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Using Rulers, Vernier Calipers, and Micrometers

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Using Rulers, Vernier Calipers, and Micrometers MEASUREMENTS AND ERROR PROPAGATION OBJECTIVE 1. To learn how to use the following measuring devices and understand the uncertainties associated with them. a) meter stick b) metric ruler c) triple-beam balance d) digital balance e) vernier calipers f) micrometer 2. Use the following general error propagation equation to analyze the errors involved in making calculations involving measurements with their own uncertainty. 222 Lab Skills: Using∂∂∂ Rulers,fff222 Vernier Calipers, and σ fxyz=++σσσ ∂∂∂xyz ∗ GeneralMicrometers Error Propagation Equation THEORY Refer to lab handout on Error Propagation. 1 Using the Metric Ruler Using the Metric Ruler Consider the followingConsider standard the following metric standard ruler. metric ruler. The ruler is incrementedThe ruler is incremented in units in units of centimeters of centimeters (cm). (cm). The The smallest smallest scale division scale is divisiona tenth is a tenth of a centimeterof a centimeter or 1 mm. or 1 Therefore,mm. Therefore, the the uncertaintyuncertainty ∆x = smallest increment/2 = 1mm/2 = 0.5mm = 0.05cm. Note that a measurement made with this ruler must be stated to a tenth ∆x = (smallest increment) / 2 = 1mm/2 = 0.5mm = 0.05cm. 1 Note that a measurement made with this ruler must be stated to a tenth of a centimeter since the uncertainty is stated to a tenth of a centimeter. In the example above, the length of the object is clearly longer than 2.7 cm and less than 2.8 cm. It looks closer to 2.8 cm, so the value would be stated as x = 2.77 cm ± 0.05 cm. (You might think it looks more like x = 2.78 cm ± 0.05 cm, that would also be fine, it is still inside the uncertainty range. Make the best estimate you can.) 2 Using the Vernier Calipers The Vernier caliper is an instrument that allows you measure lengths much more accurately than the metric ruler. The smallest increment in the vernier caliper you will be using is (1/50) mm = 0.02 mm = 0.002 cm. Thus, the uncertainty is 1 ∆x = 2 0:002 cm = 0.001 cm. The vernier scale consists of a fixed metric scale and a sliding vernier scale. The fixed scale is divided into centimeters and millimeters, while the vernier scale is divided so that 50 divisions on it cover the same interval as 49 divisions on the main scale (at least this is the way the ones De Anza has are constructed). Thus, the length of each scale vernier division ∗These notes are largely drawn from the first lab explanation of Prof Eduardo Luna. 1 of a centimeter since the uncertainty is stated to a tenth of a centimeter. In the example above, the length of the object would be stated as x = 2.77 cm ± 0.05 cm. Using the Vernier Calipers The Vernier caliper is an instrument that allows you measure lengths much more accurate than the metric ruler. The smallest increment in the vernier caliper you will be using is (1/50)mm = 0.02mm = 0.002cm. Thus, the uncertainty is ∆x = (1/2)0.002 cm = 0.001 cm. 5 1. THE VERNIER SCALE Equipment List: C 3 X 5 card C one vernier caliper C one ruler incremented in millimeters What you will learn: This lab teaches how a vernier scale works and how to use it. I. Introduction:The vernier scale consists of a fixed metric scale and a sliding vernier scale. The fixed scale A vernieris divided sca leinto (Pierr centimeterse Vernier and millimeters,, ca. 1600) while the can vernier be scale used is divided on any so that measuring 50 device with a is 49/50 thedivisions length on it of cover a mainthe same scale interval division. as 49 divisions Close on the the main jaws scale. completely Thus, the length and note that the graduated scale.of each Most scale often vernier divisiona vernier is 49/50 sc alethe length is found of a main on scale leng division.th measuring Close the jaws device s such as first line atcompletely the far and left note on that the the vernier first line at scale the far (called left on the the vernier zero scale or (called index the line) “zero” coincides with the verniezeror ca linelipers onor the“index”or microme main line) scale.coincidesters. Carefully with A vethe rnierzero compare line instrument on the main and scale. see incr that Carefullyease thes comparethe first measur vernier and seeing division precision is 0.02 beyond what itthat would the first normallyvernier divisi onbe is with0.02 mm an short or dinaryof the first measuring main scale division, scale the like second a ruler or meter stick. mm short ofvernier the division first mainis 0.04 mm scale away division, from the seco thend second main scale vernier division, divisionand so on. isIf the 0.04 mm away from the secondjaws main are slightly scale opened division, it is easy and to sotell what on. fraction If the of jaws the main are scale slightly division opened the vernier it is easy to tell II. Howwhat a fraction verindexnie ofr has sy the movedst mainem by wor noting scaleks: which division vernier the division vernier best coincides index haswith a moved main scale by noting which vernier division. divisionA ver bestnier coincides scale slides with acr a mainoss a scalefixed division. main scale. The vernier scale shown below in figure 1 is subdiA measurementvided so is madethat with ten a vernierof its caliper divisi byons closing cor ther jawsespo onnd the objectto ni tone be divisions on the main measured and then reading the position where the zero line on the vernier falls on the1 main scale.2.1 When More tenscale. vernie The details measurementr divisions about is incomplete are how compressed until a an Vernier additional into fraction the Scale spac of a maine works of scale nine division main scale divisions we say the vernis determined.ier-scale This ratio is obtained is 10:9. by no Stingo the which divisions line on the vernieron the scale vernier (0,2,4,6,8) sc ale are not of a A vernier scalecoincides slides best with across a line aon fixed the main main scale. scale. The vernier scale shown below in figure 1 is standardsubdivided length so (i.e., that inche ten ofs or its ce divisionsntimeters), correspond but the to divisions nine divisions on the on main the mainscale scale.are always (10/9 As an example, let’s consider measuring the length of the aluminum block below. someis easierstandar tod seeleng thanth like 49/50.) millimeters When ten or vernier decimal divisions inches. are A compressedvernier scale into ena thebles space an of nine unambiguousmain scale interpolation divisions we saybetwe theen vernier-scale the smallest ratio divisions is 10:9. on So the the main divisions scale.2 on the vernier scale are not of a standard length (i.e., inches or centimeters), but the divisions on the main scale are always some standard length like millimeters or decimal inches. A vernier scale enables an unambiguous interpolation between the smallest divisions on the main scale. Figure 1: A 10/9 Vernier scale. Figure 1: Ten vernier divisions in the space of nine main scale divisions, a scale ratio of 10:9. 1This section is drawn from the first lab explanation of Prof David Newton. Since the vernier scale pictured above is2 constructed to have ten divisions in the space of nine on the main scale, any single division on the vernier scale is 0.1 divisions less than a division on the main scale. Naturally, this 0.1 difference can add up over many divisions. For 6 example, after six divisions have been spanned by both scales, the difference in length between the vernier and main scale would be 6 X 0.1 = 0.6 divisions. In figure 2 below, both the vernier and main scale start evenly at the left. After a distance6 of six increments they differ in length by 0.6 increments as is indicated by the two dotted lines. example, after six divisions have been spanned by both scales, the difference in length between the vernier and main scale would be 6 X 0.1 = 0.6 divisions. Since the vernier scale pictured above is constructed to have ten divisions in the space ofIn ninefigure on 2 the be mainlow, scale,both anythe singlevernie divisionr and ma onin the scale vernier star scalet evenly is 0.1 at divisions the lef lesst. After than a distancea of division six incre on thements main they scale. differ Naturally, in leng thisth 0.1 by difference 0.6 incre canments add as up is over indicated many divisions. by the two For example, after six divisions have been spanned by both scales, the difference in length dotted libetweennes. the vernier and main scale would be 6 × 0:1 = 0:6 divisions. In figure 2 below, both the vernier and main scale start evenly at the left. After a distance of six increments they differ in length by 0.6 increments as is indicated by the two dotted lines. Figure 2: In the span of six divisions, the difference between the vernier and main scale is 0.6 divisions. In practice, the left sides of the two scales are not matched up as above. Instead, the Figure 2: In the span of six divisions, the difference between the vernier and main scale is two left sides of each scaleFig ureare 2: offset In th eb yspan an ofa msioxu dnivit csioorns,res pthoen ddiffinereg tnceo the length measured.
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