Problem 1: «Three Pulleys».

Transcription

1 Poblem : «Thee Pulleys» A light inextensible ope is theaded though thee identical pulleys Two pulleys ae fixed (cannot otate while the thid can fictionlessly otate about an immobile hoizontal axis The pulleys and the ope lie in the same vetical plane To pevent the ope fom sliding off the pulleys the latte have gooves Coefficient of fiction between a pulley goove and the ln ope is The mass of the otating pulley is M, 8 kg When evaluating its moment of inetia one can neglect the goove and the cental hole and conside the pulley to be a unifom disk The left end of the ope (see the Figue is loaded with a weight m g and the ight end with a weight m m nitially the system is held at est Evaluate the atio of tensions in the sliding ope on both sides of a fixed pulley Expess you answe via the coefficient of fiction and calculate its numeical value ( points Detemine the citical value of m equied fo setting the system in motion afte it has been eleased Expess m in tems of the quantities specified in the poblem and wite down an explicit fomula ( points Evaluate the numeical value of m found above ( poin Detemine an acceleation of m (absolute value afte the system has been eleased if its mass is geate by the facto n than the citical mass found in Expess the answe in tems of the quantities specified in the poblem and wite down an explicit fomula ( points 5 Expess the acceleation found in as a faction of gavitational acceleation g ( poin 6 Detemine an acceleation of m (absolute value afte the system has been eleased if its mass is geate by the facto k than the citical mass found in Expess the answe in tems of the quantities specified in the poblem and wite down an explicit fomula ( points 7 Expess the acceleation found in 6 as a faction of gavitational acceleation g ( poin 8 Unde the condition specified in 6 evaluate a tangential acceleation (absolute value of the otating pulley cicumfeence ight afte the system has been eleased Expess the answe as a faction of gavitational acceleation g ( points Suppose a value of m exceeds the citical mass found in and now mass M of the otating pulley can be vaied Detemine the values of M fo which a type of elative motion of the ope and the otating pulley emain the same fo any m Wite down a fomula and expess it in tems of the quantities specified in the poblem ( points Evaluate the atio of ope tensions on both sides of the otating pulley if the value of m is specified in while the pulley mass is less by the facto k than the value specified in the poblem The coefficient of fiction between the pulley and the ope emains the same Wite down the answe as a simple faction ( points TOTAL scoe of the poblem is points TABLE OF ASWES: Answe Maximum scoe m M m Tension of the «downsteam» ope section is geate by the facto of T T e / m m e m m

2 8 g a ( n m ( n m M e g ( n m ( n m M g 5 g 6 ke a ke k g g k 7 g 8 a g When M ( e e m o M 8m the ope does not slide unde the otating pulley fo any m Tension on the «ight» to the otating pulley is of the tension on the «left» Total Poposed Solution: f the weights ae in motion the ope has to slide ove the fixed pulleys Theefoe fo any ope element of a length dl in a pulley goove the absolute value of fiction foce equals dff d, whee d is a nomal foce exeted by the pulley The fiction foce balances a net tangential foce of ope tension dt The foce pessing the ope element to the pulley is the nomal component of the net tension foce Td d Hee d dl / is the angula size of the ope elemen Thus dt dff d Td and the tension of the ope sliding ove a fixed pulley vaies accoding to an exponential law T ( T e Suppose that the weight m is moving downwad and the ope tensions ae labeled in the Figue Then accoding to the exponential law T T and T T whee e, 5, ow let us wite down equations of motion of the weights Since the ope is inextensible the absolute value of a weight acceleation is the same Let the vetical axis be diected upwad Then one obtains fo tensions T, : ma T m g T ma T mg T T m ( g a T m ( g a m( g a T m( g a a T T T a

3 Concening the otating pulley, the ope does not have to slide unde it and they can otate togethe f this is the case the net tension dt fo any ope element is still balanced by a fiction foce, although now it is static fiction ( df d The net toque exeted on the otating pulley due to f fiction is df dt T T Besides, if thee is no sliding, a tangential acceleation f ( of the pulley cicumfeence equals a linea acceleation a of the weights and ope, so an angula acceleation of the pulley is a / Since the pulley moment of inetia can be appoximated by M, the equation of angula motion becomes a M ( T T, whence T T Ma Substituting hee the tensions found above one obtains the acceleation of weights and ope (poviding the ope does not slide unde the otating pulley: ( m m m m m ( g a m ( g a Ma a g g m m M m m M This equation is coect if two conditions ae met: a (we assume that the system is moving and T (we assume the ope does not slide unde the otating pulley, othewise T dt dff d Td and T T The consideed motion is ealised if m m m m 8 g (othewise fiction foce holds the system at es and if Mm 8Mm m, kg (fo a geate m the ope slides unde the «middle» [ M m ( ] M 8m pulley One can see that if m is a bit moe than 8 g the above conditions ae met and the system stats moving as soon as it has been eleased, 5 Fo m nm, 6 kg both inequalities deived above ae satisfied, so ( n m ( n m g a g g ( n m M ( n m M 6, 7 When m inceases to k m, kg the second inequality no longe holds This means that the ope is now sliding unde the otating pulley and T T The angula acceleation of the pulley is a / Using expessions fo the tensions deived above one obtains: m m k k g m ( g a m ( g a a g g g m m k k 8 Equation of angula motion fo the otating pulley now becomes: ( M ( T T ( T a m( g a M Substituting hee the expessions fo m and the acceleation one obtains: Obviously this acceleation does not exceed T T a k ( m 8km g g ( k M ( k M g a g, as it should be, otice that if M ( m ( e e m 8m, 6 kg, the condition holds tue fo any m, ie thee is no sliding Theefoe acceleation of the «lightweight» pulley can be evaluated accoding to the equation deived in Substituting m 8m and g 8 T M 6m one obtains: a Then T m g and T m g, whence T

4 Poblem : «nstantaneous switching-on» A tansfome consists of a tooidal feomagnetic coe (magnetic pemeability of a length l m and coss-sectional aea S, a pimay winding with the numbe of tuns, and a seconday winding with the numbe of tuns The tansfome is a pat of the cicuit shown in the Figue The battey emf equals V, its intenal esistance is Ohm The seconday winding is connected to a esisto of Ohm nitially thee is no cuent in the windings and the switch K is open Then the switch is apidly closed ts esistance falls to zeo vey quickly (although not instantly! duing a time much less S than, 5 sec Electical esistance of both windings is much less than Ohm l Dissipation of magnetic flux within the coe is negligible Deive an equation elating a magnetic flux in the coe and cuents and in the pimay and seconday windings Conside a positive diection of cuent flow in both windings to be the same ( poin Conside the cicuit including the pimay winding and the intenal esistance of the battey Suppose that duing switch closing the voltage acoss the cicuit vaies as U U ( Wite down a complete set of fist-ode diffeential equations fo functions ( and ( which descibe cuents in the pimay and seconday windings ( points Specify initial values ( and ( (ie initial conditions fo the set of diffeential equations deived above ( poin Detemine a atio of the cuent in seconday winding at t to the cuent at t afte the voltage U( has been switched on Evaluate an appoximate numeical value of the ( atio ( points ( 5 Detemine the maximum value of the cuent in seconday winding afte the switch has been closed Deive an equation fo in tems of the quantities specified in the poblem and max max evaluate an appoximate numeical value of ( points 6 Detemine a time dependence ( at t in tems of the quantities specified in the poblem ( points 7 Plot appoximately the dependence ( fom the moment the switch stats being closed to t T,5 sec ( points 8 What is the maximum cuent in the pimay winding afte the switch has been closed? max Deive the equation fo in tems of the quantities specified in the poblem and evaluate the numeical value ( points Deive a time dependence ( fo t in tems of the quantities specified in the poblem ( points Plot appoximately the dependence ( fom the moment the switch stats being closed to t T,5 sec ( points TOTAL scoe fo the poblem is 5 points TABLE OF ASWES: K

5 Answe Maximum scoe [ ] l S, ( dt d dt d l S dt d dt d l S t U ( (,7 ( ( e 5,6 max k k A Hee 5 k 6 t k k k t ( exp ( max А t k k t ( exp (, A t/τ

6 , A Total 5 t /τ Poposed Solution: Since the coe length l m the system can be egaded as quasistationay, so a magnetic flux confined in the coe and cuents in the windings ae elated as S [ ] l (positive diection in the windings is the same f the voltage acoss the pimay winding cicuit vaies duing the switch closue as U U (, the Kichhoff s loop ule yields (a winding esistance is negligible: t S d d U (, l dt dt S d d l dt dt Since the cuent in the windings cannot change instantaneously, the initial condition at t = (the moment the switch stats being closed must ead ( ( By multiplying the fist equation by and the second one by and then subtacting the U ( equations one obtains at any time (hee k Substituting this expession into k the second equation one obtains the following equation fo the cuent in seconday winding: d k du Theefoe, as soon as the voltage acoss the pimay winding dt k k dt du cicuit eaches the battey emf and emains appoximately constant theeafte (, the dt t dependence ( will be appoximately given by ( C exp t C exp ( k The last equation is evaluated by using numeical values of,, and k One can see that ( e,7 ( 5, 6 Howeve, the above expession does not allow one to detemine the final answe fo ( ight away since the constant C must be specified as the cuent at the end of the voltage «instantaneous switching-on» To do this one has to specify an appopiate function U( such that U( = and U(τ =, whee is a switching-on time of the battey (the time it takes the

7 t switch esistance to dop almost to zeo Fo example conside U ( duing the time inteval fom zeo to Then fo t one obtains: d k d k dt k k dt k (the tem with can be neglected since k and, whee ( Thus in this time k inteval ( t This implies that at the end of «instantaneous switching-on» the k cuent in seconday winding is opposite to the cuent in pimay winding and it eaches its max k maximum equal to, 6 A Subsequently the cuent decays as k k ( exp t k ( k T 7 Since 5 the cuent in seconday winding at T, 5 sec is less than,7% of ( k the maximum value An appoximate plot of ( is shown in the figue, A t/τ 8,, Using the expession obtained above one finds fo the cuent in pimay winding: k t, t ; k ( exp t, t k ( k Thus afte the switching-on stats the cuent in pimay winding apidly (fo inceases to k А and then exponentially appoaches the «stationay» value 6 А by almost k eaching it at the time T An appoximate plot of ( is shown in the figue

8 , A t/τ otice that the maximum cuent in pimay winding afte the switch has been closed (the answe to 8 can be found simply by analysing the «stationay» mode when the cuents ae constant and emf in both windings is zeo emak: Of couse, the linea dependence used to model U ( of the «instantaneous switching-on» is not a must: any easonable model gives the same esult poviding The t t linea model is just the simplest Fo example one can define U( fo t (a smooth cossove to U ( fo t Quite effective solution follows if one chooses U( [ exp( t / ] (this expession can be used fo any t, so the system evolution has not to be split in two stages n this case a bit moe complicated diffeential equation yields the solution: ( k t t exp exp, (* k ( k k t t ( exp exp (** k k ( k One can see that qualitatively time dependence of the cuents at is the same (eg the value of max as in the «linea» model Poblem : «Planet Conus» Let us imagine that some civilisation left an atificial planet in the Sola system The planet obit lies beyond the obit of Pluto, so let us call it «Conus» The Conus obit is vey close to a cicle of adius K = 5 au ( astonomical unit (au is oughly equal to the aveage distance fom the Eath to the Sun, the planet itself is a sphee of adius 5 km made of a solid mateial with a density g/cm and good heat conducting popeties, and an aveage heat capacity c, J/(g K A Conus «yea» lasts «sola days», the planet otates aound its axis in the same diection as its obital otation aound the Sun, and the Conus otation axis is pependicula to its obital plane (otice, that a planet «yea» is a peiod of otation of its cente of mass aound the Sun and «sola day» is the aveage inteval between two «noondays», the moments when the Sun is at maximum elevation on planet sky Conus has no satellites, it has an atmosphee consisting of nitogen, helium, neon, and wate vapou At the beginning of obsevation Conus is a athe hot place This is due to a unifom laye of adioactive mateial unde the planet suface which povides the heat The mateial half-life is 5 Conus «yeas» So, one day a human-made pobe /

9 landed on the Conus suface and measued tempeatue and elative humidity of atmospheic laye at the suface Accoding to the measuements T K and 8 % t tuns out the tempeatue within a dense pat of Conus atmosphee (which contains about % of its mass deceases with altitude as h T ( h T, whee H km This distibution emains 6H almost constant (if distubed the atmosphee etuns to this state in about «yea» The atmosphee is so «pue» that all its wate emains in vapou state and thee ae almost no clouds ecessay constants: The gavitational constant G 6,67 m sec kg Tempeatue of the Sun photosphee T 6 K, sola adius, 65 au, the 8 Stephan-Boltzmann constant 5,67 W/(m K The specific heat of evapoation of wate can be consideed to be almost tempeatue independent and appoximately equal to 66 J/g at T The mola mass of wate 8 g/mole and the univesal gas constant 8, J/(mole K The boiling point of wate unde the nomal atmospheic pessue of, kpa equals 7 K Using the above infomation you should be able to answe the following questions (indicate appopiate units of measuement at all numeical esults Detemine how many Eath yeas does a Conus «yea» have ( poin Evaluate angula velocity of the planet otation aound its axis elative to the fame of efeence of «distant stas» Evaluate a atio of the planet centipetal acceleation to the fee fall acceleation at the planet equato The answe must consist of two numbes calculated with an accuacy of % at least ( points Detemine the powe of sola adiation absobed by Conus by assuming that the planet absobs all incident adiation Expess the answe in tems of the quantities specified in the poblem ( poin Evaluate a numeical atio of the absobed powe to the powe adiated by Conus at the beginning of obsevations ( points 5 Evaluate a time fo which the suface tempeatue of Conus would decease by K if the adioactive «heating» abuptly vanished Wite down the fomula in tems of the quantities specified in the poblem and numeical value in Conus yeas ( points 6 Use the above estimates to develop a mathematical model of the planet suface cooling poviding the adioactive heating opeates all the time and the suface cools natually Detemine an appoximate time dependence of suface tempeatue T Wite down the fomula in tems of the quantities specified in the poblem ( points 7 Evaluate a time fo which the suface tempeatue of Conus would decease by K (with adioactive heating pesen Wite down the fomula in tems of the quantities specified in the poblem Obtain a numeical value expessed in Conus yeas ( points Futhe analysis elies on popeties of wate vapou Conside two isothems of wate vapou on a pv-diagam at tempeatues T and T dt Let the isothems coespond to a slow tansition fom vapou to wate and back (it is well known that vapou pessue does not change along such a tansition By «closing» these isothems nea thei edges with two small adiabatics you should be able to calculate the following quantities: a an efficiency of the obtained cycle; b a wok done duing the cycle; and c a heat taken fom a «heate» by vapou duing the cycle 8 Using a elation between the above quantities (a-c and discading a volume of the liquid phase compaed to a volume of vapou of the same mass detemine a slope of the tempeatue

10 dependence of the satuated wate vapou pessue at a given point of T and p v The answe is dp a fomula fo v as a function of T and p v ( points dt Unde the same assumptions and consideing the heat of evapoation of wate to be almost independent of tempeatue detemine a tempeatue dependence of pessue p of satuated wate vapou nea T Wite down the fomula ( points Using the obtained esults detemine the planet suface tempeatue T at which wate vapou begin to condense Wite down the fomula expessed in tems of the quantities specified in the poblem Evaluate the numeical value A loss of wate «outside» the atmosphee is negligible (5 points Detemine the time afte beginning of obsevation when wate vapou stats to condense Wite down the fomula in tems of the quantities specified in the poblem and evaluate the numeical value expessed in Conus yeas (5 points Evaluate the maximum depth of ocean on the Conus suface built up of wate vapou condensed fom the atmosphee Wite down the fomula in tems of the quantities specified in the poblem, the pessue p of satuated wate vapou at T, and a density of liquid wate v ( T l ( points Detemine (unde the same assumptions a elation between the suface tempeatue and a atio of ocean depth to the maximum possible depth at T T but befoe the complete condensation took place (ie when ocean depth is not close to the maximum value Wite down the equation in tems of the quantities specified in the poblem (6 points Detemine the time when the aveage ocean depth on the Conus suface becomes a quate of the maximum Wite down the fomula in tems of the quantities specified in the poblem and evaluate its numeical value in Conus yeas ( points 5 s it necessay to take into account the heat of vapoisation when evaluating the time of planet cooling at T T if the accuacy of calculation is 5%? And if the accuacy is,5%? Answe to both questions «yes» o «no» ( points TOTAL scoe fo the poblem is 5 points v TABLE OF ASWES: Answe Maximum scoe K ( 5,6,6 Eath yeas* 8 (,7, с, a g (,56,5 P Sun K T 5 Sun T (,, K T P P c T ad (,5 Conus yeas T

11 6 t /(/ ( T T 7 ln T T / (5 Conus yeas 8 dpv dt p v T p ( T p ( T exp T T v v T T (5,, К T ln 5 / T T/ ln ln ln (5 5 ln ln Conus yeas 5 h p v ( T g max l h l exp exp hmax T T T T 6 / T T ln ln ln / (5 5 ln ln Conus yeas 5 o, no Total 5 *An allowed deviation Δ is indicated fo any numeical answe: point is awaded if a poposed numeical value falls within the ange f the deviation exceeds Δ and is less than Δ, the scoe is down to,5 points o point is awaded fo a numeical answe if the deviation is moe than Δ Poposed Solution: Accoding to the thid Keple s law one obtains that Conus «yea» is elated to Eath yea K as 5 / 5, 6, ie K 5, 6 Eath yeas o K,5 sec E The duation of a «sola day» is due both to planet otation aound its axis and to its obital motion aound the Sun That is, the peiod of Conus otation aound its axis with espect to 8 «distant stas» («stella day» equals / «yea», so,7 sec The absolute K

12 GM value of fee fall acceleation nea the Conus suface g G and the absolute value of a centipetal acceleation at the equato a, theefoe,56 This means that g G the planet otation has almost no influence on dynamical pocesses nea the suface, in paticula, one can conside the planet atmosphee to be spheically symmetic emak: the accuacy of numeical values fo and a/g in the table of answes is chosen to be % in ode to distinguish the coect quantities fom incoect but numeically close ones which follow if one does not take into account the diffeence between the peiod of planet otation and the sola day, ie when using (the ensuing discepancy is about 5% K, f the adioactive «heate» is switched off, the only heating the planet eceives is due to themal adiation of the Sun, the heating powe is P Sun T K ; the heat loss is due to themal adiation by the planet to oute space, its powe is P K T (the Sun and Conus ae teated as black bodies, so the Stephan-Boltzmann law applies The atio of these two quantities is, K PSun T PK T 5 Planet tempeatue deceases by a small amount T K duing a time ad detemined fom the equation: T C T T T ad T ad K T K (Hee C is the planet heat capacity The second tem in squae backets was found to be ~ and is negligible compaed to This tem estimates the contibution of sola adiation to the planet themal balance Thus sola adiation has almost no influence on tempeatue dynamics aound T and can be neglected otice that this is anothe agument in favou of the assumption of spheical shape of the atmosphee (themal adiation of the Sun violates this symmety Since C c M c one obtains: c T ad,68 c Conus yeas T This time significantly exceeds the time of elaxation to «quasiequilibium» altitude pofile of the atmosphee and is significantly less that the half-life of adioactive «fuel» of the heating laye 6, 7 The above esults show that the poblem of planet cooling can be consideed to be spheically symmetic and the cooling pocess to be «quasiequilibium» (ie it is easonable to assume that at any time t a heat coming fom the planet heating laye to the suface is appoximately balanced by the heat P( T ( adiated by the suface to oute space On the othe hand P ( / is diectly popotional to the numbe of adioactive decays pe second, so P ( P( / Theefoe T ( T Then the tempeatue dops by T K fo a time detemined t /( T / T fom the equation / / 5 Conus yeas Obviously the T ln T inequality ad holds, which validates the assumption that themal adiation «tunes» the planet tempeatue in accod with the powe of adioactive decay t /

13 emak: f a conteste has not justified spheical symmety of the poblem and quasistationay natue of the pocess (by compaing ad and the maximum scoe cannot be awaded fo items 6 and 7 even if the conteste has obtained the coect dependence T (! 8, The cycle consideed is the Canot cycle with the heate tempeatue T dt and the dt coole tempeatue T The cycle efficiency The wok done pe cycle equals its aea, ie T A dpv ( Vv Vl dpvvv (hee dp v is the change of pessue of satuated vapou when tempeatue inceases fom T to T dt and a volume of liquid wate V l is negligible compaed to a volume of vapou V v The heat taken fom the heate is the condensation heat of vapou Q m, so fo the cycle efficiency one can wite: dt dpvv v dpv T dpv dpv p v T m p dt T (hee the ideal gas law p v is used fo the vapou By integating this equation one obtains: T pv ( T pv( T exp T T emak: Actually this pat of the solution is a deivation of the Clausius Clapeyon elation f a conteste efes to this elation ight away, neglects specific volume of wate, and uses the ideal gas law to evaluate specific volume of vapou, item 8 should be awaded points out of, while item should be awaded the maximum scoe t is impotant to note that since the total mass of wate above the planet suface emains mh O g constant, the pessue exeted on the planet suface by wate does not vay eithe: p Befoe condensation it completely due to the pessue of vapou in a suface laye of atmosphee Accoding to the poblem p p The condensation at the Conus suface begins when this v ( T pessue equates the pessue of satuated vapou at a new tempeatue T such that T T p pv ( T pv ( T pv ( T exp ln, T T T T whence T (5,, K otice that befoe the onset of condensation the planet T ln cooled by less than 5 K, so the above calculation of the time equied to educe the tempeatue by T K is quite ealistic Accoding to the above law of tempeatue vaiation, the condensation begins at / T T/ ln ln ln 5 Conus yeas ln ln mh O g Afte the condensation begins the pessue p is a sum of the pessue of liquid wate laye of a thickness h l and the pessue of wate vapou above its suface which is equal to the pessue of satuated vapou at a cuent tempeatue, ie v

14 p pv ( T l ghl pv ( T exp T T is the density of liquid wate The maximum ocean depth coesponds is eached when all ( l available wate has condensed, ie h m p ( T p ( T HO v v max g l l l g hmax Theefoe ocean depth at a tempeatue T T is detemined fom the equation h l exp, whence hmax T T h l exp exp hmax T T T T T Ocean depth becomes equal to a quate of the maximum at T T ln Theefoe / T T ln ln ln / ln ln Conus yeas 5 Condensation of vapou eleases heat This pocess, which takes place evenly ove the planet suface, is quite «fast», ie the heat elease «follows» the condensation ate which, in tun, is detemined by the ate of planet cooling Of couse, the elease of vapoisation heat slows down the planet cooling to some extent and this influence must be analysed Using the known pessue of satuated wate vapou at T 7 K one estimates pv ( T pv ( T exp 7 kpa, T T so the total vapoisation heat eleased is pv ( T pv ( T Q c mh O 8 J This heat is eleased fo hundeds of g G Conus yeas on the suface of a planet which mateial is a good heat conducto Taking into 7 account the heat capacity of the planet C c, J/K one can see that its waming due to this pocess is vey small, T c 5 K, even if the total amount of the eleased heat is absobed Obviously the planet cooling and the ensuing condensation of vapou is mostly due to eduction of adioactive heating The tempeatue deceases appoximately accoding to exponential law Theefoe the ates of cooling and condensation slow down almost at the same pace Since the condensation lasts no less than cooling befoe it stats (, and a tempeatue change fo the consideed pocesses is about T 5 K a decease in the ate of cooling because of the elease of T vapoisation heat does not exceed c Theefoe the influence of condensation on the T cooling is negligible both at an accuacy of 5% and,5% emak: Let us ecapitulate: the essential pat of the analysis of themal dynamics of Conus atmosphee is based on undestanding of existence of seveal time scales: a scale of weak «fast» effects (like inhomogeneity of sola adiation and elease of vapoisation hea which is about one «day» o less; such effects have almost no influence

15 on the dynamics because they ae small and because they get aveaged ove a time scale of the ode of Conus yea and moe a scale of «middle ate» elaxation pocesses (like estoation of the equilibium tempeatue pofile and the suface tempeatue lasting fom a Conus yea to decades; a scale of «slow» vaiation of extenal factos (vaiation of the heating powe taking hundeds and thousands Conus yeas