This exam is formed of three exercises The use of a non-programmable calculator is recommended

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1 وزارة التربیة والتعلیم العالي المدیریة العامة للتربیة داي رة الامتحانات امتحانات الشھادة الثانویة العامة فرع علوم الحیاة مسابقة في مادة الفیزیاء المدة: ساعتان الاسم: الرقم: دورة سنة ۲۰۰٦ العادیة This exam is formed of three exercises The use of a non-programmable calculator is recommended First exercise ( 6 ½ pts) Verification of Newton s second law puck (S) of mass M = 00 g and of center of mass G, may slide along an inclined track that makes an angle α with the horizontal so that sinα = Thus G moves along an axis x'x parallel to the track as shown in figure (). Take g = 0 m/s. P.g = 0 x G 0 i G (S) Figure G 4 α x We release (S) without initial velocity at the instant t o = 0 and at the end of each interval of time = 50 ms, some positions G o, G, G, G 5 of G are recorded at the instants t o = 0,t, t, t 5 respectively. The values of the abscissa x of G ( x = G o G ) are given in the table below. t x (cm) 0 G o G = 0.50 G o G =.00 G o G 3 = 4.50 G o G 4 = 8.00 G o G 5 =.50 ) Verify that the speed of the puck at the instants t = and t 4 = 4 are V = 0.40 m/s and V 4 = 0.80 m/s respectively. ) a) Calculate the mechanical energy of the system (puck-arth ) at the instants t 0, t and t 4 knowing that the horizontal plane through G 0 is taken as a gravitational potential energy reference. b) Why can we suppose that the puck moves without friction along the rail? 3) Determine the variation in the linear momentum P = P4 P of (S) during t = t 4 - t. 4) a) Name the forces acting on (S) during its motion. b) Show that the resultant Σ F of these forces may be written as Σ F = (Mg sinα) i. ΔP 5) ssuming that t is very small, Δt d P may be considered equal to dt second law is verified between the instants t and t 4.. Show that Newton s

2 Second exercise (6 ½ pts) Measurement of the speed of a bullet In order to measure the speed of a bullet, a convenient setup is used. The principle of functioning of this setup is based on the charging of a capacitor. - Study of the charging of a capacitor We are going to study the charging of a capacitor using a series circuit formed of a resistor of resistance R, a switch K and a capacitor of capacitance C initially neutral across the terminals of a generator of constant emf and of negligible internal resistance (figure ). K D R The switch K is closed at the instant t 0 = 0. The capacitor starts to charge. t the instant t, the circuit carries a current i and the armature of the capacitor carries the charge q. q C ) pplying the law of addition of voltages, determine the differential equation that describes the variation of the voltage u C = u across the capacitor as a function of time. e K K D D Figure ) a) Verify that u C = ( - ) is the solution of the differential equation where = RC. b) What does the time interval represent? 3) fter what time would the steady state be practically attained? Measurement of the speed of a bullet The setup used to measure the speed V of a bullet is represented in figure. ' and ' are two thin parallel connecting wires lying in a vertical plane and are of negligible resistance. and are separated by a distance L. Given: = 00 V ; R = 000 Ω ; C = 4 μf ; L = m. The capacitor being neutral, the switch K is closed. ) a) The potential difference between and is zero. Why? b) The charging of the capacitor did not start. Why? ) K being closed, we shoot the bullet normally at ' and ' with a speed V (fig.3). t the instant t 0 = 0, the bullet cuts the wire ' and the capacitor starts to charge (fig.4). The bullet continues its motion which is considered uniform rectilinear of the same speed V. t the instant t, the bullet cuts the wire ' and the phenomenon of charging stops. The voltage across the capacitor is then 45.7 V. a) Taking into consideration the study in part, determine the time interval t taken by the bullet to cover the distance L. b) Calculate V. R Figure R V Figure 3 K D R Figure 4 3) In order to measure precisely the value of V, the distance L between ' and ' must not exceed a maximum value L max. Determine the value of L max. ' ' ' C C q C V ' ' '

3 Third exercise ( 7 pts ) Measurement of the age of arth One of the questions that preoccupied man long ago since he started to explore the universe was the age of arth. s from 5, Rutherford proposed a measurement of the age of minerals through radioactivity. In 956, Clair Paterson used the method (uranium - lead) to measure the age of a meteorite assuming that it originates from a planet that is formed approximately at the same time as that of arth. I radioactive family of uranium 38 9 U Uranium 38, of radioactive period T= y (year) is at the origin of a radioactive family 06 leading finally to the stable lead isotope 8 Pb. ach of these successive disintegrations is accompanied with the emission of an α particle or a β - particle. 38 The diagram (Z, N) gives all the radioactive nuclei originating from the uranium 9 U leading 06 to the stable isotope 8 Pb. (page 4) The tables (page 4 ) give the radioactive period of each nuclide ) In the first disintegration, a uranium nucleus 9 U gives thorium nucleus Th and a particle denoted by Z X. a) Write the equation of this disintegration and calculate the values of and Z. b) Specify the type of radioactivity corresponding to this transformation. 34 ) In the second disintegration, the thorium nucleus Th undergoes a β - decay. The daughter nucleus is the protactinium Z Pa. Calculate and Z. 3) a) Referring to the diagram, tell how many α particles and how many β - particles are emitted when the uranium nucleus 9 U is transformed into lead nucleus 8 Pb. b) Write the overall nuclear equation of the decay of uranium 38 into lead 06. 4) Using the diagram (Z,N) of the figure and the tables, tell why after few billions of years we can neglect the presence of the intermediary nuclei among the products of the disintegration uranium-lead. Il- The age of arth We have studied a sample of a meteorite whose age is equal to that of arth. t the instant t, the sample studied contains g of uranium 38 and 0.88g of lead 06. We suppose that at the instant of its formation t 0 = 0, the meteorite does not contain any atom of lead. Numerical data: molar mass of uranium: 38 g/mol; molar mass of lead: 06 g/mol; vogadro s number: N = 6.0x 0 3 /mol. ) Calculate, at the instant t : a) the number of uranium 38 nuclei, denoted by N U (t), present now in the sample; b) the number of lead 06 nuclei, denoted by N Pb (t), present now in the sample. ) Deduce the number of uranium 38 nuclei N U (0), present in the sample at the instant t 0 = 0. 3) Give the expression of N U (t) as a function of N U (0), t, and T. 4) Deduce the age of arth at the instant t. 3

4 46 N = Z 38 9 U β - α Pb Z Pb i Po Rn Ra Th Pa U Nucleus 38 9 U 34 Th 34 9 Pa 34 9 U 30 Th 6 88 Ra 86 Rn Radioactive period y 4 d 6.7 h y y y 3.8 d Nucleus 8 Po 4 Pb 4 i 4 Po 0 Pb 0 i 0 Po Radioactive period 3. min 7 min 0 min s y 5d 38 d 4

5 أسس التصحیح لشھادة علوم الحیاة لمادة الفیزیاء. دورة ۲۰۰٦ العادیة First exercise: (6 ½ pts) G G ) V = 3 G G G G = ( ) 0 = = 0.4 m/s. ( 0. G V 4 = 3 G 5 G0G5 G0G3 (.5 4.5) 0 = = = 0.8 m/s. ( 0. ) a) M. = K. + P. g ; M. 0 = K. 0 + P g0 = = 0 M. = K. + P g = M(V ) - Mgh ; h = G 0 G sinα= 0.4 = 0.8 cm = m M. = 0 J. M. 4 = K. 4 + P g4 = M(V 4 ) Mgh 4 ; h 4 = G 0 G 4 sinα= = 3. cm = 0.03 m M. 4 = 0 J. ( pts) b) M. 0 = M. = M. 4 the mechanical energy is conserved during motion No friction. ( 3) Δ P = P4 P = M (V 4 i -V i ) = 0.04i ( 3 4 4) a) The forces acting on (S) : The weight W of (S) and the normal reaction N of the path b) Σ F = W + N = W + W + N where : W = Mgsinαi, W = -Mgcosα j, N = N j ; W + N = 0 (no motion along y'y) Σ F = W = mgsinαi ( d P Δ P 5) The nd Law of is given by : Σ F = =. dt Δt We have: Δ P 0.04 Σ F = mgsinαi = 0.4i and = = 0.4 i Δt 0.i ( 4 Second exercise (6 ½ pts) dq du - ) = u R + u C = Ri + u C = R + uc = RC C + u C ( dt dt ) a) du C = dt RC e = RC RC e b) is the time taken for u c = 63%. ( 3) The steady state is reached for t = 5RC. ( + ( - e ) = ( - ) a) because ' is a connecting wire of negligible resistance. ( 4 b) u C = u ' = 0 ( 4 ) a) We have u C = ( - e ). We can write : - e u = C e t = -RC ln ( - u = - C t u = ln ( - C ) RC u C 45.7 ) = ln ( - ) =.44 ms. 00 t is the time of charging which is the same as the time taken by the bullet ( pt ) b) V = L = = 40 m /s. ( t 3) ecause if this time exceeds 5RC, the steady state is attained and will no more vary. Thus t 5RC V L 5RC L 5RCV L 8.0 m L max = 8.0 m. ( The nd Law of Newton is thus verified. (

6 Third exercice : (7 pts) I-) a) 9 U Th + X Z 38 = 34 + = 4. 9= +Z Z = ( b) nucleus X is Helium The type of the radioactivity is α. ( Z 34 ) Th 0 e + Pa Z 34 = 0 + = 34 = - +Z Z = 9 ( 3) a) There are 8 particles of α and 6 particles of β b) 9 U 8 Pb + 8 He e ( ( ) The radioactive period of each nucleus of the family is too small as compared to that of the nucleus of 9 U. ( 4 II - ) a) Nu(t) = N 38 = nuclei ( b) N Pb (t) = N = nuclei ( ) N U (0) = N U (t) + N Pb (t) = nuclei ( t T 3) N U (t) = N U (0) e ( T NU (0) 4) t = ln NU ( t) The age of the arth is = years. 9 0 years ( 3 4

This exam is formed of three exercises in three pages numbered from 1 to 3. The use of a non-programmable calculator is recommended.

This exam is formed of three exercises in three pages numbered from 1 to 3. The use of a non-programmable calculator is recommended. 0 وزارة التربية والتعليم العالي المديرية العامة للتربية دائرة االمتحانات امتحانات الشهادة الثانوية العامة الفرع : علوم الحياة الدورة العادية للعام مسابقة في مادة الفيزياء المدة ساعتان االسم: الرقم: This