Experiment 1,The measurement ofelastic modulus by tensile method1Experiment 1. The measurement of elastic modulus bystretching methodIntroductionAny object under the action of external forceoccur deformation,when thedeformation doesnot exceed a certain limit,deformation candisappear after thewithdrawal of external force, and this deformation is called elastic deformation.Whenthematerial deformselastically,the internal forcewill beproducedwithinthematerial which tend to bring the material back to its original shape.Themodulus ofelasticity commonly used in engineering technology is a physical quantity thatdescribes the degree of materials deformation and internal force density,which is ameasure of material's ability to resist elastic deformation.In this experiment we willmeasure the elastic modulus by stretching method.ExperimentalObjectives(1) Determine the elastic modulus of metal wire by stretching method.(2) Understand the principle of small length changes measurement by optical levermirror method, try to learn the specific measuring technique.(3) Learn how to process experimental data by least square method.Experimental InstrumentsElastic modulus stretcher (steel wire, optical lever apparatus composed by scale,mirror, optical lever mirror and telescope, digital tension meter, and etc.), tape,micrometer caliper and vernier caliperElastic modulus stretcher shown in fig1 was mainly composed by mounting frame,telescope system, digital tension meter and measuring tools
Experiment 1. The measurement of elastic modulus by tensile method 1 Experiment 1. The measurement of elastic modulus by stretching method Introduction Any object under the action of external force occur deformation, when the deformation does not exceed a certain limit, deformation can disappear after the withdrawal of external force, and this deformation is called elastic deformation. When the material deforms elastically, the internal force will be produced within the material which tend to bring the material back to its original shape. The modulus of elasticity commonly used in engineering technology is a physical quantity that describes the degree of materials deformation and internal force density, which is a measure of material’s ability to resist elastic deformation. In this experiment we will measure the elastic modulus by stretching method. Experimental Objectives (1) Determine the elastic modulus of metal wire by stretching method. (2) Understand the principle of small length changes measurement by optical lever mirror method, try to learn the specific measuring technique. (3) Learn how to process experimental data by least square method. Experimental Instruments Elastic modulus stretcher (steel wire, optical lever apparatus composed by scale, mirror, optical lever mirror and telescope, digital tension meter, and etc.), tape, micrometer caliper and vernier caliper. Elastic modulus stretcher shown in fig1 was mainly composed by mounting frame, telescope system, digital tension meter and measuring tools
Experiment I, The measurement of elastic modulus by tensile method2UpperclampingheadScaleSteel wireVerniercalipeMicrometereaCtapeOptical leverTelescopeDigital tensionmeterTension sensForce nutFig1-1Schematic diagramofelastic modulus system1.ExperimentalframeTheexperimental frameisthemainplatformformeasuringtheelasticmodulusofthetested wire. The wire is connected to the tension sensor through a clamping head, andthe force is applied by the rotation of force nut. The signal output of tension sensorthat represents thetension value of the wire is displayed via a digital tension meterThe pivot support of reflecting mirror of optical lever is fixed to a plate, and the reartoe is freely placed on the surface ofthe clamping head.2.TelescopeThe telescope system consists of a telescope bracket and a telescope. The telescopebracket can fine tune the telescope by adjusting the screw. The telescope has amagnification of 12 and a closest line of sight of 0.3 m. It contains eye-crossing lines(vertical andhorizontal).ThetelescopeisshowninFigure2
Experiment 1. The measurement of elastic modulus by tensile method 2 Fig 1-1 Schematic diagram of elastic modulus system 1. Experimental frame The experimental frame is the main platform for measuring the elastic modulus of the tested wire. The wire is connected to the tension sensor through a clamping head, and the force is applied by the rotation of force nut. The signal output of tension sensor that represents the tension value of the wire is displayed via a digital tension meter. The pivot support of reflecting mirror of optical lever is fixed to a plate, and the rear toe is freely placed on the surface of the clamping head. 2. Telescope The telescope system consists of a telescope bracket and a telescope. The telescope bracket can fine tune the telescope by adjusting the screw. The telescope has a magnification of 12 and a closest line of sight of 0.3 m. It contains eye-crossing lines (vertical and horizontal). The telescope is shown in Figure 2. Tension sensor Force nut Upper clamping head Optical lever Steel wire Digital tension meter Scale Telescope Vernier caliper Micrometer caliper tape
Experiment I, The measurement of elastic modulus by tensile method31. Blister mirror132. Round blister3. Round blister adiustment4. Horizontal micro adiustment5.Base106. Foot screw hand wheel7. Focusing hand wheel8. DialCrossing lines9. Objective lens10. Dial indicator11.Eyepiece12. Dust cover13. Sighting devicesFig1-2 Schematicdiagram oftelescope (Telescope appearance, cross lines, components)3.DigitaltensionmeterPowersupply:AC220V±10%，50HzDisplay range: 0~±19.99kgResolution:0.001kgZero cleaning function (Short press the clear button for zero cleaning)ContainsDCpoweroutputinterface:OutputDCpowerforpoweringthebacklightThepanel of digitaltensionmeter:ZI中数字拉力计清零传感器背光源02.86Kg120102四川世纪中科光电技术有限公司Fig1-3Thepanel of digital tensionmeterExperimental principle1. Elastic modulusAn even thickness metal wire with length I, and cross section S, and fasten its upperend, m isthe weight exerted on the lower end of the wire.Thus, F=mg is the forceexerted on the wire when it is extended by /. The force per unit area F/S is referred
Experiment 1. The measurement of elastic modulus by tensile method 3 Fig 1-2 Schematic diagram of telescope (Telescope appearance, cross lines, components) 3. Digital tension meter Power supply：AC220V±10%，50Hz Display range：0~±19.99kg Resolution：0.001kg Zero cleaning function（Short press the clear button for zero cleaning）. Contains DC power output interface: Output DC power for powering the backlight. The panel of digital tension meter： Fig 1-3 The panel of digital tension meter Experimental principle 1. Elastic modulus An even thickness metal wire with length l, and cross section S, and fasten its upper end, m is the weight exerted on the lower end of the wire. Thus, F=mg is the force exerted on the wire when it is extended by Δl. The force per unit area F/S is referred 1. Blister mirror 2. Round blister 3. Round blister adjustment 4. Horizontal micro adjustment 5. Base 6. Foot screw hand wheel 7. Focusing hand wheel 8. Dial 9. Objective lens 10. Dial indicator 11. Eyepiece 12. Dust cover 13. Sighting devices Crossing lines
Experiment 1,The measurement ofelastic modulus by tensile method4to as stress, the elongation per unit length l/l is referred to as strain. The stress F/Sis proportional to strain i/l within elastic limits. HenceF-EN(1-1)s-1This principle is known as Hooke's law, the proportional constant in this formula isreferred as elastic modulus of this material.E-F/S(1-2)/1The experiment has proven that the elastic modulus E is only depend on the materialof metal wire, which is independent of the external force, length of object I and crosssection S.According to equation (1-2), we could calculate the elastic modulus if all thequantities on the right side of the equation has been measured. The external forceapplied on the metal wire, the original length of metal wire and the cross-sectionalarea could be measured by common methods. The main problem in this experiment isthat the length change is so small, usually at the magnitude of 10-' mm, which is quitedifficult to realize the accurate measurement. In this experiment, we will measure thesmall variation of length relatively accurately via optical lever method.2.Theprincipleofoptical leverAt the beginning we assume that the scale no on the scale is imaged on the alignmentline of the telescope reticle. When the wire is stretched by Al, the rear leg of theoptical lever falls subsequntally by l, which will drive the plane mirror to rotate by, at this position, the value on the scale is imaged on the alignment line, the changein scale value n = n1 - no is proportional to l, when Al < H, is very small.ThereforeN_ An/2(1-3)b-H
Experiment 1. The measurement of elastic modulus by tensile method 4 to as stress, the elongation per unit length ∆𝑙/𝑙 is referred to as strain. The stress F/S is proportional to strain ∆𝑙/𝑙 within elastic limits. Hence l l E S F = (1-1) This principle is known as Hooke's law; the proportional constant in this formula is referred as elastic modulus of this material. l l F S E = (1-2) The experiment has proven that the elastic modulus E is only depend on the material of metal wire, which is independent of the external force, length of object l and cross section S. According to equation (1-2), we could calculate the elastic modulus if all the quantities on the right side of the equation has been measured. The external force applied on the metal wire, the original length of metal wire and the cross-sectional area could be measured by common methods. The main problem in this experiment is that the length change is so small, usually at the magnitude of 10-1 mm, which is quite difficult to realize the accurate measurement. In this experiment, we will measure the small variation of length relatively accurately via optical lever method. 2. The principle of optical lever At the beginning we assume that the scale n0 on the scale is imaged on the alignment line of the telescope reticle. When the wire is stretched by ∆𝑙, the rear leg of the optical lever falls subsequntally by ∆𝑙, which will drive the plane mirror to rotate by 𝜃, at this position, the value on the scale is imaged on the alignment line, the change in scale value ∆𝑛 = 𝑛1 − 𝑛0 is proportional to ∆𝑙, when ∆𝑙 ≪ 𝐻, 𝜃 is very small. Therefore H n b l 2 = (1-3)
Experiment I, The measurement of elastic modulus by tensile method5AnScaleLnnoHRear feetTelescopeALReflectingmirrorFigure1-4Theprinciple of optical leverWhere H is the distance between optical lever mirror and scale, b is vertical distancebetween the rear leg of the mirror and twofront legs,b is called optical lever constant,thenbN/ =(1-4).An2HSubstitute formula (1-4) into formula (1-2), we get2FIHE=(1-5)SbAnSubstut the cross-sectional area SDofthesteelwire(Disthediameterof4steelwire),thentheelasticmodulusis8FIHE=(1-6)元DbnIf the force applied on the steel wire is Fi, the corresponding readings of the scale is ni,then,81H(1-7)n,F + ng元D"bEn, is proportional to Fi, the elastic modulus E can be derived from straight line's slopethat obtained by the least squares method.Experimental Procedure1)Turn on the digital dynamometerpower switchand preheatfor10minutes.The
Experiment 1. The measurement of elastic modulus by tensile method 5 n0 H O n1 θ θ 2θ b Δl Δn N Nn Reflecting mirror Telescope Scale Rear feet Figure 1-4 The principle of optical lever Where H is the distance between optical lever mirror and scale, b is vertical distance between the rear leg of the mirror and two front legs, b is called optical lever constant, then n H b l = 2 (1-4) Substitute formula (1-4) into formula (1-2), we get Sb n FlH E = 2 (1-5) Substitute the cross-sectional area 4 2 D S = of the steel wire（D is the diameter of steel wire）, then the elastic modulus is D b n FlH E = 2 8 (1-6) If the force applied on the steel wire is Fi , the corresponding readings of the scale is ni, then, 2 0 8 F n D bE lH ni = i + (1-7) ni is proportional to Fi, the elastic modulus E can be derived from straight line's slope that obtained by the least squares method. Experimental Procedure 1）Turn on the digital dynamometer power switch and preheat for 10 minutes. The
Experiment 1, The measurement of elastic modulus by tensile method6backlight is illuminated and the scale should be clearly visible. The force applied tothe wire at this time is displayed on the panel of digital tension meter.2) Loosen the locking screw on the rear leg of the optical lever and adjust the rear legof the optical lever to the appropriate length (It is advisable that the rear toe can be asclose as possible but not against the wire, and the two front legs can be placed on thesame concave platen). Use the three-foot tip to press three shallow marks on the flatpaper, draw the height of the two front legs by drawing a thin line (the optical leverconstant), and then measure the length of optical lever constant b with a verniercaliper., and the experimental data of 6 time measurements is recorded in Table 1.Place the optical lever on the platen with the back toe close to the wire and the backtoe should be directly in front of the wire.3) Rotate the force-applying nut, first make the digital tension meter display less than2.5kg, then apply the force from small to large (avoid contrarotation), apply a certainpre-tension (3.00±0.02kg) to the wire to straightened the folded place of wire4) Measure the original length L of the wire with a steel tape measure. The beginningof the steel tape is placed on the lower surface of the upper clamp head of the wire,and the other end is aligned with the upper surface of the lower clamping head.Repeatyourmeasurementsfor6times andtheexperimental data isrecorded in Table2.5) Measure the vertical distance H from the center of the reflecting mirror to the scaleby a steel tape measure. The beginning of the steel tape is placed on the upper surfaceof the scale and the other end is aligned with the center of the reflecting mirror.Repeatyourmeasurementsfor6timesandtheexperimental dataisrecorded inTable36) Measuring the diameter d 规i of the wire at different positions and in differentdirections with a spiral micrometer (atleast 6 times)and recordthe zero deviation ofthe spiral micrometer dobefore your measurement. The experimental data is recordedinTable47) Move the telescope closer to face the experimental platform plate (the distancebetween thefront edge of thetelescope and the edge of the platform plate should be inthe range of 0~30cm). Use the coarse aiming device to adjust the position of thetelescope so that it is facing the wire. Adjust the telescopic foot screw hand wheel tomake the bubble in the center.8)Adjust the telescope so that it is facing the center of reflecting mirror, then carefullyadjust the angle of the reflecting mirror until you can see the bright light emitting
Experiment 1. The measurement of elastic modulus by tensile method 6 backlight is illuminated and the scale should be clearly visible. The force applied to the wire at this time is displayed on the panel of digital tension meter. 2) Loosen the locking screw on the rear leg of the optical lever and adjust the rear leg of the optical lever to the appropriate length (It is advisable that the rear toe can be as close as possible but not against the wire, and the two front legs can be placed on the same concave platen). Use the three-foot tip to press three shallow marks on the flat paper, draw the height of the two front legs by drawing a thin line (the optical lever constant), and then measure the length of optical lever constant b with a vernier caliper., and the experimental data of 6 time measurements is recorded in Table 1. Place the optical lever on the platen with the back toe close to the wire and the back toe should be directly in front of the wire. 3) Rotate the force-applying nut, first make the digital tension meter display less than 2.5kg, then apply the force from small to large (avoid contrarotation), apply a certain pre-tension (3.00±0.02kg) to the wire to straightened the folded place of wire. 4) Measure the original length L of the wire with a steel tape measure. The beginning of the steel tape is placed on the lower surface of the upper clamp head of the wire, and the other end is aligned with the upper surface of the lower clamping head. Repeat your measurements for 6 times and the experimental data is recorded in Table 2. 5) Measure the vertical distance H from the center of the reflecting mirror to the scale by a steel tape measure. The beginning of the steel tape is placed on the upper surface of the scale and the other end is aligned with the center of the reflecting mirror. Repeat your measurements for 6 times and the experimental data is recorded in Table 3. 6) Measuring the diameter d 视 j of the wire at different positions and in different directions with a spiral micrometer (at least 6 times) and record the zero deviation of the spiral micrometer d0 before your measurement. The experimental data is recorded in Table 4. 7) Move the telescope closer to face the experimental platform plate (the distance between the front edge of the telescope and the edge of the platform plate should be in the range of 0~30cm). Use the coarse aiming device to adjust the position of the telescope so that it is facing the wire. Adjust the telescopic foot screw hand wheel to make the bubble in the center. 8)Adjust the telescope so that it is facing the center of reflecting mirror, then carefully adjust the angle of the reflecting mirror until you can see the bright light emitting
Experiment I, The measurement of elastic modulus by tensile method7fromthe scalebacklightfromthetelescope9) Adjust the dioptric adjustment hand wheel so that the cross lines is clearly visibleAdjustthefocusinghandwheel sothatthe image oftherulerinthefieldof viewisclearly visible. Rotate the telescope so that the horizontal line of cross lines is parallelto the scale line of the ruler, and then adjust the focusing hand wheel again to makethe image of the ruler inthefield of view clearly visible10) Carefully adjust the angle of the reflecting mirror again so that the horizontal lineof cross lines is aligned with the scale line ≤ 2.0cm (to avoid the exceeding of thescale range at the end of your experiment). Move the bracket horizontally so that thevertical lineof cross linesisalignedwiththecenteroftherulerThe telescope cannot be adjusted in the following steps, and try not to havevibration on the test table to ensure the stability of the telescope. After the forceand force reduction process, the force nut cannot be contrarotated.11) Click the “clear" button on the digital dynamometer, and record the scale value ofthe horizontal line of cross lines nt.12) Slowly rotate the force-applying nut, gradually increase the tension of the wire,record the scale nevery 1.00 (±0.02)kg, add force to 9kg (±0.02),add about 0.5kgafter data recording (if less than 1.Okg, no data is recorded). Then rotate the force nutback to9kg (±0.02)and record thedata. Similarly,gradually reduce the tension of thewire, and record the scale ni every 1.00 (±0.02) kg until the tension is 0.00 (±0.02)kg. Record the above data in the corresponding position in Table 3.13)After completion of the experiment, loosen the force nut to freely extend thewire and turn off the digital dynamometerData recordingTable 1 Optical constant mearuement234Sequencenumberi5Db;(mm)Table2Original length measurement2345SequencenumberiDL;(mm)
Experiment 1. The measurement of elastic modulus by tensile method 7 from the scale backlight from the telescope. 9) Adjust the dioptric adjustment hand wheel so that the cross lines is clearly visible. Adjust the focusing hand wheel so that the image of the ruler in the field of view is clearly visible. Rotate the telescope so that the horizontal line of cross lines is parallel to the scale line of the ruler, and then adjust the focusing hand wheel again to make the image of the ruler in the field of view clearly visible. 10) Carefully adjust the angle of the reflecting mirror again so that the horizontal line of cross lines is aligned with the scale line ≤ 2.0cm (to avoid the exceeding of the scale range at the end of your experiment). Move the bracket horizontally so that the vertical line of cross lines is aligned with the center of the ruler. The telescope cannot be adjusted in the following steps, and try not to have vibration on the test table to ensure the stability of the telescope. After the force and force reduction process, the force nut cannot be contrarotated. 11) Click the “clear” button on the digital dynamometer, and record the scale value of the horizontal line of cross lines n1. 12) Slowly rotate the force-applying nut, gradually increase the tension of the wire, record the scale ni + every 1.00 (±0.02) kg, add force to 9kg (±0.02), add about 0.5kg after data recording (if less than 1.0kg, no data is recorded). Then rotate the force nut back to 9kg (±0.02) and record the data. Similarly, gradually reduce the tension of the wire, and record the scale ni − every 1.00 (±0.02) kg until the tension is 0.00 ( ±0.02) kg. Record the above data in the corresponding position in Table 3. 13) After completion of the experiment, loosen the force nut to freely extend the wire and turn off the digital dynamometer Data recording Table 1 Optical constant mearuement Sequence number i 1 2 3 4 5 6 bi(mm) Table 2 Original length measurement Sequence number i 1 2 3 4 5 6 Li(mm)
Experiment I, The measurement of elastic modulus by tensile method8Table3theverticaldistancebetween optical levermirrorand scaleSequence numberiH(mm)Table4thediametermeasurement ofthe steelwirezerodeviationofthespiralmicrometerdo-mmSequence number354N1Average valueDiameterd规(mm)Table 5the scale and corresponding force during the process of force adding and reducing2345678910Sequence numberiForce Fi0.00(kg)The scale duringforceaddingprocessnt(mm)The scale duringforce reducingprocessni(mm)Average scale (mm)ni =(nit+ ni)/2Notices(1) Once the adjustment of optical system is finished, including a telescope, a lightlever mirror,a planemirror,do not disturb the setup or try to make anyadjustmentwhen you measuring the value of scale n and the distance B between the planemirror to the scale. otherwise the measured data will be invalid(2)Donottouchtheoptical surfaceof theopticsbyhand.(3) Care must be taken to prevent the optical lever mirror from falling and break themirror.(4) The experimental frame have the maximum force limiting function. The maximumactual force in the experiment should not exceed 13.00kg.DataAnalysis and Results1 Calculate the optimal value and uncertainty of optical lever constant, the original
Experiment 1. The measurement of elastic modulus by tensile method 8 Table 3 the vertical distance between optical lever mirror and scale Sequence number i 1 2 3 4 5 6 Hi(mm) Table 4 the diameter measurement of the steel wire zero deviation of the spiral micrometer d0= mm Sequence number i 1 2 3 4 5 6 Average value Diameter d 视 i(mm) Table 5 the scale and corresponding force during the process of force adding and reducing Sequence number i 1 2 3 4 5 6 7 8 9 10 Force Fi (kg) 0.00 The scale during force adding process ni + (mm) The scale during force reducing process ni − (mm) Average scale (mm) ni =( ni ++ ni − )/2 Notices (1) Once the adjustment of optical system is finished, including a telescope, a light lever mirror, a plane mirror, do not disturb the setup or try to make any adjustment when you measuring the value of scale n and the distance B between the plane mirror to the scale, otherwise the measured data will be invalid. (2) Do not touch the optical surface of the optics by hand. (3) Care must be taken to prevent the optical lever mirror from falling and break the mirror. (4) The experimental frame have the maximum force limiting function. The maximum actual force in the experiment should not exceed 13.00kg. Data Analysis and Results 1 Calculate the optimal value and uncertainty of optical lever constant, the original
Experiment 1, The measurement ofelastic modulus by tensile method9lengthofthewire,theverticaldistancefromthecenterofthemirrortothescale,andthe wire diameter. Write the result expression and the processes of your calculationThe error limits of the optical lever constant, the original length of the steel wire andthe diameter of the steel wire are 0.02 mm, 0.5 mm, 0.005 mm respectively.2 According to formula (1-7), calculate the slope of the line by least squares method(Without seeking the uncertainty of k, this experiment ignores this amount in orderto reduce the amount of calculation.), and then obtain the modulus of elasticity E anditsuncertainty.Hint: The uncertainty formula of the elasticU,UUH+(2UDy(U.modulus is calculated as follows, where the)+(EVHDb1uncertainty ofthe slopek is not consideredQuestions1. What is the principle of measuring the small change in length by the optical levermirrormethod?Whichquantities are related to its magnification?Howto improvethe sensitivity of small elongation measurement by the optical lever mirror method?Isthereanylimittothesensitivityincrease?2.Why do we need to measure multiple lengths with different length measuringinstruments? Try to analyze which of your experimental results present the largesterror, and how to improve that?
Experiment 1. The measurement of elastic modulus by tensile method 9 length of the wire, the vertical distance from the center of the mirror to the scale, and the wire diameter. Write the result expression and the processes of your calculation. The error limits of the optical lever constant, the original length of the steel wire and the diameter of the steel wire are 0.02 mm, 0.5 mm, 0.005 mm respectively. 2 According to formula (1-7), calculate the slope of the line by least squares method (Without seeking the uncertainty of k, this experiment ignores this amount in order to reduce the amount of calculation.), and then obtain the modulus of elasticity E and its uncertainty. Hint: The uncertainty formula of the elastic modulus is calculated as follows, where the uncertainty of the slope k is not considered. Questions 1. What is the principle of measuring the small change in length by the optical lever mirror method? Which quantities are related to its magnification? How to improve the sensitivity of small elongation measurement by the optical lever mirror method? Is there any limit to the sensitivity increase? 2. Why do we need to measure multiple lengths with different length measuring instruments? Try to analyze which of your experimental results present the largest error, and how to improve that? E 2 2 2 2 ( ) ( ) (2 ) ( ) U U U U U l b H D E l H D b = + + +