Problems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)

Transcription

1 Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems of the nd and the 9 th ntenational Physics Olympiad oganized in Budapest Hungay 968 and 976 and thei solutions ae pesented ntoduction ollowing the initiative of D Waldema Gozkowski [] pesent the poblems and solutions of the nd and the 9 th ntenational Physics Olympiad oganized by Hungay have used Pof Rezső Kunfalvi s poblem collection [] its Hungaian vesion [3] and in the case of the 9 th Olympiad the oiginal Hungaian poblem sheet given to the students (my own copy) Besides the digitalization of the text the equations and the figues it has been made only small coections whee it was needed (type mistakes small gammatical changes) omitted old units whee both old and S units wee given and conveted them into S units whee it was necessay f we compae the poblem sheets of the ealy Olympiads with the last ones we can ealize at once the diffeence in length t is not so easy to judge the difficulty of the poblems but the solutions ae suely much shote The poblems of the nd Olympiad followed the moe than hunded yeas tadition of physics competitions in Hungay The tasks of the most impotant Hungaian theoetical physics competition (Eötvös Competition) fo example ae always vey shot Sometimes the solution is only a few lines too but to find the idea fo this solution is athe difficult Of the 9 th Olympiad have pesonal memoies; was the youngest membe of the Hungaian team The poblems of this Olympiad wee collected and patly invented by Miklós Vemes a legenday and famous Hungaian seconday school physics teache n the fist poblem only the detailed investigation of the stability was unusual in the second poblem one could foget to subtact the wok of the atmospheic pessue but the fully open thid poblem was eally unexpected fo us The expeimental poblem was difficult in the same way: in contast to the Olympiads of today we got no instuctions how to measue (n the last yeas the only similaly open expeimental poblem was the investigation of The magnetic puck in Leiceste 000 a eally nice poblem by Cyil senbeg) The challenge was not to pefom many-many measuements in a shot time but to find out what to measue and how to do it Of couse the evaluating of such open poblems is vey difficult especially fo seveal hunded students But in the 9 th Olympiad fo example only ten counties paticipated and the same peson could ead compae gade and mak all of the solutions

2 nd PhO (Budapest 968) Theoetical poblems Poblem On an inclined plane of 30 a block mass m 4 kg is joined by a light cod to a solid cylinde mass m 8 kg adius 5 cm (ig ) ind the acceleation if the bodies ae eleased The coefficient of fiction between the block and the inclined plane µ 0 iction at the beaing and olling fiction ae negligible m m m gsin µ m g m gsin S igue igue f the cod is stessed the cylinde and the block ae moving with the same acceleation a Let be the tension in the cod S the fictional foce between the cylinde and the inclined plane (ig ) The angula acceleation of the cylinde is a/ The net foce causing the acceleation of the block: m a mg sin µ mg and the net foce causing the acceleation of the cylinde: m a m g sin S The equation of motion fo the otation of the cylinde: S a ( is the moment of inetia of the cylinde S is the toque of the fictional foce) Solving the system of equations we get: a g S ( m m ) g sin µ m cos () m m m m sin µ m () m m ( )

3 sin µ m cos mg (3) m m The moment of inetia of a solid cylinde is m Using the given numeical values: ( m m ) sin µ m cos a g 0 337g 35 m s 5m m mg ( m m ) sin µ m S 30 N 5m m ( 5µ 05sin ) m mg 09 N 5m m Discussion (See ig 3) The condition fo the systeo stat moving is a > 0 nseting a 0 into () we obtain the limit fo angle : m tan µ m m µ o the cylinde sepaately 0 and fo the block sepaately tan µ 3 f the cod is not stetched the bodies move sepaately We obtain the limit by inseting 0 into (3): m tan µ 3 µ The condition fo the cylinde to slip is that the value of S (calculated fom () taking the same coefficient of fiction) exceeds the value of µ m g This gives the same value fo 3 as we had fo The acceleation of the centes of the cylinde and the block is the same: g ( sin µ ) the fictional foce at the bottom of the cylinde is µ m g the peipheal acceleation of the cylinde is m µ g Poblem Thee ae 300 cm 3 toluene of 0 C tempeatue in a glass and 0 cm 3 toluene of 00 C tempeatue in anothe glass (The sum of the volumes is 40 cm 3 ) ind the final volume afte the two liquids ae mixed The coefficient of volume expansion of toluene β 000 C Neglect the loss of heat ( ) β a g 3 S a β S (N) igue

4 f the volume at tempeatue t is V then the volume at tempeatue 0 C is V0 V ( β t ) n the same way if the volume at t tempeatue is V at 0 C we have V0 V ( β t ) uthemoe if the density of the liquid at 0 C is d then the masses ae m V0d and m V d 0 espectively Afte mixing the liquids the tempeatue is t m m The volumes at this tempeatue ae V0 ( β t) and V ( β t) The sum of the volumes afte mixing: V 0 V V V 0 ( β t) V ( β t) V V β ( V V ) V V m m β d β d d ( β t ) V0 ( β t ) V V 0 V 0 mt m m 0 βv t V 0 t βv The sum of the volumes is constant n ou case it is 40 cm 3 The esult is valid fo any numbe of quantities of toluene as the mixing can be done successively adding always one moe glass of liquid to the mixtue Poblem 3 Paallel light ays ae falling on the plane suface of a semi-cylinde made of glass at an angle of 45 in such a plane which is pependicula to the axis of the semi-cylinde (ig 4) (ndex of efaction is ) Whee ae the ays emeging out of the cylindical suface? t 0 A O ϕ β D E B C igue 4 igue 5 Let us use angle ϕ to descibe the position of the ays in the glass (ig 5) Accoding to the law of efaction sin 45 sin β sin β 0 5 β 30 The efacted angle is 30 fo all of the incoming ays We have to investigate what happens if ϕ changes fom 0 to 80 4

5 t is easy to see that ϕ can not be less than 60 ( AOB 60 ) The citical angle is given by sin β n ; hence β 45 n the case of total intenal eflection cit cit ACO 45 hence ϕ f ϕ is moe than 75 the ays can emege the cylinde nceasing the angle we each the citical angle again if OED 45 Thus the ays ae leaving the glass cylinde if: 75 < ϕ < 65 CE ac of the emeging ays subtends a cental angle of 90 Expeimental poblem Thee closed boxes (black boxes) with two plug sockets on each ae pesent fo investigation The paticipants have to find out without opening the boxes what kind of elements ae in them and measue thei chaacteistic popeties AC and DC metes (thei intenal esistance and accuacy ae given) and AC (5O Hz) and DC souces ae put at the paticipants disposal No voltage is obseved at any of the plug sockets theefoe none of the boxes contains a souce Measuing the esistances using fist AC then DC one of the boxes gives the same esult Conclusion: the box contains a simple esisto ts esistance is detemined by measuement One of the boxes has a vey geat esistance fo DC but conducts AC well t contains a capacito the value can be computed as C ω X C The thid box conducts both AC and DC its esistance fo AC is geate t contains a esisto and an inducto connected in seies The values of the esistance and the inductance can be computed fohe measuements 5