Mechanics. Free rotational oscillations. LD Physics Leaflets P Measuring with a hand-held stop-clock. Oscillations Torsion pendulum

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1 Mechanic Ocillation Torion pendulum LD Phyic Leaflet P.5.. Free rotational ocillation Meauring with a hand-held top-clock Object of the experiment g Meauring the amplitude of rotational ocillation a function of time g Determination of the damping contant and the logarithmic decrement g Invetigating the tranition form the weekly damped ocillation cae to the limit cae Principle Ocillation (and wave) phenomena are well known due to their preence everywhere in nature and technique. Their invetigation i thu both from experimental point of view a from theoretical point of view an important topic a it allow to tudy fundamental method and concept of phyic. The rotary ocillation are a pecial cae among variou mechanical ocillator model (compound pendulum, pring pendulum etc.) which allow to invetigate the mot important phenomena which occur in all type of ocillation. dditionally, to the uual obervation of a free damped harmonic ocillator anharmonic ocillator can alo be realized. In experiment P.5..4 anharmonic chaotic rotary ocillation are examined in order to how that the harmonic ocillation are only a pecial cae. mplitude / m 0,2 time / 0,25 0,20 () 0, 0,0-0, -0, (B) mplitude / m 0,5 0,0 0,05 0,00-0, time t / 0,25 0,20 (C) Bi / Fö 0506 Fig. : Schematic repreentation of variou damped ocillation curve: () weakly damped cae: ω > (blue curve) (B) heavily damped cae: ω (red curve) in comparion with an damped ocillation of type (, blue) (C) aperiodic limit cae: ω 0 δ 0 < δ 0 = δ (green curve) in comparion with heavily damped cae (B, red). mplitude / m 0,5 0,0 0,05 0,00-0, time t / LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de

2 P LD Phyic leaflet pparatu Torion pendulum DC power upply 0 6/ mmeter, DC, I 2, e.g. LD analog Connecting lead, 00 cm, blue Pair cable, red and blue, 00 cm Stop clock However, according to equation () ocillation occur only when the angular frequency (i.e. equation (I)) ha a poitive radiant (Fig. : cae ()): If ω 0 > δ ω 0 < δ the olution ha the form δ t ω t ω t (t) = e (e + e ) (III) 0 The movement of a free damped (rotary) ocillating ytem can be decribed by the differential equation 2 d d + k + D = 0 dt dt J 2 J: moment of inertia D: directional quantity (retoring torque) k: damping coefficient (coefficient of friction) : angle of rotation With the damping contant δ = k 2 J the natural angular frequency of an undamped ocillation ω = 0 D J and the angular frequency of the damped ocillation 0 δ (I) (II) (III) ω = ω (I) equation (I) may be reolved by (t) = t e δ co ω t () 0 0 : initial angel of rotation at time t = 0 δ: damping contant ω 0 : characteritic frequency of an undamped ytem ω: angular frequency of the damped ocillation From equation () follow that the amplitude decreae by the amplitude factor e - δ t (Fig. cae ()). Thu after a time /δ the amplitude ha decreaed to /e of it initial value 0. Moreover, from equation () follow that the ratio of two ucceive amplitude n and n+ i contant n n+ = q = e δ T (I) The ocillating ytem approache the equilibrium poition aymptotically after one ocillation (o called creeping or heavily damped cae Fig. cae (B) red curve). The higher the damping contant the lower the approach to zero. If ω 0 = δ the olution ha the form (t) = ( (IX) δ t 0 + b t) e The damping i o great that there i jut no longer a croing through the ret poition. ny reduction in damping lead to an ocillation. Thi i the o-called aperiodic limit cae which i of practical importance becaue the time required to reach the zero poition i minimal. meauring intrument having a pointer of a moving-coil ytem i thu deigned with aperiodic damping (Fig. cae (C) green curve). In thi experiment a rotatable metal wheel with inertia J i ued a an ocillator. helical pring act on the wheel when it diplaced by an angle from it ret poition to produce a retoring torque M which i approximately given by M = D (X) Owing to the unavoidable frictional force (in the ball baring etc.) the amplitude of mechanical ocillation decreae in time. a reult a free damped ocillation i produced. In many (but not in all!) cae, the frictional force (torque) are proportional to the (angular) velocity in the firt order of approximation: M F d = k (XI) dt On the torion pendulum the damping according equation (XI) i realized by paing the metal wheel through the field of an electromagnet. The electron experience the Lorentz force. Thu the electron are diplayed perpendicular to the field of the electromagnet and the direction of the moving wheel. They flow back through the field free part of the wheel (Fig. 2). a reult a cloed eddy-current I eddy circuit i produced. The part of the metal wheel in the magnetic field act like a moving current carrying conductor on which a force F oppoed to the direction of motion and proportional to the velocity v act. Thi generate a deaccelerating torque M F. q: damping ratio The exponent i called the logarithm decrement B Ieddy Λ = δ T = ln n = n + lnq (II) Fig. 2: Generation of eddy current I eddy. LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de

3 Ux m OFF OFF m 00m 00µ m 00µ 00µ m 0m 00m 00m 00µ m 0m 00m 00m 0m 5 20 C DC 5 20 C DC LD Phyic leaflet - - P.5.. Setup Set up the experiment a hown in Fig.. The time i meaured by the top clock (not hown in Fig. ). Set the pointer of the metal wheel (a) to the zero poition of the cale by turning the drive wheel (e). () DC NETZGERÄT 0 6 / 0 5 DC POWER SUPPLY 0 6 / 0 5 FINE Safety note g The current through the eddy current brake hould not exceed 2 for a long time. POWER Carrying out the experiment a) Invetigating the damping of the ocillation - Set the current for the electromagnet to a mall value, e.g. I = Move the pointer of the pendulum to the limit poition and read off the amplitude on the ame ide of the cale after each ocillation T (for the cae of weak damping after 5 or 0 ocillation). - dditionally, meaure everal time the time for 0 ocillation to determine the ocillation period T. Hint: If the pendulum achieve an equilibrium in le then 0 ocillation meaure the time everal time to obtain the mean value. - Repeat the experiment in the ame way for a larger current (i.e. I = 0.4 ). a (B) 00m 0m m e b) Invetigating the tranition from ocillation to the limit cae DC NETZGERÄT 0 6 / 0 5 DC POWER SUPPLY 0 6 / Increae the current until the pendulum perform an ocillation depicted by the blue curve in Fig. (B). - Move the pointer of the pendulum to the limit poition and meaure the time taken for an ocillation until the equilibrium poition i reached. Determine the ocillation period a mean value from everal meaurement. - Increae the current until the pendulum perform an ocillation depicted by the green curve in Fig. (C). - Meaure the time taken by the pendulum when releaed from the limit poition. Determine mean value from e.g. 5 meaurement. FINE POWER 00m Fig. : Experimental etup (wiring diagram chematically) for oberving damped rotational ocillation. LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de

4 P LD Phyic leaflet Meauring example a) Invetigating the damping of the ocillation Note: The experimental data may differ from pendulum to pendulum due to inevitable tolerance between the eddy current brake and the tiny difference in the mechanical et up. Table. : Meaured ocillation amplitude a function of time n T (n-time the ocillation period) for I = 0.8 and I = t Scd t Scd Table. 2: Ocillation period (mean value determined by 5 meaurement) for different eddy current. Eddy current I Ocillation period b) Invetigating the tranition from ocillation to the limit cae I =.. Meaured ocillation period: 2.4. I =.5 Meaured ocillation period:.9 Evaluation and reult a) Invetigating the damping of the ocillation Fig. 4 ummarize the reult of Table. The damping contant δ can determined for intance by fitting equation () to the experimental data. lternatively, the fit of a traight line to data plotted in Fig. 5 give the damping contant δ from which the logarithmic decrement Λ can be determined (Table.). Table. : Ocillation period T (from Table 2). damping contant δ (determined by a fit to the experimental data plotted in Fig. 5) and logarithmic decrement Λ for variou eddy current I. mplitude / Scd I T δ ,0 7,5 5,0 2,5 0,0 7,5 5,0 2,5 I = , time / T Λ I = 0.8 Fig. 4: mplitude a function of time. The olid line correpond to a fit according equation (). LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de

5 LD Phyic leaflet P.5.. mplitude / Scd 0 I = 0.25 I = time / Fig. 5: mplitude a function of time. The olid line correpond to a fit of a traight line yielding the damping contant δ lited in Table. b) Invetigating the tranition from ocillation to the limit cae The pendulum reache the equilibrium after one ocillation for I =.. The meaured ocillation period i 2.4. For I =.5 the pendulum reache the equilibrium in.9 without ocillating over the zero poition. In thi o-called aperiodic cae the adjutment time required by the ytem to return to the equilibrium i a minimum. Supplementary information The ocillation with a retoring torque decribed by equation (X) are called harmonic ocillation. The harmonic ocillator i only a pecial cae among ytem which are capable of ocillation. Mot of the real ocillation are not harmonic, i.e. relation (X) i not trictly atified. However, many ocillation can be conidered a harmonic ocillation at leat in the firt approximation by developing the retoring torque (force) a function about the ret poition in erie and neglecting nonlinear term. The equation of motion (I) of uch on ocillating ytem can generally not be olved analytically. The anharmonic ocillator i invetigated in experiment P LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de

6 P LD Phyic leaflet LD Didactic GmbH. Leyboldtrae. D-5054 Huerth / Germany. Phone: (022) Fax: (022) info@ld-didactic.de