Physics 2001/2051 Moments of Inertia Experiment 1

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Physics 001/051 Momens o Ineia Expeimen 1 Pelab 1 Read he ollowing backgound/seup and ensue you ae amilia wih he heoy equied o he expeimen. Please also ill in he missing equaions 5, 7 and 9.

1 Physics 001/051 Momens o Ineia Expeimen 1 Pelab 1 Read he ollowing backgound/seup and ensue you ae amilia wih he heoy equied o he expeimen. Please also ill in he missing equaions 5, 7 and 9. Backgound/Seup The momen o ineia, I, o a body is a measue o how had i is o ge i oaing abou some axis. The momen I is o oaion as mass m is o anslaion. The lage he alue o I, he moe wok mus be done in ode o ge he objec spinning. This is analogous o he lage he mass, he moe wok mus be done in ode o ge i moing in a saigh line. Alloy im wheels on a bicycle hae a lowe momen o ineia han seel im wheels, heeby making hem easie o se spinning and, as a esul, making i easie o acceleae he bicycle. The momen o ineia o a body is always deined wih espec o a paicula axis o oaion. This is equenly he symmey axis o he body, bu i can in ac be any axis een one ha is ouside he body. The momen o ineia o a body abou a paicula axis is deined as: I m i i, (1) Figue 1 A oaing disk is composed o many paicles, wo o which ae shown. i whee he sum is oe all he body pas (o index i), m i is he mass o pa I and i is he disance om pa i o he axis o oaion. This sum is easy o peom i he objec consiss o discee poin masses (Figue 1). I he body is a coninuous objec o abiay shape, peoming he sum equies using inegal calculus. In his pacical we will deemine he momen o ineia o a numbe o objecs and compae he alues obained using wo dieen mehods. Fo a disk wih an axis hough is cene o symmey (Figue ) he momen o ineia is gien by: 1 I m. () mass, m Axis o oaion Noe ha he hickness o he disk has no inluence on he alue o I, which depends only on he adius, and he oal mass, m. Figue Disk wih axis hough is cene. 1.1

2 PY001/051 Momens o Ineia Expeimen 1 In his expeimen you will deemine I o wo dieen sysems: (i) a disk and axle olling down an incline and (b) a ball-beaing oscillaing on a concae spheical suace. Fo boh sysems you will deemine I in wo ways. Fis, you will measue he mass and adii o he sysems unde inesigaion and compue I om gien omulae. Then you will compue I by expeimenal inesigaion and using he pinciple o conseaion o enegy. You will be expeced o compae he alues ha you obain. Pelab Read he es o his lab descipion and se-up ables o you daa collecion. You should hae ideniied he goals o he expeimen and he measuemens ha you mus make beoe coming ino he lab. Ensue ha you hae deemined expessions o he eo in each o you esuls, simila o ha shown in Appendix A. Pa 1 Momen o Ineia o a Disk and Axle In his expeimen you mus measue I o a disk mouned on an axle. The axle can be hough o as a ey hick disk and you use he same expession o compue and I axle. The oal I o he disk axle is he sum I I. disk axle I disk R adius m axle m disk Mehod 1: Calculae he momen o ineia o he disk and axle abou he axis o symmey using he equaion: 1 1 I I wheel I axle mdisk R maxle. (3) Measue he masses and adii o he disk and he axle and hen compue I using he expession aboe. Noe ha we ae assuming ha he disk is complee i.e. we ae ignoing he missing secion hough which he axle passes by assuming ha he dieence is negligible since << R. m R disk m axle Ensue you include he maximum eo in each o you measuemens, e.g. m axle ( 300 ± 0.05) g. All daa ha you ecod should show he measued alue and is associaed maximum eo (and, o couse, unis). 1.

3 PY001/051 Momens o Ineia Expeimen 1 Resul using mehod 1 o axle disk I Mehod : In he second mehod you will compue I by iming he wheel as i olls down inclined ails and using he pinciple o conseaion o enegy. Conside a wheel consising o disk and axle, olling down an inclined se o ails ae saing om es a he op. The wheel will moe down he plane wih consan acceleaion and is oal enegy will consis o he sum o he anslaional kineic enegy, he oaional kineic enegy and he gaiaional poenial enegy. 1 1 Enegy KEans KEo PE M Iω Mgh. (4) Hee, M is he oal mass o disk axle, is is anslaional speed, ω is is angula elociy and h is he heigh o he cene o mass. I he wheel sas om es a posiion A and olls downwads o posiion B hen he loss in poenial enegy mus equal he gain in kineic enegy. A h l B Zeo PE heigh α Figue 3: Expeimenal aangemen A poin A, he enegy is eniely poenial since he wheel is a es: Enegy iniial PE (5) A poin B, he enegy is eniely kineic: Enegy inal 1 1 KEans KEo M Iω, (6) w hee is he inal anslaional speed and ω is he inal angula speed. 1.3

4 PY001/051 Momens o Ineia Expeimen 1 By assuming ha icion is ey small, we can assume ha he oal enegy is consan as he wheel olls down he ails and so he iniial enegy is equal o he inal enegy. (7) Fo an axle o wheel ha olls wihou slipping, he angula elociy and he anslaional speed ae elaed by: ω. (8) Noe ha hee, is he adius o he axle, NOT he adius o he lage disk. Using equaions (7) and (8) one can ind I in ems o M,, g, and h. I (9) Since he body sas om es and moes wih a consan acceleaion we can deemine in ems o he disance aelled l and he ime aken. Newon s equaions o moion gie ha: i a a 1 l i a 1 l i i 0 l (10) Subsiuing his ino he expession o I in (9) we ge ha: gh I M 1 (11) l Looking a igue 3 and ensuing ha he slope o he plane is kep consan a an angleα wih he hoizonal we can ewie expession (11) as: g sinα I M 1 (1) l since sin α h l. Keeping he slope o he ails consan, measue he ime i akes he wheel o moe hough dieen disances l along he ails om es using a sopwach. The same peson should use he sopwach and elease he wheel and make a ew ial uns o deemine he bes pocedue. Fo each disance l ake a numbe o measuemens o in ode o deemine he aeage ime and esimae he unceainy in. Plo a gaph o esus l and ind 1.4

5 PY001/051 Momens o Ineia Expeimen 1 he slope. Subsiue his alue ino (1) and calculae he momen o ineia o he disk plus axle. Ree o Appendix A o he eo analysis. Resul using mehod o axle disk I How does he answe you obained o mehod 1 compae wih ha obained o mehod? Menion possible souces o eo ha would accoun o his discepancy. 1.5

6 PY001/051 Momens o Ineia Expeimen 1 Pa : Momen o Ineia o a Ball-Beaing In his expeimen you mus measue I o a ball-beaing. Mehod 1: Measue caeully he mass m and adius o he ball-beaing and deemine is momen o ineia using he expession: I m (13) 5 Ensue you answe conains esimaes on he eo in I. Resul using mehod 1 o ball-beaing I Mehod : In his mehod you will deemine I o he ball-beaing using he pinciple o conseaion o enegy and eeing o Appendix B. Conside a uniom sphee o mass m and adius olling back and oh wihou slipping on a concae spheical suace o adius o cuaue R, i will execue small ampliude oscillaions in a eical plane. By showing hese oscillaions ae simple hamonic in naue i is possible o deemine an expession o he peiod and, hence, he momen o ineia o he sphee. As wih he peious mehod, we will ignoe any icional eecs and assume ha enegy mus be conseed. I we conside he schemaic o he poblem shown in igue 4, we can assume ha when he oscillaions each maximum ampliude (posiion A) he enegy o he sphee is eniely poenial. When i eaches he equilibium posiion (posiion B) he enegy will be eniely kineic. O R y x A B 1.6

7 PY001/051 Momens o Ineia Expeimen 1 A R O C y x D The cene-o-mass o he sphee moes in a eical cicle o adius R. Applying he geomeical heoem AC x CB CD, we see ha: ( ( R ) y) y x ( R ) y y x ( R ) y x, y << R. (14) Applying conseaion o enegy we hae ha: B Figue 4: Expeimenal paamees whee ω. 1 1 E m Iω mgy consan, (15) Dieeniaing equaion (15) wih espec o ime we obain: d m dω Iω dy mg d m d m I I d d mg mg d x ( R ) x ( R ) 0 (16) I d I d x g 1 1 x (17) m m R Equaion (17) has he om & x γ x, whee g m γ. The moion is R m I heeoe simple hamonic in naue. We can deine he peiod o he moion as T π ω π γ. 1.7

8 PY001/051 Momens o Ineia Expeimen 1 T I ( R ) m π. (18) mg Hence he momen o ineia is gien by: gt I m 1. (19) 4π ( R ) Time he peiod o, say, 10 oscillaions o he sphee on he cued suace and, hence, deemine he aeage peiod o he oscillaions, T. Make sue you answe is in he SI uni o ime, i.e. he second. Measue he adius o he ball-beaing and deemine he adius o cuaue o he suace R using he spheomee as descibed in Appendix B. Ensue you include he eos on each alue and make an esimae on he eo in I deemined using his mehod. Resul using mehod o ball-beaing I Compae he alues obained o he momen o ineia o a sphee using boh mehods compae and commen on whee addiional souces o eo may aise. 1.8

9 PY001/051 Momens o Ineia Expeimen 1 Appendix A Example o Eo Analysis o he Disk Axle Expeimen I M M g sinα l [ XS 1], 1 whee X g sinα and S is he slope. l ln I ln ( M ) ln( XS 1) ln M ln ln ( XS 1) I I M M M M ( XS ) XS 1 S X X S XS 1 Hee g sinα g X, X cosα α NB: The eo α mus be expessed in adians. 1.9

PY001/051 Momens o Ineia Expeimen 1 Appendix B Deeminaion o he adius o cuaue o a conex suace using a spheomee Appaaus Spheomee, la suace, cued suace Mehod Sep 1 The spheomee is is inspeced o deemine

10 PY001/051 Momens o Ineia Expeimen 1 Appendix B Deeminaion o he adius o cuaue o a conex suace using a spheomee Appaaus Spheomee, la suace, cued suace Mehod Sep 1 The spheomee is is inspeced o deemine how a (as measued on he eical scale), he scew moes when oaed hough one complee eoluion o he cicula scale. In geneal his will be 0.5 o 1 mm. The alue o one diision on he cicula scale is hen known. Sep The spheomee is nex placed on a slab o glass and he cene leg is adjused unil is poin jus ouches he suace o he glass. This is bes deemined by obseing he image o he leg in he glass suace when iewed a an angle. In his posiion he zeo on he wo scales should align. I no, he zeo eo mus be deemined by aking he aeage o seeal seings. Sep 3 The cene leg is now scewed upwads and he spheomee is placed on he cued suace so ha he hee legs ae in conac wih he suace. The cene leg is again scewed downwads unil i jus ouches he suace (bes deemined when iewed opically as aboe) and he eadings o he wo scales ae aken. This pocedue is epeaed seeal imes and he aeage o he eadings is aken. Sep 4 Finally he spheomee is pessed ono a piece o pape and he aeage disance, l, beween he cene poin and oue legs is deemined. Theoy On placing he spheomee on he cued suace, he hee oue legs sand on he cicumeence o a cicle o adius l (diamee AB). Le he heigh hough which he cene leg is aised be h. I R is he adius o cuaue o he suace, om he popeies o inesecing chods, we hae ha: AC CB DC CE A l D h C B ( ) In ohe wods: l h R h. l h On eaanging we ge ha: R. h O R E 1.10

11 PY001/051 Momens o Ineia Expeimen 1 Resuls #1 # #3 #4 Aeage Value Zeo eo eadings Aeage zeo eo Readings on cued suace Aeage Readings o l Aeage h R cm 1.11

Physics 2107 Moments of Inertia Experiment 1

Physics 2107 Moments of Inertia Experiment 1 Physics 107 Momens o Ineria Experimen 1 Prelab 1 Read he ollowing background/seup and ensure you are amiliar wih he heory required or he experimen. Please also ill in he missing equaions 5, 7 and 9. Background/Seup