Experiment 1: Errors, Uncertainties, and MeasurementsLaboratory Report
Charles Sanchez, Geminesse Sianghio, Ferguie Solis Department of Chemistry College of Science, University of Santo Tomas España Street, Manila Philippines Abstract In this experiment experiment measuring instruments instruments were used. When measuring quantities, it is assumed that the values consists of two parts: the recorded value and its uncertainty. These measurements lead to error leading to the uncertainty of the result. Errors are not defined as mistakes rather the gross errors sometimes usually yield results that are not expected. Accuracy is the goal goal we aim for this experiment. experiment. 1. Introduction Laboratory experiments involve taking measurements of physical quantities, and the process of taking any measurements measurements always involves some experimental uncertainty or error. For accuracy, the use of the significant figures are to be applied, The importance of the significant figures is that the number of digits present in a result signifies the precision of the result. The students will also be able to imply in this experiment the different rules in significant figures and the Vernier’s principle. In this experiment, the students will be using measuring devices such as the Vernier caliper, Micrometer caliper, and the Foot rule in order to observe and practice the precision of the length of the measurements as they differ in each instruments used.
In this experiment, the group should be able to achieve the following objectives:(1) to study errors and how they propagatein simple experiment, (2) to determine theaverage deviation of a set of experimentalvalues, (3) to determine the mean of a set of experimental values as well as set of averagedeviation of the mean, (4) to familiarize thestudents with the vernier
caliper, micrometer caliper, and foot rule, (5) to compare theaccuracy of these measuring devices, (6) andto determine the density of an object givenits mass and dimension. 2. Theory
Given that all numbers used involved integers, it is nearly quite impossible to obtain the exact value of the quantity during investigation. From this the margin of error in a measurement is indicated clearly by the indicating the number of significant figures, the meaningful digits in a measured or calculated quantity, such that the last digit is understood to be uncertain. There are rules in determining thesignificance of a digit. The digits from 1-9 are significant. The zeroes between two other significant digits are significant. One or more additional zeroes to the right of both the decimal place and another significant digit are significant. Zeroes used solely for spacing the decimal point are not significant. In propagation of errors, measuring some quantities might be recorded with uncertainties. Therefore calculating the uncertainty propagates to the uncertainty of the value is crucial. Assume we measure two values A and B, using some apparatus. We know these values are uncertain. By physical reasoning, testing, repeated measurements, or manufacturer¶s specifications, we estimate the magnitude of their uncertainties. u{A} ist he absolute error in A, and u{B} is the absolute error in B. The relative errors are [1] u{A}/A and u{B}/B. Least count is the highest degree of accuracy of measurement that can be achieved.
The Vernier caliper is a tool from the caliper family allowing users to measure the inner or outer dimensions of items, and step or hole depths. The large jaws at the bottom of the tool have flat faces that each other when the vernier caliper is in closed position The outer caliper jaws wrap around objects and are used to measure outside distances, such as an egg or the length of a square. The inside caliper jaws, on the top of the tool, appear as a smaller version of the outer caliper jaws. The inner caliper jaws’ flat edges face away from each other when the vernier caliper is opened and are used to measure inner distances, such as the inside of a tube. The depth probe is a long, flat, thin piece of metal that runs through the center of the caliper and moves out from the body of the vernier calipers when the jaws are opened. The depth probe is used to measure step or hole distances. By placing the flat end of the caliper flush against the upper face of the object being measured, then moving the caliper jaws to lower the depth probe into the object's hole, you can use the scale to read the depth of the step or hole. Vernier calipers have main scales running along the length of the tool. The scale along one edge of the tool is in inches, while the other side has increments in centimeters. The main scales can be used as a simple ruler. Reading a vernier caliper is a multistep process. First, lightly place the jaws or depth probe against the object being measured. As the jaws move along the length of the caliper, a smaller scale called a vernier travels with them. The number on the main scale opposite the zero on the vernier scale is the first part of the measurement. Next, look at the marks, which are in either millimeters or fractions of an inch, along the length of the vernier scale. By eye, identify the mark on the vernier scale that lines up most accurately with the opposite mark on the main scale. This number is the rest of your measurement. For example, if the vernier scale's zero lines up with 5.6cm on the main scale, and the 2.4-mm increment aligns most accurately with its opposite main scale mark, the final measurement will be 5.624cm A micrometer is a caliper-like measuring device resembling a C-clamp designed to precisely
measure the lengths, diameters and thicknesses of solid objects. It consists of two measuring rods with a movable jaw operated by a thimble and friction screw or barrel, a calibrated cylinder and a locking lever, and can measure dimensions within tolerances of several microns. Read the value just exposed by the thimble on the central line of the cylinder. This value is in millimeters. Typically, there is a mark every half-millimeter, with the millimeter marks rising above the central line and the half-millimeter marks going below it. Read the mark on the thimble aligned with the central line on the cylinder. This mark is in hundredths of millimeters. There are 50 such marks, meaning that each turn of the thimble corresponds to half a millimeter, the distance between the upward and downward marks on the cylinder central line. Add these values together. This is the measurement of the object between the measuring rods. 3. Methodology
In this experiment, measuring devices such as Vernier Caliper, Micrometer Caliper, Foot rule and the Electronic gram balance are to be used. Before anything else, the researchers checked the measuring devices if some parts are broken. If so, add or subtract it from the measurements gathered. In this experiment, The researchers will each make ten independent measurements for the diameter of the sphere using the foot rule, Micrometer caliper and the Vernier caliper. Afterwards, calculations are needed to be able to acquire the different set of experimental values. The values needed are as follows: mean diameter of the sphere, the deviation ( d ) of each measurement of the diameter from the mean diameter, the average deviation ( a.d )( the average deviation is the sum of the deviations (d ) divided by the number (n) of observations. n in our case = 10 )
a.d. =
The average deviation ( A.D) of the mean diameter,
√
| |
The % error for the diameter , and the volume of the sphere. After measuring and computing for the experimental values, weigh the sphere using the electronic gram balance. Using the mass and the volume of the sphere, calculate the density of the sphere. For the % error, the researchers asked instructor for the accepted value of the density of the sphere to verify the results.
4. Results and Discussion
The researchers did an experiment about errors, uncertainties and measurement. The objective of the said experiment was to study errors, to determine the average deviation, to determine Table 1. Results of Measurements Trial Foot Rule 1 1.50 0.05 2 3 4 5 6 7 8 9 10 Mean Diameter Average Deviation(a.d) Average Deviation of the Mean (A.D) %Error of Diameter Volume (cm3) Mass (g) Experimental Value of Density (g/cm3) Accepted value of 3 Density (g/cm ) %Error for Density
1.60 1.55 1.57 1.60 1.55 1.57 1.55 1.50 1.53 1.55 0.026
0.05 0.00 0.02 0.05 0.00 0.02 0.00 0.05 0.02
the mean of a set of experimental values, to familiarize with equipments named as Vernier Caliper, Micrometer Caliper and Foot rule.
Vernier Caliper 1.535 0.097
Micrometer Caliper 1.5800 .0000
1.600 1.600 1.670 1.675 1.680 1.680 1.600 1.600 1.680 1.632 0.045
1.5800 1.5800 1.5800 1.5775 1.5800 1.5797 1.5815 1.5815 1.5800 1.5800
0.032 0.032 0.038 0.043 0.048 0.048 0.032 0.032 0.048
0.008
0.014
0.005
5%
9%
3%
1.95 16.27 8.34
2.28 16.27 7.14
2.07 16.27 7.86
7.8
7.8
7.8
6.9
8.5
.77
Table 2. Measurement of width of thumb
.0100 .0000 .0000 .0025 .0000 .0018 .0015 .0015 .0000
Group Member Width of thumb (in)
1 17/20
5. Conclusion
The researchers might have conclude that using different measuring device particularly with foot rule, vernier caliper, micrometer caliper can lead to different measure with minimal difference. Also, in the table 1. The researchers got small percentage of error with some reason like having small amount of time to do the experiment, not familiar on how to use the instruments. The result that the researchers did was not accurate in different instrument however it has minimal difference which can consider as the measure of the sphere. The average deviation, the mean of experimental values and the average deviation of the mean was determined by the researchers. The researchers also had the chance to familiarize the instruments (foot ruler, vernier caliper, micrometer caliper) they used. 6. Applications 1.) Which among the three measuring devices give the least % error? Is the accuracy of the measurement affected by least count of the measuring device?
Among the three measuring devices used, the micrometer caliper gave the least percent error. Upon calculation it only showed2.748% error compared the foot rule that gave 10.802% and the vernier caliper giving5.992%. All measuring equipments have a least count which is the smallest quantity that can be measured accurately using that instrument. The least count indicates the degree of accuracy of measurement that can be achieved by the measuring instrument. Thus, the least count of an instrument is indirectly proportional to the accuracy of the instrument. 2.) What do you mean by error? What are types of errors? What are the errors encountered in the experiment?
An error is a deviation from accuracy or correctness and from standard or accepted value.
2 4/5
3 4/10
Measurement errors may be classified as either random or systematic, depending on how the measurement was obtained. Random error is always present in measurements. It is the statistical fluctuations in the measured data due to the precision limitations of the measurement device. Systematic errors are caused by imperfect calibration of measurement instruments or imperfect methods of observation, or interference of the environment with the measurement process and always affect the results of an experiment in a predictable direction. One does not always get the same result in making a series of measurements. This one is unavoidable because there will always be some uncertainty in the measurements and there is no perfect measurement. This is an example of random error the group encountered in the experiment. Another error encountered is the inadequate calibration of the vernier caliper and micrometer caliper which resulted to a systematic error. 3.) Sketch a) a vernier caliper that reads 5.08cm b) a micrometer caliper that reads 2.55mm
4.) A student weigh himself using a bathroom scale calibrated in kilograms. He reported his weight in pounds. What are the percentage errors in his reported weight if he uses this conversion: 1kg = 2.2 pounds ? The standard kilogram is equal to2.2046 pounds.
Suppose the student weighed 65 kg. With his reported weight using the conversion 1kg=2.2 lbs, he weighed 143 lbs. But with the standard kilogram equal to 2.2046 pounds, his weight would be 143.2990000 lbs. Hence, in calculation of percentage error given the formula, %error=|acceptedvalue-experimentalvalue|x100/ Accepted value % error=|143.2990000 lbs -143 lbs |x100/ 143.2990000 lbs % error= 0.2086546 Hence, the 0.209% is the percent error 5.) In an experiment on determination of mass of a sample, your group consisting of 5 students obtained the following results: 14.34g, 14.32g, 14.33g,14.30g and 14.23g. Find the mean, a.d. and A.D. Suppose that your group is required to make only four determinations for the mass of the sample. If you are the leader of the group, which data will you omit? Recalculate the mean, a.d. and A.D. without this data. Which results will you prefer? The group consisting of 5 students obtaining the results : 14.34g, 14.32g,14.33g,14.30g and 14.23g
Mean: (14.34g + 14.32g + 14.33g + 14.30g+ 14.23g)/5 = 14.3g Average Deviation (ad): 14.34g - 0.026 cm 14.33g - 0.016 cm 14.32g - 0.006 cm 14.30g - 0.014 cm 14.23g - 0.034 cm =(0.026cm + 0.016cm + 0.006cm + 0.014cm + 0.034 cm)/5 =0.019 cm Average Deviation of the Mean (A.D.): =0.004 cm Ommitting 14.23 Mean:(14.34g + 14.32g + 14.33g + 14.30g)/4 = 14.323g Average Deviation (ad): 14.34g - 0.017 cm 14.33g - 0.007 cm 14.32g - 0.023 cm 14.30g - 0.043 cm 0.017 cm+0.007 cm+0.023 cm+0.043 cm / 4 =0.023 cm 7. References [1] Errors, Uncertainties and Measurements. Retrieved August 1, 2014 from http://www.scribd.com/doc/44118327/E rrors-Uncertainties-And-Measurements
[2]Dunn,. 2005 Measurement and Data analysis for Engineering and Science International Edition. Mc-Graw Hill USA: 1221 Avenue of the Americas [3] Wilson and Hernandez-Hall. 2010 Physics Laboratory Experiments Cengage Learning USA : 20 Channel Center Street Boston [4] Eaknen. 1950 Industrial Instrumentation John Wiley and Sons USA
