Experment 1 Mass, volume and densty Purpose 1. Famlarze wth basc measurement tools such as verner calper, mcrometer, and laboratory balance. 2. Learn how to use the concepts of sgnfcant fgures, expermental uncertanty (error) and some methods of error and data analyss n your expermental measurements. Equpment A steel ball, a rectangular alumnum block, a brass cylnder, an alumnum annular cylnder, verner calper, mcrometer, laboratory balance. Basc measurement tools 1. Least Count of an Instrument Scale Least Count of an nstrument s the smallest subdvson marked on ts scale. Ths s the unt of the smallest readng that can be made wthout estmatng. Fgure 1 Least count Meterstck s commonly calbrated n centmeter (numbered major dvson) wth a least count, or smallest marked subdvson, of mllmeter. 2. Verner Calper Internal measurng jaws Lockng screw Man scale (nch) Fgure 2 Verner Calper Fne adjustment Verner scale (nch) Verner scale (cm) External measurng jaws Man scale (cm) Depth probe The verner calper n Fg. 2 conssts of a rule wth two man engraved scales (n nch and cm respectvely) and movable jaws wth engraved verner scales (.e., small movable graduated scales for obtanng fractonal parts of subdvsons on a fxed man scale of a measurng nstrument). The span of the lower external measurng jaw s used to measure length and s partcularly convenent for measurng the dameter of a cylndrcal object. The span of the upper nternal measurng jaw s used to measure dstances between two surfaces (e.g. the nsde dameter of a hollow cylndrcal object). In ths experment we only use the cm man scale whch s calbrated n cm wth a mm least count, and the movable verner scale has 10 dvsons that cover 9 dvsons on the man scale. Fgure 3 shows an example of readng the verner scale on a calper. 9
Man scale Verner zero mark 1 cm + 0.2 cm + 0.03 cm + 0.000 cm = 1.230 cm (major dvson) (mnor dvson) (algned mark) (estmated of doubt) (a) Man scale Verner zero mark 1 cm + 0.2 cm + 0.025 cm = 1.225 cm (major dvson) (mnor dvson) (phase change for 2 and 3 marks) (b) Fgure 3 Verner scale. Examples of readng the verner scale on a calper. The frst two sgnfcant fgures are read drectly from the man scale. The verner zero mark s past the 2-mm lne after the 1-cm major dvson mark, so we have 1.2 cm. The next sgnfcant fgure s the fractonal part of the smallest subdvson on the man scale. Ths s obtaned by referrng to the verner scale markngs. If a verner mark concdes wth a mark on the man scale, then the mark number s the fractonal part of the man-scale dvson. In Fg. 3a, ths s the thrd mark to the rght of the verner zero, so the thrd sgnfcant fgure for a readng s 3 (0.03 cm). Fnally, snce the 0.03-cm readng s known exactly, a zero s added as the doubtful fgure for a readng of 1.230 cm or 12.30 mm. Note how the verner scale gves more sgnfcant fgures or extends the precson. However, a mark on the verner scale may not always lne up exactly wth one on the man scale (Fg. 3b). In ths case, there s more uncertanty n the 0.001-cm or 0.01-mm fgure. In Fg. 3b, the second verner mark after the zero s to the rght of the closest man-scale mark and the thrd verner mark s to the left of the closest man-scale mark. Hence, the marks change phase between 1.22 cm and 1.23 cm. Most verner scales are not fne enough to make an estmate of the doubtful fgure, so a suggested method s to take the mddle of the dgt for a readng of 1.225 cm. Before makng a measurement, the zero of the verner calper should be checked wth the jaws completely closed. It s possble that through msuse the calper s no longer zeroed. And thus gves erroneous readngs (systematc error). If ths s the case, a zero correcton must be made for each readng. In Fg. 4 (b), the zero readng s +0.05 cm and ths amount must be subtracted from each measurement readng. Smlarly, f the zero readng s negatve, or the verner zero les to the left of the man-scale zero, the measurements wll be too small and the zero correcton must be added to the measurement readngs. 10
Man scale Verner zero mark (a) 0.00 Properly zeroed Man scale Verner zero mark 0.05 (b) c Postve error + 0.05 cm (subtracted from measurement readng) Fgure 4 Zerong and systematc error (zerong the verner calper wth the jaws closed) 3. Mcrometer Calper Fgure 5 shows a mcrometer and an example of a mcrometer readng. The mcrometer provdes for accurate measurements of small lengths and s partcularly convenent n measurng the dameters of thn wres and the thckness of thn sheet. It conssts of a movable spndle (jaw) that s advanced toward another, parallel-faced jaw (called an anvl) by rotatng the thmble. The thmble rotates over an engraved sleeve (or barrel ) mounted on a sold frame. Most mcrometers are equpped wth a ratchet (ratchet handle to far rght n Fg. 5) whch allows slppage of the screw mechansm when a small and constant force s exerted on the jaw. Ths permts the jaw to be tghtened on an object wth the same amount of force each tme. Care should be taken not to force the screw, so as not to damage the measured object and/or the mcrometer. The axal man scale on the sleeve s calbrated n mllmeters, and the thmble scale s calbrated 0.01 mm. The movement mechansm of the mcrometer s a carefully machned screw wth a ptch of 0.5 mm. The axal lne on the sleeve man scale serves as a readng lne. Snce the ptch of the screw s 0.5 mm and there are 50 dvsons on the thmble, when the thmble s turned through one of ts dvsons, the thmble moves 1 50 of 0.5 mm or 0.01 mm ( 1 50 * 0.5 mm = 0.01 mm). One complete rotaton of the thmble (50 dvsons) moves t through 0.5 mm, and second rotaton moves t through another 0.5 mm. 11
Fgure 5 Mcrometer (a) Ths partcular mcrometer has the 1.0 mm and 0.5 mm scale dvsons below the readng lne. (b) In ths dagram, as on some mcrometer, the 1.0 mm dvsons are above the readng lne and the 0.5 mm dvsons are below t. (b) The thmble n the dagram s n the second rotaton of a mm movement, as ndcated by ts beng past the 0.5 mm mark. The readng s 5.500 + 0.285 or 5.785 mm, where the last 5 s the estmated fgure. 4. Laboratory Balance Laboratory balance s used to balance the weght of an unknown mass m aganst that a known mass m 1,.e. mg = m 1g or m = m 1. Mass, volume and densty of an object The densty of a substance s defned as the mass m per volume V,.e. = m V. Ths may be determned expermentally by measurng the mass and volume of a sample of a substance and calculatng the rato m V. The volume of a regular shaped object can be calculated from length measurements; for example: Sphere V D 3 /6 D, dameter Rectangular block V L L L L 1 2 3 1, length; L 2, wdth; L 3, thckness Cylnder 2 V D L /4 D, dameter; L, heght Annular Cylnder 2 2 V D D L D 1, outer dameter; D 2 nner dameter; L, heght 1 2 /4 12
Procedure Use approprate measurement tools to do followng measurements on 4 objects. 1. For each object take fve measurements to determne the average dmensons. Notce the sgnfcant fgures of the readng. Remember to make a zero correcton for each readng f t s necessary. 2. Calculate the volume of each object (V ), where V s the mean of volume and dv s the mean devaton of volume. 3. Usng laboratory balance to determne the mass (m) of each object. 4. Calculate the densty ( ) of the materal of each object, where s the mean of densty and d d s the mean devaton of densty. 5. Compare the measured wth accepted of each object and calculate the percent %. dv Table 1 Steel ball D (mm) V D 3 /6 d V V V m (g) (g/mm 3 ) d 1 2 3 4 5 average Compare the measured wth accepted Fe (7.8x10 3 kg/m 3 ) and calculate the percent % error. Table 2 Alumnum block L 1 (cm) L 2 (cm) L 3 (cm) V L L L 1 2 3 d V V V m (g) (g/cm 3 ) d 1 2 3 4 5 average Compare the measured wth accepted Al (2.7x10 3 kg/m 3 ) and calculate the percent % error. 13
Table 3 Brass cylnder D (mm) L (cm) 2 V D L /4 d V V V m (g) (g/mm 3 ) d 1 2 3 4 5 average Compare the measured wth accepted brass (8.9x10 3 kg/m 3 ) and calculate the percent % error. Table 4 Alumnum annular cylnder L (cm) D 1 (cm) D 2 (cm) L V D D 4 d V V d V 2 2 1 2 m (g) (g/cm 3 ) 1 2 3 4 5 average Compare the measured wth accepted Al (2.7x10 3 kg/m 3 ) and calculate the percent % error. Work to be done: 1. Put measurement tools and objects n proper places. 2. Let your TA check your data Tables. If they are OK, your TA wll sgn them. 3. Clean up your bench. Lab report on Experment 1 1. Your lab report should be n the requred format descrbed n the Introducton of the lab manual. 2. Tables 1 to 4 should be ncluded n your lab report. 3. It s requred that the answers to the followng questons should be ncluded n your lab report. 4. You can tear those pages out of the lab manual as a part of your lab report, whch contan measured (raw) data and analyzed data, answers to questons. The data sheets must be checked and sgned by your lab TA. 14
Queston and exercses 1. Wrte down the readngs on the sde of Fgs. 6 (a), (b) and (c) respectvely. What s the least count of nstrument scale for each of the three measurement tools? 10 20 Fg. 6 (a) Meterstck (cm) 1 2 Fg. 6 (b) Verner Calper (cm) Fg. 6 (c) Mcrometer (mm) 2. What s the dfference between the measured values 1.05 m and 1.050 m? What factor of a measurement tool determnes the sgnfcant fgures of a measured value? 15
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