Physics71.1 Activity Manual

2007 Edition r '!.'::' j Prepared by Leilani Torres Elise Stacey Agra Maricor Soriano .:t - , ..r ri .oil. i.: l.l .; HTTIONAL IS$TrUrT OF?HrMs CCILLEGE rlr$fiBlfrF urutfER$rY 0r fl.fi Pf|lLFflEs ' mJilA$rr QUEZ0T{ CtfY I t0,}frrfto -uaMt6 :. :' : j , '.; .;': . ;'.,. ,ir': tu Hr= i ! i ao ?g pur- 0 a 6 z 6il (nt l"n7 The 2007 Lab Manual Authors Elise Stacey Agra Junius Andre F. Balista Mary Ann B. Go Margie Olbinado Athena Evalour Paz Leilani Torres Coordinator Maricor Soriano I [' F @2007 and2004 Lab Manual Authors All rights reserved. No part of this publication may be reproduced or transmitted or by any means, including photocopy, without written permission from the 2007 and2004 *ilHttlijS,'i -iiil,.r,1111]/i?;5 l0 ii Y'I11, i4tji;i ri{rj&t^[0 &. Jll,lAi.l- 0:1Ii.{ Published by the Philippine Foundation for Physics, Incorporated efpD for the exclusive use of the National lnstitue of Physics, uP Diliman .,*i, t:' ' i 'r "ii' ' -l I , . :,1 Table of Contents '.l.nlg.. , *vg- tvg- tv" 'a$ 30 ':-1.t l, Preface The 2007 Physics 7l . I Activity Manual is the 4'h outing of the Elementary Physics I Lab Manual series. This year,s volumehas l0 experiments. The concepts covered by theselxperiments ur. (t1 Experimental skills in FundamenLt rhysics I(Measurement, (Jncertainty and Deviation,Graphical Analysis, Using Calipers, Vectors) , p) Motion in 2D or 3D(Untformly Accelerated Linear Motion, Projectile Motion) , (3) Conservation Laws (Conservation of Energt andMomentum) , (4) Torque (static Equilibrium), (5) Simple Harmonic Motion (Simple Harmonic Motioi: sprifi-MassSystem), and (6) Mechanical waves (Sound) Of the l0 experiments in the current volume, 6 are new or revised. In Measurements, [Jncertainty and Deviation, rules forhandling significant figures and propagation of uncertainty are made more explicit. rn (Jsing Calipers, the use of the depthprobe of the Vernier caliper is explained and incorporated in the experiment. Instructioni oo Lo* to create plots usingMicrosoft Excel have been includedin Graphical Analysis while expiriments to demonstrate two conservation laws havebeen merged into one experiment in Conservation of Energlt and-Conservation of Momentum. T\e Simple HarmonicMotion experiment is totally new in that a spring-mass system replaces the simple pendulum which had been used in thepast 3 volumes. Finally, sound explores the properties mechanical waves. Tlte 2007 version makes increased usage of the Vernier LabPro computer interface system. If in the 2004 volume, there was one experiment that req-uires a computer interface, in the 2007 ,rolu-" there are three. Besides Untfurmly AcceleratedLinear Motion (UALM, Simple Harmonic Motion (SHltl)and Sound requires the use of tre photogate and Vemier microphone respectively. Because of the increased use of computers, we recommend the following flow of experimts fur parallel sections inPhysics 7l.l to avoid overlap in the use of interfaces. i I i ii i:,: I I l! i f: !. :: t, i' r I. ll !. ii f: I I. i ir, ii:lii! gii ili. Ii [l: [.:,[: ffi Hli,l!' Ii, $ $ lti" ffi fi H ffi ffi ftiE !lr EI h t,. m r ii i,- II v' t li.t f tl [, fr ! t lr T l h r t I, I Hi $ nh Section 1 Section 2 Experimental Skills in Elem Physics 1 Expermental Skills in Ebm Physics 1 UALM (computer) Torque Projectile Motion Sound(computer) Conservation Laws SHM (computer) Torque UALM (computer) SHM (computer) Projectile Motion Sound (computer) Conservation Laws The lab and lecture topics of Physics 7l need not be synchroni zed. ln case a class follows thc Sctftn 2 plan, topics coveredin the lab _may even be ahead of the lecture. This should not be a problem because fre iuo&ctory text 'su_f.flciently discusses the necessary concepts for the experiments. Stadents are required ,o @ra b *.c ptryd by reading the rext cnd procedures prior to engagement in the lab. The prelab exercises have been dooc !r.y wift in this -.,oli,irn" butinstmctors may give a quiz before the experiment to check on the student's readiness. . .. .,., The 2007 Physics 7l.l Activity Manual was pilot tested in the second semest€r and slrllrg.of AY Z(/)f.-2007. We aregrateful to the students who participated in the pilot testing and to the instnrctors wto cilrH-odii.d lhe text during the General Physics Committee workshop in June 2, ZOO7. Itfuicor N. Soriano Elise Stacey G. Agro Ith- Leilani Y. Tones Measu rement, U nceftai nty and Deviation Objectives lntroduction At the end of this activity you should be able to: Report the best estimate of'observables and quantiff it. Determine if a theoretical prediction is acceptable given the precision and deviation of an experimental data. Report the final data in terms of the proper degree of precision. Appreciate the role of measurements in scierrtific activity. 1. 2. J. 4. In a scientific endeavor, experiments involve collection of information or data through measarement.Datasets are presented to gain empirical knowledge about a phenomenon, validate or invalidate an existing theoretical model and demonstrate that a proposed method works. The measurement of certain variables called observables allows us to achieve this goal. Observables are also called parameters. It is usually the quantity being controlled during the experiment. Since measurement involves unknown quantities, there is always an uncertainty in the measured values. This uncertainty is not always due to personal mistakes. The degree of uncertainty is mainly due to the precision of the measuring device used and the quantity'that is measured. These uncertainties determine the signif,rcance of the measurement. Hence, proper handling of uncertainties must be known. @ 2007 Lab Manual Authors Measurement, Uncertai nty and Deviation Physics 71.1 This activity deals with the analysis of uncertainties; that is, proper judgment of their magnitude, their conventional description and calculation of numerical values based on individual measurements. Precisiofi'and'AcCUracy : ' - ,,1 . Individual measurefllents do not yield the same result. Hence, measurements become uncertain and deviate from true value. The agreement among repeated measurements or the closeness of these measurements with each other is defined as precision. The measuae of precision is called uncertaingt On the other hand, if an accepted value is present, the closeness of the measured value to the accepted one is termed as accuracy and is presented in terms of deviation. To understand more clearly the difference between precision and accuracy,let us consider arrows shot into a bull's eye. Precision and accuracy are two independent terms. Figure 1 (a) shows that most of the stars,are on one location only but far from the center target. Hence, this case is high precision but low accuracy. Figure 1 (b) is low in precision but the average of the location of the stars is close to the bullseye center, hence it has higher accuracy compared to Figure l(a). Figure I (c) shows that most of the stars are on one location only and is at the center target and is the ideal case. While Figure 1 (d) shows the worst case scenario where the marks are both low in accuracy and precision. Uncertainty is not only due to mistake or sloppiness. It is brought upon by the ambiguity of the real value of the quantity being measured. The variation in each measurement may be due to the fluctuations in the quantities measured such as temperature, current or light intensity. It is also dictated by the qualrty of the measuring device or the fineness of its scale. For example, one digital balance may have a reading of 2.13 kg while another reading is 2.134 kg. The latter (a) (c) (d) Figure l. Arrows on a bullseye. Four- point stars mark their landing. Arrows on(a) shows high'piecision but low accuracy,(b) low precision but high accuracy, (c) high precision and accuracy, (d) low precislon and accuracy. j I l (b) @ 2007 Lab Manual Authors Physics 71.1 Measurement, Uncertainty and Deviation measurement has more certainty. Deviation maybe minimized by properly calibrating the measuring device. For example, a weighing scale should read zero if there is nothing on it. The limits of the instrument must also be checked. A body:-filerrnorneter cannot be used for measirring the temperature of a,,boiling water while a l2-inch ruler cannot be directly used to measure the Oarth.moon distance. During the measurement proOess, deviation may also occur due to mistakes, improper use of devices, and, most commonly, due to parallax. Parallax can be removed by ensuring that the eyesight is perpendicular to the scales. Figure 2 shows a reading with parallax. In manual time measurements, the finite human rbaction time (in the order of milliseconds) may greatly affect the accuracy of the result. Hence, it is not advisable to have manual timers for highly precise time measurements. 15 16 t7 18 19 20 mmReporting and handling of uncertainty can be categorized into four approximations. The use of each category depends on the level of uncertainty the experimenter requires. Figure 2. Measurement with parallax. What do The first level of handling you expect the observer will read? What should uncertainty is called zeroth ordi the readins be? approximation which deals with the order of magnitude of the value. The next level involves the use of the significant figures (SF) which is the jirst upproximation. The second approximation deals with the maximum and minimum range of measured quantities. The third approximation involves the rules of probability and statistics which will not be discussed here. Order of Magnitude The first order of approximation is done by estimating the measurement by powers of 10. Fermi questions are answered by thinking of reasonable assumptions followed by simple calculations that narrow down the range of values where the answer lies. Hence, Fermi questions are answered in terms of order of magnitude. The order of magnitude is the power of ten at which a quantity is expected to fall in. For example, in calculating for the number of seconds in the year which is exactly 3 x107 s/yr, order of 106 to 107 is sufficient @ 2007 Lab Manual Authors Measurement, UncertainU and Deviation an approximate. Physics 71.1 is the least Significant Figures The,significant figures in an experimental measurement include the numbers that can be directly read from the instrument scale plus an additional estimated number. Some of the rules in counting the number of SF are listed below. 1. The leftmost nonzero digit is the most significant. 2. If there is no decimal point, the rightmost nonzero significant. 3. If there is a decimal point, the rightmost digit even if it is zero is the least significant. 4. All digits between the least and the most significant digits are considered to be significant. Example 1: Numbers and the number of digits that is significant r. 1200-2sF 2. 13.20-4SF 3. 112000.-6SF 4. 0.003456 -4 SF Problem may arise if the decimal point is omitted and the rightmost digit is zero. This maybe solved by presenting the data in scientific notation. For example, 3560 has 3 SF but the zero may be significant. Thus, the number may be wriffen in the powers of ten, that is, 3.560 x 103 which shows that the last zero digit is significant. Multiplication and Division In multiplication and division of trro or more measurements, the number of SF in the final answer is equal to the least number of SF in the measurements. Example 2: Multiplying two measurements: 2.34 x2.2: 5.148: 5.1 Since, the least number of SF is two, the answer should be reported as 5.1. An experimental data cannot be made more signilicant by' a mathematical operation. @ 2OO7 Lab Manual Authors Physics 71.1 Measurement,llncertainty and Devialion Addition and Subtraction In addition or subffaction, the sum or difference has SF only in the decimal places where the digits of the measurement are both signifrcant. Hence, we report the sum or difference which corresponds to the least number of decimal plaoe,of the addends. Example 3: Adding two measurements: 6.56 + 3.1 :9.66 = 9.7 Since, the least number of decimal place is one, the answer should be reported as 9.7 . Rounding off Nonsignificant digits are removed if they are at the right of the dpcimal point. The rightmost significant digit is retained and rounded off. The rules for rounding off are as follows: l. If the fraction is less than 0.5, the last SF is leftunchanged. 2. If the fraction is greater than or equal to 0.5, the SF is increased by l. Example 4: Best estimate of repeated measurenients. A student did a repetitive measurement of length and obtained the following data: 215 rt" 222 m,219 m,231 m,224 m The expectation value, : 215+ 222+ 219+ 231+ 224 (q)= zzzm To obtain the uncertainty, A q, subtract the from the maximum value and the minimum value from A q. The larger of the two differences is the uncertainty of the data. t 231 m_ 222m:9 m , 222 m-215 m: 7 m The difference 9 m is greater than 7 m, hence the best estimate ofthe data is reported as (9) = 222mt gm 3. In cases of multi-step mathematical operation, only the final result should @ 2007 Lab ManualAuthors Measu rement, Uncertainty and Deviation be rounded off. Physics 71.1 (1) Absolute and Relative Uncertainty The second approximation to uncertainty analysis is based on maximum pessimism. This implies that a measurement cannot be expressed as a single, exact value but is a range of values wherein the true measurement lies, called the best estimate of the measurement. The best estimate of an experimental data set is usually presented as (q)*.A q where (q) is termed as the expectation value or the central value, which can be used for further calculations. For repeated measurements, the expectation value is usually obtained by computing the mean value of the measurement trials. The ubsolute uncefiainty of the measurement is denoted by lq. The absolute uncertainty gives us the quality of the measurement process, and its value can be used in continued calculations on uncertainties. Note that, as the name implies, the absolute uncertainty represents the actual amount, or range by which the expectations value is uncertain. For single measurements, the absolute uncertainty is defined as the least count of the measuring device divided by two. The least count of a measuring device is the smallest division in that device. For example, in Figure 2, the least count of the device is 1 mm, because that is the smallest division in the device that you can obtain. To calculate the absolute uncertainty of repeated measurements, refer to Example 4. For example, in measuring the length of a table, a best estimate of 35 cm t 2 cm implies that the true length lies within the range of 33 cm to 37 cm. Example 4 shows how to obtain the absolute uncertainty of a data set. To determine the signilicance of the uncerlainLy, we have to extend its definition. For example, if you obtained an absolute uncertainty of +0.1 cm, how do you explain its significance? When we measure the length of a book, or perhaps a table, the value of this absolute uncertainty is significant to some extent. However, if we are to measure the distance between two provinces, or interplanetary distance, an absolute uncertainty of +0.1 cm is highly insignificant. On the other hand, an absolute uncertainty of +0.1 cm becomes meaningless if we are to measure the size of microscopic organisms such as viruses' Obviously, the significance of an uncertainty value depends on the magnitude of the measurement itself. Hence, it is desirable to compare an absolute uncertainty L L @ 2007 Lab Manual Authors U ncertai nty Propagation The rules for compounding uncertainty of measurements are still based on maximum pessimism. For most laboratory work, the following rules are sufficient: Let v:(v)+Ax , y=(y)tAy , z=(z')*A, l. Addition and Subtraction. In addition or subtraction , the absolute uncertainty of the sum or dffirence is the sum of the absolute uncertainties of the terms. Eg. z=x+ ! (z)=(x)+ (y) Az=Ax*Ay 2. Multiplication: ' uncertainty of the factors. Eg, z= xy (z)=(x).(y) Az (r/,1: Ax Physics 71.1 .Measurement, Uncerlalnty and Deviation with the acfual value of the measurement. For this purpose, we define a quantity called the relatiue uncertainty, /q(o/o), of the measurement. It is defined by Aq%_# (2) The relative uncertainty is often quoted as a percentage so that in Example 4, the relative uncertainty is h : 4.05 % . Therefore, the best estimate in terms of relative uncertainty may be reported in the form 222 m + 4%o. Note that the number of SF in the absolute uncertainty is equal to the number of SF in the relative uncertainty The relative uncertainty gives us a much better feeling for the quality of the measureftrent, and we often refer it the precision of the .measurement. The absolute uncertainty has the same dimensions aqd units as the expectation value of the measurement, whereas the relative uncertainty, being a ratio, has neither dimensions nor units and is a pure number. ,: If'"'twb numbers are"'being'multiplied, the relative product is the sum of the relative uncertainties of the @ 2007 Lab Manual Authors (%)+ ^y (%) M6asu rement, Uncefiatnty a nd Deviation Physics 71.1 3. Power. If a number is raised to a power, the relative uncertainty of the result is the product of the relative uncertainty of the number and the absolute value of the power to which the number is raised. Eg. t , =x" where a is any number L\-1.\o\-/\""/ Az (%): lalAx (%) From rules 2 and 3, it is obvious that for division, the relatiue uncertainty of the quotient is the sum of the relative uncertainties of the numbers being divided, as in multiplication. The assignment of uncertainff bounds depend on the judgment of the experimenter based on many factors such as fte measuring device, the quantity to be measured and the precision needed. Deviation If a set of experimental data is compared to an aeceptable measurement of the variable being measured to determine the accuracy of the measurement, it is necessary to define a lew quanfity called d&iutior. ' The absolute deviation of a measarement is the absolute difference between the accepted value and the experimental value of the measurement. ab s o lu t e dev i at ion = | ac c ep t e d v alu e - exp er iment al v aluel To determine the significance of the absolute deviation, we define the relative deviation of a measuri:ment as the'ratio betweerf the absolute deviation and the accepted value: relative deviation= absolute deviation x 100% accepted value Acceptability of Measurement Results \ '' To determine the ,acceptability of a measurement fesult, we follow the following rules: 1. If the accepted value of measurement is given, a measurement is acceptable if the absolute deviation is less than the absolute uncertainty. 2. If a maximum percent error is given, a measurement is acceptable if the @ 2007 Lab ManualAuthors rg-lqtiue uncertainty is less tl4an the maximum peycent eff9r given. Note that if both the accepted -value.of the medsurement and a maximum percent error are given, then a measurement is acceptable only if both the above conditions are satisfied. a I ..1 Reference o D.C. Baird, Experimentation: en tntroOubiion'tot'Meai*eil.nt'Theory and Experiment Design, 3rd Edition, Prentice-Hall,Inc., USA, 1995. O-2OOZ Lab l*anUal /6gdhoB Measurement, Uncertainty and Deviation Physics 71.1 @ 2OO7 Lab Manual Authors10 I It- l{rm. Date Submittod Data' Pedomed Scorc Group Mombera lnalructor Sec{lon Worksheet: Measurement, Unceftainty and Deviation A. Scientific notation and rounding off. Round off the following numbers up to three significant figures and express them in scientific notation. Table I B. Rules on significant figures on operation. Perform the following operations. Write your answers in correct number of sisnificant figures. Table 2 C. Acceptability of measurement results. Compute for the best estimate of the observables presented in Table 3, given a number of its corresponding estimates. Write your best estimate in the form qlAq Complete Table 4 based on data from Table 3. Briefly answer the questions that follow. Use proper units. Indicate if additional sheet/s is/are used. 0.000 856 400 10.562 3 3 26 500 8.595 00 56.450 001 96.442 s 90 523.5 4 646.56 146 500 000 001 10.050 000 96.895 + 4.65 26.45312 x 6.500 265.239 008 + 86 000 958 54.2 t26.598 5.610 257 - 2.5 962.581t25 88.264 4 -15 26.53 x 12.5 + 6.98 -2.1 / 0.905 13.265 x 4.1 53.24 + 15 x2.3615 -7.625 x26 @ 2007 Lab Manual Authors 11 Meaisu reiient; lln;iertaiity aid Deivtdtion Physics 71.1 Table 3. Best estimates of observables obseruable trlal Eesf estimate Accepted valueI 2 3 4 5 T( 3.1514 3.1421 3.1416 3.',|420 3.1501 3.1416 length (crh) 6.544 6.555 " 6.s23 6',.520 '', 6575- 6.61 volirme ( rn' ) i.045 '1.203 1.158 1.009 1.001 1.100 mass (g) 5.5 5.3 5.1 5.3 5.5 5.2 speed (m/s) 1.507 1.601 1.512 1.514 1.500 1.6 Table 4. Absolute and relative deviation observable Absolute deviation Relative deviation T( length (cm) volume ( 773 ) mass (g) speed (m/s) Questions 1. How did you estimate the value of the uncertainty for the best estimate? Explain why this is valid. Based on Table 3 and Table 4, which observable has absolute deviation greater than the uncertainty obtained? 3. Which of the observables can be considered to have an acceptable experimental proofl whv? 12 @ 2OOl Lab Manual Authors Physics71.1 M€abu rermeht; Uhceila inty ahd DCvi atian D. Uncerta'inty' of caleulated' values.,' Compute for the minimurn possible value for each of the quantities given below. Given are the best estimates of the yariables needed. Olserve proper units. o Square of time f if t: 50.00+ 2.0 s. expectation value (t') : _; minimum t' : _; maximum i : fr . Period of pendulum f =Zntl(L) if /: 100.00 + 2.OO cm and g : 9.81 + 0.10Ig mlsz E. Problems o In measuring the volume: 19.6+0i.2m3 and the mass uncertainty qf the density ( I : ir' ' :''; : : of a metal sample, the volume (n ' -, '(m) obtained was 2.45 + 0.15 kg. What ) calculatbd usingithe eciuation, ,e=f expectation value : Calculation: Final Answer: ; minimum T: ; maximum T : obtained was is the absolute ,l i: @2007 Lab Mddual Authors 13 H F, [. l@arrorrerf,' U n ee fiai n&t. an d, u9ev iall q O Physics 71.,.1 o A simple plndulum is used to measllr,g the"pqpelergiro3 due,to gtpvity.using , : ..1; , : rT T =2n tl: . The period 7 was measured to be I .34 * .02 s and the length to be 0.58,1Ig + 0.002 m. Whatis the resulting value for g with its absolute and relative uncertainty? Calculation: Final Answer: o An experiment to measure the density , d, of acylindrical object uses the e{uation .md =- , where rn is''the niass, r is thb radius and / is the length of thb cylindricalTtr I i :".. ,r.. I object. The dimsnsions of the object is listed below. m : 0.033 + 0.005 kg, r: 8.0 + 0.1 mm l: 14.6 +b.t mm. What is the absolute uncertainty of the calculated value of the density? Calculation: Final Answer: 14 @ 2007 Lab,Manual Authors Using Calipers Objectives At the end of this activity you should be able to: 1. Appreciate the role of the available measurement precision to the practical choice of measuring device. 2. Measure the dimensions of an object using a ruler, aVernier caliper and a micrometer caliper. 3. Identifu a metal sample based on its density. lntroduction Calipers are devices that can measure dimensions of small objects and hard to ' measure observables. The main advantage of usingrone is it allows user to find the very small fractional measurelnents (up to micrometer scale). This activity teaches the use bf calipers and the application of uncertainty and precision in measuring devices. Main and Fractional Scale A measurement of a specific device consists of two parts (a) main scale reading ( x us ) and (b) fractional scale ( xrs ). The main scale reading is determined by reading the largest measurement the device can provide. On the other hand, the fractional scale is the fraction of the least count (smallest possible measurement) of the device or may be estimated by the experimonter. In the end, the final measurement is found by adding the niain scale reading and the fractional scale reading, that is @ 2007 Lab Manual Authors x=xrr*xo, (1) 15 Using Calipers estimated fraction : 0.2512: O.13 cm x = x,s+ xrJ estimated fraction *: +.fS ". Figure 1. The length of an object is meqsured u$ng a ruler. The estimated fraction is approximated afier visually dividing the ruler's least count. Vernier Galiper Physics 71.1 Figure 1 shows how the length of an object may be measured using a ruler with least count of 0.25 cm and an estimated fraction part of 0.13cm. The experiment may report 4.38 + 0.07cm or 4.4 + 0.1 cm as his or her best estmate as long as the pnge of lhe reportiqg is" practical and consistent with maximum pessimism or ,.iounding'off prineiples. Also, the reporting of uncertainty should also be consistent. In adding the main scale, fractional scale and estimated fraction 0.I3 cm is reported instead of 0.125 cm since adding all these make the 0.005 insignificant. xrr: 4.00 cm 0.25 cm i I I I ir I Ii I lt li I lt The French mathematician Piorre Vernier (1580-1637) invented the Vernter caliper in 1631, a device that can measure outer and inner diameters or lengths as well as depths. Figure 2 shows the parts of a Vernier caliper. Fignre 2. A;picture of a typical Vernier c'aliper showing'the main scale (4 for metric and 5 for English systeru), Vernier scale (6 for metric.ani 7 for Englisk system), clamping mouth (l for outer diameters and 3 for outer), locking screw (8 ) and depth probe (3).. (graphics by Joaquim Alves Gaspar) 16 O 2007 Lab ManualAuthors Physics 71.1 How to Using Calipers The parts of the Vernier caliper are ' main'scale (4 and 5) - reads the main scale reading obtained by taking the last mark of the main scale before the zero of vemier scale (edge of zero mark). Vernier scale (6 and 7) - estimates the fractional scale reading by taking the order of the Vernier scale mark that literally aligns with the main scale mark. clamping mouth (1) - used to measure diameters, opposite to this mouth (2) is used to measure inner diameters of pipes. depth probe (3) - used to measure depths. locking screw (8)- used to lock the caliper after sbtting it. The caliper is set after applying enough pressure (avoid squeezing the object) as the clamping mouth spans the diameter of the object. The zero reading of the Vernier scale is obtained by closing the mouth completely and getting the reading. If the main scale reading is to the left of zero, the least count of the main scale should be subtracted from the fractional reading. Before ,rirg measuring devices be sure that they are properly calibrated and are in good working condition. Calibration of instrumentsr,,imrolves ensuring they work well within the range of values being measured and are properly zeroed. The Vernier caliper is properly zeroed if the zero mark of the rnain, scale coincides with the Vernier scale when the clamping mouth is closed. In using a vernier caliper the clampirrg *outh is,set after applying enough pressure to keep the object in place but not enough to defonn or squeeze it. The lock may be turned to ensure that the clamping mouth will not move even if the measured object is removed. Use a Vernier Caliper A Vernier caliper allows better estimation of the fractional part of a length measurement by the use of its VERNIER SCALE (VS). To read the vernier scale, the LEAST COLTNT (LC) or the precision of the caliper must be known. This is obtained by counting how rnany subdivisions the VS will make on the main scale. The caliper in Figure 2 has the smallest reading on the main scale at 0.1 cm. Meanwhile, the Vernier scale can create 20 subdivisions. Hence LC is obtained using tr=*=olo5cm @ 2007 Lab Manual Authors (2) 17 Using Calipers Physics 71.'1 Figure 3 shows an example of Vernier caliper reading. The caliper has 50 (including the smaller tick marks) Vernier divisions and its smallest reading on the main scale is I mm. Hence the LC of the caliper is '!! :o.o2mm.50 In reading a Vernier scale measurement, take the main scale reading at the left of the zero mark of the VS, not the edge. In Figure 3, the main scale reading is 26 mm. Next, take the VS scale line which is coincient with the Vernier scale. Note that in Figure 3, the VS mark coincides at the 17'h line. From these values we can determine the measurement of the Vernier caliper: ::t?il{i?1rl 4i #{ *rt,ittl Figure 3. A close up view of the Vemier caliper. What is the least count of the caliper? What is the reading of the caliper? The uncertainty in readings is subjective. Its value must be given by the experimenter. As a rule of thumb, the uncertainty should be half of the least count as long as no other technical reason interferes with the measurement process. Try out the simulation in http://www. physics. smu.ed u/-scal ise/apparatus/caliper/tutorial/ practice reading a Vernier caliper. to 18 @ 2OO7 Lab Manual Authors Physics 71.1 Micrometer Caliper Using Calipers Figure 4 shows the parts of a typical micrometer caliper. Figure 4. Micrometer caliper showing its main (barrel) scale (M), thimble scale (T) for its fractional scale, lock (L),jaw (J), and rachet (R). A micrometer caliper estimates the fractional scale using a screw mechgpism. The displacement of the barrel is proportional to the number of turns of the thitnBf# For example, if the thimble moves at a distance of 0.5mm per rotation, then dividing the thimble into 50 equal parts would make the least count to be 0.01mm. Figure 4 shows the parts of a typical micrometer caliper: (J) jaw -partthat actually spans the diameter/length/width of the sample. (B) barret - used to read out the main scale (M) reading (the last mark the edge of the thimble has passed), in case of ambiguity, look at the value of the thimble reading (if less than half a revolution it means the thimble has just passed the mark). (T) thimble - rotated to make the jaw clamp the object , this part is divided equally along the edges so that the fraction of revolution can be obtained. (R) rachet - this is a knob that is tightened or loosened to set the strength of clamping to the object. (L) lock - this is used to keep the setting of the instrument for reading (used if the sample is hard-to-reach and the micrometer need to be removed from site to read the measurement). The micrometer screw must be turned at the rachet while closing the jaw to prevent the screw mechanism from wearing off and to avoid excessive clamping of the sample to be measured. One or two clicks from the rachet should indicate enough tightness of the clamp. @ 2007 Lab Manual Authors 19 0, 1r Psnl u0+ tJtP- 0 Using Calipers Physics 71.1 ilf#.1",i;,,'"?,T"""iJ:'",i"i1.il:H:J:ii;1,:'Ji"1#:[.Tiff:l'i;: -- turning the Each complete rotation is divided and marked into equal subdivisions which make reading of the fractional part straightforward. Arbitrary further division (user - rrr dependent) in the thimble reading can be done. See for example in Figure 6. The0 6 2 6W main scale reading is x^ : l3.5mm. since the upper marks correspond to lmm ' and the lower to 0.5mm marks. This particular micrometer caliper has 50 divisions in the circular scale. One fullturn moves it 0.5 mm. Therefore, the least count of the fine scale is 0.50mm/50 : 0.01mm. The fine scale has passed the 21" notch therefore xr"=0.01 mmx2l:0.21mm However, as can be noticed, we can still make the reading finer by having fractional reading within the thimble's least count - the zero barrel mark is near the 22d notch, say, it may be around 8/10 of 0.01mm or 0.008mm. The fine scale reading plus estimate will then be 0.218mm. So the final reading would be: x : t3.5mm + 0.2l8mm or x : l3.7l8mm. Figure 6. An example of micrometer reading. The marks show a reading of 13.71Smm. 20 @ 2OO7 Lab Manual Authors Physics 71.1 Materials Using Calipers :. ,. ,$uler, Vernier caliper, micrometer screw, dlgital balance€nd metal samples . i i ,. ' :. I l: :' r. Proc6dUfe,' 1. Calibrate the ruler, Vemier caliper, micrometerj caliper and the digital balance by noting the least count of these equipment, the least count of bo. Input your data in Table 1 of the worksheet. 2. Measure the mass of the metal samples using the digital balance. Use Table 2 to record the masses. Compute for the relative uncertainty by using the expression a m(N1=4 ?-x I oo % (3) \m) 3. Measure the dimensions of the sample and tabulate in Table 3. @ 2007 l-ab Manual Authors 21 Using Calipers Physics 71,1 Figure 8. Use the depth probe to measure the depth or the inner height of the metal sample. Compute for the volume of the samples. Assume a specific shape for each sample. Write out your computed volumes Table 4. Finally, compute for the density o/orho of the samples using Mp=T Compute the relative and absolute uncertainty of the density values. Write them down in Table 5. Identifu what type of metal the samples are made of by comparing your computed densities with densities of different metals. 4. 5. 6. 7. 22 @ 2OO7 Lab Manual Authors Name Dats Submitt d Date Pertomed Scolt Group ilembarg lnstructor Slectlon Worksheet: Using Calipers l. Galibration of measuring devices Complete the table below to determine the least count and estimated uncertainty of the vernier caliper and the weighing scale. Data Table I. Least count and estimated uncertainty of the measuring devices used. Weighing scale Ruler Vernier Caliper micrometer caliper Main scale (least count) Number of fractional divisions Least count Estimated uncertainty . Based on the least count of ruler, Vernier caliper and micrometer caliper, which of the devices is most precise? ll. Calculation of the density of the sample A. Mass measurement Data Table 2. Masses of the metal samples. The relative uncertainties are based on the estimated uncertainty in Data Table l. Itetal sample Mass (g) Re I ati ve U n ce rta i n ty (/o) A B A 20AT Lab Manual Authors 23 . .i. 11 Using Calipers B. Volume measurement Data Table 3. Measured dimensions of the metal samples using the ruler (R),Vernier caliper (yC) and micrometer caliper (MC). The relative uncertainties A x are based on the relative Physics 71.1 @,2007 Lab ltrlanual Airthorsa Measuring device Sample V (mm3) aY ("/;) AV (mms) Ruler A B Vernier caliper A B micrometer caliper A B Physics 71.1 Using Calipers Data Table 4. Yolume of the samples. The uncertainties are calculatedfrom the absolute uncertainties in Data Table 3.Write out your solution in a separate sheet of paper. Data Table 5. Sample identification. From the values of the mass and volume found in Data Tables 2 and 4, calculate the best estimate of the density of the samples. Reseorchfor the densities of these samples. Identifu what element comprised sample A and B. B: Measuring device Sample q (ilcm') Acp W Acp (g/cm') Ruler A B Vernier caliper A B micrometer caliper A B @ 2OO7 Lab,Manual Authors 25 U@r:.Ga.fipers ,Physics'1,'t.1 ri , 1, Shat assrrmption(p), if any, inthe shapeof-the samples is/are most likely 1o1_ry3li,ze$? , : !':i -r i--. -- :-1. ,-', ;.' :.i. l- ..- ,:,i 2. Would the use of a more precise length measuring device improve the performance of the method used to determine the density of the sample metals? 3. Can this method accurately identiff the major percent composition of analloy? Try this out by identiffingthe m.4jor element composition oJa 5 centayo cgin. 26 923;gr Lab Maaual Authors G,raphical Analysis At the end of this activity you should be able to: 1. Create a graphical representation of a given set of data thatwill best show its purpose; 2. Formulate a theory or a model based on the parameters from a graph of experimental data using linear fit and trendlines. , 3. Leam how to use spreadsheets (Microsoft Excel) and some of its basic functions. lntroduction Theory The most convenient way of presenting a dataset is through graphical presentation. A graph'is defined as the pictorial representation of a set of data which could be'in 2 or 3 dimensions. trt allows the experimenter to understand the relationship between 2 or'more parameters Graphs may involve shapes, curves and symbols. Some types of graphs are pie, bubble, scatter, bar and line graphs. Figure I shows an example of two dimensional scatter graph which is most commonly used,as a way to present the relationship between two variables. @ 2007 Lab Manual Authors 27 Graphical Analysis Physics 71.1 \ Figure {. The plot shows a linear relationship between the squad of the period of a simple pendulum and the length of the string. \ f-uur ur rrrs surrrs' L | ""eil;l Graphs have basic parts that need some attention before they could express their purpose well. " Shown in 'Figure I is a Sample graph with parts described below. a) Title - This part is usually placed at the top of each graph. It tells a specific thought about what the graph shows. Since a caption is usually included, this part can be omitted due to redundancy i b) Axes - This is ,the part that shows the values of the variables involved. The x-axis usually contains the parameter values (independent variable) and y-axis contains the observable in ques.tion (dependent variable). The range of values in the'axes should be reasonably enough for the range of data concerned be shown. Oftentimes, the maximum and the minimum scale should also be adjusted to give the best display (the numbers are well spaced and readable). c) Labels - Labels are words or phrases that best describe Jhe quantity being represented by 1n axiq, Thus there are two labels for a 2- dimensional graph since there are two variables (thus two axes) involved. It should be noted that a label includes the unit used in measurement. d) Symbols - These could be filled circles, squares, triangles, and other shapes that represents a point or a thought about a datapoint. These 28 @ 2007:Lab Mdnual Authors Physics 71.1 Graphical Analysis symbols should be clear enough (not too big but not too small) so that other datasets plotted in the same set of axes can be easily differentiated. Color atdlor shading should be utilized to maximize this effect. Shadows and other o'special" or "aesthetic" effects should be avoided specially for graphs with technical or formal purposes. Legend - This describes each of the dataset used in a graph. Using a word or a short phrase, the legend differentiates different symbols used. This is not necessary for graphs that shows only one dataset. Caption - This is used to briefly describe the idea being presented by a graph by clearly pointing:out salient parts in the presentation (e.g. skewed points, alignment of points, trends, similarities). Important parameters not in any of the axes should be mentioned and described in this part. It is a challenge for the presentor to make captions as short as possible. Captions may include titles which may prove useful for quick glances. e) Graphing Procedures Error Bars Variables are commonly plotted in a rectangular coordinate system. The dependent variable is placed on the y-axis and the independent variable is placed on the x-axis.. The location of a point on a graph is defined by its x and y coordinates, written (x,y), with respect to a specific origin. In plotting a dataset, the axis scales should be chosen such that the plot is easy to understand. With axis scales that are too small, the points will bunch together, making the plot incomprehensible. Collection of data involves measurement; hence, this implies that uncertainties are present. In plotting a set of data which includes the expectation value and its corresponding uncertainties, the expectation value is plotted and the corresponding uncertainty is presented as an error bar. Error bars show the possible range of values of one or more variables in a data point. This is useful since it allows the experimenter to know the range of possible values under the @ 2A07 Lab Manual Authors 29 Graphical Amlysis influence of a certain variable. Trendlines and linear fit Physics 71.1 (3) Hence, Equation (3) Data points in an x-y scatter plot should not be individually connected by lines. In the event that the experimenter is certain about the relationship of the variables being presented, a smooth line or curve called the best fit line can be drawn to represent the known relationship. The word'osmooth" does not imply that the line or curve must pass exactly through'each point. But the best fit line should best represent the data set. This type of plotting is called eyeball method. The main criterion for.this method is to minimize the distances of all data points from the line drawn. Once linearized, the variables'can be represented:using the equation y=mx'+b where m and b are constants that represent the slope and the y-intercept of the plot respectively. Slope is an algebraic relationship of the line and is given by the equation Ax Ay Any set of intervatr may be used to determine the:slope of a linear plot. But for best results , points, showld be chosen within the best fit line. lf the data point is not included in the best fit line, it should not be used to calculate the slope of the graph. Other forms of nonlinear functions may also be represented as a linear plot. For example, the equation (l) (2) y- gx'+b may be reduced to a linear equation if we let x =x reduces to the form .r. y,=.gx'*b which is just equivalent to Equation l. 30 @ 2007 Lab ManualAuthors l J,. Physics 71.'1 Graphical Analysis Graphing using a spreadsheet The following is a step by step way to plot data using Microsoft Excel. 1. Input your x and y data in two separate columns. Try this out using the sample data from your worksheet. 2. Highlight these two columns and click "chart wtzard" icon on your toolbars. 3. Choose x, y scatter on your Chart type and click next. 4. You will see a preview of your plot. Ensure that the option you choose is series in 'column'. 5. click 'next' and enter the chart title, x -axis label and y-axis label. You may also click on the tabs to modifii the axes, gridlines, legend and data labels. Just continue clicking next and you have your plot. g& 4Bw r]hHt FE@ Idc [*6 ffitu EeIF li.l i.]S$t L} ElidJ irh,1F'Ht, l( *:e,J&: {fl,'rr " ,r" -lo - E / !t,E#gfig$iry#% i#, a : fl,al :ffijjom - r{S g r tdg ;1$ i AF EF, g: t$r - A -ffi To include error bars on your plot, just type half of the magnitude of your error bar on a column beside your y-data points. Right-click your data points on the plot and choose 'Format data series'. i I 6. 7. @ 2007 Lab Manual Authors 31 Graphical Analysis Physics 71.1 After choosing 'Format data series', click the y-error bar tab. Choose 'Custom'. You may opt to type the series on the + and - space or you may click the icon beside the space and highlight the corresponding series. To add a trendline, right-click again the data points and choose the option 'Add trendline'. Choose the corresponding best fit curve for your plot. To insert the equation of your trendline, click on the 'Options' tab and check the box on ' Show equation'. 065 0ffi, 1_Dt' !,n. 15 8. 9. 11.57 i71 q.si 1.11 1 :lE '.r -,i. *.rd.&-=;=d wt" 32 A 2007 Lab Manual Authors Xama . Date $ubmittod Dat6 Perioamed Scorc Group tlemberc lnstructor Section Worksheet: Graphical Analysis A. PreSenting data'set graphically During an experiment, a physics student obtained the following data: x + 0.10 v -5 384.5 -4 208 -3 96.5 -2 35 -l 8.5 I 0.5 2 -l I 3 47.5 4 -124 5 -255.5 The variables x and y are the independent and dependent variables of the experiment, respectively. o Ploty as a function of x, Can you conclude with certainty lhat.the plot is linear? Explain your answer. You may try to fit a line using the eyeball method and argue from there. o Ploty as a'function of x2 , . Canyou conclude with certainty that the plot is linear? @ 2047' Lab Manual Atrthors 33 Graphical"Analysis Physics 71.1 Explain your answer. You may try to fit a line using the eyeball method and argue from there. Ploty as a function of *' . Can you conclude with cgrtainty that the plot is linear? Explain your answer. You may try to fit a line using the eyeball method and argue from there. I I I i I I i I I I I I I I t N i : IL From your answers in items 1-3, determine the degree (in x) of the equation relatingy and x. (Recall that the equation y=ax2 *bx-lc has a degree of 2 in x.) B. Problem solving using graphical analysis 1. The Chronicles of Narnia: The King, the Prince and the Heirloom. 'On his King-father's deathbed, Prince Caspian of Narnia was mandated to find the mass (M of the royal family's heirloom. After days of sleepless nights, he was reminded of a very important lesson from the great Professor Digory: The Parallel-Axis Theorem. This states that a body rotating about an axis parallel to and at a distance d from the center-of-mass axis has a moment of inertia I p about that axis written as I ,= I "^*Md2 where I "^ is the moment of inertia about the center of mass. By the Prince's command Regpicheep, {he ,commander, of the Army, conducted a series of 34 @ 20Ol Lab Manual Authors Physics 71.1 Graphical Analysis experiments using Vernier LabPro@ that eould determine d and I p at precisions (least counts) of 0.10 cm and 0.50 g.cm2 respectively.Reepicheep was V great wa.r.rior, but so poor physicist, that he tabulated his data so horrendously: A. Re-tabulate Reepicheep's data correctly by writing the expectation value of the moment of inertia and the distance from the center of mass based on the given precisions. (1,)(s'cm') (d)(cm) (I o)G'cm') (d)(cm) IJ 2.51 2.s2 3.6 4.010 7 8.1200 8.667 11.010 9.41 A 2007 Lab Manual Authors 35 Gruphical Analysis Physics 71.1 B. Plot I ovs. d2 and paste it' on the space below. Calculate the best estimate of the mass of the mysterious heirloom. Solution: 36 Final Answer: M: @ 2007' Lab Manual Authors Physics 71.1 Graphical Analysis 2. Off to the moon! The accepted value for the acceleration due to gravity of the lunar surface gmoon, is 1/6 that of the earth, gno,,^=9.8m1s2 You decided to go to the moon and I conduct experiments to verify this value. However, because of your busy schedule, you have no time to go to the moon and decided to send your younger brother instead. He conducted free fall experiments, measuring the time it takes for a freely-falling ball to ,reach the lirnar surface upon release from an initial height h. He used a timer with 0-001 s precision (least count) and a meterstick with a least count of I mm. His estimated fraction for the meterstick is 0.5 mm.He obtained the following data below. However, he has no Physics 71.1 training when it comes to reporting measured data. t(s) h(m) 0.34 0.1 0.58 0.27 I 0.85 1.3410 1.6s r.604 2.5 time and the initial heisht based on the si A. Retabulate your younger brother's data'correctly by writing the expectation value of the @ 2OO7 Lab.Manual Authors 37 Graphical Analysis Physics 71.1 B. If / and hare related by n=)S t' , obtain the best estimate for gmoon. Solution: 38 Final Answer: I *oon : @ 2007 Lab Manual Authors Vectors a'nd Force Table Objectives At the end of this activity you shoutrd be able to: L Show that the sum of forces acting on a system in zero. equilibrium is 2. Obtain the equilibrant of two or more forces using the concept of equilibrant. 3. Obtain the orthogonal components of a force. lntroduction Vectors are mathematical representation of physical quantities that involve a rnagnitude and a sense of 'direction. Examples of physical quantities that can be represented by vectors are: position, velocity, force, and electric fields. These quantities follow rules of addition and multiplication just as vectors do . The magnitude and direction of vectors do not necessarily need to be real. A vector can be represented by an affow in space. A two-dimensional vector needs an arrow in a planar surface. On the other hand, a three-dimensional vector is represented by an arrow with three-dimensional direction. Oftentimes it is difficult to imagine the graphical representation of vectors making graphical approach impractical and analytic representation comes handy. Analytically vectors can be decomposed into its orthogonal (graphically perpendicular; physically' independent) components. Since vectors are mathematical entities, they follow certain rules of combinations. The simplest static @ 2AO7 Lab Manual Authors 39 Vectors and Force Table Physics 71.1 means of combination are addition (and subtraction) and multiplication (division is not possible for vectors). This activity.deals with comparing theoretical (graphical and analytic) approaches in dealinglwith combining physical vectors, force in particular, including about the concept of resultant and equilibrant. Theory Vector addition (and subtraction) Just like the physical quantities vectors represent, they can be added (or subtracted) to (or from) each other. It should be emphasizedthat only vectors that represent the same physical quantity. can be added or'subtracted. This translates to the idea that only vectors with same units can be addgd together or subtracted ' from each other. Thus the vectors i , E , and e should have the same uriit so that t=Z+B (1) has a physical meaning. The magnitude of the vectors follows the inequality below ileil Physies 71.1 V*tors and F.orce Table every time a physical vector is drawn either a scale (say,"lcm 's"ro 100N'?) is indicated or the vector is labEled with its corresponding magnitude. Taking the sum of two vectors would then involve drawing them as in Figure I and measuring.the length of and scaling it back to the actual magnitude value. Mathematically, Equation 1 can be rearralged. tq become a subtraction: B=t-i , just as Figure I can be rearranged into a similar figure shown in Figure 2 via the concept of translation . It.,should be obvious that the sum of a vector and its negative is zero (null vector, 0 ) with the negative of a vector represented by the same vector but pointing towards the opposite direction. Figure 1. Head-to-tail method of how two vectors Z and B add up to 1aC : C = A+ B . The same figure represents the difference of two vectors: b =t -2 . The vectors 2 , b , and e represent the same rype of physical quantities. Note that the "head" of b A is placed onto the "tail" of tu "-*l Figure 2. Head-to-tail method addition of the negative of a vector, - 2 , to a vector e is considered the subtraction process that yields b . Again, the vectors must represent the same type of physical quantities. 4 A @ 2007 Lab Manual AUthors 41 Vectors and ForceTable Physics 71.1 ,4 Figure 3. Parallelogram formed by the vectors can be obtained using trigonometric concepts' + and B and B .Themagnitude is the angle between 7 2 e The same relation can be obtained trigonometrically from Figure Equation 2. Note that the angle O between e and i cosine law: c2+ A2-E i cosO=-- ,;a- 1 consistent with is related by the Parallelogram method Figure 1 can atso be viewe{lvi.o.iTrr,*: :ti,t1^::"-1. ;Jl, ,f. ,.Til:t;parallelogram as shown in Figure 3' In the same mannel method, the magnitude of e can also be obtained by measuring the length and converting it into the actual magnitude. Trigonometrically, with the angles O and e as ,shown, the magnitudes of the vectors obey the relation (obtained from the cosihe law ): cz = Az + 82 + 2ABcos o (3) Resultant and equilibrant A set of forces (or other physical quantities that can be represented by vectors) can be replaced by a single force - the net force, which renders the same effect. In terms of vectors, a set of vectors can be replaced by one vector called the resultant. The resultant is merely th9 sum of all the vectors it will replace. In the .*u*pi.'uUor., d is the resultant of the vectors 7 and b The concept of resultant is commonly used in engineering where a set of forces acting on an object can be replaced by a resultant without changing the overall effect. Lr*. 42 @ 2A07 Lab Manual Authors Physics 71.1 Vectors and Force Table The action of a set of forces (again, force is just an example) can be countered/nullified. Indeed, a particular single force introduced into this system can produce a zerolntll overall effect. In terms of vectors, this particular vector is a vector that will cancel the resultant of the set and is called equilibrant. The sum of the resultant and equilibrant is therefore zerolnull vector making equilibrant and resultant a negative of each other.Symbolically , E=-h. (s) with E as the equilibrant and fr as the resultant. Figure 4 shows the graphical relation of E and fr ' Figure 4. Graphical representation of the equilibrant E . Note that the dashed vector (tfanslated ) placed beside fr shows that it can cancel fr and thus fr 's effect. Unit vectors Vectors with unit length or magnitude are called unit vectors. Unit vectors are used to indicate direction and are represented by symbols with a hat, e.g. 2 Thus the direction of 2 is along 2 The unit vector can be obtained by rescaling a vector into one unit length/magnitude. This involves multiplying a vector with a (unitless, dimensionless) scalar, say s, changing only its magnitude not its direction . Thus to get the unit vector of i , we scale it with a scalar equal to the reciprocal of its magnitude: r.) Ft 2=tltz (6) Orthogonal vector components Vice versa to the problem of finding the resultant, a set of vectors can be sought so that the given vector will be their resultant. The vectors belonging to this set is called the components of the vector. This is the same as asking what forces should be combined to yield an effect equal to the single given force. Additional @ 2007 Lab Manual Authors Vee&rs atld FordeTable Physics 71,1 Condition however; isiimposed for these contponents: they should be orthogonal to eaohiother. ,This physically, moalls'tfiat thebe compondrts have to be directed along a fixed set,of directions. F E Figure 5. The component F " of F along the direction A The unit vectors corresponding to orthogonal vectors are called orthonormal vectors or basis vectors. Finding the components of a vector along a given a set of unit direetions (frame of referehce) inv-olves'finding the oomponent of a vector along a gi$en direction. See Figure 5 for'an example' The direction of F " extends along the direction of o and ends at the point where a perpendicular line dropped from the "head" of crosses the direction of F ' If the angle between O and F' is e , then the magnitude of F " is'grVe,lr by.l i ,' ! "=l c9se (7) Thus, if 'we take 1 and ' i' as the urtit airecti Physics 71.1 Vectors and Force Table (10) (12) lih.# * rnr- rr Figure 6. The components of F along F, between F and, i is e Fy .Theangle Using trigonometry, we immediately See in Figure 6 that the magnitude of F is related to the magnitude of its components by the Pythagorean relation: p+= fi+ F2, The vector F canalso be written as F=F,i+rri (ll) The angle between F and x-axis, e is related to the components by the equation: tane=L-F, Equation 10 can also be obtained from Equation 3 by replacing the angle between the components with gO' A vector with known components can now be normalized by scaling it with the reciprocal of its magnitude derived from Equation 10. The axis directions are arbitrarily chosen and each chosen set of axes results to a different set of component vectors . These components however, still add up to the same vector (the resultant of the components). This is advantageous in cases when the axes have to be oriented so that most components of the vectors lie along one direction only, making trigonometric analysis (as well as other mathematical arguments) straightforward. @ 2AOV Lab Manual Authors 45 Vectors and.Fsrce Table Physics 71:1 Adding vectors using its orthogonal components becomes straightforward. Each of the components of the sum of two vectors, say i and h , are simply the sum of the corresponding components of the vectors . Symbolically, C,=A*+8, C ,--'Ar+ B, As an example, consftler i=5i+6i and h=i-7 y straightforward to see that their sum is t=i+B=6i-i Reference o D.Halliday, R. Rgsnick, and J. Walker, Fundament4ls of Physics 6th Ed. (John Wiley & Sons, !nc: Singapore,200l). i Materials Force table and the accompanying weights and a ring, level, digital balance (for the total mass), graphing papers, rulei, protractor, pencil, calculator or equivalent. Procedure The experiment utilizes a force table to examine the effect of forces acting on a ring. The forces are supplied by hanger with weights pulling towards directions controlled by the position of thg pulleys as indicatedby q large 350' protractor printed on the foroe table. The pulleys are much lighter,than.the loads and can be assrrmed to have insignificant effects compaled to forces. The magnitude of the force applied to the string (and therefore to the ring) is equallo the weight of the hanging mass (the container included). Since the weight is the product of the corresponding total mass M and the acceleration due to gravity g (considered constant all around the experirnental proa) M,may be, considered to be the force magnitude. To reeover the, actual force strength, we just have to multiply it with and (13) (14) Then it should be I 46 @ 2007 LabManual Authors Physics 71.1 Vectors and Force Table Figure 7. The force table and its accessories. Shown are the weights (W) with hooked hanger, pulley (P) and its locking screw (L), the ring (R), string (S), and the balancing screws (B). The force direction is read from the angular scale (C) marked along the perimeterof the table like a big 360-degree protractor. A complete setup of the force table is shown in Figure 7. The ring serves as the object at which the forces act together. The sum of these forces becomes the net force acting on the ring. Once the net effect to the ring is null, it is expected to stay on the center. The aim of adjusting the masses and their directions is to place the ring at the center indicative that the effects of the forces (provided by the strings) on it have been canceled out. The hooked hangers afid a set of masses are shown in Figure 8. These may be replaced by other unconventional weights like water bottles, sand and cups. The actual weights just have to be weighed using a (digital) balance. Figure 8. Hooked hangers (H) and a set of masses (M). These may be replaced by other weights like water in bags or sand in cups, etc. @ 2OO7 Lab Manual Authors 47 Vectors and Force Table Physics 71.1 A. Setting up the force table To avoid systematic errors introduced by the weight of the ring, the entire set-up must be leveled. A level is a device that uses water (or other liquid) to indicate leveled surface. A bubble resting in the center of the markings indicate level surface only along the direction of the level, so it will be advantageous to take two level readings - the second reading perpendicular to the first. The force table can be balanced by adjusting its three balancing screws. There are other things to be kept in mind to avoid erroneous readings. The strings have to virtually pass the center ofthe force table so that all forces (vectors) intersect through one point at the center of the table. The pulleys have to be made sure to rotate freely about their axles so that they offer insignificant added tension to the strings. The orientation of the pulleys should also be aligned with its strings' direction of pull. B. Reading the angular position The angular position is read from the mark on the force table (C in Figure 7) that is aligned with the string. Make sure that you read directly above the string, pelpendicular to the table otherwise, parallax error will be committed (see Figure e). Consider 0o as the positive x-axis direction and 90o as the positive y-axis direction. All calculations for the angle should be measured or determined relative to the 0' mark. Three basic cases will be studied, each case trying to show that the vector representation of forces is valid. All experimental value should have uncertainties given by the experimenter. C. (Case l) Resultant and equilibrant r I i I l t x I I t Assume that there is already a given resultant force with magnitude of about 2009 directed towards the 2lO' direction. Locate the equilibrant (the force that will nulliff the effect of this resultant) for this force. @ ,tttu"f, masses (total of about 2009 including its hanger) on one string, tie it to the ring, and pull it over a pulley. The pin placed in the center of the 48 @ 2OO7 Lab Manual Authors Physics 71.1 Vectors and Force Table force table should pass trough the ring holding it in place while there is a nonzero net force. @/eaiust the position of the pulley by loosening its lock and sliding it along the circumference of the table until the string aligns with the 2lo" ; mark. once in place, lock the pulley again. This pull serves.as the given ':i resultant.3) i i ,.,,.,i"""i', @ril-i.rthe entries for F, inTable I oftheActivitySheet. I ";'t ; @) r"another string on the ring, pass it oytranother pulley, and then o';t ,*i" i. matching combination of mass and,ldnger to be attached to the end of tliis stringtocounterorba1ancethepullofthefirststring-f,.'.,:Theposition of this pull may also have to be adjnsted by movin!-ihi: pulley lik.'th." ' '"J first. This pull has completely countered F t if the ring is at the center of the table (indicated as the pin passes though center of not touching, the ring as shown in Figure 10). The total mass and,position of this pull corresponds to the magnitude and direction of the equilibrant of the given resultant. @ or.. the ring is at the center, record the experimentally obtainedv equilibrant as F, in Table I of the Activity Sheet. 6. Record the expected (theoretical) magnitude and direction of the equilibrant in Table II as E along Case I row. Determine the angle 0o, between the predicted and experimental results of the equilibrant and fill in the corresponding column (case I). This angle is simply the absolute of the difference between the angles0t and e2 Complete Case I row by computing for the percent deviation A F (%) . The percent deviation is computed using the following formula: 7. 8. AF=@ and AFAF (%): ;t7 This formula is derived from the difference: (l s) @ 2007 Lab Manual Authors AF=E-Ft 49 F, Vectors and Force Table Physics 71.1 D. (Gase 1l) Equilibrant of two forees Two known forces are given: one pull, F, , clirected towards the 100' direction and the other, F, , towardd 2oO;';. The equilibrant; assigned as F, ', willbe sought. Clear the fotie tablb of the\ previous pulls and ieplace them with new pulls as follows. than 2009. in the directions in CaseJ in setting up th$e and F2 in Jaflsl.t.;I 6 ) 0,u.. two forces with net masses greater l0cf and 200' Follow the directions pulls. Fill in the column for F, correspondingly. td, r * rer ) 2a9o ;H:fn",:,?_TJ,T;,ff1;11',"f,,};;X :;::J* #Itr*r,, : (given) resultant in the previous case: Find the equilibranti;ofkhese pul'lq, just as what is done in Case I. 6) On.. the ring is placed at the center, record the obtained magnitude,a!rd\/ direction of the e[uilibrant as F, in the Activity Sheet's Table I. 4. Graphically determine the theoretically predicted restrltant in the space provided in Question la of the Activity Sheet. ..,Label the vectors correspondingly as well as the angle eR it makes with the x-axis. Indicate the scale used. A scale of 40g to 1cm is rdcommended. 5. Based on the results of the resultant, recordlhe expected magnitude and direction of the equilibrant in Table II as F r along the row of Case II (graphical). The equitribrant can be determined using the concept of Case I (Equation 5). 6. Compute for 0 oo as in Case I and complete the row for Case II on , Graphical method (also filI in its A F (%) oolumn, use Equation 15).. 7 . Label the angle between F, and F 2 as in the drawing in Question la. Use Equation 3 to compute for the magnitude of the resultant.Utilize the space provided in Question lb for the computation from rt to E using trigonometry. Fill in the row for Case II (Trigonometric). 8. Determine the angular position e E of the solved b (equal to 180'-0, ) using Equation 4. Complete the trigonometric method row by computing for 0 o, ahd A F (d/o). : 9. For the prediction using component method, compute for the component of the vectors in Table III along the x-axis and the y-axis. The components of F are just the sum of the corresponding cornponents of F 1 and 50 @ 2A07 Lab Manual Authors Physics 71.1 @ g6v Fr Veetors and Force Table F, . Use Equation 5 to get the components of h 10. Using Equations 10 and 12, calculate the magnitude and angular position of and complete Table II. Use the space below Table III for calculations. IV. E. (Gase lll) Orthogonal components of a force The equilibrant of a given force F , can be decomposed into two other forces that are perpendicular (orthogonal forces) to each other. The magnitude and direction of the equilibrant of a single force is already established from Case I so now, the case deals with decomposing an unknown equilibrant into two orthogonal components by showing that a given resultant can be countered by two orthogonal pulls. Again, clear the force table of the previous pulls and replace them with new pulls as follows. 1. Similar to Case I, place a pull with magnitude about 2009, directed towards 200' 2. Record the magnitude and direction of this pull as F, in Table I. 3. Now place two other pulls directed towards 0' and 90' so that they are perpendicular to each other. 4. Adjust the magnitudes of these two forces such that they counter the pull of the first. Note that these two forces should cancel their components along the direction perpendicular to the given first pull. 5. Once the ring is placedron the center, record the magnitudes and directions of the two pulls as F 2 and F 3 in Table I. Note that these vectors are actually the experimentally determined components of h=-F along the i and i directions, E, and E y respectively. ( 0. yCgmnute for the theoretically predicted magnitude of the components of\-/ R=F, using Equations 8 and 9. Record R, and Ry in Table G, "ur"d on Equation 5, determineV rubl" IV. E * and E y Record your results in (QCo*plete the table by calculating the percent deviation of the predictedV value from the experimentalvalue. @ 2007 .Lab Manual Authors 51 Vec{ors a nd Farce, Tdhle Physics 71.1 '' ,.t '' a- .---"' &,* 52 @2O07 Lab Manual Authors Nams Dato Submitted Date Pedomed Scorc Group llemb€E lnalluclor Sectlon WorkSheet: Vectors and Force Table Data Summary Data Table I. Experimentally obtained equilibrants (Cases I ond II) and orthogonal components (Case III). Uncgrtainty in magnitude : Uncertainty in position: Comparison of theoretical predictions and the experimentally obtained,vectors, .1 Data Table 2. Theoretically predicted equilibrant in comparison to the experimental result. Case Fl F, F3 Magnitude (g) Position e1 Magnitude (s) Position 02 Magnitude (g) Position e3 I 210. N/A N/A II 100" 240', m 200o' 00 900 case method i oo, * aF (o/o)Magnftude, E Direction, eE I expected II graphical trigonome'tric component * 0 o, is the angle between the theoretically predicted and the experimentalty obnined equitibraiil @ 2OO7 Lab,Manual Authors 53 vectors' aid,Folee rloit Physics 7'1,1 Data Analysis* ) When can we say that the predicted : ' ' tt i ! E is close enough to its experimentally determined i.:.' ,"' ' :,i i ,,!i A. Graphl66l p6ffigsJ.:. 'ri . i..3. Show youlcosputatrons necessary for determining the theoretical prediction of equilibranf E' - ln tho spacls provided below usinglgraphical, trigonometric and component method ; Indicate if-a separate sheetis attached' - Oo- :'j 2700 5/+: scale: _ cm: _g @,20O7 Lab.Mdriural Authtks'' u Physics 71.1 Veclotsand Force TaNe B. Trigonometric method Computation of fr ( is just equal to,this): Computation of e R (relative to the Oaposition, can be sho$m'graphically):' Computation of 0 E (still relative to the 0" position): C. Gomponent method Data Table 3; Coi,mponents of theforces invblved in obiaining the components of magnitude s (component method, able 3): E vector x-component (g) y-component (g) F, F2 n E Calculations for the masnitude E and the ansle e I)afe T @ 2007lab Manual Authors Veator componen{ Prdtctton (g) ',t, oawas*6h.,W). R,,3F1* N/A N/A 'R'-F"' N/A N/A Er=- R, E ,=- R, :,lffiorsafr&Fo*pe.:GWe I F.hlsbs:7tll Dare Tshle.A Cowparisons of the components of F, alo@rlrall+? t arl:d ,9*drped&ais.':i ,l l;. tl i 2: Summarize your"eonclusiofls iii llne-with'the obje0tive -clf the Sctivify.- l '.j ..: -..- .-:',,*.., .-, .. , -. ;I ,'illl: !- i:i.r'.i ..r,r;-ri:';.i:i ;!:J,i{..:{iLir}.;} i: ;;1'1;9i1i 1"; ' i'r l ' '} :'ti'':rt;ill1g r; :iii: s56 ,@ffiS7,E&,.fit'anu&liAilth6rs U niformly Accelerated Li nea r Motion (Ball) Objectives At the end of this activity you should be able to: 1. Determine experimentally the niagnitude of acceleration of an object undergoing un i form ly accel erated I inear m oti on. 2. Plot experimentally the graphs illustrating the position and velocity as a function of time for an object undergoing uniformly accelerated linear motion lntroduction , An object moving in one dimension with a constant acceleration is said to be undergoing uniformly accelerated linear motion. One example of type of motion is an object which is dropped from an initial height h which is allowed to fall freely to the ground. At all times, its acceleration is constant (with a magnitude of 9.8mls2 ) and is directed downward. Based on this knowledge, we will observe a freely falling object and, with the help of computer interfaces, determine the graphs that illustrate the position, velocity and acceleration of this object as functions of time. It turns out that the graph of the position is quadratic and for the velocity, it is a straight line with a negative slope, having a value very close to the predicted value of the acceleration due to granity g which is 9.8mlsz Theory Consider an object undergoing free fall. This object may either be dropped from a 57@ 2047 Lab Manual Authors Uniformly Accelerated Linear Motion (Ball) Physics 71.1 height above the ground or tossed into the air and allowed to fall back down again. Assuming that the object is moving in a uniform gravitational field and that there are no other forces present, the only force acting on the object is the gravitational,forc,e, whigh impartS.,an acceleration gfmaSmtude 9.8ru1s' to the objebt. Th'b magnitude of this acoeleration is cbnsttint'''and'' is always directed downward; hence, since the object is traveling in one dimension only, it is a perfect exarrtpfe' pf uniformly accelerated linear motion. Since the magnitude and direction of the acceleration of the object (hereby represented as g) is constant, from the definition of acceleration (which is the first derivative with respect to time of the object's speed and the second derivative with respect to time of the object's position), we find that: a) the object's speed is expected to be a linear function of time (speed is directly proportional to the time), and b) the object's position is expected to be a quadratic function of 'time (position is directly proportional to the square of time). Specifically, the equations describing the object's velocity v(t) and position y(t) with respect to time are v(t)=vo- 8t 1y(t)=y"-v,rt-)Bt2 where vo is the object's initial velocity and !o is the object's initial position. Note that the acceleration here is denoted by g. Hence, from the form of equations (1) and (2), the graph of the velocity is a straight line slanting downward with a slope of g : 9.8mls2 and the graph of the position is a parabola opening downward. Materials Vernier LabPro@ computer interface, motion detector, large round ball (basketball, soccer ball, volleybalt) Procedure 1. Connect the Vernier LabPfo@ interface to the computer. Follow the instructions in Appendix A illustrating how to carry out this procedure. Connect the Vemier Motion Detector to DIG/SONIC 2 of the LabPro or PORT 2 of the Universal Lab Interface. Place the Motion Detector on the (1) (2) il 2. I 58 @ 2007 Lab Manual Authors Physics 71.1 Uniformly Accelerated, Linear Motion (Ball) table. Open the file in the Experiment 6 folder of Physics with Computers. Three graphs will be displayed: distance vs. time, velocity vs. time, and acceleration vs. time. Toss the ball straight upward above the Motion Detector and let it fall back toward the Motion Detector. This step may require some practice. Hold the ball directly above and about 0.5 m from the Motion Detector. Click "collect" to begin data collection. You will notice a clicking sound from the Motion Detector. Wait one second and then toss the ball straight upward. Be sure to move your hands out of the way after you release it. A toss of 0.5 to 1.0 m above the Motion Detector works well. This can be achieved by tossing the ball in such away that it will reach the tip of your nose. You will get best results if you catch and hold the ball when it is about 0.5 m above the Motion Detector. Examine the distance vs. time graph. Repeat Step 4 if your distance vs. time graph does not show an area of smoothly changing distance. Check with your teacher if you are not sure whether you need to repeat the data collection NOTE: The COM, USB and other cables are indicated on the box containing the Vernier LabPro@ interface. l. Connect the COM or USB cable to the COM or USB ports on the LabPro@ interface and the computer. 2. Connect one end of the power supply to the corresponding outlet on the LabPro@ interface. The other (socket) end will be connected to the plug of the power supply. 3. If the LabPro@ device is connected properly, after a few seconds a tone will be heard and the lights in front of the device will blink. 4. Double click the Vernier LabPro@ icon on the desktop. If the device is properly connected, you should see the WELCOME screen immediately. If the SCAN screen is seen, click the SCA.N button to giVe the computer aJ. 4. 5. APPENDIX A. CONNECTING THE VERNIER LABPRO@ INTERFACE TO THE COMPUTER @ 2007 Lab Manual Authors APPENDIX.B. TROUBLESHOOTING GUIDE FOR THE LABPRO@ INTERFACE Unifomrly Accelerated. Linear Motion (Ball) Physics 71.1 more time to connect. If the connection still fails, consult the troubleshooting guide. PROBLEM:, The SCAN, not the YruLCOME, screen is seen after double clicking the LabPro@ icon 'C1ick'the SCAN button. If, after a few minutes, the WELCOME screen is seen,' proceed with the'experiment. If after clicking the SCAN button, the SCAN screen is still seen, close the LabPro@ window and remove the COMruSB cable attached to the computer. Reconnect the cable to another COMruSB port on the computer and repeat steps 1- 4 in Appendix A. 3. If, after the previous step, the SCAN screen is still seen, replace the COMruSB cable with another COMruSB cable, and repeat steps 1 4 in Appendix A. I. 2. 4; If ,after thg,,previous step,.the SCAN, sprgen is,still seen,.replace COMruSB'cablb with-a USB/COM bable, and repeat steps 1 4 Appendix A. the in 5. If, after the previous step, the SCAN screen is still seen, check the - connection of the power supply. If it is not connected properly, reconnect it and repeat steps 1 4 in Appendix A. 6. If, after the previous step, the SCAN screen is still seen, the CPU may have a problem interfacing with the unit. Replace the CPU or, if the CPUs are not enough for the class, merge with another group whose unit is functioning properly. PROBLEM: No tone is heard andlor the lights on the LabPro@ unit do not light up after beiong connected to the power supply 1. Re check the connection of the power supply with the unit. If the connection is faul.ty, recoqnect it aqd repeat step 2 in Appendix A. 2. If, after'the previous step, no tone is heard and/or the lights do not blink, 60 @ 2007 Lab Mdnual Authors Physics 71.1 Uniforqly Accelerated Linear Mation (Batt) the power supply may be defective. Replace it with a new one or a fully functional one, and repeat step 2 in Appendix A. If after the previous step, no tone is heard and/or the lights do not blink, electrical outlet may be defective. Move to another table with functional electrical outlets, and repeat step 2 in Appendix A. If after the previous step, no tone is heard and/or the lights do not blink, the CPU or the unit itself is defective. Replace it with a new one or a fully functional one. PROBLEM: No data is being collected by the motion sensor/photogate after the COLLECT button is clicked. 1. The device may be connected to the wrong port on the LabPro@ unit. Re check the port where it is supposed to be connected, and reconnect the device. 2. If, after the previous step, the problem still persists, close the LabPro@ window then, after a few minutes, double click the LabPro@ icon on the computer and repeat the experiment. 3. If, after the previous step, the problem still persists, the device may be faulty. Replace it with a new or fully functional one. PROBLEM: The graph produced by the motion sensor/photogate is truncated 1. Check the calibration of the motion sensor/photogate and the settings on the graph. Adjust them in such a way as to produce an untruncated graph. If, after the previous step, the problem still persists, the motion sensor or the CPU is faulty. Either replace the CPU or refer to the previous section and carry out steps I 3 there. aJ. 4. 2. 61@ 2007 Lab Manural Authors Lhtitutnly'Ais,el€rated l[,i'nwi *5666,,6Bla t0 Physios'7f .1 @2OA7 Lab Mdnual Authors Name Date Submitted Date Perfomad Scotr Grcup MembeB ln3tructor S6ction Worksheet: Uniformly Accelerated Linear Motion (Ball) For this portion, sketch or attach the printouts of the graphs produced by the interface on the spilces provided and answer the questions listed below each graph. Graph 1. Distance vs. time On the graph above, identiff and mark the region where the ball is being tossed but still in your hands. On ttre graph above, identiff and mark the region where the ball is in free fall. From the graph, what is the maximum height that the ball reaches? a a 63@ 2007 Lab Manual Authors ,,, IrdLlairr.tu*'-.,^.u . *^i!di lJniformly Accelerated Linear Mofron (Ball) Physics 71.1 . Click and drag the mouse across the portion of the distance vs. time grapn'that is - parabolic, highlighting the.free-fall portion. Click the Curve Fit button, select Quadratic fit f.o* the list of models and click utry frt". Examine the fit of the curve to your data and click "ok" to return to the main graph. Now consider the value of your "a" tetrn on the. graph (as co_mputed by the interface). Compute for the percent difference with respect Percent Difference: Graph 2. Velocity vs. time mark the region where the ball is being tossed but still in your hands. o On the graph above, identifu and mark the region where the ball is in free fall. . From the graph, what is the maximum velocity that the ball reaches? @ 2007 LabrManual Authors Physics 71.1 Uniformly Accelerated Linear Motion (Ball) . From the graph, what is the velocity of the ball at the highest point of its motion? o Click and drag the mouse across the free-fall region of the motion. Click the Regression button. Now consider the value of the slope (as computed by the interface). Compute for the percent difference of the slope with respect to the theoretical value ofg, which is 9.8 m/s2. Percent Difference: Graph i. Accelerationvs. time . Is your graph for the acceleration as a function of time perfectly straight? If not, what could be the reasons why it is not perfectly straight? @ 2007 Lab ManualAuthors 65 Ua ffid Phyeice.71,1 o Click and*ry6o:BQ*rQ9rasress fre ftee:fall.sporion o[thernption,qs,clic{c rhe Statbtics button. How olossly does the mean acceleration value compare to the values ofg found in the previous,steps? I ll 1l l - ,: -_ i itl .r *i.. j i ! I i 1. .,1 ri : I {. .i;ii , t . ,1it ., .,. i,1;,, , i... ii. U n iform ly Accelerated Li nea r Motion (Picket fence) Objectives At the end of this activity you should be able to: 1. Determine experimentally the magnitude of acceleration of an object undergoing uniformly accelerated linear motion. 2. Plot experimentally the graphs illustrating the position and velocity as a function of time for an object undergoing uniformly accelerated linear motion. motion Introduction This experiment extends the previous experiment (Unifomly Accelerated Linear Motion: Ball) by considering a picket fence released fiom rest. The same theory applies and the main difference is found in the procedure. The student is also advised to consult the appendices of the previous experiment which detail how to connect the sensors and how to troubleshoot the setup. In this experiment, the ball is replaced by a plastic bar (the picket fence) with equally-spaced black-painted strips. The motion detector is replaced by a Photogate which is an infrared light source coupled to a detector. When infrared light to the detector is blocked, the detector records the time. In this manner, dropping the'plastic picket fence through the photogate allows us to measure the time between dark bands as the picket fence accelerates, @ 2007 Lab Manual Authors 67 l! niformly Accelerated Li near Motion (Picket fence) Materials Proced0ie vernier LabPro@ computer interface, photogate, picket fence 1.,. 1 Attach the vernicr LabPro@ interface. .to . tht computer. Follow the instructions in the APPendtx. fu.t", the Photogate rigidly to a ring stand so the arms are extended horizontally. The entire length of the Picket Fence must be able to fall free'ly through the Photogate. To avoid damaging the Picket Fence, make sure it has a soft surface (such as a carpet) to land on' Connect the Photogate to the DIG/SONIC 1 input of the LabPro or the DG Physics 71.1 a -r. 1 input on the ULI. . : 4. Open the file in the Experiment 5 folder of Physics with computers'Two graphs will appear on the screen., The top graph displays distance vs' time, and the lower graph, velocity vs. time' 5. Observe the reading in the status bar of Logger Pro at the bottom of the screen. Block the Photogate with your hand; note that the Photogate is shown as blbcked. Remove your hand and the display should change to unblocked. This means that the photogate detector is ready. 6. Click "collect"to prepare the Photogate. Hold the top of the Picket Fence and drop it through the Photogate 5 to 8 seconds after the "collect" butlon is clicked, releasing it from your grasp completely before it enters the Photogate. Be careful when releasing the Picket Fence. It must not lylt the sides of the Photogate as it falls and it needs to remain vertical. click "stop" to end data collection' 7. , ixamina your graphs. The slope of a velocity vs. time graph is a measure of acceleratioo. ti the vplocity graph is approximately a straight line of constant slope, the acceleration is constant. If the acceleration of your Picket Fence appeafs constant, fit a straiglrt line to your data. To do this, click on the velociry graph once to select it, then fit the line y : mx * b to the data. Record the.slope in the data table' 8. To determine the shape of the distance vs time curve, click and drag the mouse across the graph. Click the Curve Fit button, select Quadratic fit from the 1ist of models. Examine the fit of the curve to your data and return to the main graPh. 68 @ 2OO7 Lab Manual Authors Physics 71.1 Uniformly Accelerated Linear Motion (Picket fence) If you are not satisfied with just one trial, you may repeat steps 5 and 6 as many times as you want to obtain an average value of the slope. Do not use drops in which the Picket Fence hits or misses the Photogate. Record the slope values in the data table. 9. @ 20,07 Lab Manual Authors 69 .t .il tl I Uniform ly Accdl6raled Llnea r M'iltio n;(Picket fe nce) Physics'71.1 @ 2007 Lbb Manual Authors ,l t L l{tme Dre Submftf.d o# F.rfomld Scdr GEup ltembeE lnstruc{or Sacdon Worksheet: Uniformly Accelerated Linear For this portion, sketch or attach the printouts of the graphs produced by the interface on the spaces provided and answer the questions listed below each graph. Graph 1. Distance vs. time o What is the shape of your distance vs. plot? Using the curve fitting tool, vnite out the equation describing your graph. What is the value of the acceleration due to gravity ? Motion (Picket Fence) @ 2007 Lab Manual Authors lJniformly Accelerated Linear Mofrfli (Plcket fence) Physics 71.1 o What is the shape of your velocity vs. plot? Using the curve fitting tool, write out the pquation describing your graph. What is the value of the acceleration due to gravity? Obtain several measurementsof the acceleration due to gravity using this setup. Determine the best estimate and use this as your experimental value. Calculate your percent deviation using 9.81 m/szas your theoretical value, which is the accelaration due to gravity at the Earth's surface. 72 @ 2A07 Lab Manual Authors Kinematics of Projectile Motion Objectives At the end of this activity you should be able to: Veriff that in a projectile motion, the horizontal and vertical motions are independent with each other. Determine the trajectory of a projectile motion. 1. 2. lntroduction Theory The most common example of two dimensional motion is projectile motion. Consider for example a tall thrown at an angle less th*' 90' rror" irr. horizontal. Assuming that only the gravitational force significantly acts on the ball, the trajectory or path observed is parabolic. Many bodies in motion exhibits projectile motion. Some examples are) a cannon shot,a ball thrown upwards and an affow shot by a bow. Projectile is the motion of an object that has an initial velocity vo moving undgr the influence of graviff. In the absence of air resistance, gravity is the only force that acts on the object which acts only along the vertical rnotion. Since there is an absence of horizontal force to affect the horizontal motion of the object, the magnitude of the component of velocity along the horizontal does not change. Hence the acceleration alongx ( ox )and,y ( an )are givenby equation l. @ 2OO7 Lab Manual Authors 73 Knematics of Proiectile Motion ar=0 and or=-B Figure 1. An object in a projectile motion has an initial velocify v o Physics 71.1 (1) of initial velocity is given by the (2) . . ,. (,:) (4) The path followed by the motion of an object is called a trajectory. To derive this mathematically, thg trajectory of a body in projectile motiorr, we consider an $qjeet.rtrlthaainitidt veloc ; 'vr1;tnqwa;at an auglq '0. .iias shown in Figure :i t ill lI lu The horizontal and vertical expresslons component y= yorcos0 v=vorsin? Since the acceleration of the object along the x-axis is zero, at every point in the path of the objeg! the horizontal component of yelo-city i.s alyays equal to the ro, . The horizontal'dis'tance x, traveled by the objeit is given by While the vertical position, y canbe described by the equation lI '.y=yo+v,sin0t-;gt' I (s) : it willGenerally, the projectile may not be released at the same height at which land. The initial height of felease'is expressedtas:' !o in expression 5. By obtaining the expression of timb, / from equation 4 and substituting it to 5, we derive the form 12!=lo* *tanT-!-L- " v|"cos2 e (6) @ 2007 Lab ManualAuthors Physics 71.1 Kinefiatics otf"Projectite Motion The curve ex'hibited by equation 6 is an inverted parabola. In this experiment, we ' will experimefltally.obtain the trajectory of a body under a projectile motion. Reference Tipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H. Freeman and Company, USA, 1999. Materials inclined plane, protractor, ruler, metal ramp, carbon paper, marble , : : , ,, : ; : Procedure . A. Galculation of the initial velocity of the projectile. ) Figure 2. The experimental setup. 2. Set the inclined plane to a specific angle. Use angles less than 20' @ 2OO7 Lab Manual Authors 7i K nematics of Prsjestile MOlisn Drop dhe marble at the high end of the B, the marble will,undergo projectile marble land (point C). Physics 71.1 metal ramp (at point A). After point rnotion. Mark the range where the ,. J. 4. Figure 3. The schematic diagram in obtaining the initial velocity of the projectile Choose four (4) angles and write ouf your ddta in Table 1. Do not forget to obtain several trials for the measured range and present your data in terms of the best estimate. B. Determination of the trajectory of the projectile l. Set up the inclined plane, 4. Carbonppu (-r* -: ',: l]l Figure 4. The schernatic diagram in obtaining the trajectory of the projectile metal ramp and ca-r,bon paper as shown in Figure 76 @ 2007 Lab Manual Authors Physics,71.1 Kl n em atles of. Proj eati I e M oti o n using the same angles in Table 1, set the horizontal distance x from the carbon paper. Drop the marble at point A and, obtain the corresponding vertical position y. Vary the horizontal distance and obtain the corresponding vertical position. You should have a minimum of five (5) data points to observe the trajectory of the marble. Again, do not forget to obtain several trials for the measured vertical position. Write out your best estimate of y in Table 2. compute the theoretical value of vertical position y by substituting the variable.r on expression (6) and write it out on Table 2. Plot the vertical position as a function of the horizontal distance using Excel and superimpose the theoretical y'on the graph. Make sure that the size of your graph is not too small. Use one sheet of paper for every graph. Compare the theoretical and experimental hajectory. What is the general shape ofthe curves for each angle? 2. 3. 4. 5. 6. @ 2007 Lab,Manual AUthors Tf K n e ma ti c s of . P rol ec ti ! p,l/. I ati sn Physies 71.1 @ 2007 Lab Manual Authors'78 Name D.te Submitted Date Podomed Scor? Grcup MembeB lnstructot Section Worksheet: Kinematics of Projectile Data Summary A. Galculation of initial velocity of projectile Data Table I. Range and initial velocity of a projectile at dffirent angles of release Angte of release (d"S) Range (cm) lnitial velocity (cm/s) . ril/hat do you observe about the projectile's range Motion @ 2007 Lab Manual Atdthors Kinematics of Proj*tile Motion Physics 71.1 B. Projectilels trajectory Dato Table 2. Horizontal and vertical components of the trajectory of a projectile. Angle of release (deg) Angle of release (deg) Initial height (cm) .: Hortzontal'Illstance x (cm) Horizontal Distance x (cm) Vertical Distance Vertical Distance Angle of release (deg) lnitial height (cm) ! theo Veftical Distance Angle of release (deg) lnitial height (cm) ! theo Veftical Distance lupr I *eo / theo . On a separate sheet of paper, paste your graphs of the superimposed theoretical and experimental plot of the trajectory of the marble. Use one sheet of paper for every plot. @ 2OO7 Lab Manual Authors Conseruation of Energy and Momentum Objectives At the end of this activity you should be able to: o Illustrate the law of conservation of energy and momentum using a pendulum-proj ectile system. lntroduction One of the basic laws of Physics is the law of conservation of mechanical energy (COME). It states that f,or a physical system where the only internal forces acting on it are conservative, the total mechanical energy, i.e., the sum of the kinetic and potential energies is constant. In the presence of external and dissipative forces, the law becomes more general. In this system, some mechanical energy may be lost and transformed into another form of energy, ensuring that the total energy is constant. The law of conservation of momentum (COM) is important in situations where we have two or more interacting bodies. This conservation law is valid when the vector sum of all extemal forces acting on the system is zero. In these type of systems, the momentum before and after collision of two objects is constant. The pendulum-projectile experiment allows us to'simultaneously veriff the energy and momenfum conservation laws. The collision of the pendulum bob and @ 2007 Lab Manual Authors Conseruation of Energy and Momentum Physics 71.1 the marble can be analyzedusing the momentum conservation. On the other hand, the dependence of the angle of release of the bob with the range of the marble can be understood by means of energy conservation. Consider the system in Figure l. A length / is raised at some angle 0 pendulum bob ( m B ) attached to a string of .+x + Figure 1. A pendulum-projectile setup After releasing the;bob, it hits a marble ( m* ) which is initially at rest. The velocity of the bob just before'it collides with the marble ( vrl ) is obtained by'applying'COME from point Ato B. We find that the expression for the velocity of the bob before colliding with the marble is given by :' i I i . ,ur=J2g( 1- coso) (1) Upon collision of the bob with the marble and noting that the marble is initially at rest and hence v*t : 0, we apply COM to obtain the equation mBv Bt=-mov szlm*y*z (2) where v az and 'v -z are the velocities of the bob and the marble just after the collision. We may express equation (2) in terms of v*2 and vsr since for elastic collision, the speed of approach is just equal in magnitude but opposite in direction with the speed of recession. Hpnce, we may vsz is given by the following expression 82 @' 2007 Lab:Mdnual Authors Physics 71.1 Reference V sy=V 12- | Bt Substituting equation (3) to (2), we obtain an expression of in mm and vat : Conseruatlon of Energy and Momerilum , , (3) termsof mB , '"'(4) After collision, the marble then undergoes projectile motion and will land at some distance x from its initial position. By applying conservation of energy from point B to point C, we obtain the expression ,, -2mou u,Y m2- lTlst lll- 1p2 2 LLoth,i V ^r=V^rl-- Zgnmru The expression above maybe fttoie concretely shown in terms the projectile by substitutiirg'expression (4) and (6): (5) where E o,0", ,7s the energy diqsipate&in the systern.,The velocity v*3 in the right-hand side of the equation is the velocity of the marble as it hits the ground, v ^3=x !tr (6) By manipulating equatiori ($), wre obrtainp final form for v *2 : (7) ofx, the range of 2 vnt g(m r+ *-)' *' (mu+m^)2 (E *o*-m^gh) (8)8m2rh Zm'rm- Therefore, from the law of COE and COM, we were able to derive an equation for the velocity of the bob ( vat ) just before it hits the rnarble as a function of the range of the projectile (r). A plot of ,'r, vs. *' will give us a slope of s(rypl?^)' and a y-inrercept of ,?\*^Y (Eo,n",_m,gh)8m"uh zm Bmm Tipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H. @ 2007 Lab Manual Authors 83 Conseruation of Energy and Momentum Freeman and Company, USA, 1999. pendulum setup (bob, string, tripod digital balance, carbon paper Physics 71,.1 stand), protractor, ramp, meterstick, marble, Materials Procedure Figure 2. The experimental setuP. Measure the mass of the marble and the bob using the digital balance. Setup the pendulum such that the length of the string is just right for the bob to hit the r,narble at different set angles. Attach a sheet of paper under a carbon paper where the marble will most likely land. Place the marble on top of the ramp and displace the pendulum bob at some angle. Measure the eorresponding range (x) traveled by the'marble. Carefully take note of the uncertainty in your measurements. 1. 2. 3. 4. 84 @ 2OA7 Lab Manual Authors Physics 71.1 Conservation of Energy and Momenturn Figure 3. To determine the angle of release, place a protractor with the 90-degree angle aligned to the string. The angle is then measured from the vertical angle. Repeat step 4 for five (5) different angles. Plot 4, vs. * and answer the questions provided in your answer sheet. 85@ 2007 Lab Manual Authors ff Conservation of Energy and Momentum Physics 71 .1 I I I r l. 86 @ 2OO7 Lab Manual Authors Name Date Submitted Date Perfomed Score Group Members lnstruct6r Section Worksheet: Cohservation of Energy and Data Summary l. Measurement of Constants Tabulate below the results you obtained from steps 4 and 5 of the experiment. Table l. Range and velocity of the marble corresponding to the angles of release. Table 2. Square of the range andvelocity of the marble corresponding to the angles of release. Angle of Release ( 0 ) Square of the range of the marble ( *' ) Sguare of the velocity of the bob ( v2u, ) Momentum Range of the Marble (x)Angle of Release ( 0 ) Velocity of the bob ( v r, 1 A 2OA7 Lab.Manual Authors 87 Conservation of Energy and Moryq1tytm Physies 71.1 Questions l. On i graphing paper, plot a graph of v'r, vi. *' fiom the values recorded in Table 2. The y-axis of your plot should correspond to the square of the initial velocity of the bop ( ,tr, ), yhil-eu tle x_, q.Iil lltogtq cg{rg.:pgn.$ lp ttrp sggale qf.lhg -rar1gg .9f,the,rnarble - ' ,lY .x )' ;rlomiute th6 flop6 bna thb f-inierdept ortnr plot.and rvli'te dbwn"tH v'dhres in ,:.;: , ,; the;paQe nroVided below. These will correspond to the experimental slope and y- ' intercept. Slope: y-intercept : ,i To obtain the theoretical value for the slope, compute the value of the quantity . / , .. r2 using the measured value of the height of the ramp. Write down yourg\mB+ mn) -@; calculation and the final value in the space provided below. Theoretieal slope: Calculate the relative deviation between the theoretical and experimental values of the slope. Write down your calculations below. Relative Deviufion: 88 O 2007 LtbManual Authors Physics 71.1 C,onliCruafrdn' of [email protected] Momentu m o ' What does this deviation s'ignifu? Can fiisib€ aprodfdf the'conservatibn of eirergy in the system? Why or why not? To obtairt'the valiie'of the dissipated energy' E o*n , follow the step:6r-.r* calculation: 1. Using the expression of your slope from your v'r, vs. x' plot, obtain the value for the h of the ramp corrosponding to the experimental slope 2. Using this value of h,wite the expression for the y-intercept of the vs. x2' plot 3. Equate the expression obtained above to the experimental y-intercept 4. Compute for the value of Eotnn Amount of energt dissipated: .2, vst @ 2WT Lab,Manuhi Alttltors 89 Consgrvation of Energy and.Mo" mefitu m Physics.71.1 o . Cornpare thE arnoqnt of enorgy dissipate.{go the perqent diff,erence of your slope. How can you relate these two quantities? Is the rrrechanioal-energy of the system conserved? E5plain'by using data obtained in the experiment o Is the total energy of the system conserved? Explain by using data obtained in the experiment. 90 O 2007 Lab,MaQual A't4hors Static Equilibrium Objectives At the end of this activity you should be able to: 2. lntroduction Determine experimentally where an object must be suspended (center of gravity) and the conditions which it must satisff (conditions of equilibrium) for it to be in static equilibrium. Apply the conditions of equilibrium in finding the mass of an object. Theory t. Whenever we see an object perched or mounted on any surface and is perfectly still, we say that the object is "balanced". Examples of these are a bird perched on a wire or a person standing on a ledge. How do these objects maintain their balance or state of equilibrium? In this experiment, we aim to answer this question by observing an object which is suspended at a point. We find that these objects must satis$r certain conditions to remain balanced, or what is known in physics as in a state of static equilibrium. First, nothing must be causing the object to move (the net force acting on the object must be equal to zero). Second, the object must not rotate or tip over (the net torque due to the forces acting on the object must be equal to zerc). To achieve the second condition, the object must be supported or suspended at apoint which we call the object's center of gravity. An object must satis$r two conditions for it to be in static equilibrium. The first condition is based on Newton's law and the second condition on the dynamics of rotation of rigid body. A body"iatisfiing the first and the second conditions of @ 2007 Lab Manual Authors Static Equilibrium Physics 71.1 equilibrium is said to be in static equilibnum. When a rigid body is in equilibrium, it does not accelerate. This is often called the first condition of equilibrium. That is, the vector sum of all the (external) forces acting:ron thg bPdY:is zero, or ' ,l' 'l) r,=o When the vector sum of all the torques acting on a rigid body is zero, it does not rotate. The sum of the torques due to all the external forces acting on the body, with respect to any specified point, must be zero. This is the second condition of equilibrium, or in equation form, I r:o where the torque T (which is a vector quantity) is defined as i:7 xF where 7 is the radius vector pointing from any axis point to the point at which the force vector F acts on the object. The magnitude of torque is given by r=rF sin? (1) (2) (3) (4) Materials where e is the angle between the vectors F and i There is a particular point in a rigid body where the sum of the torques. due to its weight elements is zero. This point is called the center of gravity of the object. We can think of the center of gravity as the point where the weight effectively acts. An object suspended along a line through its center of gravity will not rotate. plastic beam, ruler or tape measure, metal pans, a set of standard masses, digital balance, hanger or beam holder 92 @ 2007-Lab Manual Authors Physics 71.1 Procedure Static Equilibrium Figure 1. Some of the equipment for this activity. The beam suspended by the hanger is the setup for Part I in the Procedure. Part l. Determining the center of gravity of a uniform object. Insert'the beam into the holder then slide the holder along the beam until it reaches an arbitrary point on the beam. Tighten the screw in the holder then suspend the beam from this point. Observe how the beam moves as you release it after suspending it. Determine the forces acting on the beam in this case. Note the distance of the point where the beam was suspended from the right side of the beam. Locate the center of gravity of the beam by' first moving the beam along the holder to another point on the beam then releasing it until, upon release, the beam is no longer moving and is almost parallel to the horizontal. Suspend the beam at this point and determine the forces acting on.the beam in this case. Again, note the distance of the point where the beam was suspended from the right side of the beam in this case. Part ll. Determining the mass of a'uniform object using the conditions of static equilibrium 1. Support the beam at a point 30 cm from its left end. Put a 1009 mass on the shorter end of the beam and restore equilibrium by putting masses on the other side. Take note of their position with respect to the point of suspension of the beam. Using the values of the masses and their position, ' calculate the mass of the beam by means of the conditions of static equilibrium. l. 2. @ 2007 Lab Manual Authors g3 Static Equilibrium Physics 71.1 2. Determine the mass of the beam using the electronic balance. Use this as the reference value for the mass of the beam. Calculate the percent difference with respect to the value calculated in the preceding part. Part lll. Finding the center of gravity and mass of a nonuniform object using the conditions of static equilibrium. 1. Attach an arbitrary mass to any point of the beam. Once the mass is attached, consider it to be part of the beam. : Locate the center of gravity of the beam masq system using the same procedure in number 2,Pafi I of this experiment, then suspend the beam from this point. Note the distance of the point of suspension from the right side. Suspend the nonunifofm beam from a point not at the center of gravity, then restore equilibrium by adding masses to the left,and right sides of the beam. Note their positions with respect to the point of suspension. Using these values, apply the conditions of static equilibrium and calculate for the mass of the beam. Using the electronic balance, obtain the mass of the beam and use this as your reference value for the beam's mass. Calculate the percent difference of the reference mass against the mass calculated in the previous number. 2. aJ. 4. Figure 2. The experimental setup for Part II. 94 @ 2007 Lab, Manual Authors Name Date Submitted Date Perfomed ;Score Group Members lnstructor Section Worksheet: Static Equilibrium I Center of gravity of a uniform object . Initially, suspend the beam at some arbitrary point. Write it down as your initial point of suspension. Observe the direction of rotation. Based on this, locate the final position of suspension of the beam. At this position, the beam is not rotating. Initial point of suspension (from the right of the beam): Direction of rotation: (clockwise/counterclockwise) Final point of suspension (from the right of the beam): cm . Draw a free body diagram of the beam when it is supported at the final point of suspension. . Is the net force and the net torque on the beam equal to zero? Why or why not? . From your observations in Part 1, would it be better tb simply weigh the beam digital balance than to obtain its mass indirectly using the principles of static equilibrium? Why or why not? using the 95@ 2007 Lab Manual Authors ,, Static Equilibrium Physics 71.1 ll. Determining the mass of a uniform obiect using the conditions of static equilibrium Left of pivot Right of pivot Location of the added mass from the point of support Mass Draw the schematic diagram of the setup with corresponding measurements.Write out your solution in determining the mass of the beam. . :' 100 g Analytical mass of the beam: Measured mass of the beam : Percent deviation : (}E g (measured using digital balance) % and2,what are the conditions that a body must satisfu o List down all the forces acting on the beam in this case: . Are the net torque and net force acting on the beam both equal to zero? Why or why not? From your observations in parts I for it to be in static equilibrium? 96 @ 2007 Lab Manual Authors Physics 71.1 Static Equilibrium lll. Finding the center of gravity and mass of a nonuniform object using the conditions of static equilibrium *Indicate wether CoG is to the left or to the right of the point of support. Draw the schematic diagram of the setup with corresponding measurements.Write out your solution in determining the mass of the beam. Write out your solution in determining the mass of the non-uniform beam. Suppose now that the mass which is part of the non-uniform beam be moved to another portion of the beam. Would the center of gravity of the mass-beam system change? Location of center of gravity of the non-unfform beam, measured from the right end (cm) Added arbitrary mass,right of support (g) Added arbitrary mass, left of support @) Distance of the point of support to the center of gravity of the non-uniform beam (cm)* Distance of the point of support to the arbitrary added mass, right of support(cm) Distance of the point of support to the arbitrary added mass, left of support(cm) Measured mass of the non-uniform beam (g) Analytical mass of the non-uniform heam (g) Percent devi atio n (o/o) @ 2OO7 Lab Manual Authors 97 Static Equilibrium Physics 71.1 lr ( ; I, !rr n,, r,ix 98 @ 2007 Lab Manual Authors Simple Harmonic Motion: Spring Mass System Objectives At the end of this activity you should be able to: 1. Determine the dependence of the period of a simple harmonic motion on the amount of displacetnent and mass of the object. 2. Obtain the best estimate of the elastic (spring) constant for the vertical spring-mass system. 3. Determine the mass of an object using the concept of simple harmonic motion using a spring-mass system. Many types of motion are repetitive. A ship bobbing up and down in water, a swinging pendulum of a clock and the vibrations of guitar strings - these types of motions are called periodic or oscillatory. The simplest form of periodic motions is called simple harmonic motion. This occurs when the restoring force is directly proportional to the displacement from the equilibrium. The classic examples of this type are the simple pendulum and the spring-mass system. In this activity,we shall study the motion of the spring-mass system, and examine the parameters that affect its motion. lntroduction @ 2007 Lab Manual Authors 99 Theory Simple llarmonic Motion: Spring Mass System Physics 71.1 (1) Newton's 2dlaw states that F, is also relatedto the mass of the object and its acceleration such that Consider a system consisting of a spring with spring constant k also termed as the stiffness constant, and an object with mass m attaghed to the qnd of the spring FigqG 1). Whgrr rhe, Qbjeet iso disp'lacel Of s'bla? fistance .r, a force, F* is exerted by the spring on the object, given by Hooke's law: F "=ffia, where ax is the acceleration of the object with mass la displacement of the object.r given by d2xa*=m7/ Substituting equations I and 3 into 2,we obtain" : d2xr , _t*=mlt And finaily an expression for a " 'the' in terms of the spring constant k and the displacement of the object from the equilibrium position d'x -kn :-=-vxdim The acceleration of the object is proportional to and opposite in direction from its displacement. This is a characteristic of an object in simple harmonic motion. The period T of a simple harmonic motion is the amount of time required for the object to completely oscillate back and forth about its equilibrium position ( (2) and is related to the (3) (4) (s) Figure 1. A spring-mass sYstem 100 @ 2OO7 Lab Manual Authors Object on a Vertical Spring When an object hangs vertically fiom a gpring, in addition to the restoring force F : - k, exerted by the spring on the object, there is a force equivalent to mg directed downward., Chooslng the downward.direction to be positive, Newton's Phlsics 71.1 Simple Harm on lg, M oti on /$gri ng':Mass,sysfem x= xo ). It is related to the niass of the blopk aud the spring constant, and the relationship is given by r=2nE (6) (7) How do we handle this extra term? ; ; I If y,=T is the distance the spring is stretched when the object is added and the system'is in equilibrium, then making'a change of varidble in the form second law reads I ,d'vm':=-krl mg This differs from eqtibltiofl (2)'o*V the addition of the constant mg. which reduces equation 3,into ;12, aytn*=-lsY'dt' which is now similar to equation (2). .1 *l .i (8) (e) @ 2047 Lab, Manual Authors 1.B{ Simple Harmonic Motion: Sprrhg,613s5 Sysfem Physics 71.1 Figure 2. An object suspended from a vertical spring. (A) Equilibriumposition of the spring when the object is not yet attached. (B) Equilibrium position of the ryatem when the object is attached. The spring is sffetched by an amount of ye:mgllc (C) The object oscillates about the equilibrium position with a displacement of y':y-yo. The effect of the gravitational force mg is simply to shift the equilibrium position by an amount lo :mglk, fromy:0 to y':9. When the object is then displaced by an amount y', the spring exerts a restoring force of -lry'on the object. The object oscillates about this equilibrium position with a period equal to equation 6 the same as that for an object on a horizontal spring. Hence, even in the presence of gravitational force, the spring-mass system also undergoes simple harmonic motion. Materials Vernier LabPro@ computer interface,Photogate, A set of similar springs, Pendulum setup, Set of masses, Object of unknown mass, Digital balance Procedure Determination of the spring constant of a single spring 1. Connect the Vernier LabPro@ computer interface to the computer. 2. Fasten the Photogate rigidly to a ring stand (using the pendulum set-up) such that the arms are suspended horizontally. Make sure that the masses are able to pass freely through the Photogate as shown in Figure 3 142 @ 20i07 Lab Manual Authors Physics 71.1 Simple Harmonic Motion: Spring lUass Sysfem Connect"the Photogate to DIG/SONIC 1 input of the computer interface.In the ffi icon, double click Physics with Computers, open the folder marked Experiment 14: Pendulum Periods then the file Photogate. A plot of the Period as a function of the Trial Number will appear on the screen. Attach the spring-mass system to the ring stand. Make sure that in equilibrium position, the mass is blocking the Photogate (as in Figure 3). This can be seen in the status bar of the Logger Pro at the bottom of the screen - if the mass is blocking the Photogate, the status is noted as blocked, otherwise it is unblocked. 5. Displace the spring-mass system from its equilibrium position by a distance y'. 6. Click the ffi button, and release the mass. The mass then oscillates about the equilibrium position, and the period of oscillation is shown in the graph. Note that you may have to wait for a -few seconds before the period of oscillation is registered by the Logger Pro. 7. After 10 trials, click the Stop button. Highlight the plot, then click the 3. 4. .Figure 3. Simple Harmonic Motion setup. 103@ 2007 Lab Manual Authors $irnpl€ Harmonic Motion: Spring /lfass Sysfem Physics 71.1 button to obtain the mean period of oscillation of the system. 8. Obtairr tho'period for varying values of the mass displaced and the amount of displacement. 9. From the period of oscillation obtained, calculate the experimental spring constant. This value will be used to obtain the mass of the unknown object. Determination of the mass of an unknown object Using a set of springs of the same spring constant, create five setups with different resultant spring constants. 1. Obtain the period of oscillation for each of the setups created. 2. From the,period of oscillation obtained, calculate the mass of the unknown object, ., ':" Reference . Tipler, Paul A., Physics for Scientists and Engineers, Fourth Edition, W.H. Freeman and Company, USA, 1999. 104 @ 2007 l-ab Manual Authors Name Date Submitted Datd. Performed Score Group Members lnsiructor Section Worksheet: Simple Harmonic Motion A. Period dependence on the angle 6f release Object's mass : Table 1. Period dependence on the amount of displacement . How does the period depend on the amount of displacement of the object? B. Period dependence on mass of the object Amount of displacement : Table 2. Period dependence on mass of the object Mass (g) Period T(s) @) Amount of displacement (cm) Period T(s) ol5 Difference Experimental Theoretical @ 2007 Lab Mdnual Authors Srmple Harmonic Motion: Spring /l{ass System Physics 71.1 . How does the period depend on the mass of the object? lll. Spring constant calculation Plot the square of the experimental period ( T' ) as a function of the mass (z) obtained from Table 2,then answer the following questions: o What is the slope of the plot of f vs. m? o From this slope, calculate the experimental value of the spring constant. {,06 @ 2407 Lab Mdnual Authois Physics 7't.1 Sr:mple Harunonia Motion * Spring Mass Sysfem lV. Period dependence on the spring constant Amount of displacement : km) setups and calculation. Galculations o How does the period depend on the spring constant? Table 1. Period dependence on the amount of displitcement Spring constant (dyne/cm) Period T(s) %o Difference Experlmental Theoretical . How did you obtain the different values for the resultant spring constant? Show all @ 2007 Lab Manual Authors 107 Simple Harfiionic Motlon : Spiing ltfassisyb(em Physics 71.1 V. Unknown mass calculation Plot the square of the experimental period ( r ) as a function of the inverse of the spring constant(i/k)obtained from Table 3, then answer the following questions: . What is the slope of the plot of I vs' 1/k? . From this slope, calculate the experimental value of the unknown mass' r08 @ 2007' Lab Mdnual Authors Physics',7{.1 s t m pb : narhiarffEllt'dfioif r spriiigr uessqrc'rerri: . Measure the mass of the object using a digital balance. What is the percent deviation of the experimental mass to the acbloilmass of the object? What are the possible sources of error in the experiment? @ 2'007' Lab,Ma n ual Atrttlorb {oe gimile,HarmonicMation:$-pr-pgMass,Sys{efi :. Physics 71.1 O 2007 Lab ManualAuthors Sound At the end of this activity you should Ue:iUle to: Measure the speed of sound. understand and observe interference and beats using sound waves. Measure the beat frequency of two tuning forks. 1. 2. aJ. lntrsduction Sound waves are longitudinal wavds passing through any medium such as air, solid or liquid that have frequencies within the range of human hearing. Sound waves may alsq be in the form not audible gnopgh to be perceived by humans. For example, medical practitioners usq ultrasound waves to form an image of a fetus inside a pregnant woman's womb. Sound waves have also been used to detect oil in the earth's crust. Ships cany with them sound emiuing equipments called SONARS to detect underwater objects. - In this experiment, we will measure the speed of sound by detecting the echo or ieflected sound of a finger snap. Also, we will study the interference of two sound waves with slightly different frequencies called beais using two tuning fo.ks and a Vemier microphone. Sound is a form of mechanical wave that is understand how sound waves are produced, produced by a vibrating object. To consider a loudspeaker. When its Theory @ 2007 Lab'Manual Authors Sound PhYsics 71.1 diaphragm moves outward, the air in front of it is compressed and will cause an increase in air pressure. This region with increased pressure is called a condensation. After producing a condensation, the diaphragm immediatety reverses its motion and moves inward. The inward motion produces a region knows as rarefaction,withpressure less than the ambient surrounding air' These oscillatory changes in pressure propagate and arrive at the ear' It forces the eardrum to vibrate with the same frequency as the loudspeaker' The vibration of the eardrum is sent to the brain as sound. Keep in mind that the change in pressure is the one propagating. The air molecules are disturbed, moving back and forth parallel to the disturbance. The sinusoidal behavior of the pressure shown in Figure 1 can be measured by the Vernier microphone. The microphone converts the pressure signal to an electric signal that is recorded by the interface' Figure 1. The oscillatory motion of pressure amplitude as a function of time' The points of condensation (C) and rarefraction (R) are labeled' Hence, the Vernier microphone can provide us some measurements in order to calculate the speed of sound. Theoretically, the speed of sound (v) in air is related to the tempelature of air which can be approximated by the equation vo33l.4+0.67 "mls (1) where T " is the temperature of the air in celsius. Remember that the speed of sound is dependent on the properties of the medium and not on the properties of the wave. From Equation 1, the speed of sound increases with air's temperature' When two or more sound waves are present, the resulting sound is due to the summation of the waves. This is called interference.In a special case where in you have two tuni4g forks with slightly different frequencies -f , and 'f ' such that the oscillatory equations of pressure for both tuning forks are given by Pr=Acos2nf ,t und Pz=Acos2n.f zt , where A is the amplitude of the sound waves. If the two sounds reached your ear at the same time, the resulting wave willbe a sum of the waves such that P = A(cosrr f ,t +cosn f ,t) Using triginometric identities cosa*cos fr=2cosf,'"-P)cos lO- A (2) (3) 1',12 @ 2007 Lab Manual Authors Physics 71.1 Sound This will allow us to write the resultant wave in the form of P=2Acos(wt+oo't) t, Eo E o o o E 0- where the angular frequencies ur and or' are given by the expressions *=){zn fi2'n .f ,\ and, *,=}yzn f r2n.f ,) (s) 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 Loud TimG (s) Figure 2. A plot of two interfering waves that form beats. Notice that resulting wave have periodic loud and faint sound with variation in intensity. This phenomenon is what we call as beats. The frequency that reaches our ears is the average of the two frequencies. Physically this is manifested as an alternate loud and faint sound that repeat at a certain beat frequency, .f uot given by .fru,=ft-f, (4) (6) Musicians often used beats to tune their instruments. They tune their instruments in comparison to a certain reference tone. once the beat disappears, the instrument can be said to be in tuned with the standard. @ 2007 Lab Manual Authors 113 Physics 71.1 Speed of Sound Materials Procedure computer, vernier LabPro, Logger Pro,Vernier microphone, meterstick, PVC pipe, thick hardbound books Measure the room temPerature. I 1.' Connect the VemierlahPrd@ computer interface to the computer' l'r . ' . Z. Connect the Vernierimicrophone to the Vernjer LabPro@ interface. In the ' i"or, double click Physics with Computers, open the fiilder marked Setup by the materials by covering one end of a PVC pipe with a thick (hardbound) book to avoid great loss of sound. Place the microphone near the entrance of the PiPe. click the "collect" ,button to begin the data collection, and snap your finger near the opening of the tube. This will trigger the interface to start collecting the data. 3. 4. ++iiffil*#j.*,+;fut,;::,r.;,.:,r'ir.i=I li,:,.i 114 @ 2007 Ldb ManualAuthot's Physics 71.1 Sosnd 5. Ybu should'be able to distihguish the incident wave and the reflected (echo)" Determine the time interval between the lst wave and the 2nd wave. You may use the examine button on your toolbar. 6. Repeat the measurements aa.d'obtain several trials and calculate the best estimate of the speed of sound. Sound waves and beats Materials Procedure computer, vernier LabPro, Logger pro,vernier microphone, two (2) tuning forks Using the tuning'fork produce a sound and hold it close to the microphone and click "collect" . The data plot should be a sinusoidal curve. In your data plot, count and record the number of complete cycles shown after the first peak in your data. . . click the "Examine" button. Drag the mouse across the graph and record the times for the first and last peaks of the waveform. Divide the time difference by'the number of cycles to deterihine the period of the tuning l. 2. J. @ 2007 Lab Manual Authors Sound Physics 71.'t fork. Calculate the frequency of the tuning fork in Hz and record it in your data table. Drag the mouse across the graph and record the maximum and minimum y values for an adjacent peak and trough. Calculate'fie amplitude of the waVe. Record the values in your data table. Plot the data using excel. Calculate the wavelength of this sound. Record on the graph the information rbgarding the sound such as wavelength, amplitude, period, and frequency. Repeat Steps 3 - 9 for the second frequency. To observe beats, the tuning forks must be struck at the same time. Listen for the combined sound on the tuning fork. Beats is observed when there is a variation of intensity or an emergence of a third pitch. Figure 3. Equiptment for investigating sound beats and waves. A rubber mallet, not pictured, is used to strike the tuning forlcs. 10. Collect data plot of this waveform. Strike the tuning forks equally hard and hold them the same distance from the Microphone. ll.Count the number of amplitude maxima after the first maximum and record it in a data table. 12. Click the "Examine" button. Drag the mouse across the graph and record the times for the first and last amplitude maxima. Divide the time difference by the number of cycles to determine the period of beats (in s). Calculate the beat frequency. in Hz from the beat period. Record these 4. 5. 6. 7. 8. 9. 116 @ 2007 Lab Manual Authors FhysieT4rtl' &rmd. values in your data table. Reference Tipler, Paul A., Physics fgr,scjegJislg an{ Engineers, Fourlfu Edition, W.H. FreemanandCom$airy,uSAj,rgg9..'...:]1':::]ii..) @ 2007...L4b i Mdri uhl AUft rors. 1flI Physics 71.'1 @ 2007 Lab Manual Authors lfeim Date Submitted Date Petbmed ScoE Grcup temb€E lnatruc{or Setion Worksheeil Sound waves Data Table 1. Temperature of the room and length of PVC pipe Room tgmperature.f,C) Length of PVe p:ipe (m) Graph 1. Amplitude of soundwaves as afunction of time A. Speed of sound @20OT Lab Marual Authors '1 19 Physics 71.1 Duta Table 2. Tirne measurement of.sound waves . Using the room temperature measured what should be the theoretical speed of sotrnd? Calculate the percent deviation between the experimental and theorctical speed of sound, Calculations Data Table 3. sound Experimental speed of sound (m/s) Theoretical speed of sound (m/s) Percent devlation (/o) 120 @ 2007 Lab Manual Authors Physics 71.1 Sound Questions 1. If you use a longer or shorter pipe will the calculated speed of sound change? 2. What happens to the plot when you gradually move the book away from the end of the pipe? You could try this. What is the difference between the open ended pipe and closed end pipe? ll. Sound waves and beats Observe the markings on the tuning forks. Usually, the frequency of sound emitted by the tuning forks is specified and carved on the side of the tuqing forks. Data Table 7. Frequency of tuningforl Physics,71.1 sound waves as a Graph 2. Amplitude of sound wavrjs as afuntction of time (tuntingfork B) O 2007 Lab ManualAuthors Physics 71.1 Data Table 2. Tuningforl Souad Physics 71,1.: Data Table 3. Results F,rqwtffbf tuning fork A (Hz) Frequency of tqning fork B (Hz) Experlmental wave frequbncy (Hz) Tieoretical wave frequency (Hz) Experimental beat frequency (Hz) Theorctical beat frequency (Hz) Percent devlatlon f/o) . Explain the beat pattern by noting at which points is the beats loud antlwliete it is faint? What happens to the sum of the trigonometric functions at these points? !r.4 124 @ mOTrLab. Manual Authors