A we connect it in series with a capacitor of capacitance C 160 F. C The circuit thus carries an alternating sinusoidal current i.

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Marlene Fitzgerald
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1 I-(7 points) Deterination of a characteristic of a coil In order to deterine the resistance r of a coil of inductance 0 03 H, A we connect it in series with a capacitor of capacitance C 160F across the terinals of a low frequency enerator (FG) deliverin an alternatin sinusoidal voltae Take: 03 1 C The circuit thus carries an alternatin sinusoidal current i B An oscilloscope is connected so as to display the voltae u on the G channely 1, and the voltae across the coil u on the channely We see on the screen of the oscilloscope a display of the wavefors represented in fiure et S v 5V / div be the vertical sensitivity on channel Y 1 1 D and Sv be the vertical sensitivity on channely Fiure 1 The voltae across the coil is iven by: u 10sin100 t (u in V & t in s ) 1Redraw fiure (1) showin on it the connections of the oscilloscope a)find the vertical sensitivity S u v on channely b)find the frequency of sinal delivered by the FG u 3a)Calculate the phase difference between ug and u Which of the leads the other b)deduce the expression of the voltae u A D across the terinals of the enerator as a function of tie 4By supposin that uc u AB U sin 100 t Applyin the law of addition of voltaes, and ivin the tie two particular values Find the values of U and 5a)Deduce that the instantaneous expression of the Fiure current i as a function of tie is iven by: i 05sin100 t 6 b)usin two expressions of the averae power consued across the coil, the expression justified in u, for w 100 rad / s part 5 and that of Find the value of r 6We keep the axiu value of u constant but we vary its frequency f ; the effective value of the current in the circuit is axiu for certain value f 0 a)what is the nae of the physical phenoenon thus observed? b)find the value of f 0 c)the circuit is now equipped by a fuse that cannot withstand a current whose effective intensity exceed 400 A Show that it is not possible to visualize the precedin physical phenoenon

2 II-(6 points) Photons and vision Accordin to Planck-Einstein s Hypothesis: «An electroanetic radiation of frequency, is coposed of enery particles called photons» Given: 9 1n eV J The speed of liht in vacuu 8 c 310 / s 1a)What aspect of liht, the previous stateent, does it show evidence of? b)state two physical properties of the photon c)what is eant by duality of liht? Give a physical phenoenon that is interpreted basin on each aspect To study the relationship between the enery E of a photon and its frequency, we easure the photon s enery of any electroanetic radiations that are placed in the followin table Hz E ev a)plot the raph representin the variations of the enery of a photon as a function of the frequency, by takin as a scale: 14 On the abscissas axis: 1c 110 Hz On the ordinate axis : 1c 0 5 b)justify, basin on the raph plotted and by the eans of two evidences, that the enery of a photon is proportional to the frequency c)deterine, in SI units, the value of this constant of proportionality, called Planck s constant h d)knowin that the spectru of visible liht in vacuu extends in the interval : 400n 750n i)find the rane of frequencies of visible liht ii)indicate to which doain the radiations entioned in the previous table belon, the visible, ultraviolet or infrared spectru? ev In what follows we consider that the Planck s constant h is equal to: 34 h J 3To start up a visual excitation, 100 photons at least ust reach the retina whose area is durin a duration of 1s 0 The retina s surface holds 0 rods a)what is the nuber of photons needed to provoke an excitation of a rod durin this duration b)i)write the expression of a photon s enery in ters of h, c& 0 ii)what is the power absorbed by a rod receivin a radiation whose wavelenth is n c)what ust be the iniu power of a luinous source eittin this sae radiation uniforly in all directions of the space in order to be visible fro10 k

3 III- (7 points) Horizontal elastic pendulu and Earthquakes Part A Theoretical study The horizontal elastic pendulu of the fiure below is fored of a solid sprin of constant k 10N / The center of ass G of S of ass 100 and a S ay ove alon a horizontal axis O, i as shown in Fiure1 The abscissa of G at any instant t durin oscillations is x and its velocity is v v i The horizontal plane containin G is taken as the ravitational potential enery reference The pendulu thus perfors free undaped oscillations of aplitude x 4c 1 Write the expression of the echanical enery ME of the pendulu [ S, sprin] as a function of, k, x and v Derive the differential equation that describes the otion of the center of ass G 3 a) Deterine the expression oft 0 so that x x cos t is the solution of this differential equation T0 b) Deduce the value of the proper period T 0 4 Applyin the principle of conservation of echanical enery, find the speed of the box when it passes throuh the equilibriu position Part B Functionin of a seisoeter A seisoeter is a device used to detect and record any disturbance of Earth It functions on the principle of a daped oscillator solid-sprin connected to a buildin fixed to round The seisoeters are sensitive to vertical and horizontal vibrations We are interested to the functionin of a horizontal seisoeter In fact a seisic sinal can hold waves, that are produced at a point called epicenter, of which we study in particular two types P and S, havin periods that extends fro few hundredths of second to any inutes and the recordin obtained is shown in Fiure The seisoeter enters in resonance for a particular frequency It acts equally as a dapin syste, necessarily to obtain a reliable restitution of the round otion Fiure

4 1 Indicate in this syste: the exciter? The resonator? By adittin that the dapin is weak; for what period of the exciter, the resonance phenoenon takes place? Note: without the phenoenon of resonance, the relative otion is very weak and the recordin is not reliable 3 To siplify the study we consider that the waves P and S are periodic as indicated in fiure a) Which wave propaates faster P or S basin on the previous recordin? Justify b) Observin the recordin, specify the rane of anitude of the periodst, shown in the previous recordin, of waves P and S is10 s, 1s or 01s fro fiure above c) Calculate an approxiate value of the solid ass, durin this Earthquake 1 The constant of sprin k 100N 4 Durin the seis, certain waves have frequencies very sall to that of the waves P and S How should we odify the ass in order to obtain a reliable recordin? IV- Variations of the period The objective of this study is to find the paraeters that effect on the duration of oscillation of a echanical oscillator (elastic pendulu) whose oscillation takes an interval of tiet A-Experiental study In order to show the effects of the oscillations aplitude x, the ass of the solid and the constant k of the sprin on the duration of one oscillation of a free undaped horizontal elastic pendulu We perfor any experients, in each one factor is only odified x in the first, k in the second and k in the third then we easure each tie, the duration t for 10 oscillations usin a stopwatch 1Why we easure the duration of 10 oscillations instead of 1 only directly? The curves below are obtained: What is, with justification, the conclusion that can be drawn concernin the dependence of the proper period T on: Fiure 1 Fiure Fiure 3 a)the aplitude x fro fiure 1? b)the ass fro fiure? Deduce the expression of T in ters of c)the constant of elasticity k fro fiure 3? 3By supposin that the proper period can be written in the fort A x k where A,, & constants a)what ust be the value of the constant so that the expression of the period T is independent of the aplitude x? are

5 b)verify that the unit of the constant of elasticity k in SI units is k s c)basin on diensional study ( units study) applied on the expression of the proper period 1 Show that the value of Deduce the value of? d)knowin that the values of fiure are obtained when the constant of the sprin k 10N / Deduce the value of the constant A B-Theoretical study Consider a horizontal elastic pendulu fored of a solid S of ass attached to a sprin of constant k and of neliible ass; the other end of the sprin is fixed to a support The forces of friction are supposed to be neliible and the solid of center of ass G can ove on a horizontal axis Ox When the solid is at rest, G coincides with the pointo taken as oriin of abscissa x' Fi1 The solid is pulled fro its equilibriu position by a distance x, and then released without initial velocity at the instant t 0 0 The horizontal plane passin throuh G is taken as a ravitational potential enery reference dx At any instant t, the abscissa of G is x and the alebraic easure of its velocity is v dt 1Write down the expressions of the echanical enery of the syste (Solid, sprin, Earth) as a function of, k, x and v a)derive the second order differential equation that overns the otion of S b)knowin that x x cos t is a solution of this differential equation, deterine the expression of T the natural period T as a function of and k c)copare this result to that obtained in A3-d O G x

Dynamics - Midterm Exam Type 1

Dynamics - Midterm Exam Type 1 Dynaics - Midter Exa 06.11.2017- Type 1 1. Two particles of ass and 2 slide on two vertical sooth uides. They are connected to each other and to the ceilin by three sprins of equal stiffness and of zero