Using Rulers, Vernier Calipers, and Micrometers

Home , Calipers, Micrometer, Ruler, Vernier scale

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.

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 scalemm 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. In 0.6 divisions. between the vernier and main scale is 0.6 divisions. figure 3, the two scales are still off by 0.6 divisions as in figure 2 above, however in figure 3 the scales match up along the dotted vertical line on the right side instead of matching up on When using the Vernier calipers for a measurement, the object in the jaws will cause an left side asoffsetI nin prac fig inuretice, the two2. the I scalesn le figft onsidesure the 3, ofleft w theeedge would two of thesca say scales.les theare In twonot figure matched dotted 3, the twover up scalesticalas above. arelines still I nstead,on off the by leftthe side of thetwo fi leftgur0.6 sidese a divisionsre ofse paeach asra intescd figureale by a re0. 2 above,6offset divi howeversioby nsan aofm in leo figureungnt thc 3o ri thenre sth scalespeo nsadime matchng tseo t upnshee alongl ethangt theh th me dottede twasuor elidne. Ins in figurefig ure2 a 3,revertical thesepara two lineted scale on by thes are0.6 right stilldivisions. side off instead by 0.6 S oftudy divisions matching and compa up as on in leftfigurere sidefigur 2 as above,es in figure2 and how 2. 3 In everto figure unde in fig 3,rstandure 3 how we would say the two dotted vertical lines on the left side of the figure are separated by a verthenier sca sylesstem match up along the dotted vertical line on the right side instead of matching up on left side0.6 asdivisions in figure of 2. length In fig inure the 3, same we sensewould that say the the two two lines dotted in figure ver 2tical are separatedlines on the by 0.6left side works. divisions. Study and compare figures 2 and 3 to understand how a vernier system works. of the figure are separated by 0.6 divisions of length in the same sense that the two lines in figure 2 are separated by 0.6 divisions. Study and compare figures 2 and 3 to understand how a vernier system works.

Figure 3: The oﬀset on the left is 0.6 divisions, so now the 6th line aligns exactly with the main scale. Figure 3: A length at the left has a 0.6 division difference also.

Figure 3: A length at the 3left has a 0.6 division difference also. 2.2 Making a measurement A measurement is made with a vernier caliper by closing the jaws on the object to be measured and then reading the position where the zero line on the vernier falls on the main scale. The measurement is incomplete until an additional fraction of a main scale division is determined. This is obtained by noting which line on the vernier scale (0,2,4,6,8) coincides best with a line on the main scale. As an example, lets consider measuring the length of the aluminum block below.

Note that the zero line on the vernier scale falls between the 4.4 cm and 4.5 cm mark on the main scale. Thus, the first significant digits are 4.4 cm. The remaining two digits are Note that theobtained zero by linenoting on which the line vernier on the vernier scale sc fallsale (0,2,4,6,8) between coincides the 4.4 best cm with and a line 4.5 on cm mark on the main scale.the Thus,main scale. the Looking ﬁrst signiﬁcantclosely at the picture digits below are indicates 4.4 cm. that Thethe 46remaining line lines up the two digits are closest. Therefore, the reading is 4.446 cm. Or in standard form 4.446 cm ± 0.001 cm. obtained byNote noting that the whichzero line lineon the on vernier the scale vernier falls between scale the (0,2,4,6,8) 4.4 cm and 4.5 coincides cm mark on best the with a line on the main scale.main scale. Looking Thus, the closely first significant at the digi picturets are 4.4 below cm. The indicates remaining two that digits the are 4.6 line lines up the closest withobtained a mark by noting on thewhich main line on scale. the vernier Therefore, scale (0,2,4,6,8) the coincides reading best is with 4.446 a line cm. on Or in standard the main scale. Looking closely at the picture below indicates that the 46 line lines up the form 4.446closest. cm ± Therefore,0.001 cm. the reading is 4.446 cm. Or in standard form 4.446 cm ± 0.001 cm.

3 4 Using The Micrometer Caliper

The micrometer caliper has a linear scale engraved on its sleeve and a circular scale engraved on what is properly called the thimble. The linear scale is divided into divisions of 1 mm and is 25 mm long. Half-millimeter marks are provided below the linear scale. The circular scale has 50 divisions. One complete revolution of the circular scale moves the thimble 0.5 mm along the linear scale so that the distance between the jaws is also changed to 0.5 mm. Since the circular scale has 50 divisions, rotating it through one circular-scale division, will cause the rod to move through a distance equal to 1/50 of 0.5 mm, or 0.01 mm. Thus the numbers on the circular scale represent hundredths of a millimeter. A 3 Usingmicrometer The Micrometer caliper can be used to Caliper measure lengths directly to 0.01 mm, and by estimating tenths of a circular scale division, it can be used to estimate lengths to 0.001 mm. Measurements made with a The micrometermicrometer caliper has caliper a linear can scalebe estimated engraved to thousandths on its sleeve of anda millimeter. a circular Th scalee total engraved micrometer caliper reading on what is properlyis the sum called of the the readings thimble. on Thethe main linear and scale circular is dividedscales. into divisions of 1 mm and is 25 mm long. Half-millimeter marks are provided below the linear scale. The circular scale has 50 divisions. One complete revolution of the circular scale moves the thimble 0.5 In the figure to the left, notice the main scale mm along the linear scale so that the distance between the jaws iswhich also changed is marked to with 0.5 mm. a 0 and a 5. These Since the circular scale has 50 divisions, rotating it through one circular-scaleindicate millimeters. division, Below will the main scale cause the rod to move through a distance equal to 1/50 of 0.5 mm,are or marks 0.01 at mm. the Thus half way the points between numbers on the circular scale represent hundredths of a millimeter.mm A micrometer marks. These caliper l/2 mm can marks are present be used to measure lengths directly to 0.01 mm, and by estimatingbecause tenths one of arevolution circular scaleof the thimble moves the thimble only 1/2 mm down the barrel. division, it can be used to estimate lengths to 0.001-0.002 mm. Measurements made with a micrometer caliper can be estimated to thousandths of a millimeter. The total micrometer caliper reading is the sum of the readings on the main and circular scales. In the ﬁgure to the left, notice thePossible main scale settings which and is readings marked of with the micrometer a 0 and a 5. caliper These are indicate shown millimeters. here. In the first example, the sixth main scale mark is visible just to the left of the circular scale. This means we have a reading Below the main scalesomewhere are marks between at the6.0 halfand 6.5 way mm. points The between line on the mm main marks. scale Thesepoints to l/2 the mm barrel about halfway marks are presentbetween because 19 and one 20 revolution divisions. ofTherefore, the thimble the micrometer moves the calliper thimble reading only is 1/2 6.0 mmmm + 0.19 mm + 0.005 down the barrel.mm Possible = 6.195 settings mm = 0.006195 and readings meters. of the micrometer caliper are shown here.

In the second example, the half-millimeter mark to the right of the sixth main scale mark is visible. So In the example,the reading the half-millimeteris somewhere between mark 6.5 to theand 7.0 right mm. of The the line sixth on mainthe main scale scale mark points is to a point of the visible. So thebarrel reading just isslightly somewhere past the between 41 mark. 6.5 So andwe can 7.0 estimate mm. The the last line place on the of the main reading scale to be 0.002 mm. points to a pointThe ofreading the barrel is then just 6.5 slightlymm + 0.41 past mm the + 410.002 mark. mm So= 6.912 we can mm estimate = 0.0069 the 12 lastmeters. This shows place of the readingthat we to can be estimate 0.002 mm. micrometer The reading readin isgs then to one 6.5 thousandth mm + 0.41 of mm a millimeter. + 0.002 mm = 6.912 mm = 0.0069 12 meters. This shows that we can estimate micrometer readings to −3 one thousandth of a millimeter. The ﬁnal value is (6.912 ± 0.005) × 10 m.