lab report we might report the slope as (-1.52 ± .08) millimeters/Newton. Because R2 is near 1, or equivalently, because the estimated error in the slope, 0.08 is small compared to the slope, 1.52, we would say the model which predicted the straight line is a good model. Later today we will learn to fit higher order polynomials, such as quadratics.

6. A student measured the velocity of a glider on an airtrack at several times and recorded the following measurements.

t / v(t)
0 / 0
0.75 / 3.56
2.1 / 9.1
3.2 / 15.3
4.1 / 18.1
4.9 / 20.5
5.9 / 27.61
7.1 / 31.22
8.2 / 36.71
9.1 / 41.67
10.3 / 47.01

He thinks that v(t)= at. Make a plot of v(t) versus t in EXCEL and then paste into this report. Using regression analysis, determine a and the estimated error in a.