Rotational Inertia of a Point Mass

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1 Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha his value corresponds o he calculaed heoreical value. EQUIPMENT - pulley - paper clips (for es < 1 gram) - calipers - es and weigh hanger se - phoogae imers wih 3 phoogae heads - riple beam balance - meersick g (black, square) - roaing plaform - A base - hread/sring SKETCH 8/06 1

2 Figure.3 The se-up shown above, wih he phoogae in Pendulum mode, enables one o deermine wheher he hanging is falling a an approximaely consan rae. Figure.4 When arranging he wo phoogaes as shown o he lef, be sure he sring and hook (a he op of he weigh hanger) do no break he phoogae beam. 8/06

3 THEORY Theoreically, he roaional ineria, I, of a poin is given by and R is he disance he is from he axis of roaion. I = MR, where M is he To find he roaional ineria of a poin experimenally, a known orque is applied o he objec and he resuling angular acceleraion is measured. Since " = I!, rearranges o " I =! (1) where! = a () r and! = rt (3) (where r = radius of cylinder abou which he hread is wound and T is he ension in he hread when he apparaus is roaing.) Subsiue equaions () and (3) ino equaion (1) above and wrie he resul in he box below. (4) Apply Newon s Second Law o he hanging, call i m for now, and hen solve he resuling equaion for he ension in he hread, T. Subsiue his value for T ino equaion (4) above and you have a formula yielding he experimenal roaional ineria in erms of he linear acceleraion of he, m. Call his new equaion (5) and wrie i below. (5) The linear acceleraion, a, of he, m, can be found wih he kinemaic equaion below: 1 a! y = v + o (6) where o v is he iniial velociy of he hanging (zero), and is he (average) ime for he hanging o fall from res hrough a displacemen! y. NOTICE: You will be solving he above equaions for wo differen cases o find he experimenal roaional ineria of he poin. To find I you mus find in I hen subrac ou he roaional ineria of he apparaus alone (i.e. po in & apparaus wih no poin ). The mahemaical relaionship is seen below: I = I in po & apparaus po in - apparaus alone po I (7) 8/06 3

4 PROCEDURE Equipmen Seup 1) Leveling he Roaing Plaform: This may ake 30 minues o do correcly, bu i is ime well spen! Before leveling he apparaus you MUST se enire apparaus in he posiion shown in Figure.4. Noice ha he hanging off he pulley canno hi he able and he sring wound around he cylinder mus pass sraigh over he pulley. ONCE APPARATUS IS LEVEL, YOU CANNOT MOVE IT! ) Se up your apparaus as shown in Figures.1 and. (above), making sure he hanging almos reaches he floor when he sring is compleely unwound. Par I: Measuremens for he Theoreical Roaional Ineria 1) Deermine he Mass of he square black, M (he poin ) and record i in able.1. ) Aach he square black o he rack on he roaing plaform a a large radius. 3) Measure he disance from he axis of roaion o he cener of he square and record his radius, R, in Table.1 8/06 4

5 Theoreical Roaional Ineria Mass, M Radius, R Table.1 Par II: Measuremens for he Experimenal Roaional Ineria ACOUNTING FOR FRICTION Because he heory used o find he roaional ineria experimenally does no include fricion, i will be compensaed for in his experimen by finding ou how much over he pulley i akes o overcome kineic fricion and allow he o drop a a consan speed. Then his fricion will be subraced from he hanging ha was used o accelerae M and he resuling will be m, see calculaions. 1) To find he required o overcome kineic fricion, place a phoogae (se on PENDULUM ) so ha he roaing plaform will break he beam and ime one complee cycle of he roaing plaform. ) Tie he sring hrough he hole in he middle cylinder and wind i evenly abou ha cylinder. 3) Place jus enough over he pulley (using paperclips) so ha he roaing plaform moves a a consan angular speed. You migh need o lighly nudge he hanging or pulley o help he sysem overcome saic fricion. Jus eyeball he speed a firs hen when you hink he speed is nearly consan, use he phoogae (se on Pendulum mode) o ime a revoluion early on and o ime a revoluion oward he end of he moion (see Figure.3 for se-up). Clearly hese wo periods should be he same if a consan angular speed is aained. Try o ge he periods o mach o hree significan figures. Record he fricion in able., under he appropriae column. Finding he Acceleraion of he Poin Mass and Apparaus 1) You will no use he above phoogae unil you repea he procedure wih he Apparaus Alone. Carefully clamp/ape he oher wo phoogae heads ono a verical sand abou 50+ cm apar and connec hem o a phoogae imer se on PULSE and 1 ms (which means he precision of he phoogae imer is o he neares millisecond, bu he unis of he reading are sill seconds). See Figure.4 for se-up. The op phoogae beam mus be broken by he hanging immediaely when i begins o fall, since he iniial velociy of he hanging is assumed o be ZERO in equaion (6) of he heory. Measure he separaion beween he phoogae heads,! y, and record i in able.. Make sure he sring suspending he hanging does no pass hrough he beams of he phoogae heads, as his can break he beam premaurely! When he phoogae imers reach 0 seconds hey rese o 10 seconds and hen coninue o coun every 10 seconds hereafer. Unforunaely hey are inended o coun shorer inervals. Do a pracice run and jus wach he phoogae imer o learn how o correcly read and record he ime of fall. ) To find he acceleraion, place abou 50 g (record he hanging in able. under he appropriae column) over he pulley and wind he hread around he 8/06 5

6 plaform. Using he verical phoogaes, ime he hanging as i falls hrough he phoogaes and record he ime in he appropriae column in able.. Do his for a oal of 15 rials. Measuring he Radius, r Using calipers, measure he diameer of he cylinder abou which he hread is wrapped and calculae he radius. Record his in Table.. Finding he Acceleraion of he Apparaus Alone Since in he Par II procedure Finding he Acceleraion of he Poin Mass and Apparaus, he apparaus is roaing as well as he poin, i is necessary o deermine he acceleraion and roaional ineria of he apparaus by iself so ha his roaional ineria can be subraced from he oal, leaving only he roaional ineria of he poin. 1) Take he poin off he roaional apparaus and repea he Par II procedure Finding he Acceleraion of he Poin Mass and Apparaus for he apparaus ALONE! ) Record he daa in Table. under Apparaus Alone. 8/06 6

7 Table. Poin Mass & Apparaus Apparaus Alone Fricion Mass Hanging Mass Mass, m (for use in equaions) Radius, r (of cylinder) Acceleraion, a (of hanging ) Disance beween Phoogaes, Time, (o fall hrough phoogaes) ! y Average ime, Poin Mass & Apparaus Apparaus Alone 8/06 7

8 Calculaions 1) Subrac he fricion from he hanging used o accelerae he apparaus o deermine he, m, o be used in he equaions and record in Table.. ) Use he average ime of fall o deermine he acceleraion of he sysem from equaion (6). Record he acceleraion in he appropriae column of Table.. 3) Calculae he experimenal value of he roaional ineria of he poin and apparaus ogeher and record in Table.3. 4) Calculae he experimenal value of he roaional ineria of he apparaus alone. Record in Table.3. 5) Apply equaion (7) in he heory o ge he roaional ineria of he poin alone. Record in Table.3. 6) Calculae he heoreical value of he roaional ineria of he poin. Record in Table.3. 7) Compare he experimenal value o he heoreical value. Record in Table.3. I po in I apparaus & alone apparaus Resuls I in (Experimenal Value) po I in (Theoreical Value) po % Difference in I po in Table.3 8/06 8

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