For a 1-weight experiment do Part 1. For a 2-weight experiment do Part 1 and Part 2

Transcription

1 Page of 6 THE GYROSCOPE The setup s not connected to a computer. You cannot get measured values drectly from the computer or enter them nto the lab PC. Make notes durng the sesson to use them later for composng an electronc report. Recommended readngs: For a -weght experment do Part. For a -weght experment do Part and Part. Physcs for Scentsts and Engneers by Serway and Jewett. V., 9 th ed. Chapter.5, pp The Feynman Lectures on Physcs, Chapter 0 (ths has a very nce, ntutve descrpton of the operaton of the gyroscope) Copy avalable at the Resource Centre. 3. Operaton and nstructon Manual for the MTAC Gyroscope (avalable n a hard copy at the Resource Centre or onlne at NTRODUCTON One of the most nterestng areas n the scence of rotatonal dynamcs s the study of spnnng sold objects, tops, hoops, wheels, etc. From the gyrocompass (whch ndcates true North, rather than Magnetc North) to an understandng of how a cyclst turns corners, the applcatons of ths feld of study are both practcal and fascnatng. Ths experment s desgned to ntroduce you to some of these nterestng and often counterntutve propertes of rotatng bodes. The apparatus conssts of a.7 kg cylndrcal rotor that s spun at a constant rate by an electrc motor. The rotor s mounted n a double gmbal arrangement whch allows t to assume any orentaton. THEORY (a) Angular Moton. The basc equatons for angular moton can often be obtaned smply from those for lnear moton by makng the followng substtutons: Lnear varables Angular varables Force, F Torque, Mass, m Moment of nerta, Velocty, v Angular velocty, Momentum, p Angular Momentum, L Acceleraton, a Angular acceleraton,

2 Page of 6 NB. The analogy needs to be treated wth cauton; e.g. s not a constant property of the body, as s mass, snce ts value depends the axs around whch t s measured. Thus Newton s Law, dp dm v F dl d m a, becomes So we see that a torque appled to a body rotatng wth angular momentum L produces a change n that angular momentum accordng to the relatonshp: dl where s the appled torque, L s the angular momentum, s the angular velocty of rotaton of the body and s the moment of nerta of the body about the axs of rotaton (.e., about the axs). f ths torque s appled perpendcular to the drecton of L (.e., perpendcular to the drecton of ) then from equaton (), L does not change n magntude but does change n drecton (f you do not understand why ths s so, check wth a demonstrator before gong any further). Ths change n drecton of (and thus also of ) s called precesson and appears as a rotaton of the drecton of the vector n space wth a precesson angular velocty of. Then t can be shown that p L L () L () p (b) Sprng Constants. You wll need to measure the sprng constants, k, of the sprngs whch are used n the measurement of the moments of nerta of the rotor. Consder a sprng whch s hangng vertcally, wth a mass m attached to ts lower end. Suppose ts extenson under the acton of the force mg s d. f t s now made to oscllate up and down t wll acqure an adonal extenson or contracton. The net force on the mass due to ths adonal extenson (postve or negatve) wll be gven by F = ma = - k, where k s a constant called the sprng constant, determned by the physcal propertes (rgy, etc.) of the sprng. The negatve sgn ndcates that the force always acts n such a way as to reduce the sze of the extenson. But a, the acceleraton of the mass can be wrtten as d, so, rearrangng, we have : d k m (3) Ths s the equaton of Smple Harmonc Moton, whch has snusodal or cosnusodal solutons. t yelds for the perod of the oscllaton:

3 Page 3 of 6 T m k, or k T (4) m (c) Moments of nerta. n the stuatons shown n Fgures, a, b and 3, the torque appled to the rotor-plus-gmbal s l F, by defnton of torque. Here, s the sum of the forces actng wth arm about the axs of rotaton. These forces are provded by the two sprngs. As, the magntude of torque can be rewrtten as l F l F l F FG.: FG.: a FG.: b FG.: 3 f k and k are the sprng constants of the two sprngs, d and d the extensons from the unstressed length when the rotor s at rest, and the extenson of one sprng and the concomtant compresson of the other as the rotor oscllates (vertcally n Fgures a and b, horzontally n Fgure 3), we have, F [ k ( d ) k ( d )], whch, f k k k and d d, becomes : k Now,, where s the angular dsplacement of the rotor-plus-gmbal. Ths angle d undergoes an acceleraton, so that: d k k (5) Ths s agan the equaton of Smple Harmonc Moton, smlar to Equaton (3) above. Analogously to (4) t has a soluton for the angular frequency o f oscllaton of: k (6)

4 Page 4 of 6 EXPERMENT Study the gyroscope and ensure that you can dentfy the outer frame, the outer gmbal (whch has a vertcal rotaton axs), the nner gmbal (wth horzontal rotaton axs), and the cagng screw whch locks the outer gmbal to the frame, along wth the assocated equpment, the weghts, the sprngs and ther supports, and the scale. Part Start by balancng the gyroscope. Lock the outer vertcal gmbal to the frame. Start the motor and allow a mnute or so for the rotor to attan a constant operatng speed (NB - mportant!). Then adjust the weghts on ether the head or tal weght of the arrow by screwng them n or out untl the spn axs stays horzontal. Frst measure the spn angular velocty of FG.: 4 the rotor (n radans per second) and the moment of nerta of the rotor around the spn axs,. The moment of nerta can be calculated from the dmensons of the rotor arm, consdered as the dfference between two sold cylnders. M M r r M r M r M r t t t 4 4 The result s : ( r ), t r t where the densty of the materal of the rotor 6.7 g cm -3 ; r s the outsde radus of the rotor wheel : r s the nsde radus of the rotor wheel; t s the outer thckness of the rotor wheel : t s the depth of the nner cylnder wall. The angular momentum of the rotor around the spn axs, L, can now be found analytcally wth values of and.

5 Page 5 of 6 Before takng more measurements, famlarze yourself wth some of the propertes of the gyroscope. Remove the cagng screw so that the gyroscope has now two degrees of freedom (t can rotate n both a vertcal and a horzontal plane). Wth the rotor runnng, notce what happens f you move the base by rotatng t, lftng t, or otherwse movng the whole gyroscope. Then apply a torque to the gyroscope, by pushng gently on the spn arrow n any drecton; observe the subsequent precesson. Can you explan the drecton of precesson from a study of Equaton ()? Be sure that you understand the drectons n space of the quanttes n Equaton (). Now apply a very gentle vertcal torque to the outer gmbal; notce that even f the outer gmbal does not rotate, you can stll nduce precesson of the spn arrow. Explan why. Agan lock the outer gmbal to the frame, and repeat the above experments. Explan why the gyroscope now appears to have lost ts nertal propertes. You are now ready to check the valy of Equaton () usng the weghts whch clp on to the notches on the spn arrow to provde a horzontal torque (wth two weghts and two notches, t s possble to take up to sx measurements - count them!). A plot of torque versus frequency of precesson should yeld a straght lne passng through the orgn - does t? - and a value for L, the angular momentum of the rotor. How does ths value of L agree wth that found earler? Part Another nterestng property of the free gyroscope s nutaton, the oscllaton of the spn arrow as t precesses. (See Feynman, n Recommended Readngs for a good dagram and dscusson). Ths may be observed by gvng a downward tap to the head of the spn arrow as t precesses; a damped oscllatory moton wll be observed. The angular frequency of the nutaton depends on the nertal propertes of the spnnng rotor. f s the moment of nerta of the nner gmbal-plus-rotor-plus-weght about ts horzontal axs, and o the moment of nerta of the outer gmbal-plus-rotor-plus-weght about ts vertcal axs, then t can be shown (see the MTAC booklet) the angular frequency of the nutaton of the rotor s gven by: L n f n (7) and o can be measured by observng oscllatons n the vertcal and horzontal planes respectvely of the gmbal-plus-rotor when two sprngs are attached to the system. Fgures - 4 show the setup. For both measurements, remove the weght, and SWTCH OFF the rotor; for the measurement of, the outer gmbal should be locked to the frame. Values for l used n the dervaton of Equaton (6) are l = 8.5 cm for the nner gmbal, and l = 4.0 cm for the outer. The sprng constants can be found from the apparatus manual provded n the lab. (Queston: how accurate s the assumpton of k = k?). We suggest that t s best to frst measure the frequency of nutaton by removng the weght, settng the rotor n moton (wth the outer gmbal unlocked), dsplacng the arrow o

6 Page 6 of 6 head n the vertcal drecton, and then releasng t. Measure the frequency of the resultant moton and compare t to the result of Equaton (7). Measure fn whle the rotor s actually precessng. Unlock the rotor and clp on one of the weghts to the spn arrow (to observe a change n fn t s a good dea to use the maxmum torque avalable). n order to compare ths result to Equaton (7), both moments of nerta and o have to have a term added to them to take account of the extra contrbuton of the added mass. One lmtaton of the apparatus arses from frcton n the bearngs of the gmbals. Ths s partcularly notceable wth a large weght hung on the rotor axs. f frcton dd not exst, the weght would not drop wth tme, but rather would just move n a horzontal plane due to precesson. f the weght drops t s the result of some frctonal torque. You should fgure out whch bearng s causng ths. You mght work out how you mght move the base to elmnate ths torque whle you perform your experment. One of the most useful features of a gyroscope whch you wll have observed s the stablty of ts drecton of spn n space. Ths feature made gyroscopes to completely replace magnetc compasses. The modern devce s called gyrocompass. Stable drecton of spnnng s used by footballers when they mpart a spn to the ball as they throw t. Modern guns have rflng n the barrel whch mparts a spn to the bullets whch helps mantan ther drecton of flght. Gyroscopc stablty can be understood by a consderaton of Equaton () above. Let L be a change n the angular momentum of a spnnng and f beng the ntal and fnal angular speed (spn) of the object, respectvely, and t be the tme over whch a torque τ s appled, we can rewrte Equaton () as t. Thus f τ s small, or appled for a short tme, t wll f have only a small effect n changng the angular momentum of the object. Now, explan why, when you want to turn a corner on your bcycle, you cannot smply turn the front wheel n the drecton you want to go? What n fact, do you do n ths case - and why? f (dh - 974, jv - 988, tk - 995) Last updated by N. Krasnopolskaa n 03