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 The momen o ineria, I, o a body is a measure o how hard i is o ge i roaing abou some axis. The momen I is o roaion as mass m is o ranslaion. The larger he alue o I, he more work mus be done in order o ge he objec spinning. This is analogous o he larger he mass, he more work mus be done in order o ge i moing in a sraigh line. Alloy rim wheels on a bicycle hae a lower momen o ineria han seel rim wheels, hereby making hem easier o se spinning and, as a resul, making i easier o accelerae he bicycle. The momen o ineria o a body is always deined wih respec o a paricular axis o roaion. This is requenly he symmery axis o he body, bu i can in ac be any axis een one ha is ouside he body. The momen o ineria o a body abou a paricular axis is deined as: I m r i i, (1) Figure 1 A roaing disk is composed o many paricles, wo o which are shown. i where he sum is oer all he body pars (o index i), m i is he mass o par I and r i is he disance rom par i o he axis o roaion. This sum is easy o perorm i he objec consiss o discree poin masses (Figure 1). I he body is a coninuous objec o arbirary shape, perorming he sum requires using inegral calculus. In his pracical we will deermine he momen o ineria o a number o objecs and compare he alues obained using wo dieren mehods. For a disk wih an axis hrough is cenre o symmery (Figure ) he momen o ineria is gien by: 1 I mr. () mass, m Axis o roaion r Noe ha he hickness o he disk has no inluence on he alue or I, which depends only on he radius, r and he oal mass, m. Figure Disk wih axis hrough is cenre. 1.1
PY107 Momens o Ineria Experimen 1 In his experimen you will deermine I or wo dieren sysems: (i) a disk and axle rolling down an incline and (b) a ball-bearing oscillaing on a concae spherical surace. For boh sysems you will deermine I in wo ways. Firs, you will measure he mass and radii o he sysems under inesigaion and compue I rom gien ormulae. Then you will compue I by experimenal inesigaion and using he principle o conseraion o energy. You will be expeced o compare he alues ha you obain. Prelab Read he res o his lab descripion and se-up ables or your daa collecion. You should hae ideniied he goals o he experimen and he measuremens ha you mus make beore coming ino he lab. Ensure ha you hae deermined expressions or he error in each o your resuls, similar o ha shown in Appendix A. Par 1 Momen o Ineria o a Disk and Axle In his experimen you mus measure I or a disk mouned on an axle. The axle can be hough o as a ery hick disk and you use he same expression o compue I disk and. The oal I o he disk + axle is he sum I + I. I axle disk axle R + radius r m axle m disk Mehod 1: Calculae he momen o ineria o he disk and axle abou he axis o symmery using he equaion: 1 1 I I wheel + I axle mdisk R + maxler. (3) Measure he masses and radii o he disk and he axle and hen compue I using he expression aboe. Noe ha we are assuming ha he disk is complee i.e. we are ignoring he missing secion hrough which he axle passes by assuming ha he dierence is negligible since r << R. m R disk m r axle Ensure you include he maximum error in each o your measuremens, e.g. m axle ( 300 ± 0.05) g. All daa ha you record should show he measured alue and is associaed maximum error (and, o course, unis). 1.
PY107 Momens o Ineria Experimen 1 Resul using mehod 1 or axle + disk I Mehod : In he second mehod you will compue I by iming he wheel as i rolls down inclined rails and using he principle o conseraion o energy. Consider a wheel consising o disk and axle, rolling down an inclined se o rails aer saring rom res a he op. The wheel will moe down he plane wih consan acceleraion and is oal energy will consis o he sum o he ranslaional kineic energy, he roaional kineic energy and he graiaional poenial energy. 1 1 Energy KErans + KEro + PE M + Iω + Mgh. (4) Here, M is he oal mass o disk + axle, is is ranslaional speed, ω is is angular elociy and h is he heigh o he cenre o mass. I he wheel sars rom res a posiion A and rolls downwards o posiion B hen he loss in poenial energy mus equal he gain in kineic energy. A h l B Zero PE heigh α Figure 3: Experimenal arrangemen A poin A, he energy is enirely poenial since he wheel is a res: Energy iniial PE (5) A poin B, he energy is enirely kineic: Energy inal 1 1 KErans + KEro M + Iω, (6) here w is he inal ranslaional speed and ω is he inal angular speed. 1.3
PY107 Momens o Ineria Experimen 1 By assuming ha ricion is ery small, we can assume ha he oal energy is consan as he wheel rolls down he rails and so he iniial energy is equal o he inal energy. (7) For an axle or wheel ha rolls wihou slipping, he angular elociy and he ranslaional speed are relaed by: ω. (8) r Noe ha here, r is he radius o he axle, NOT he radius o he larger disk. Using equaions (7) and (8) one can ind I in erms o M, r, g, and h. I (9) Since he body sars rom res and moes wih a consan acceleraion we can deermine in erms o he disance l raelled and he ime aken. Newon s equaions o moion gie ha: i + a a 1 l i + a l 1 i i 0 l (10) Subsiuing his ino he expression or I in (9) we ge ha: gh I Mr 1 (11) l Looking a igure 3 and ensuring ha he slope o he plane is kep consan a an angleα wih he horizonal we can rewrie expression (11) as: g sinα I Mr 1 (1) l since sin α h l. Keeping he slope o he rails consan, measure he ime i akes he wheel o moe hrough dieren disances l along he rails rom res using a sopwach. The same person should use he sopwach and release he wheel and make a ew rial runs o deermine he bes procedure. For each disance l ake a number o measuremens o in order o deermine he aerage ime and esimae he uncerainy in. Plo a graph o ersus l and ind 1.4
PY107 Momens o Ineria Experimen 1 he slope. Subsiue his alue ino (1) and calculae he momen o ineria o he disk plus axle. Reer o Appendix A or he error analysis. Resul using mehod or axle + disk I How does he answer you obained or mehod 1 compare wih ha obained or mehod? Menion possible sources o error ha would accoun or his discrepancy. 1.5
PY107 Momens o Ineria Experimen 1 Par : Momen o Ineria o a Ball-Bearing In his experimen you mus measure I or a ball-bearing. Mehod 1: Measure careully he mass m and radius r o he ball-bearing and deermine is momen o ineria using he expression: I mr (13) 5 Ensure you answer conains esimaes on he error in I. Resul using mehod 1 or ball-bearing I Mehod : In his mehod you will deermine I or he ball-bearing using he principle o conseraion o energy and reerring o Appendix B. Consider a uniorm sphere o mass m and radius r rolling back and orh wihou slipping on a concae spherical surace o radius o curaure R, i will execue small ampliude oscillaions in a erical plane. By showing hese oscillaions are simple harmonic in naure i is possible o deermine an expression or he period and, hence, he momen o ineria or he sphere. As wih he preious mehod, we will ignore any ricional eecs and assume ha energy mus be consered. I we consider he schemaic o he problem shown in igure 4, we can assume ha when he oscillaions reach maximum ampliude (posiion A) he energy o he sphere is enirely poenial. When i reaches he equilibrium posiion (posiion B) he energy will be enirely kineic. O R r y x r A B 1.6
PY107 Momens o Ineria Experimen 1 A R r O C y x D B Figure 4: Experimenal parameers The cenre-o-mass o he sphere moes in a erical circle o radius R r. Applying he geomerical heorem AC x CB CD, we see ha: ( ( R r) y) y x ( R r) y y x ( R r) y x, y << R. (14) Applying conseraion o energy we hae ha: where rω. 1 1 E m + Iω + mgy consan, (15) Diereniaing equaion (15) wih respec o ime we obain: d m dω + Iω + dy mg d m d m + I r + I r d d + mg + mg d x ( R r) x ( R r) 0 (16) I d I d x g 1 + 1 + x (17) mr mr R r Equaion (17) has he orm x γ x, where g mr γ. The moion is R r mr + I hereore simple harmonic in naure. We can deine he period o he moion as T π ω π γ. 1.7
PY107 Momens o Ineria Experimen 1 T I ( R r) m + r π. (18) mg Hence he momen o ineria is gien by: gt I mr 1. (19) 4π ( R r) Time he period o, say, 10 oscillaions o he sphere on he cured surace and, hence, deermine he aerage period or he oscillaions, T. Make sure your answer is in he SI uni o ime, i.e. he second. Measure he radius o he ball-bearing r and deermine he radius o curaure o he surace R using he spheromeer as described in Appendix B. Ensure you include he errors on each alue and make an esimae on he error in I deermined using his mehod. Resul using mehod or ball-bearing I Compare he alues obained or he momen o ineria o a sphere using boh mehods compare and commen on where addiional sources o error may arise. 1.8
PY107 Momens o Ineria Experimen 1 Appendix A Example o Error Analysis or he Disk + Axle Experimen I Mr Mr g sinα l [ XS 1], 1 where X g sinα and S is he slope. l ln I ln ( Mr ) + ln( XS 1) ln M + ln r + ln ( XS 1) ΔI I ΔM M ΔM M ( XS) Δr Δ + + r XS 1 Δr SΔX + XΔS + + r XS 1 Here g sinα g X, ΔX cosαδα NB: The error Δα mus be expressed in radians. 1.9
PY107 Momens o Ineria Experimen 1 Appendix B Deerminaion o he radius o curaure o a conex surace using a spheromeer Apparaus Spheromeer, la surace, cured surace Mehod Sep 1 The spheromeer is irs inspeced o deermine how ar (as measured on he erical scale), he screw moes when roaed hrough one complee reoluion o he circular scale. In general his will be 0.5 or 1 mm. The alue o one diision on he circular scale is hen known. Sep The spheromeer is nex placed on a slab o glass and he cenre leg is adjused unil is poin jus ouches he surace o he glass. This is bes deermined by obsering he image o he leg in he glass surace when iewed a an angle. In his posiion he zero on he wo scales should align. I no, he zero error mus be deermined by aking he aerage o seeral seings. Sep 3 The cenre leg is now screwed upwards and he spheromeer is placed on he cured surace so ha he hree legs are in conac wih he surace. The cenre leg is again screwed downwards unil i jus ouches he surace (bes deermined when iewed opically as aboe) and he readings o he wo scales are aken. This procedure is repeaed seeral imes and he aerage o he readings is aken. Sep 4 Finally he spheromeer is pressed ono a piece o paper and he aerage disance, l, beween he cenre poin and ouer legs is deermined. Theory On placing he spheromeer on he cured surace, he hree ouer legs sand on he circumerence o a circle o radius l (diameer AB). Le he heigh hrough which he cenre leg is raised be h. I R is he radius o curaure o he surace, rom he properies o inersecing chords, we hae ha: AC CB DC CE A l D h C B ( ) In oher words: l h R h. l + h On rearranging we ge ha: R. h R O E 1.10
PY107 Momens o Ineria Experimen 1 Resuls #1 # #3 #4 Aerage Value Zero error readings Aerage zero error Readings on cured surace Aerage Readings o l Aerage h R cm 1.11