Download Simple Measurements and Error Estimation - Assignment 3 | PHY 191 and more Assignments Physics in PDF only on Docsity!
Experiment 3
Simple Measurements and Error Estimation
Reading and problems:
Homework 3: turn in as part of your preparation for this experiment.
Read sections 3.1-3.10 of Taylor (you can skip 3.2). Read the handout on the important things in uncertainty calculations. Look carefully at the definition of independence in the handout.
Do the problems below. They and the analysis and discussion requested below will help prepare you for the uncertainty calculations needed for this lab, and indeed for uncertainty calculations all term. Using a spreadsheet is usually easier than a calculator. If you decide to do the additions in quadrature on a calculator, note that the conversion from rectangular to polar coordinates automatically calculates √(x 2 + y 2 ) for given x and y.
1) If x has been measured as 4.0 ± 0.1 cm, what should I report for x^2 and x^3? Give percent and absolute uncertainties, as determined by rule (3.10) for a power.
2) A student measures a = 50 ± 5, c = 60 ± 2, e = 5.8 ± .3, all in cm, and calculates the sums a+c and a+e. Assuming the original errors were independent and random, find the uncertainties in her answers (using rule 3.16, “errors add in quadrature”). If she has reason to think the original errors are not independent, find the uncertainties in her answers (using rule 3.17, “errors add directly”). Summarize your calculations in a table. Useful headings might be Sum, Value, δq(independent), δq(not independent). Indicate with an asterisk those cases in which the second uncertainty (in c or e) can be entirely ignored, assuming the uncertainties are needed with only one significant figure. Comment on the comparative sizes of the uncertainties in each case.
3) A student makes the following measurements
a = 5 ± 1 cm, b = 18 ± 2 cm, c = 12 ± 1 cm, t = 3.0 ± 0.5 s, m = 18 ± 1 gm
Calculate the fraction uncertainty of each quantity (q=a,b,c,t,m) and put this in a table with headings
q value δq δq /q (%).
Compute the quantities q=a+b+c and q=m b/ t , their uncertainties, and percentage uncertainties for the two cases of independent uncertainties, and not independent uncertainties. Before you start, predict which errors will be most important in each case. Show the formulas you will use, and arrange your results in a table with headings as below.
Independent not independent q value δq δq /q (%) δq δq /q (%) most important
Explain whether the uncertainty of a+b-c will be the same, or different from, the uncertainty of a+b+c. From this answer, explain whether the fractional uncertainty of a+b+c will be larger or smaller than the fractional uncertainty of a+b-c, and why. Explain which uncertainty was most important in each case. Why did the importance of δb change from the q=a+b+c case to the case q = m b/ t case?
4) A student is studying the properties of a resistor. She measures the current flowing through the resistor and the voltage across it as
I = 2.10 ± 0.02 amps and V = 1.02 ± 0.01 volts.
a) What should be her calculated value for the power delivered to the resistor, P = IV, with its uncertainty? b) What for the resistance R = V/I? (Assume the original uncertainties are independent. With I in amps and V in volts, the power P comes out in watts and the resistance in ohms.) Start by evaluating the fractional uncertainty of I, and V in percent, then calculate δP/P in percent following the example on p 62. Show algebra, not just numbers. Finally, derive δP from δP/P. For b), see if you can avoid repeating the entire calculation.
5) In an experiment on the conservation of angular momentum, a student needs to find the angular momentum L of a uniform disc of mass M and radius R as it rotates with angular velocity ω. She makes the following measurements: M = 1.10 ± .01 kg, R = .250 ± .005 m, ω = 21.5 ± 0.4 rad/s And then calculates L as L = ½ MR^2 ω. (The factor ½ MR^2 is just the moment of inertia of the uniform disc.) What is her answer for L with its uncertainty? (Consider the three original uncertainties independent and remember that the fractional uncertainty in R^2 is twice that in R.) For this and any calculations in the lab involving products proceed in the same way as you did for 4). A very useful step is to pause and explain if any of the terms are obviously negligible before performing the final calculation of the fractional uncertainty. Also, explain why the fractional uncertainty of R^2 is twice that of R.
6) (a) According to theory, the period T of a simple pendulum is T = 2 π√ (L/g) , where L is the length of the pendulum. If L is measured as L = 1.40 ± 0.01m, what is the predicted length of T? (b) Would you say that a measured value of T = 2.39 ± 0.01 s is consistent with the theoretical prediction of part (a)? Again, show algebra as well as numbers.
1.Introduction
The main purpose of this experiment is to introduce you to methods of dealing with the uncertainties of the experiment. The basic procedures to correctly estimate the uncertainty in the knowledge of the measured value ( the error of the measurement ) include:
- Estimation of the uncertainty in the values directly measured by, or read from, the measurement device ( directly measured quantities, Taylor, Chapter 1 );
- calculation of the errors of the quantities which are not measured directly ( the propagation of errors, Chapter 3 );
- rounding off the insignificant digits in the directly measured and calculated quantities (Chapter 2 and the Appendix to Experiment 2).
3.1.4 If we are going to use the results of our measurements to calculate some other quantities (e.g., calculate the density of the rod using the measurements of its dimensions and the mass), which formulas will we use to calculate the mean values and the uncertainties of these quantities?
3.1.5 In our calculation, the calculator (computer) will typically return the results with as many digits as possible, including digits well beyond our measurement uncertainty. What procedure will you follow to systematically get rid of these insignificant digits?
3.1.6 From the homework problems, it was evident that much labor can be saved by judicious simplification of the uncertainty calculations. If you choose to do so, how will you justify approximations to the uncertainty calculations?
3.1.7 The measurement of the pipe poses special problems, which only begin with obtaining a mathematically correct formula for the volume. Discuss which instrument(s) would be best for this measurement. Record your choices and your reasons. As part of the discussion, consider how your answer would change if the diameter of the pipe were much larger, or much smaller; and if the wall was much thicker or thinner.
Density Measurements
4. Introduction
Your text (Sec. 1.3, p. 5) describes how Archimedes was able to determine the composition of a king's crown by measuring its density. We will attempt to perform a similar exercise, but in an effort to limit tuition increases, we shall use copper instead of gold. Copper has a density of 8.91 g/cm^3 at 20 degrees Celsius (C). We will consider later what to do if the temperature is not exactly 20 degrees C. Your task today is to measure the density, calculate the appropriate uncertainties and decide whether your measurement agrees with the given value. Then you will measure the density of some more complex objects. You will attempt to identify the materials of which these objects are made by comparing a list of known densities for various metals with their measured density and uncertainty.
Report both % and absolute uncertainties for your final values; using % uncertainty in your uncertainty calculation tables will usually make things much clearer, both for you and the grader.
In our lab, we will use rulers, vernier calipers and micrometers. Discuss in your group the following questions:
4.1 Which of these instruments is the most precise; the least precise? How do you know? In particular, is the caliper more precise than the ruler? If you don’t see how to use the caliper, refer to the Appendix. 4.2 Refer to the appendix on Alloy Densities. If your goal is to distinguish whether a sample is made of copper, or an alloy nearest to copper in density, with what accuracy will you need to measure the density?
4.3 What tables will you need for the measurements below? For the uncertainty calculations? Be aware that you will be expected to make your own tables to organize calculations and summary in the future, so think hard about this.
Now begin your measurements:
4.4 Write down in your lab notebook the sample code for the unknowns you are measuring, and the number of the table at which you are working ( do for EACH experiment ).
4.5 Write down the uncertainties of the length measurements with each of these instruments. Use the ruler to measure the three dimensions of the block. Repeat these measurements with the vernier caliper and the micrometer. For each instrument, measure the length with the highest precision possible. Assign uncertainties to your measurements. Each time you assign an uncertainty of a measurement, you must justify your choice in your lab notebook.
4.6 Measure the mass of your block and estimate its uncertainty.
4.7 For the ruler measurement, compute the volume of the block and its uncertainty. To retain your sanity, we recommend you use Excel to carry out the calculations. To organize your calculations, refer to the suggested spreadsheet at the end of the writeup. IF you do the calculations by hand, then consider the following hint for calculating fractional uncertainties with small error bars. Say you measure 1.0 cm ± .001mm. That’s a fractional uncertainty of 10 -4^ m or .01%. If all your uncertainties are this small, record the uncertainties as δx = .04% δy = .03% etc, but to do a hand calculation, take out a big factor like this before doing the square roots. Effectively it’s a unit change in the uncertainties, by factoring out the scale of .01%: Q = x y δQ/Q = √ [ (.01%)^2 { 4^2 + 3^2 } ] = .01% √{4^2 + 3^2 } = .01% √25 = .05%
Pay attention to the units. In the calculation, follow the rules for rounding off the insignificant figures, particularly for the final density result. If the calculation is done correctly, the smallest significant figure in the final value will be of the same order as the final uncertainty.
4.8 Make two estimates of your uncertainty using alternatively Eqs. 3.18 and 3.19 on p. 61 of the text.
4.9 Which uncertainty is more appropriate for this calculation? Why?
4.10 Compute the density and its uncertainty. Use Eq. 3.18 on p. 61 of Taylor.
4.11 Calculate the density and its uncertainty using the data obtained with the help of the caliper and micrometer. Compare the three measurements (and their inputs and uncertainties) in a table!
Answer the following questions:
4.12 Based on your best measurement, is your value consistent with the density of pure copper? Could the sample be one of the copper alloys listed in the appendix on Alloy
Appendix 1: Commercial Metal and Alloy Densities
Table of density (specific gravity) of alloys.
SG = Specific Gravity; the units are either g cm
CE = Coefficient of fractional linear Expansion ( 10 −6^ / °F)
Common name and classification SG CE Aluminum alloy 380 ASTM SC84B 2.7 11. Aluminum alloy 3003, rolled ASTM B221 2.73 12. Aluminum alloy 2017, annealed ASTM B22 2.8 12. Hastelloy C 3.94 6. Cast gray iron ASTM A48-48. Class 25 7.2 6. Ductile cast iron ASTM A339, A395 7.2 7. Ni-resist cast iron type 2 7.3 9. Malleable iron ASTM A47 7.32 6. Cast 28-7 alloy (IID) ASTM A297-63T 7.6 9. Aluminum bronze ASTM B169, alloy A;ASTM B124, B150 7.8 9. Ingot iron (included for comparison) 7.86 6. Plain carbon sheet AISI-SAE 1020 7.86 6. Stainless steel type 304 8.02 9. Beryllium copper 25 ASTM B194 8.25 9. Inconel X, annealed 8.25 6. Yellow brass (high brass) ASTM B36, B134, B135 8.47 10. Copper ASTM B152, B124, B133, B1, B2, B3 8.91 9. Haynes Stellite alloy 25 (L605) 9.15 7.
Appendix 2. THE VERNIER CALIPER
A vernier caliper consists of a high quality metal ruler with a special vernier scale attached which allows the ruler to be read with greater precision than would otherwise be possible. The vernier scale provides a means of making measurements of distance (or length) to an accuracy of a tenth of a millimeter or better. Although this section will be devoted to the use of the vernier caliper, it should be noted that vernier scales can be used to make accurate measurements of many different quantities. In the future, you will also use an instrument with a vernier scale to make precise readings of angular displacements.
JAWS
SAME DISTANCE AS
BETWEEN JAWS
INCH
CM
VERNIER SCALE
RULE
0 2 4 6 8 10
0
0 5 10 15 20 25
1 2 3 4 5 6 7 (^1 ) 1 2 3 4 5 6 7 8 9
Figure 1: Vernier Caliper
Looking at the vernier caliper in Fig. 1, notice that while the units on the rule portion are similar to those on an ordinary metric ruler, the gradations on the vernier scale are slightly different. The number of vernier gradations is always one more than the number on rule for the same distance. The line on the vernier which is aligned with one on the rule tells us the fraction of the units on the rule. For example, in Fig. 1 the vernier reads 1.440 cm or 0.567 in.
To use the vernier caliper: (1) Roll the thumb wheel until the jaws are completely closed (touching each other). Now check whether the caliper is reading exactly zero. If not, record the caliper reading, and subtract this number from each measurement you make with the caliper. (2) Use either the inside edges of the jaws, or the outside edges of the two prongs at the top of the caliper to make your measurement. Do not use the tips of the prongs. Roll the thumb wheel until these surfaces line up with the end points of the distance you are measuring. (3) To read the caliper: (a) record the numbers which correspond to the last line on the rule which falls before the index line on the vernier scale. On the following page, this would be 32 since the index line falls just after the 32 cm line. (b) count to the right on the vernier scale until you reach a vernier line which lines up with a line on the rule and record the number of this vernier line as your last digit. on the following page it is the ninth vernier line which is aligned with one on the rule, so the whole distance is 32.9 cm.
The following pages show six vernier scales, similar to that on the vernier caliper, which will allow you to test your ability to read a vernier caliper.
Recommended Tables for Experiment 3
Density Measurements d is used for uncertainty δ for simplicity Inputs for Block (lengths in cm) Object method l dl dl(%) w dw dw(%) h dh dh(%) Block ruler Block caliper Block micrometer
Density of Block volume in cm^3, mass in grams Object method v dv dv(%) m dm dm(%) density ddensity ddensity(%) Block ruler Block caliper Block micrometer
Inputs, cylinder & pipe (lengths in cm) Object method Do dDo dDo(%) Di dDi dDi(%) h dh dh(%) solid cyl Silver tube
Density, cylinder & pipe volume in cm^3, mass in grams Object method v dv dv(%) m dm dm(%) density ddensity ddensity(%) solid cyl Silver tube expected (Cu), or best fit alloy
Summary density in g/cm^3 density
best alloy density 2nd best
Object method density ddensity ddens(%)
best alloy name t best
2nd best name t 2ndbest Block ruler Copper Block caliper Copper Block micrometer Copper Solid cyl Silver tube
