; 2) diffraction should not be taken into account.

Transcription

1 Problem of the IV International Olympiad, Mocow, 197 The publication i prepared by Prof. S. Kozel & Prof. V.Orlov (Mocow Intitute of Phyic and Technology) The IV International Olympiad in Phyic for choolchildren took place in Mocow (USSR) in July 197 on the bai of Mocow State Univerity. Team from 8 countrie participated in the competition, namely Bulgaria, Hungary, Poland, Romania, Czecholovakia, the DDR, the SFR Yugolavia, the USSR. The problem for the theoretical competition have been prepared by the group from Mocow Univerity tuff headed by profeor V.Zubov. The problem for the experimental competition ha been worked out by B. Zvorikin from the Academy of Pedagogical Science. It i pity that marking cheme were not preerved. Theoretical Problem Problem 1. A long bar with the ma M 1 kg i placed on a mooth horizontal urface of a table where it can move frictionle. A carriage equipped with a motor can lide along the upper horizontal panel of the bar, the ma of the carriage i m.1 kg. The friction coefficient of the carriage i μ.. The motor i winding a thread around a haft at a contant peed v.1 m/. The other end of the thread i tied up to a rather ditant tationary upport in one cae (Fig.1, a), wherea in the other cae it i attached to a picket at the edge of the bar (Fig.1, b). While holding the bar fixed one allow the carriage to tart moving at the velocity V then the bar i let looe. Fig. 1 Fig. By the moment the bar i releaed the front edge of the carriage i at the ditance l.5 m from the front edge of the bar. For both cae find the law of movement of both the bar and the carriage and the time during which the carriage will reach the front edge of the bar. 1

2 Problem. A unit cell of a crytal of natrium chloride (common alt- NaCl) i a cube with the edge length a m (Fig.). The black circle in the figure tand for the poition of natrium atom wherea the white one are chlorine atom. The entire crytal of common alt turn out to be a repetition of uch unit cell. The relative atomic ma of natrium i and that of chlorine i 5,5. The denity of the common alt ρ. 1 kg/m. Find the ma of a hydrogen atom. Problem. Inide a thin-walled metal phere with radiu R cm there i a metal ball with the radiu r 1 cm which ha a common centre with the phere. The ball i connected with a very long wire to the Earth via an opening in the phere (Fig. ). A charge Q 1-8 C i placed onto the outide phere. Calculate the potential of thi phere, electrical capacity of the obtained ytem of conducting bodie and draw out an equivalent electric cheme. Fig. Fig. 4 Problem 4. A pherical mirror i intalled into a telecope. It lateral diameter i D,5 m and the radiu of the curvature R m. In the main focu of the mirror there i an emiion receiver in the form of a round dik. The dik i placed perpendicular to the optical axi of the mirror (Fig.7). What hould the radiu r of the receiver be o that it could receive the entire flux of the emiion reflected by the mirror? How would the received flux of the emiion decreae if the detector dimenion decreaed by 8 time? Direction: 1) When calculating mall value α (α<<1) one may perform a ubtitution α 1 α 1 ; ) diffraction hould not be taken into account.

3 Experimental Problem Determine the focal ditance of lene. Lit of intrument: three different lene intalled on pot, a creen bearing an image of a geometric figure, ome vertical wiring alo fixed on the pot and a ruler. Solution of the problem of the IV International Olympiad, Mocow, 197 Theoretical Competition Problem 1. a) By the moment of releaing the bar the carriage ha a velocity v relative to the table and continue to move at the ame velocity. The bar, influenced by the friction force F fr μmg from the carriage, get an acceleration a F fr / M μmg/m ; a. m/, while the velocity of the bar change with time according to the law v b at.. Since the bar can not move fater than the carriage then at a moment of time t t it liding will top, that i v b v. Let u determine thi moment of time: t v v M a µ mg 5 By that moment the diplacement of the Sb bar and the carriage Sc relative to the table will be equal to v M Sc vt µ mg, S at v M. b µ mg The diplacement of the carriage relative to the bar i equal to S S v M µ mg c Sb.5m Since S<l, the carriage will not reach the edge of the bar until the bar i topped by an immovable upport. The ditance to the upport i not indicated in the problem condition o we can not calculate thi time. Thu, the carriage i moving evenly at the velocity v.1 m/, wherea the bar i moving for the firt 5 ec uniformly accelerated with an acceleration a. m/ and then the bar i moving with contant velocity together with the carriage. b) Since there i no friction between the bar and the table urface the ytem of the bodie bar-carriage i a cloed one. For thi ytem one can apply the law of conervation of momentum: mv + Mu mv (1)

4 where v and u are projection of velocitie of the carriage and the bar relative to the table onto the horizontal axi directed along the vector of the velocity v. The velocity of the thread winding v i equal to the velocity of the carriage relative to the bar (v-u), that i v v u () Solving the ytem of equation (1) and () we obtain: u, v v. Thu, being releaed the bar remain fixed relative to the table, wherea the carriage will be moving with the ame velocity v and will reach the edge of the bar within the time t equal to t l/v 5. Problem. Let calculate the quantitie of natrium atom (n 1 ) and chlorine atom (n ) embedded in a ingle NaCl unit crytal cell (Fig.). One atom of natrium occupie the middle of the cell and it entirely belong to the cell. 1 atom of natrium hold the edge of a large cube and they belong to three more cell o a 1/4 part of each belong to the firt cell. Thu we have n /4 4 atom of natrium per unit cell. In one cell there are 6 atom of chlorine placed on the ide of the cube and 8 placed in the vertice. Each atom from a ide belong to another cell and the atom in the vertex - to even other. Then for one cell we have n 6 1/ + 8 1/8 4 atom of chlorine. Thu 4 atom of natriun and 4 atom of chlorine belong to one unit cell of NaCl crytal. The ma m of uch a cell i equal m 4(m rna + m rcl ) (amu), where m rna and m rcl are relative atomic mae of natrium and clorine. Since the ma of hydrogen atom m H i approximately equal to one atomic ma unit: m H 1.8 amu 1 amu then the ma of an unit cell of NaCl i m 4(m rna + m rcl ) m H. On the other hand, it i equal m ρa, hence ρa 7 mh kg. 4 m + m ( ) Problem. Having no charge on the ball the phere ha the potential rna rcl 1 Q ϕ 45V. R 4

5 When connected with the Earth the ball inide the phere ha the potential equal to zero o there i an electric field between the ball and the phere. Thi field move a certain charge q from the Earth to the ball. Charge Q`, uniformly ditributed on the phere, doen t create any field inide thu the electric field inide the phere i defined by the ball charge q. The potential difference between the ball and the phere i equal 1 q q ϕ ϕ b ϕ, (1) r R Outide the phere the field i the ame a in the cae when all the charge were placed in it center. When the ball wa connected with the Earth the potential of the phere φ i equal Then the potential of the ball 1 q + Q ϕ. () 4 πε R 1 q + Q q q 1 Q q ϕ b ϕ + ϕ + + () R r R R r Which lead to r q Q. (4) R Subtituting (4) into () we obtain for potential of the phere to be found: 1 ϕ Q Q r R 1 Q ( R r) R R The electric capacity of whole ytem of conductor i Q C ϕ πε R r 4 R V. F 44pF The equivalent electric cheme conit of two parallel capacitor: 1) a pherical one with charge +q and q at the plate and ) a capacitor phere Earth with charge +(Q-q) and (Q q) at the plate (Fig.5). Fig. 5 Fig. 6 5

6 Problem 4. A known, ray parallel to the main optical axi of a pherical mirror, paing at little ditance from it after having been reflected, join at the main focu of the mirror F which i at the ditance R/ from the centre O of the pherical urface. Let u conider now the movement of the ray reflected near the edge of the pherical mirror of large diameter D (Fig. 6). The angle of incidence α of the ray onto the urface i equal to the angle of reflection. That i why the angle OAB within the triangle, formed by the radiu OA of the phere, traced to the incidence point of the ray by the reflected ray AB and an intercept BO of the main optical axi, i equal to α. The angle BOA and MAO are equal, that i the angle BOA i equal to α. Thu, the triangle AOB i iocele with it ide AB being equal to the ide BO. Since the um of the length of it two other ide exceed the length of it third ide, AB+BO>OAR, hence BO>R/. Thi mean that a ray parallel to the main optical axi of the pherical mirror and paing not too cloe to it, after having been reflected, croe the main optical axi at the point B lying between the focu F and the mirror. The focal urface i croed by thi ray at the point C which i at a certain ditance CF r from the main focu. Thu, when reflecting a parallel beam of ray by a pherical mirror finite in ize it doe not join at the focu of the mirror but form a beam with radiu r on the focal plane. From Δ BFC we can write : r BF tg β BF tg α, where α i the maximum angle of incidence of the extreme ray onto the mirror, while in α D/R: R R R 1 coα BF BO OF. coα coα 1 coα in α Thu, r R. Let u expre the value of co α, in α, co α via in α taking coα co α into account the mall value of the angle α: Then in coα 1 in α 1, α inα inαcoα, coα co α in α 1 in α. r R in α R D 1 in α 16R in α Subtituting numerical data we will obtain: r m mm.. 6

7 From the expreion D 16R r one can ee that if the radiu of the receiver i decreaed 8 time the tranveral diameter D of the mirror, from which the light come to the receiver, will be decreaed time and thu the effective area of the mirror will be decreaed 4 time. The radiation flux Φ reflected by the mirror and received by the receiver will alo be decreaed twice ince Φ S. Solution of the Experimental Problem While looking at object through lene it i eay to etablih that there were given two converging lene and a diverging one. The peculiarity of the given problem i the abence of a white creen on the lit of the equipment that i ued to oberve real image. The competitor were uppoed to determine the poition of the image by the parallance method oberving the image with their eye. The focal ditance of the converging len may be determined by the following method. Uing a len one can obtain a real image of a geometrical figure hown on the creen. The poition of the real image i regitered by the parallax method: if one place a vertical wire (Fig.7) to the point, in which the image i located, then at mall Fig. 7 diplacement of the eye from the main optical axi of the len the image of thi object and the wire will not diverge. We obtain the value of focal ditance F from the formula of thin len by the meaured ditance d and f : df + ; F1, F d f d + f. 1, In thi method the bet accuracy i achieved in the cae of f d. The competitor were not aked to make a concluion. The error of meauring the focal ditance for each of the two converging lene can be determined by multiple repeated meaurement. The total number of point wa given to thoe competitor who carried out not le fewer than n5 meaurement of the focal ditance and etimated the mean value of the focal ditance Fav: 7

8 F av 1 n Fi n 1 and the abolute error F or root mean quare error Frm 1 n Fi n 1 F, Fi Fi Fav 1 n ( ). Frm F i One could calculate the error by graphic method. Fig. 8 Determination of the focal ditance of the diverging len can be carried out by the method of compenation. With thi goal one ha to obtain a real image S of the object S uing a converging len. The poition of the image can be regitered uing the parallax method. If one place a diverging len between the image and the converging len the image will be diplaced. Let u find a new poition of the image S. Uing the reveribility property of the light ray, one can admit that the light ray leave the point S. Then point S i a virtual image of the point S, wherea the ditance from the optical centre of the concave len to the point S and S are, repectively, the ditance f to the image and d to the object (Fig.8). Uing the formula of a thin len we obtain ; F fd F f d d f <. Here F < i the focal ditance of the diverging len. In thi cae the error of meauring the focal ditance can alo be etimated by the method of repeated meaurement imilar to the cae of the 8

9 converging len. Typical reult are: F (,,4)cm, F (1,,) cm, F ( 8,4,4) cm. 1 ± ± ± Acknowledgement The author would like to thank Profeor Waldemar Gorzkowki (Poland) and Profeor Ivo Volf (Czech Republic) for their providing the material of the IV IPhO in the Polih and Czech language. Reference: 1. O.Kabardin, V.Orlov, International Phyic Olympiad for Pupil, Nauka, Mokva W.Gorzkowki, Zadania z fiziki z calego wiata lat. Miedzyna rodowych Olimpiad Fizycznych, WNT, Warzawa V.Urumov, Proventno Delo, Skopje D.Kluvanec, I.Volf, Mezinarodni Fyikalni Olympiady (metodycky material), MaFy, Hradec Kralowe 199 9