MC-1b  Errors and the Density of a Solid

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Introduction

Micrometer

vernier

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Density

OBJECTIVES:

To learn about systematic and random errors; to understand significant figures; to estimate the reliability of one's measurements; and to calculate the reliability of the final result.

NOTE: This experiment illustrates the earlier sections on Errors and Significant Figures. The actual density of the metal is incidental. However, the accuracy of your estimate of reliability will show whether you have mastered the material in the earlier sections.

APPARATUS:

Metal cylinders of varying sizes, micrometer and vernier calipers, precision gauge blocks, precision balance.

Pre-lab Quiz

You should be able to complete this brief quiz before proceding.

• Expt. M01 Quiz

PRECAUTIONS:

INTRODUCTION:

First read the material on Errors and Significant Figures in the Manual Introduction. Since density is the mass per unit volume, you must measure the mass (on a balance) and compute the volume ( $h \pi r^2 = h \pi d^2 / 4$) from measurements of the cylinder's dimensions where $d$ is the diameter and $h$ is the height. Any one of three length measuring devices may be used. These include a micrometer, a vernier caliper and/or a simple metric ruler. The micrometer will permit the highest precision measurments but using one can be cumbersome, especially when reading the vernier scale. All methods will demonstrate the aforementioned objectives. Your instructor will give you guidance in choosing an appropriate measurement device.

THE MICROMETER:

Record the serial number of your micrometer. Then familiarize yourself with the operation of the caliper and the reading of the scales: work through Appendix on the Micrometer. Note that use of the ``friction'' head in closing the jaws insures the same pressure on the measured object each time. Always estimate tenths of the smallest division. Some micrometers have verniers to assist the estimation.

THE VERNIER CALIPER:

Work through Appendix on the Vernier Caliper. You should also try the java vernier at java vernier . Experiment with one of the large verniers in the lab until you are sure you understand it. Note that verniers need not be decimal: for many inch scales the vernier estimates 1/8's of the 1/16 inch division, i.e. 1/128's of an inch. However vernier calipers divide the inch into 50 divisions and the vernier estimates 1/25 of the 1/50 inch divisions, i.e. 1/1000 inch or 1 mil. The vernier was invented 1631 by Pierre Vernier.

Precautions on use of the calipers:

1. Unclamp both top thumbscrews to permit moving caliper jaws.

2. Open caliper to within a few mm of the dimension being measured.

3. Close right thumbscrew to lock position of lower horizontal knurled cylinder which executes fine motions of caliper jaw. Never over tighten!

CALIBRATION OF THE MICROMETER (or VERNIER, ETC.):

1. OPTIONAL: Wipe the micrometer caliper jaws with cleaning paper. Then determine the zero error by closing the jaws. Make and record five or six readings. The variation of these repeated readings gives you an estimate of the reproducibility of the measurements. (For those using the micrometers they have been given a small zero error. Thus a zero error correction is necessary.) In general any measurement device can have a zero error.

2. Measure one or four (OPTIONAL) calibration gauge blocks (6, 12, 18, and 24 mm): Set the gauge blocks on end, well-in from the edge of the table, and thus freeing both hands to handle the caliper. Record the actual (uncorrected) readings and take several readings (3 to 5) of each gauge block used. If measuring only a single block choose one with a thickness intermediate to the diameter and height of the metal cylinder.

3. If using a single precision block obtain a correction factor you will use to minimize systematic measurement errors. Otherwise, plot a correction curve for your micrometer, i.e. plot errors as ordinates and nominal blocks sizes (0, 6, 12, 18, and 24 mm) as abscissa. Normally the correction will not vary from block to block by more than 0.003 mm (for the micrometer). If it is larger, consult your instructor.

DENSITY DETERMINATION:

1. Make five measurements (should be in millimeters) of the height and five of the diameter. Since our object is to determine the volume of the cylinder, distribute your measurements so as to get an appropriate average length and average diameter. Avoid any small projections which would result in a misleading measurement. If not possible to avoid, estimate their importance to the result. Record actual readings and indicate how you distributed them.

2. Calculate the average length and average diameter reading, and then use your correction value (curve) to correct these average readings. If you were to use the uncorrected values, how much relative error would this introduce?

3. Calculate the standard deviation for your length and diameter averages.

4. Weigh the cylinder a few times on the electronic balance; estimate to 0.1 mg.

5. From the average dimensions and the mass, calculate the volume and density. Make a quantitative estimate of the uncertainty in the density. You should use, as your starting point:
   
   $$\mbox{Density}(\rho\pm \Delta \rho) =\rho(h\pm\Delta h,d \pm \Delta d, m \pm \Delta m)=\frac{m}{\pi (d/2)^2 h}$$
   
   

6. Compare the density with the tabulated value. Tabulated values are averages over samples whose densities vary slightly depending upon how the material was cast and worked; also on impurity concentrations.

   
   Link to a web-based periodic table.

7. In your notebook or lab form summarize the data and results. Also record your result on the blackboard. Is the distribution of blackboard values reasonable, i.e. ``normal'' distribution?

OPTIONAL: To test how accurately you can estimate a fraction of a division, estimate the fractions on the vernier caliper before reading the vernier. Record both your estimate and the vernier reading.

Related facts and URL links:

Question: Why are there are exactly 25.4 mm in 1 inch?

Answer(not verified): The Treaty of the Meter (Convention du Metre) in the late 19th century established the first centralized international system of metrology. This defined the meter.

In 1959, the countries of the world that were using Imperial units defined them uniformly based on the metric units. The inch was simply defined that way and agreed to by all. Before 1959, different countries related inches to meters in other ways. Among them was the United States. The Metric Act of 1866 defined the meter in terms of inches (i.e., before the Treaty of the Meter), and that relationship had continued to been used even after the Treaty fixed the length of the meter.

Changing the definition in the U.S. in 1959 caused very little problem, except for the U.S. Geological Survey. When you deal with things 10$^5$ meters big (like the sizes of the states), even 1 part in 10$^5$ changes affect the specifications of boundary lines in significant ways. So, to this day the U.S. has two different systems of inches, feet, and yards (in the ratio of 36:3:1, for both). There's the usual inch, foot, and yard; and there's the survey foot (and inch and yard), which is based on the pre-1959 definition. (The ambiguity goes away, of course, when metric specification is used. Newer USGS maps are metric.)

Links to metrology site(s):
National Institute of Science & Technology