Rotational Motion and Moment of Inertia

Transcription

1 Pupose: To detemine the ottionl ineti o disc nd o ing nd to compe these with the theoeticl vlues. Equipment: Rotting Tble, Disc, Ring Hooked Mss Set Long Rod Right Angle Clmp Cylindicl Rod Clmp Tble Clmp Smt Pulley/Photogte Stop Wtch -mete Stick Venie Clipes Cpente s Level Sting Theoy: By now you ve pobbly igued out tht evey vible tht ws deined when tlking bout line motion hs n nlogue in ottionl motion. Insted o distnce tveled, d, we hve ngle tuned, θ. Rthe thn speking simply o velocity, v, we tlk o ngul velocity, ω. Evey spect o line motion hs its ptne in ottionl motion. And not only e they nlogous, but they e elted, too! usully by some powe o the dius o the cicle. In ct, hee s how it goes 1 o 10

2 Line Vible Symbol Angul Vible Symbol Reltionship distnce d ngle θ d = θ velocity cceletion v ngul velocity ngul cceletion ω α v = ω = α oce toque τ τ = mss momentum m p ottionl ineti ngul momentum I dm L L = p igue 1 Reltionships between Line nd Angul (Rottionl) Vibles The sme bsic equtions o line motion cn be used o ottionl motion with simple chnge o vibles: x v = x + v t + = v i o o + t 1 t θ = θ + ω t + i i o ω = ω + αt The dieence between ngul nd line motion, nd the eltion between them, is something tht most students e ble to undestnd ily esily. Almost eveyone hs plyed on plygound mey-go-ound, nd hs noticed the dieence in velocity between stnding t the cente nd stnding t the edge. The conusion begins to set in with discussions o toque nd ottionl ineti, the questions being speciiclly why do we need to dieentite between the simple oce nd Mss? When discussing line motion, it is ssumed tht ll objects e point objects whose mss is centeed symmeticlly bout single point, nd they hve no size to spek o tht is, ll objects e shped essentilly like vey smll blls tht don t spin. Most objects, howeve, don t ctully it this ctegoy. Most objects (even most blls) e extended objects they hve mesuble size nd they my not be totlly symmetic. When oce is exeted on n extended object, it mttes not only how big the oce is, but lso whee on the object it is pplied. Imgine b on tble: 1 αt Eq. 1 v igue The ppliction o oce, nd the esulting motion o b o 10

3 I the oce is pplied diectly to the cente o the object, it will tnslte linely coss the tble. Howeve, push on eithe side o the cente will cuse ottion o the object. When oce is pplied wy om the xis o ottion (in this cse, the cente o mss) it cuses the object to otte. This is wht we cll toque. The toque is deined s τ = Eq. whee is the oce pplied nd is the distnce om the xis o ottion to whee the oce is pplied. Remembe tht the deinition o coss poduct is: A B = A B sinθ whee θ is the ngle between vectos A nd B. Thus, τ sinθ = Eq. 3 This mens tht the lgest toque occus when nd e pependicul to ech othe (sinθ = 1) nd the smllest toque is when nd e pllel (sinθ = 0). As n exmple, think o using wench to tun bolt: Lgest toque τ = sinθ whee sinθ = 1 so τ = Smlle toque τ = sinθ whee sinθ <1 so τ < Smllest toque τ = sinθ whee sinθ = 0 so τ = 0 igue 3 Toque: Chnging the Diection o the Applied oce Similly, we cn chnge the dius o ottion: lgest lgest τ smllest smllest τ igue 4 Toque: Chnging the Rdius o Rottion I you ve eve tied to tun tightened bolt, you pobbly ledy know tht the best esults come when the oce is pplied to the end o the hndle, in pependicul diection. Wht then, bout ottionl ineti? 3 o 10

4 Remembe Newton s ist Lw o Motion, sometimes clled The Lw o Ineti : Any object will continue in its stte o est o stightline motion unless cted upon by n outside oce. Any object elly mens ny object with mss, which is usully se ssumption. Pcticlly, this mens tht ny object tht hs mss esists chnges in its motion. This esistnce is wht is eeed to s ineti. Since ineti is popety o ll objects hving mss, some go so s to sy tht ineti is mss (o, mss is ineti). Now think bck to the wench gin. Imgine holding one end in you hnd nd otting you wist to swing the othe end in smll c. Now switch ends. Is it esie one wy thn the othe? It should be! You should ind tht it is esie to swing the wench when you e holding the ounded (clmping) end. It is hde when you llow tht sme (ounded) end to swing. NOTE: I you hve neve ctully done this, o hve ogotten wht it s like, sk you lb instucto o lb tech o wench to ty it out. Why would this be? The wench hs the sme mss no mtte how you hold it, nd i you exet the sme oce, it seems tht it should ccelete in the sme mnne. You hve just discoveed the eson o deining n object s Rottionl Mss!!! Rottionl ineti tkes into ccount not only the totl mss o n object, but how the mss is distibuted ound n xis o ottion, nd just s mss tells us how much esistnce oce will meet, ottionl mss (o moment o ineti) tells us how much n object will esist otting (o chnging its ottion). A single object s ottionl ineti will chnge depending on whee the pivot point (o xis) is plced. It is esie to tun n object when most o the mss is locted ne the pivot point, thus this conigution hs lowe ottionl ineti. I most o n object s mss is locted wy om the pivot point, the ottionl ineti is lge, nd it will be moe diicult to otte. Rottionl mss cn be deined in two (equivlent) wys: m dv o Eq. 4 nd b The expession you use depends mostly on wht poves esiest o the pticul sitution. Most oten, you will not need eithe, o the geomety o the system will be simple enough to use some combintion o the pe-deined moments o ineti. dm b 4 o 10

5 Hoop o cylindicl shell MR R Hollow cylinde ( R ) + 1 M R 1 R 1 R Solid cylinde o disc 1 MR R Rectngul plte ( ) + 1 b 1 M b Long thin od Long thin od 1 ML 1 L 1 ML 3 L Solid sphee Thin spheicl shell MR 5 R MR 3 R igue 5 Vious Solid igues nd thei Moments o Ineti When we wnt to quntiy the eltion between the oce exeted on n object nd its mss (o ineti) we use Newton s Second Lw o Motion: = m. Eq. 5 To quntiy the eltion between the toque on n object nd its ottionl mss (o ineti), we simply eplce ech pt o Newton s Second Lw with its ottionl nlogue: τ = Iα. Eq. 6 Thus, we cn peom n expeimentl test. The equtions in the cht bove llow clcultion o the theoeticl ottionl mss o pticul disc o ing. By pplying known toque to the object nd mesuing the ngul cceletion, we cn detemine n expeimentl ottionl mss, nd compe the two. 1 1 The theoy o this lb ws witten by Jennie LK Whlen 5 o 10

6 Expeiment: 1. Set up equipment s shown in igue 6. Let m = 00 g. smt pulley-photogte m to LbPo h smt pulley Rotting Tble nd Disc igue 6 Bsic Expeimentl Set-Up. o the ist un, (m = 00 g, empty tble) mesue the distnce, h the mss lls nd the time, t, (using stopwtch) it tkes the weight to ll tht distnce. Detemine the cceletion om the eltion: h = ½ t. 3. Connect the AC dpte to the LbPo by inseting the ound plug on the 6-volt powe supply into the side o the intece. Shotly te plugging the powe supply into the outlet, the intece will un though sel-test. You will he seies o beeps nd blinking lights (ed, yellow, then geen) indicting successul sttup. 4. Attch the LbPo to the compute using the USB cble tht is Velco-ed to the side o the compute box (do not unplug the USB cble om the compute!). The LbPo compute connection is locted on the ight side o the intece. Slide the doo on the compute connection to the ight nd plug the sque end o the USB cble into the LbPo USB connection. 5. Connect the Smt Pulley/Photogte to the DIG/SONIC1 pot o the LbPo. I you e using one-piece Smt Pulley/Photogte, PASCO o vey old Venie Photogte, you will need to use the digitl dpte. I you e using newe Venie (with emovble cble), simply emove the cble with the Phono plug, nd connect the Photogte Cble with Bitish-Telecom plug on one end. 6. Open the ile ottionl_ineti.xmbl (o.cmbl) in the Expeiments olde on the desktop. This will stt the pogm Logge Po3.3 nd bing up the ppopite dt ile. I you do not hve n uto-id senso (which is the likely cse), dilog box will pop up sking you to conim the sensos being used. I you hve the suggested senso ttched to the LbPo in the suggested pot, click OK. I the OK button is not ctive, sk you instucto o help. 6 o 10

7 7. Once Logge Po 3 is open, click on Expeiment > Set Up Sensos > LbPo 1. Click on the photogte icons, nd veiy tht Motion Timing is selected unde Cuent Clibtions. In the sme menu, choose Set Distnce o Length, mke sue tht Smt Pulley (10 Spoke) in Goove is selected. The pogm clcultes the cceletion nd velocity o the lling mss by teting the pulley s picket ence with the pope spcing. 8. Now pess Collect nd elese the mss. Check the esults o the compute mesuements ginst you own mesuements o cceletions (om height nd time). The "by hnd" vlues cn die by s much s 0% om the Logge Po vlues. 9. When you e stisied tht the compute is mesuing cceletion coectly, use Logge Po mesue the cceletion o the mss s it lls nd tuns the empty tble (Conigution I). Repet this mesuement two moe times nd ecod the thee dieent cceletion vlues (thee dieent slopes o v vs. t). You will lte use the nge in these vlues to detemine you uncetinty nge o I exp. 10. It will lte be necessy to detemine the ictionl toque cting on the system. To do this, eplce the 00 gm mss with tiny mss (you my wish to use ppe clips). Stt with 5 o 10 gms nd incese the mss until the tble stts to tun. The minimum mss which will just stt the tble otting is m o. Recod this vlue. 11. Add the disc to the tble (this is Conigution II.) nd epet Steps 5 nd 6. To ovecome the gete ineti o the tble, use m = 500 gms. Don't oget to mesue the minimum mss m o tht just stts the tble + disc otting. 1. Remove the disc nd dd the ing to tble (This is Conigution III). Repet Steps 5 nd 6 (500 gms my wok well o this conigution). Be sue to ecod the new vlue o m nd m o. 13. Mesue = (dimete o the sht o the otting tble/). Anlysis: 1. o ech conigution, clculte the vege cceletion,, o the mss, m, o tht conigution. Also clculte uncetinty (devition) in the expeimentl cceletion.. Using Newton s nd lw: = m (pplied to the hnging mss) nd it s ottionl nlog = I α = τ Eq. 7 7 o 10

8 (whee is the oce pplied by the sting), solve o I by eliminting (show you wok) to get I m = ( g ) exp Eq Clculte the ottionl ineti I, using Eq. 8 o the thee conigutions: I 1 = I tble I = I tble + I disc I 3 = I tble + I ing 4. Clculte: I disc = I - I 1 I ing = I 3 - I 1 5. Clculte the theoeticl vlues o I disc nd I ing using the omuls in the text. Note tht the disc hs weighted "plug" t the cente to eplce the missing mss in the hole (so it cn be consideed solid disc). Also note tht the "ing" is ctully hollow cylinde with inne nd oute dii (Do you think i mkes ny dieence in the nswe in this cse i you use the omul o ing the thn tht o hollow cylinde?). 6. Now, i I exp does not equl I theoy then pt o the poblem might be ictionl toques cting on the system. Assuming tht this ictionl oce pevents the sht om otting, we cn mesue it diectly by hnging smll weight m o. This weight will be the mximum vlue tht cn be hung without the tble otting. = m 0 g Sht igue 7 inding the oce o Rotting iction Since the system is now in equilibium, we ind tht τ net = 0 nd m0 g = 0. Also, i we cll = τ, then: m g = τ 0. Eq Detemine the vlue τ o ech conigution nd ecod. Now go bck to Step o the nlysis nd include τ in you detemintion o I exp. Show tht the Coected vlue is: 8 o 10

9 Whee τ = m o g. m( g ) I exp = - τ 8. Clculte the coected vlues o I exp o ech conigution. 9. Using the coected vlues epet Step Detemine the totl uncetinty in the expeiment: i.e., theoy + exp. 11. Compe the theoeticl vlues to the expeimentl vlues o I disc nd I ing (e the % dieences within the uncetinty?) I not, why not? 1. The expeiment ile lso clcultes o you the ngul cceletion, α. The ottionl ineti cn lso be ound diectly using Eqs. 3 nd 6: τ = sinθ nd τ = Iα. Solve these equtions o I, nd use this method to clculte I 1, I nd I As beoe, ind I disc nd I ing. Compe these vlues to those ound in Step 4 bove. Ae they the sme? Should they be? Which method do you think is moe ccute? Why? (Some things to conside: how is detemined? How is α detemined? Ws α detemined coectly? [Check the eqution!]) Results: Wite t lest one pgph descibing the ollowing: wht you expected to len bout the lb (i.e. wht ws the eson o conducting the expeiment?) you esults, nd wht you lened om them Think o t lest one othe expeiment might you peom to veiy these esults Think o t lest one new question o poblem tht could be nsweed with the physics you hve lened in this lbotoy, o be extpolted om the ides in this lbotoy. 9 o 10

10 Clen-Up: Beoe you cn leve the clssoom, you must clen up you equipment, nd hve you instucto sign below. How you divide clen-up duties between lb membes is up to you. Clen-up involves: Completely dismntling the expeimentl setup Removing tpe om nything you put tpe on Dying-o ny wet equipment Putting wy equipment in pope boxes (i pplicble) Retuning equipment to pope cbinets, o to the ct t the ont o the oom Thowing wy pieces o sting, ppe, nd othe detitus (i.e. you wte bottles) Shutting down the compute Anything else tht needs to be done to etun the oom to its pistine, pe lb om. I cetiy tht the equipment used by hs been clened up. (student s nme),. (instucto s nme) (dte) 10 o 10