Acceleration. Part I. Uniformly Accelerated Motion: Kinematics & Geometry

Transcription

1 Acceleraion Team: Par I. Uniformly Acceleraed Moion: Kinemaics & Geomery Acceleraion is he rae of change of velociy wih respec o ime: a dv/d. In his experimen, you will sudy a very imporan class of moion called uniformly-acceleraed moion. Uniform acceleraion means ha he acceleraion is consan independen of ime and hus he velociy changes a a consan rae. The moion of an objec (near he earh s surface) due o graviy is he classic example of uniformly acceleraed moion. If you drop any objec, hen is velociy will increase by he same amoun (9.8 m/s) during each one-second inerval of ime. Galileo figured ou he physics of uniformly-acceleraed moion by sudying he moion of a bronze ball rolling down a wooden ramp. You will sudy he moion of a glider coasing down a iled air rack. You will discover he deep connecion beween kinemaic conceps (posiion, velociy, acceleraion) and geomeric conceps (curvaure, slope, area). A. The Big Four:, x, v, a The subjec of kinemaics is concerned wih he descripion of how maer moves hrough space and ime. The four quaniies, ime, posiion x, velociy v, and acceleraion a, are he basic descripors of any kind of moion of a paricle moving in one spaial dimension. They are he sars of he kinema. The variables describing space (x) and ime () are he fundamenal kinemaic eniies. The oher wo (v and a) are derived from hese spaial and emporal properies via he relaions v dx/d and a dv/d. Le s measure how x, v, and a of your glider depend on. Firs make sure ha he rack is level. The acceleraion of he glider on a horizonal air rack is consan, bu is value (a = 0) is no very ineresing. In order o have a 0, you mus il he rack. Place wo wooden blocks under he leg of he rack near he end where he moion sensor is locaed. Release he glider a he op of he rack and record is moion using he moion sensor. [Click on Logger Pro and open file Changing Velociy 2]. The graph window displays x, v, and a as a funcion of ime. Your graphs should have he following overall appearance: Good Daa Region x parabola v a linear consan

2 Focus on he good daa region of he graphs where he acceleraion is consan. To find his region, look for ha par of he graphs where he x, v, a curves ake on smooh well-defined shapes: x = parabola, v = linear (sloping line), a = consan (fla line). In he bad daa region, he acceleraion is changing because he glider is experiencing forces oher han graviy, such as your hand pushing he glider or he glider hiing he bumper. Change he scales on your graphs so ha he gooddaa region fills mos of he graph window. PRINT your x, v, a graphs (wihou he daa able). Remember o wrie a shor ile. Label he Good Daa Region. Have your insrucor check your graphs and your good-daa region before you move on o he nex par of he lab. B. Acceleraion = Curvaure of x() Look a your x() graph and noe: The worldline of your glider is curved! Recall ha in he Consan Velociy lab, all graphs were sraigh. Changing Velociy is synonymous wih a Curved Worldline: Acceleraion Changing Velociy Curving Worldline a = dv/d = d 2 x/d 2. The amoun of bending in a curve he deviaion from sraighness is measured by how much he slope changes. Acceleraion he rae of change in he slope of x() measures he curvaure of spaceime. x big a small a zero a Fla Spaceime Curved x() Warped Spaceime x Level Track x Tiled Track a = 0 a 1 Black Hole a The Imporance of Curvaure in Theoreical Physics The graviaional force of he earh is he cause of he curved worldline of your glider. Remove he earh and he worldline would become sraigh. Two Hundred and Fify years afer Newon, Einsein formulaed his celebraed Field Equaions of General Relaiviy which sae he precise mahemaical relaionship beween he amoun of mass (he source of graviy) and he curvaure of spaceime. Force causes x() o curve. Mass causes spaceime o warp. Measuring Spacewarp Here you will measure he curvaure of your glider s worldline x(). Click on he Slope Icon [m=?] and find he slope of he angen line a hree differen poins on your x() curve. Make sure ha he hree poins are wihin your good-daa region and no oo close ogeher. Noe ha he slope dx/d of he posiion curve x() is equal o he insananeous velociy v of he glider. Record your daa below: 2

3 1 = s 2 = s 3 = s slope 1 = m/s slope 2 = m/s slope 3 = m/s Noe how he slope increases wih ime, i.e. he velociy v dx/d of he glider increases as i falls down he rack. Calculae he rae of change in he slope: slope 2 slope 1 slope 3 slope = m/s = m/s In heory, he rae of change in he slope of x() is equal o he acceleraion of he glider: d 2 x/d 2 = a. Since he acceleraion due o graviy is consan, your wo values of slope/ lised above should be equal o each oher wihin experimenal error. Repor he average of hese wo values (o wo significan figures) as your experimenal esimae of d 2 x/d 2 : Curvaure of x(): d 2 x/d 2 = 0. m/s 2. Now find he value of he acceleraion of your glider direcly by looking a he enries in he Acceleraion column of he moion-sensor daa able. Like all experimenal quaniies, he values of a() will flucuae around some average value. Esimae he average value (o wo significan figures) simply by looking a he able and noing he number(s) ha occur mos ofen. Average of a(): a = 0. m/s 2. % Difference beween d 2 x/d 2 and a is %. If % Diff > 10%, hen see your insrucor. C. Acceleraion = Slope of v. Displacemen = Area under v. In he previous secion, you found a from he x() graph via he relaion a = d 2 x/d 2 (curvaure). In his secion, you will find a from he v() graph via he relaion a = dv/d (slope). You will also find x from he v() graph via he relaion x = vd (area). Pick wo poins on he v() line wihin he good-daa region ha are no oo close o each oher. Find he values of and v a hese poins using he Examine Icon [x=?] or from he daa able. Also find he posiion x of he glider a hese same wo imes. 1 = s v 1 = m/s x 1 = m. 2 = s v 2 = m/s x 2 = m. 3

4 PRINT your v() graph window (no x and a). Label he poins 1 and 2 wih your pen. Wrie he coordinae values ( 1, v 1 ) and ( 2, v 2 ) nex o each poin. Calculae he following wo geomeric properies of he v() graph: 1. Slope of he line. 2. Area under he line beween 1 and 2. Show your calculaions (rise-over-run, base-imes-heigh, ec.) direcly on your prined graph. Repor your slope and area resuls here: Slope of v() line = (m/s) / s. Area under v() line = (m/s) s. Mahemaical Facs: 1. The raio dv/d is he rise (dv) over he run (d) of he v() line. 2. The produc vd is he area of he recangle of base d and heigh v. Physical Consequences: 1. a = dv/d Acceleraion a = Slope of v() graph. 2. dx = vd Displacemen x = Area under v() graph. a = Slope x = Area v v You already found a (from he a values in he daa able). Wrie his value of a again in he space below. From your measured values of x 1 and x 2 (lised above), you can find he displacemen of he glider: x = x 2 x 1, i.e. he disance moved by he glider during he ime inerval from 1 o 2. a = m/s 2. x = m. Compare his value of a wih your value of Slope of v() line. Compare his value of x wih your value of Area under v() line. % diff beween a and slope = %. % diff beween x and area = %. Physics & Calculus The problem of finding slopes and areas is he essence of he whole subjec of Calculus. Newon invened Calculus o undersand Moion. In Calculus, finding slopes (acceleraions) and finding areas (displacemens) are inverse operaions called differeniaion and inegraion, respecively. In he language of mahemaics, a = dv/d and x = vd. 4

5 D. Compuaional Physics : Finding he Bes Value of a Le s use he full compuaional power of he compuer o find he acceleraion of he glider by analyzing all he daa colleced by he moion sensor. Saisical Analysis of a() Selec he good-daa region of your a() graph. Remember he daa selecion procedure: click and drag from he lef end o he righ end of he good region. To perform a saisical analysis of he seleced daa, click on he Saisics Icon [STAT]. The compuer will find he average (mean) value and he sandard deviaion. Recall ha he average value provides he bes esimae of he rue value of he quaniy, while he sandard deviaion is he uncerainy he spread in he measured values around he average due o he experimenal errors. The averaging procedure smoohes ou he up and down flucuaions in he measured a(). There are several sources of experimenal errors ha cause he acceleraion of he glider o flucuae over ime. These errors include a bumpy rack, dir on rack, a ben glider, dir on glider, surface fricion, air fricion, and he fac ha he moion sensor approximaes he coninuiy of moion he smooh flow of ime by collecing and analyzing daa in discree ime seps. Average of a() ± Uncerainy: a = ± m/s 2. Linear Analysis of v() The acceleraion of he glider is equal o he slope of he v() line. Selec he good-daa region of your v() graph. Click on he Curve-Fi Icon [f(x)=?] and perform a Linear Fi o find he bes-fi line hrough he v- daa poins. The compuer will give he equaion of he line as y = mx + b, which in velociy-ime language is v = a + v o. The slope of he bes-fi line gives he bes value of a. Equaion of Bes-Fi v() Line: v() =. Firs Derivaive (Slope) of v(): dv/d = m/s 2. Quadraic Analysis of x() Equaion of Bes Fi x() Curve: x() =. Second Derivaive (Curvaure) of x(): d 2 x/d 2 = m/s 2. Compare Resuls Compare your hree values of acceleraion based on your analysis of a(), v(), and x(). 5

6 Par II. The Physics of Free Fall Consider an objec of mass m ha is released from res near he surface of he earh. Afer a ime, he objec has fallen a disance d and is moving wih velociy v. The free-fall equaions relaing d,, and v are d = ½ g 2, v = g, v 2 = 2gd, where g = 9.8 m/s 2 is independen of m. In his experimen, you will es hese imporan properies of free-fall moion by sudying he moion of a glider on a iled air rack. Sricly speaking, free fall refers o he verical moion of a body ha is free of all forces excep he force of graviy. A body moving on a fricion-free inclined rack is falling freely along he direcion of he rack. I is non-verical free fall moion. The rack simply changes he direcion of he fall from verical o diagonal. This diagonal free fall is a slowed-down and hus easier-o-measure version of he verical free fall. The acceleraion along he rack is he diagonal componen of he verical g. This acceleraion depends on he angle of incline. I ranges from 0 m/s 2 a 0 o (horizonal rack) o 9.8 m/s 2 a 90 o (verical rack). In oher words, he rack merely reduces he poency of graviy. A fricionless inclined plane is a graviy diluer. A. Experimenal Tes of he Squared Relaion d 2 In heory, he worldline of he glider is a parabola. Hence he disance d raversed by he glider along he rack is proporional o he square of he ime elapsed (afer saring from res). This means ha if you double he ime, 2, hen he disance will quadruple, d 4d. More specifically, if i akes ime 1 o move disance d 1 and ime 2 o move disance d 2, hen he proporionaliy d 2 implies he following equaliy of raios: d 2 /d 1 = ( 2 / 1 ) 2. This raio relaion says if 2 = 2 1, hen d 2 = 4d 1. Sar wih he iled rack wih wo blocks under he end of he rack. Use a sopwach no he moion sensor o measure he ime i akes he glider, saring from res, o move a disance of 25 cm down he rack. Repea hree more imes and find an average ime. Nex measure he ime i akes, saring from res, o move a disance of 100 cm. Experimenal Techniques: (1) The ime measuremen will be mos accurae if you sar he glider a a poin ha is 25 cm away from he rubber band a he lower end of he rack. Seeing and hearing he glider hi he rubber band ells you he precise momen o sop he sopwach. (2) The same person should release he glider and ime he moion in order o minimize reacion ime error. Average Time (d = 25 cm) (s) (d=100 cm) (s) Are your experimenal resuls consisen wih he heoreical relaion d 2? consrucing raios. Hin: Calculae d 2 /d 1 and 2 / 1 and compare. Explain carefully by 6

7 B. Experimenal Tes of v 2 H Physics Fac: The speed v of an objec, saring from res and falling down he fricionless surface of an inclined plane, depends only on he verical heigh H of he fall and no on he lengh of he incline. Furhermore, he square of he velociy is proporional o he heigh: v 2 H. This squared relaion implies ha he speed will double if he heigh quadruples. H v 4H 2v Since you are esing he proporionaliy, v 2 H, and no he equaliy v 2 = 2gH, you only need o sudy how v depends on he number of blocks ha you sack verically o elevae he rack. The heigh H can be measured in dimensionless unis, simply as he number of blocks. Place one block (H = 1) under he moion-sensor end of he rack. Posiion he glider a he poin ha is 20 cm away from he sensor. Release he glider from res and measure is velociy v (using he sensor) when i is 100 cm away from he sensor. Simply read he value of Velociy from he daa able when he Disance value is 1.0 m. Repea hree more imes and find an average velociy. Now quadruple he heigh by placing four blocks (H = 4) under he end. Once again, release he glider a 20 cm and measure is velociy a 100 cm. Average Velociy v (H = 1) (m/s) v (H = 4) (m/s) Do your experimenal resuls suppor he heoreical relaion v 2 H? Explain carefully by consrucing raios. 7

8 C. Experimenal Tes of he Universaliy of g One of he deepes facs of Naure is his: The acceleraion of an objec due o graviy does no depend on he size, shape, composiion, or mass of he objec. In he absence of fricion, all bodies fall a he same rae! Use wo blocks o incline he rack. Use he moion sensor o record he moion of he glider as i falls freely down he rack. Remember o carefully selec he good daa (consan a) region of he graph before you analyze he daa. Find he acceleraion of he glider by averaging he a versus daa: click on he saisics icon [STATS]. Repor your resuls in he able below. For example, if he saisical analysis of he acceleraion daa gives he average value m/s 2 and he sandard deviaion m/s 2, hen you would repor your measured value of acceleraion o be 35 ± 2 cm/s 2. The range of his a is cm/s 2. Add wo weighs or meal donus (one on each side of he glider) and measure he acceleraion. Add four weighs (wo on each side) and measure he acceleraion. Mass a ± uncerainy (cm/s 2 ) Range of a (cm/s 2 ) 0 added weighs ± 2 added weighs ± 4 added weighs ± Your values of a may look close, bu can you conclude ha hey are equal? The word close is no par of he language of science. When are wo experimenal values Equal? To answer his quesion, he role of uncerainy is vial. A measured value such as 15 ± 2 is really a range of numbers Two experimenal values are equal if and only if heir ranges overlap. Suppose you are given wo rods (A and B) and measure heir lenghs o be L A = 15 ± 2 cm and L B = 18 ± 3 cm. Since he wo ranges overlap, and 15 21, you can conclude ha hese wo rods are equal in lengh. A range diagram provides an excellen visual display of he experimenal values of measured quaniies. The following range diagram for L A and L B clearly exhibis he amoun of overlap: L A L B cm 8

9 Plo your hree measured values of a he acceleraion of graviy (along he rack) on he following range diagram: a 1 a 12 a 3 cm/s 2 Now you can rigorously answer he imporan quesion: Do your measured values of a provide an experimenal proof of he deep principle ha he acceleraion of graviy is independen of mass? Explain. Einsein, Curved Space, Black Holes, Warp Drive In a graviaional field, all bodies fall wih he same acceleraion. We have said his is a deep law of naure. Indeed, Einsein used his law as he basis for his general heory of relaiviy. All bodies fall in he same way because hey are merely coasing along he same downhill conours of he curved space ha hey happen o occupy. Einsein s field equaions ell you precisely how o calculae he curvaure of four-dimensional space-ime. Graviy is no a force i is he shape of space. The idea ha graviy is curvaure is he basis for warped space, bending ligh, graviy waves, black holes, and wormholes. In essence, your experimenal proof ha g is independen of m is a proof of he exisence of black holes and graviy waves! When warp drive is invened, you will appreciae ha i is a consequence of he universaliy of g. 9

10 Par III. Designing a Dilued-Graviy Sysem In verical free fall, an objec released from res moves 60 m in 3.5 s. You need o slow his moion dilue graviy so ha he objec only moves 1.5 m in 3.5 s. Your goal is o find how much he rack needs o be iled o achieve his slowed-down moion. Firs work ou he heory and hen perform he experimen. The Theory Archiecure Diagram. H = heigh of blocks. L = disance beween rack legs. θ = angle of incline. rack leg glider H L blocks θ leg able Acceleraion Diagram. g = full srengh graviy. a = dilued graviy. a g θ θ Noe: The componen of g along he rack direcion is gsinθ. As an example, for θ = 30 o, he diluion facor is sinθ = ½ and hus he acceleraion is a = g/2. For θ = 72 o, a = 0.95g. Two Sep Soluion 1. Calculae he acceleraion ha he glider mus have in order o saisfy he Design Specs: glider released from res and moves 1.5 m in 3.5 s. a = m/s 2. 10

11 2. Calculae he heigh H of he blocks ha is necessary o achieve his amoun of acceleraion. Hin: H is relaed o he diluion facor sinθ and he lengh L (see archiecure diagram). H = cm. The Experimen 1. Again, make sure ha he rack is level. Raise he end of he rack by he heigh H prediced above. To achieve his value of H (o wihin enhs of a cenimeer), you will mos likely need o sack hin square meal plaes on op of he wooden block(s). 2. Release he glider from res. Use a sopwach o measure he ime i akes he glider o move 1.5 m along he rack. To achieve greaer accuracy in iming, measure he 1.5 m disance from he elasic cord (bumper) ha he glider his a he lower end of he rack. Seeing/hearing he glider hi he cord provides a well-defined signal for you o sop he wach. Repea his measuremen five imes. Lis your five values of below and compue he average ime and he uncerainy in he ime. Esimae he uncerainy (deviaion from average) from he half-widh spread in your five values of ime: Uncerainy = ( max min )/2. (s) = ± s. 3. Compare his measured value of ime wih he design goal of = 3.5 s. Wha is he percen difference? Clearly show wheher or no he heoreical ime of 3.5 seconds falls wihin your experimenal range? Wha source(s) of error could accoun for any discrepancy? 11