LAB 03 One-Dimensional Uniformly Accelerated Motion

Transcription

1 PHYS 154 Universiy Physics Laboraory Pre-Lab Spring 18 LAB 3 One-Dimensional Uniformly Acceleraed Moion CONTENT: 1. Inroducion. One-Dimensional Kinemaics A. In general B. Uniformly Acceleraed Moion 3. Bodies in Free Fall A. Hisory B. Analysis 4. Pre-lab: A. Aciviies B. Preliminary info C. Quiz When he world hrows you oo much informaion, he only way you can say sane or survive is o look for paern recogniion. Amids all he blurs, is here a consellaion ha emerges, is here a sraigh line ha's emerging? Douglas Coupland 1. Inroducion In general, courses of inroducory physics sar by inroducing conceps of one-dimensional kinemaics: he science of how paricles move in a sraigh line. Delving ino he science of Physics via mechanics is hardly surprising, because wondering abou he naure of moion is one of he mos ancien inellecual inquiries, and i was a heory of moion Newonian mechanics ha heralded he earlies concepual breakhroughs of modern science. Indeed, we are immersed in a world in incessan moion which sems from he mos fundamenal mechanisms of he universe, so he sudy of mechanics is he firs sep owards undersanding oher branches of physics and a key for he pedagogical efficacy of inroducory physics courses. In our nex lab we are o reinforce he maerial we sudied in he lecure abou uniformly acceleraed one-dimensional moion. To illusrae he associaed model you are o use objecs in free fall. This is a subjec ineresing per se, because bodies in free fall move idenically irrespecive of heir mass much o he amazemen of he pioneering physiciss ha observed his behavior in he sixeenh cenury. Therefore, our experimen will be more han a sudy of one-dimensional kinemaics, inasmuch as i alludes o how he moion of freely falling is driven by graviy in a very peculiar manner ha acually inspired Newon when he proposed his heory of universal graviaional aracion.. One-Dimensional Kinemaics A. In general We inroduced one-dimensional ranslaional moion in class: you may recall ha he respecive kinemaics employs a se of equaions of moion as models of how objecs move. These are mahemaical expressions which wihin he limis assumed by he model are expeced o predic he posiion x, velociy v, and acceleraion a of a moving body a any ime, provided ha hey are given a an earlier momen of ime (hese values are called consans of moion because hey don change wih ime). Convenienly, hese kinemaic quaniies are inerdependen. Thus, if he posiion and velociy are x and v a an iniial ime, any of he hree can be wrien in erms of he oher wo: The insananeous velociy is he rae of change of posiion in an infiniesimally shor ime inerval: dx v x x vd d (1) In urn, he insananeous acceleraion is he rae of change of velociy in an infiniesimally shor ime inerval: dv a v v a d d () Based on hese definiions, i is pracical o noe ha he wo quaniies ha are ime raes (ha is, velociy and acceleraion) can be seen as slopes of he graphs represening he ime variaion of he respecive changing quaniy: x() vs. for velociy and v() vs. for acceleraion. 1

2 PHYS 154 Universiy Physics Laboraory Pre-Lab Spring 18 When formulaed in his manner, he equaions of moion are jus mirrors of how he objecs move. A more fundamenal way o build a model of moion is by providing he cause ha yields he kinemaics as an effec. In PHYS 154 we learn ha he characerisics of moion are shaped up by he paricular pushes and pulls experienced by objecs ha is, he ineracions wih heir surroundings concepualized ino forces. Thus, we see ha he acceleraion is proporional o he ne force (or he sum of all ineracions) aced on he objec. B. Uniformly Acceleraed Moion Therefore, he equaions of moion can be raher complex if he forces vary in a complex manner. However, while briefly ouching upon some mehods o sudy any moion, in our class we mosly focus on moions driven by consan forces: we call his sor of moion a uniformly acceleraed one, because he acceleraion ensuing from a consan force will be consan. Noe ha in his paricular case acceleraion iself is a consan of moion, so i is relaively easy o derive he equaions of moion, simply because one doesn have o accoun on how velociy and acceleraion depend on ime simulaneously. So, if an objec moves in a sraigh line a consan acceleraion a saring a an iniial posiion x wih velociy v, one can wrie he posiion and velociy afer a ime inerval as following x x v a 1, (3) v v a. (4) These expressions sae ha posiion depends on ime parabolically (because x depends on via a quadraic polynomial), whereas v depends on linearly naurally, considering ha acceleraion is he slope of v vs. graph. 3. Bodies in Free Fall A. Hisory Free fall is defined as he moion of objecs aced solely by heir weigh. I is an emblemaic opic in he hisory of science since i was one of he firs phenomena sudied sysemaically using experimenal esing. Thus, as Renaissance was waning in sixeenh cenury Ialy, Galileo Galilei arguably he faher of modern science quesioned he rigid deducive Arisoelian ideas abou naure so dear o medieval scholasics in favor of an inducive approach o scienific inquiry based on experimenal validaion. In paricular, Galileo ook a ask Arisole s inference ha heavier bodies would fall faser han heavier ones. No only ha Galileo demonsraed ha as long as graviy overwhelmingly dominaes all objecs fall he same irrespecive of mass, bu he also showed ha hey fall wih consan acceleraion and exrapolaed his ideas ino an embryonic heory of moion. No wonder ha Isaac Newon he creaor of classical mechanics who was born he year when Galileo died noed ha his science was buil on he shoulders of gians B. Analysis In modern parlance, he consan acceleraion of freely falling objecs is called graviaional acceleraion, denoed g. As we shall learn laer, he acceleraion is no really consan, as i does depend on he disance o he cener of graviaional aracion (Earh). However, o observe is variaion, one would have o measure g across disances comparable o he radius of our plane, so he dependency is hardly observably in he viciniy of Earh s surface where i has a magniude of abou 9.8 m s. The fac ha he graviaional acceleraion is independen of he mass of he falling objecs is due o wo aspecs: firs, he acceleraion if given by force per mass, and second, he srengh of he graviaional force is proporional o he mass. So, he acceleraion won depend on mass. To analyze verical free falls, we can use he equaions from he previous secion. I is cusomary o align he moion wih a verical y-axis, so in his case he downward graviaional acceleraion is a = g = 9.8 m s. Therefore, if an objec is in verical free fall (downwards or upwards) from aliude y where i has velociy v, is aliude and velociy afer a ime inerval are given by equaions (1) and () which become: y y v g 1, (5) v v g. (6)

3 Velociy (m) Aliude (m) PHYS 154 Universiy Physics Laboraory Pre-Lab Spring 18 Example 1: A suden osses an apple in he air wih an iniial speed of abou 9.8 m/s. If she considers a verically upwards axis wih he origin a he posiion where he apple leaves her palm, he consans of moion will be as following: Iniial aliude: y = Iniial velociy: v = 9.8 m s Acceleraion: a = g = 9.8 m s. Thence, according o he kinemaic model of free fall described above, he ime dependence of aliude and velociy will be represened for he firs almos wo seconds by he adjacen graphs. Noe how he slope variaion of he y vs parabola reflecs in he linear rend of he velociy: posiive decreasing o zero for one second, and negaive increasing from zero afer ha Free Fall - Aliude vs. Time Time (s) Free Fall - Velociy vs. Time Time (s) Pre-lab A. Aciviies 1. Read carefully he inroducory maerial provided above. Make noe of he wo aspecs of free fall: I is a hisorically relevan example of uniformly acceleraed moion I pinpoins a propery of graviaional aracion ha deermines all bodies o fall idenically irrespecive of mass. Review how graphical represenaions can be used o represen moion. 3. Answer he quesions on he quiz a he end of his documen. The quesions are also available on he Blackboard sie associaed wih he PHYS 154 lecure. B. Preliminary informaion The subjec of LAB 3 is One-Dimensional Uniformly Acceleraed Moion. This is he firs experimen coordinaed wih he maerial covered in class (one dimensional kinemaics). As explained above, we use free fall as an archeypal case of a uniform moion driven by he fairly uniform weigh of bodies close o he surface of he Earh. The preex for he experimen is o demonsrae convincingly ha, indeed, he weighs of arbirary objecs will produce idenical moions wih consan acceleraion provided ha oher forces are negligible. The aim of he experimen is wofold: you have o show ha he free fall moion is 1. independen of mass. uniformly acceleraed, and compare he graviaional acceleraion o he sandard value. The sraegy is o ackle hese asks separaely: 1. Formulae a hypohesis abou he mass dependency of free fall. Then release objecs of differen masses (seel and brass balls) from he same heigh and es ha hey ouch he ground a approximaely he same ime. For now assuming ha he graviaional consan, use he ime and iniial heigh o esimae he graviaional acceleraions.. Formulae a hypohesis abou he consancy of graviaional acceleraion. Then verify he saemen by dropping he same objec from increasing heighs. Noe ha in his case you canno use he equaions of uniformly acceleraed moion o calculae anyhing, simply because you canno assume hem rue a priori (ha is, before experimen). However, you can assume hem as forming a enaively rue model and check ha he experimenal daa conforms o is predicions. In our experimen you will have o perform a bes fi on a plo involving he raw heighs and squared imes o see if i reproduces he behavior prediced by he model. In each of he wo pars you will have o specify he naure of he variables used (dependen, independen, or conrol), idenify various ypes of errors, and provide meaningful discussions of he resuls. 3

4 PHYS 154 Universiy Physics Laboraory Pre-Lab Spring 18 Technical Commens: In he second par of he lab you will have o drop a seel ball from increasing heighs and measure he falling imes o emulae measuring he ime necessary o ravel hrough he respecive heighs coninuously in one single fall. You have o ake a minue o mediae abou how dropping a ball from res from successive heighs can model a moion where due o acceleraion shorer and shorer displacemens are raveled saring a higher and higher iniial speeds. The diagram below should help explain he idea: The ball is acually dropped from differen heighs y along a rod, and falling imes are measured y 5, 5 y 4, 4 y 3, 3 y, How he daa are o be inerpreed: he pairs (y,) are similar o he posiions a differen successive imes raveled by a freely falling ball released from res g y 1, 1 y Subsequenly, he daa pairs (y,) are o be represened graphically in Excel. You will provide wo graphs assuming he y-axis verically downwards wih he origin in he highes heigh esed: 1. y vs., so you can observe and commen on he parabolic shape of he curve. y vs., so you can provide a linear fi using an Excel funcion. From he slope of he linear fi you can exrac an experimenal value for he graviaional acceleraion comparing he fi parameers o he free-fall equaion of an objec released from res in he origin: 1 y g, (7) where g is posiive because he axis is oriened downwards. Moreover, o illusrae he lineariy in ime of he free-fall velociy, you will have o calculae he velociies a imes using he following expression v y, (8) and hen use he (v,)-daa o plo a v vs. graph. (Quiz on he nex page) 4

5 PHYS 154 Universiy Physics Laboraory Pre-Lab Spring 18 C. Quiz 3 Name: Based on he pre-lab readings, please answer he following quesions on he Blackboard sie associaed wih our PHYS 154 lecure. The numbers in square brackes indicae he poins alloed for he respecive quesion. Q1. [] Which of he following is an accepable definiion for free fall? a) A verically downward moion. b) A verical moion wih negaive acceleraion. c) The moion of bodies overwhelmingly aced by heir weigh. d) A moion wih a uniform acceleraion of 9.8 m/s. e) A moion wih a uniform acceleraion of 9.8 m/s. Q. [] Which of he following is rue abou he moion described in Example 1? a) For abou one second he apple ravels wih a negaive acceleraion, hen wih a posiive one. b) In velociy vs. ime represenaion, he velociy doesn change a ime = 1 sec because he slope is coninuous. c) A 1.4 sec afer he apple was ossed, is posiion is abou 4.1 m and velociy abou 3.9 m/s. d) A.4 sec and 1.6 sec he speed of he apple is abou he same. e) None of he above is rue. Q3. [] Wha is he goal of our nex lab? a) To show ha objecs in free fall he heavier hey are he faser hey fall. b) To show ha, while hey accelerae a he same rae, ascending bodies will have a differen acceleraion han falling ones. c) To show ha, when in free fall, heavy bodies accelerae he same as ligh ones. d) To show ha, when in free fall, bodies accelerae a an approximaely consan rae. e) Boh (c) and (d) above are rue. Q4. [] In he firs par of our nex lab, which of he following will be a conrol variable? a) Mass b) Iniial heigh and velociy of he falling ball c) Falling ime and final velociy d) Graviaional acceleraion e) None of he above Q5. [] When compared o he y- graph in Example 1, in he second par of our nex lab you should expec ha he parabola represening he heigh vs ime looks a) similar o he curve in he -1 sec range. b) similar o he curve in he sec range. c) similar o he curve in he -1.8 sec range. d) unlike any inerval of he shown curve e) Acually, he curve will be linear raher han parabolic. 5