90 Cl von Ossietzky Univesity Olenbug Fculty V - Institute of Physics Moule Intouctoy lbotoy couse physics Pt I Moment of ineti - Steine's theoem Keywos: Rottionl motion, ngul velocity, ngul cceletion, moment of ineti, ottionl moment, ngul momentum, STEINER's theoem. Mesuing pogm: Mesuement of the moment of ineti of cicul isc, etemintion of the xis of gvity of n iegul shpe boy. Refeences: /1/ EICHLER, H. J., KRONFELDT, H.-D., SAHM, J.: Ds Neue Physiklische Gunpktikum, Spinge-Velg, Belin, mong othes 1 Intouction The im of this expeiment is to impove the unestning of the nlogy between tnsltionl n ottionl motion. Fo this pupose, set-up is use which enbles the mesuement of moments of ineti of boies with espect of optionl xes. Fist, the coesponing quntities of the tnsltionl n ottionl motion e clle to memoy by mens of Tble 1. Tble 1: Compison of tnsltionl n ottionl motion. Tnsltionl motion Rottionl motion Nme Symbol Unit Nme Symbol Unit Position vecto m Angle 1 ϕ 1 Velocity v = m s -1 Angul velocity 1 Acceletion = v m s - Angul cceletion 1 Mss m kg Moment of ineti Momentum Foce Theoy p= mv kg m s -1 Angul Momentum F= m= p N Toque ω = φ s -1 ω s - I = R m kg m L = Iω L= p= m v kg m s -1 T ω L = I = N m T= F We consie oty isk D of the ius, oun which thin the hs been woun ccoing to Fig. 1. The the is connecte to mss m vi pulley R. The isk is hel t est by the pin T of the mgnet B. Afte closing the switch S, cuent flows fom the powe supply U though the coil of the mgnet. The 1 The iection of the xil vectos ϕ, ωn ω/ is by efinition the iection of the xis of ottion. The sign obeys the ight-hn ule: the incuve finges show the iection of ottion, so the thumb shows the iection of ϕ, ωn ω/. Pol vectos (noml vectos), s e.g. the position vecto () n the velocity vecto (v), chnge sign upon pefoming point invesion of the coointe system, whees xil vectos (lso clle pseuo-vectos) o not. R is the istnce of mss element m fom the xis of ottion.
91 holing pin T is pulle bck by the esulting mgnetic fiel, theeby unlocking the isc. The flling mss m then cuses n ccelete ottion of the isk bout the oty xis H. R F H ω D T B l m S = U Fig. 1: Roty isk fo mesuing moments of ineti. Refe to the text fo lbels. Now we equie n eqution by mens of which we cn clculte the moment of ineti I D of the oty isk fom known o mesuble quntities. Fo this pupose we fist set up the eqution of motion fo the ottion of the oty isk. It is vey simple in this cse: the oty isk hs the ngul cceletion ω/ ue to the ottionl moment F. In nlogy to NEWTON s lw F = m we thus obtin (cf. Tble 1): (1) F = I D ω t Then it follows fom the chosen geomety ( F) fo the bsolute vlues: () F = I D ω t In this eqution we hve to eplce F n ω/ by known o mesuble quntities. In oe to fin n expession fo ω/, we fist obseve the motion of the mss m. If the time t is neee fo flling though istnce l, we obtin fo its cceletion : (3) l = t Becuse m n the oty isk e connecte vi the the, this must lso be the tngentil cceletion of mss point on the ege of the oty isk. Bse on the well-known eltionship between tngentil n ngul cceletion with Eq. (3), we thus obtin fo such point: (4) ω l = = t t Inseting Eq. (4) into Eq. () yiels: (5) l F = ID = I t D We still nee eltionship fo the foce F, which cceletes the isk, since it cnnot be mesue iectly. Fo this we look t the net foce cting on the set-up. The cceleting foce of gvity G = mg (g: gvittionl cceletion) must ccelete the mss m, ovecome fictionl foces t pulley the R n the oty isk D, n set the pulley n oty isk into n ccelete ottion. Fo this, the following foces e necessy:
9 F m : Acceleting foce fo m F RR: Fictionl foce t the pulley F R : Acceleting foce fo the pulley F RD: Fictionl foce t the oty isk F: Acceleting foce fo the oty isk Thus we obtin: (6) G = mg = Fm + FRR + FR + FRD + F The foce which cceletes m, F m = m, is theefoe consiebly smlle thn the foce of gvity G = mg. To simplify mttes we now ssume tht the foce of fiction n the cceleting foce e eplce by one foce cting on the pulley, which is necessy fo the tnsltionl cceletion of n equivlent mss m e (hee: m e. g): (7) F + F : = m R RR e We theefoe obtin fo the equie foce F fom Eq. (6): (8) F = mg ( m + m ) F Inseting this eqution into Eq. (5) we obtin: e RD (9) mg ( m + m ) = I + F e D RD Fo bette ebility we intouce foce (10) F : = mg ( m + m ) E e with the mesuble quntities m n n the known quntities m e n g such tht Eq. (9) becomes: (11) F = I + F E D RD The unknown quntity F RD which cnnot be mesue iectly is still botheing us in this eqution fo etemining I D. If we ssume, howeve, tht the fiction t the oty isc is olling n sliing fiction inepenent of the velocity (the so-clle COULOMB fiction), which only epens on the mss of the oty tble incluing the boies on it, then F RD cn be consiee time-inepenent constnt. In this cse Eq. (11) epesents simple line eqution of the fom (1) y = cx + b with (13) y = F, x=, c= I, b= F E D RD Plotting the elte quntity F E (to be clculte ccoing to Eq. (10)) ginst / (with fom Eq. (3)) fo constnt n iffeent cceleting msses m (Eq. (11)), we obtin line with the slope I D. Thus we hve foun wy to mesue the moment of ineti without knowing the quntity F RD. We now obseve the cse in which n itionl boy is plce on the oty isk. Suppose I K is the moment of ineti of this boy (mss m K) when it ottes bout one of its gvity xes (pincipl xis); if this gvity xis coespons with the oty xis H of the oty isk, then the ovell moment of ineti I of the system oty isk/boy is:
93 (14) I = ID + IK If the xes H n C un pllel t istnce s we obtin ccoing to STEINER's theoem 3 : (15) I = I + I + m s D K K Eq. (11) then es: (16) FE = I + F RD Using Eq. (3) it follows: I FE FRD FE FRD t l (17) = ( ) = ( ) We cn use this eltionship to etemine the position of gvity xis unning pllel to the oty xis of the isk fo boy of bity shpe lying on the oty isc. We tke the following steps: ccoing to Eq. (15) I is miniml when s = 0, i.e., fo the cse tht the gvity xis of the boy is ienticl to tht of the oty xis of the isc. Accoing to Eq. (17) minimum of I is equivlent to minimum of the fll time t n t, espectively. Shifting the boy on the oty isc (vying s), the fll time t must theefoe show minimum t cetin position. The elte function t = f(s) escibing this behviou will now be etemine. Fo this we inset Eq. (15) into Eq. (17), solve fo t n obtin fo t s function of s: (18) t ( ID + IK) l lmk ( ) ( ) = + s FE FRD FE FRD K1 K o in cle wy with the uxiliy quntities K 1 n K : (19) t = K + K s 1 Question 1: - Which function (cuve) oes Eq. (19) epesent? (Hint: Conic sections) In oe to etemine the position of the equie gvity xis C by mens of Eq. (19) we pocee s follows: Choose coointe system XY on the oty isc, the oigin of which coincies with the xis of ottion H (cf. Fig. ). A line of holes is cete long the y-xis of the oty isk. A pin is fixe t n optionl point P on the boy, fo which we fin the position of the gvity xis. The pin n line of holes e plce such tht the boy cn be shifte in the Y iection on the oty isk without chnging its oienttion with eg to the coointe system XY (cf. emks t the en of Chp. 3.). Let point P (the pin) hve the coointes (0, y P) fte plcing the boy on the oty isc. Fo the istnce s of the gvity xis C fom the oty xis H we then obtin: (0) ( ) s = x + y y P Accoing to Eq. (19) the fll time t fo the cceleting mss m hs minimum when s is miniml, which, ccoing to Eq. (0) with fixe x, is the cse fo y P = y. If we shift the boy in y iection on the oty isk n plot the fll time t ove the shift y P, we cn etemine the quntity y by fining the minimum in the pouce cuve. In n nlogous wy, the quntity x cn 3 JAKOB STEINER (1796-1863)
94 be etemine n poceeing fom the optionl point P, we cn stte the position of the esie gvity xis. y Pobeköpe C s x P H y y P x Fig. : Roty isc (yellow) with smple boy (white, top view). H is the xis of ottion, C the gvity xis of the smple boy 4 n P is the smple boy s point of fixtion long the veticl line of holes on the isc. s is the istnce between C n H. 3 Expeimentl poceue Equipment: Roty isc on tipo, 5 cceletion msses (m = (1,00 ± 0,01) g) with plte (m ccoing to impint, eo negligible), bss isk with locking pins, iegully shpe smple boy with locking pins, powe supply (PHYWE (0-15 / 0-30) V), mgnetic hole, stn mteil fo mgnetic hole, switch, light bie, electonic univesl counte, igitl oscilloscope TEKTRONIX TDS 101 / 101B / 01C / TBS 110B, pecision spiit level (ccucy 0.1 mm on 1 m), blnce, metl mesuing tpe, sliing cllipe, eceletion o, the. Attention: The oty iscs hve vey sensitive pecision beings which e esy to mge though impope hnling. Only move the oty iscs with ceful finges! Tke ce tht the the oes not get entngle in the being by timely eceletion! Only ecelete the iscs using the smll o vilble! Hint: Usully the oty iscs e levelle exctly by the technicl ssistnt using pecision wte level pio to the lb couse. 3.1 Moment of ineti of isc The moment of ineti I K of bss isk (ius K, mss m K) otting bout its symmety xis C (Fig. 3) is to be etemine by mens of the set-up in Fig. 1. It is then clculte ccoing to Eq. (14) s follows: (1) IK = I ID In oe to obtin I K, fist the moment of ineti of the oty isc (I D) hs to be etemine by mens of Eq. (11) n then the moment of ineti of the oty- n bss isks togethe (I) by mens of Eq. (16). Fo this pupose ) fo the oty isc b) fo oty isk with bss isc the fll time t (men vlue fom t lest fou single mesuements ech) is mesue fo five iffeent cceletion msses n fo peetemine istnce l (to be mesue!). The fll time is mesue by mens of n electonic univesl counte. The counte is stte by the impulse, which cuses the elese of the holing pin of the mgnetic hole, which is esponsible fo keeping the oty isk in the stting position. The stopping impulse fo the univesl counte is given by light bie, which the ccelete msses pss t the en of the specifie istnce l. 4 Note tht the white e epesents the top view of the smple boy. Fo this eson, C oes not nee to be locte t the cente of gvity of the white e.
95 ω K Fig. 3: Rottion of isc of ius K n mss m K bout its symmety xis C. C Subsequently F E is plotte ginst / fo ) n b) ccoing to Eq. (11) n Eq. (16) in one igm n the egession lines e clculte (mesue cefully using metl mesuing tpe) 5. An eo nlysis fo the iniviul vlues of F E n / is not equie. The fiction foces F RD on the oty isc s well s the moments of ineti I D n I e clculte fom the pmetes of the egession line (incluing eo) n fom tht I K ccoing to Eq. (1) (lso incluing eo). Question : - How cn the moment of ineti I of isk with the mss m K n the ius K otting bout its symmety xis C (cf. Fig. 3) be clculte fom the eltionship I = R m (cf. Chpte 1)? How lge is the theoeticlly expecte moment of ineti fo the bss isk use (mesue K n m K!)? Wht e the possible souces of evitions between theoy n expeiment? 3. Detemining the position of gvity xis of n iegully shpe boy Accoing to the explntions given fo Eqs. (18) - (0) the position of gvity xis C unning pllel to the oty xis H of n iegully shpe smple boy shll be etemine. Fo this pupose the pin mounte on the boy is put into ten iffeent holes of the hole ow long the y-xis of the oty isk n the coointe y P is etemine 6. At ech position, the vege fll time t (men of 4 single mesuements) fo peetemine istnce l is mesue fo one mss m ech. Aftews, t is plotte ginst y P incluing eo bs (stn evition of the men) n the vlue y is gphiclly etemine, whee t hs minimum. Altentively, the position of the minimum of t my be etemine by non-line fit 7. The tget function is, ccoing to Eq. (19), given by: t = K + K y y () ( ) 1 P with the fit pmetes K 1, K n y. This fit iectly povies the vlue y P = y fo which the fll time t is miniml. Anlogously, it woul be possible to etemine x n to stte the position of the cente of gvity C in the xy-plne eltive to the point P. In oe to sve time, howeve, we will confine ouselves to mesuing only the istnce y between P n C. Remks: In oe to mke sue tht the oienttion of the smple boy oes not chnge when shifting long the y-xis, two pins e mounte on the boy. Theefoe, it hs to be etemine fist, which of the two pins mks the position of point P. 5 The cceletion is in the oe of mgnitue of 10 - ms - n thus smll compe to g. Only smll iffeences theefoe ise fo F E (Eg. (10) in the cses ) n b). 6 The istnce between two holes on the isc is 10 mm (eo fee). 7 Nonline fits e elt with in pt II of the bsic lbotoy couse in the SoSe, see http://physikpktik.uniolenbug.e/ownlo/gpr/pf/nichtlinee_fits.pf. Hee the ppliction is optionl.