1. Physics for Scientists and Engineers by Serway and Jewett. V.1, 9 th ed. Chapter 11.5, pp

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1 Page of 6 THE GYROSCOPE The setup s not connected to a compute. You cannot get measued values dectly fom the compute o ente them nto the lab PC. Make notes dung the sesson to use them late fo composng an electonc epot. Recommended eadngs: Fo a -weght expement do Pat. Fo a -weght expement do Pat and Pat. Physcs fo Scentsts and Engnees by Seway and Jewett. V., 9 th ed. Chapte.5, pp The Feynman Lectues on Physcs, Chapte 0 (ths has a vey nce, ntutve descpton of the opeaton of the gyoscope) Copy avalable at the Resouce Cente. 3. Opeaton and nstucton Manual fo the MTAC Gyoscope (avalable n a had copy at the Resouce Cente o onlne at NTRODUCTON One of the most nteestng aeas n the scence of otatonal dynamcs s the study of spnnng sold objects, tops, hoops, wheels, etc. Fom the gyocompass (whch ndcates tue Noth, athe than Magnetc Noth) to an undestandng of how a cyclst tuns cones, the applcatons of ths feld of study ae both pactcal and fascnatng. Ths expement s desgned to ntoduce you to some of these nteestng and often countentutve popetes of otatng bodes. The appaatus conssts of a.7 kg cylndcal oto that s spun at a constant ate by an electc moto. The oto s mounted n a double gmbal aangement whch allows t to assume any oentaton. THEORY (a) Angula Moton. The basc equatons fo angula moton can often be obtaned smply fom those fo lnea moton by makng the followng substtutons: Lnea vaables Angula vaables Foce, F Toque, τ Mass, m Moment of neta, Velocty, v Angula velocty, ω Momentum, p Angula Momentum, L Acceleaton, a Angula acceleaton, α

2 Page of 6 NB. The analogy needs to be teated wth cauton; e.g. s not a constant popety of the body, as s mass, snce ts value depends the axs aound whch t s measued. Thus Newton s Law, dp d( m v) dl d( ω) F = = = m a, becomes τ = = = α So we see that a toque τ appled to a body otatng wth angula momentum L poduces a change n that angula momentum accodng to the elatonshp: dl τ = () whee τ s the appled toque, L = ω s the angula momentum, ω s the angula velocty of otaton of the body and s the moment of neta of the body about the axs of otaton (.e., about the ω axs). f ths toque τ s appled pependcula to the decton of L (.e., pependcula to the decton of ω ) then fom equaton (), L does not change n magntude but does change n decton (f you do not undestand why ths s so, check wth a demonstato befoe gong any futhe). Ths change n decton of L (and thus also of ω ) s called pecesson and appeas as a otaton of the decton of the L vecto n space wth a pecesson angula velocty of Ω. Then t can be shown that p τ = Ω L () p (b) Spng Constants. You wll need to measue the spng constants, k, of the spngs whch ae used n the measuement of the moments of neta of the oto. Consde a spng whch s hangng vetcally, wth a mass m attached to ts lowe end. Suppose ts extenson unde the acton of the foce mg s d. f t s now made to oscllate up and down t wll acque an addtonal extenson o contacton Δ. The net foce on the mass due to ths addtonal extenson (postve o negatve) wll be gven by F = ma = - k Δ, whee k s a constant called the spng constant, detemned by the physcal popetes (gdty, etc.) of the spng. The negatve sgn ndcates that the foce always acts n such a way as to educe the sze of the extenson. But a, the acceleaton of the mass can be wtten as d Δ, so, eaangng, we have : d Δ = k m Δ (3) Ths s the equaton of Smple Hamonc Moton, whch has snusodal o cosnusodal solutons. t yelds fo the peod of the oscllaton:

3 Page 3 of 6 T m k = π, o ω k = π T = (4) m (c) Moments of neta. n the stuatons shown n Fgues, a, b and 3, the toque appled to the oto-plus-gmbal s τ = l ΣF, by defnton of toque. Hee, Σ F s the sum of the foces actng wth am l about the axs of otaton. These foces ae povded by the two spngs. As l ΣF, the magntude of toque can be ewtten as τ = l ΣF FG.: FG.: a FG.: b FG.: 3 f k and k ae the spng constants of the two spngs, d and d the extensons fom the unstessed length when the oto s at est, and Δ the extenson of one spng and the concomtant compesson of the othe as the oto oscllates (vetcally n Fgues a and b, hozontally n Fgue 3), we have, τ = ΣF l = [ k ( d + Δ) k ( d )] l, Δ whch, f k k k and d d, becomes : FG.: a τ = k lδ FG.: 3 Now, Δ = lθ, whee θ s the angula dsplacement of the oto-plus-gmbal. Ths angle d θ undegoes an acceleaton α =, so that: d θ α = = klδ = kl θ (5) Ths s agan the equaton of Smple Hamonc Moton, smla to Equaton (3) above. Analogously to (4) t has a soluton fo the angula fequency o f oscllaton of: kl ω = (6)

4 Page 4 of 6 EXPERMENT Study the gyoscope and ensue that you can dentfy the oute fame, the oute gmbal (whch has a vetcal otaton axs), the nne gmbal (wth hozontal otaton axs), and the cagng scew whch locks the oute gmbal to the fame, along wth the assocated equpment, the weghts, the spngs and the suppots, and the scale. Pat Stat by balancng the gyoscope. Lock the oute vetcal gmbal to the fame. Stat the moto and allow a mnute o so fo the oto to attan a constant opeatng speed (NB - mpotant!). Then adjust the weghts on ethe the head o tal weght of the aow by scewng them n o out untl the spn axs stays hozontal. Fst measue the spn angula velocty of FG.: 4 the oto (n adans pe second) and the moment of neta of the oto aound the spn axs,. The moment of neta can be calculated fom the dmensons of the oto am, consdeed as the dffeence between two sold cylndes. M M = M = M = M = ρπ t t t 4 4 The esult s : = πρ(, t t) whee ρ the densty of the mateal of the oto 6.7 g cm -3 ; s the outsde adus of the oto wheel : s the nsde adus of the oto wheel; t s the oute thckness of the oto wheel : t s the depth of the nne cylnde wall. The angula momentum of the oto aound the spn axs, L, can now be found analytcally wth values of and ω.

5 Page 5 of 6 Befoe takng moe measuements, famlaze youself wth some of the popetes of the gyoscope. Remove the cagng scew so that the gyoscope has now two degees of feedom (t can otate n both a vetcal and a hozontal plane). Wth the oto unnng, notce what happens f you move the base by otatng t, lftng t, o othewse movng the whole gyoscope. Then apply a toque to the gyoscope, by pushng gently on the spn aow n any decton; obseve the subsequent pecesson. Can you explan the decton of pecesson fom a study of Equaton ()? Be sue that you undestand the dectons n space of the quanttes n Equaton (). Now apply a vey gentle vetcal toque to the oute gmbal; notce that even f the oute gmbal does not otate, you can stll nduce pecesson of the spn aow. Explan why. Agan lock the oute gmbal to the fame, and epeat the above expements. Explan why the gyoscope now appeas to have lost ts netal popetes. You ae now eady to check the valdty of Equaton () usng the weghts whch clp on to the notches on the spn aow to povde a hozontal toque (wth two weghts and two notches, t s possble to take up to sx measuements - count them!). A plot of toque vesus fequency of pecesson should yeld a staght lne passng though the ogn - does t? - and a value fo L, the angula momentum of the oto. How does ths value of L agee wth that found eale? Pat Anothe nteestng popety of the fee gyoscope s nutaton, the oscllaton of the spn aow as t pecesses. (See Feynman, n Recommended Readngs fo a good dagam and dscusson). Ths may be obseved by gvng a downwad tap to the head of the spn aow as t pecesses; a damped oscllatoy moton wll be obseved. The angula fequency of the nutaton depends on the netal popetes of the spnnng oto. f s the moment of neta of the nne gmbal-plus-oto-plus-weght about ts hozontal axs, and o the moment of neta of the oute gmbal-plus-oto-plus-weght about ts vetcal axs, then t can be shown (see the MTAC booklet) the angula fequency of the nutaton of the oto s gven by: L ωn = πf n = (7) and o can be measued by obsevng oscllatons n the vetcal and hozontal planes espectvely of the gmbal-plus-oto when two spngs ae attached to the system. Fgues - 4 show the setup. Fo both measuements, emove the weght, and SWTCH OFF the oto; fo the measuement of, the oute gmbal should be locked to the fame. Values fo l used n the devaton of Equaton (6) ae l = 8.5 cm fo the nne gmbal, and l = 4.0 cm fo the oute. The spng constants should be measued usng the method suggested n the THEORY secton above. (Queston: how accuate s the assumpton of k = k?). We suggest that t s best to fst measue the fequency of nutaton by emovng the weght, settng the oto n moton (wth the oute gmbal unlocked), dsplacng the aow o

6 Page 6 of 6 head n the vetcal decton, and then eleasng t. Measue the fequency of the esultant moton and compae t to the esult of Equaton (7). Measue f n whle the oto s actually pecessng. Unlock the oto and clp on one of the weghts to the spn aow (to obseve a change n f n t s a good dea to use the maxmum toque avalable). n ode to compae ths esult to Equaton (7), both moments of neta and o have to have a tem added to them to take account of the exta contbuton of the added mass. One lmtaton of the appaatus ases fom fcton n the beangs of the gmbals. Ths s patculaly notceable wth a lage weght hung on the oto axs. f fcton dd not exst, the weght would not dop wth tme, but athe would just move n a hozontal plane due to pecesson. f the weght dops t s the esult of some fctonal toque. You should fgue out whch beang s causng ths. You mght wok out how you mght move the base to elmnate ths toque whle you pefom you expement. One of the most useful featues of a gyoscope whch you wll have obseved s the stablty of ts decton of spn n space. Ths featue made gyoscopes to completely eplace magnetc compasses. The moden devce s called gyocompass. Stable decton of spnnng s used by footballes when they mpat a spn to the ball as they thow t. Moden guns have flng n the bael whch mpats a spn to the bullets whch helps mantan the decton of flght. Gyoscopc stablty can be undestood by a consdeaton of Equaton () above. Let ΔL = ω f ω be a change n the angula momentum of a spnnng ω and ω f beng the ntal and fnal angula speed (spn) of the object, espectvely, and Δt be the tme ove whch a toque τ s appled, we can ewte Equaton () as ω f ω = τδt. Thus f τ s small, o appled fo a shot tme, t wll have only a small effect n changng the angula momentum of the object. Now, explan why, when you want to tun a cone on you bcycle, you cannot smply tun the font wheel n the decton you want to go? What n fact, do you do n ths case - and why? (dh - 974, jv - 988, tk - 995) Last updated by N. Kasnopolskaa n 03