aaMPROBLEMSin GENERALPHYSICSV. S. WOLKENSTEINMIR PUBLISHERS MOSCOWProblems in GeneralPhysicsB. C. BOnbKEHllITEflHCBOPHI1K 3AJlA4noOSIUEMYKYPCY =wtw= consta=OatlCP=CJ)ot+T(0=6)0 +ata = const1.1. A cartravelsat a velocity of 80 km/hduring thefirst half ofits funning time and at 40 km/h during the other half. Find the averagevelocityof thecar. .1.2. A car covers half adistance at a velocity of 80 km/h and theother at40 km/h. What is theaverage velocity of the car?1.3. A ship goes fromAtoBat vl=IOkm/handfrom BtoA at 16 km/h. Find: (I)theaverage velocity of theship, and (2) thevelocity of therivercurrent.1.4. Determine thevelocitywithrespect totheriver bankof: (1)aboat going downstream, (2) aboat going and(3) aboat1.5-1.14)PHYSICAL FUNDAMENTALSOFMECHANICS21travelingat an angle of a=90 to the current. The velocity of the cur-rent V1 =1 mIs, thevelocity of the boat withrespect to the water v.==2 m/s.1.5. An airplaneis flying withthevelocityofVI =BOO km/hrela-tive to theair. A wind witha velocity ofV1= 15mlsis blowing fromwest toeast. What is thevelocityof theairplane withrespect totheEarth, and what should theangle withthe meridianbe tofly theair-plane: (1)southward, (2) northward, (3) westward and(4) eastward?1.6. An airplaneflies from Ato B ata distance of 300 km eastward.Find thedurationof theflight if: (1) thereis no wind, (2) thewindblows from southto north, and(3) the wind blows from west to east.The velocity ofthe windVI =20mls andthat oftheairplane withrespect totheair [1,=600km/h.1.7. Aboat movesperpendicular tothe bankwith avelocityof7.2 km/h. The current carries it 150 m downstream. Find: (1) the velo-cityof the current, (2) thetimerequiredto cross theriver. The riveris 0.5kmwide.1.8. A body thrown vertically upward returns to theEarthin 3 se-conds. (1) Whatwas theinitial velocity of thebody.?(2) Whatheightdidthe body reach? Disregard the resistance of theair.1.9. Astoneis thrownupwardtoaheight of 10 metres. (1) Afterwhat time will it fall onto theEarth?(2) What height .can be reachedbl the stone if its initial velocityis doubled? Disregard the resistanceo the air.1.10. A stone is dropped from,8balloon atan altitude of 300metres.How muchtimeis required for the stone to reach theEarth if: (I) theballoonis ascending withavelocityof 5 mIs,(2) theballoonis des-cendingwithavelocityof5 mIs, (3) theballoonis stationary?Dis-regardtheresistance of theair.1.11. Draw adiagram showing therelationshipbetween theheightn, velocityv andtimet for abody thrown vertically upward withani-nitial velocityof9.8m/s. Plot the diagramforthe time intervalfrom0 to 2 seconds, i.e., for ~ t ~ 2 s after every0.2 s.Disregardtheresistance of theair.1.12. A body falls vertically from the height h=19.6 metres withthe initial velocity equal to zero. Whatdistance will be traveled by thebody: (I) duringthe first0.1 second ofmotion, (2) duringthe last0.1 second of motion?Disregard theresistance of theair.t.13. Abody falls vertically from theheight h=19.6 metres withitsinitial velocityequal to zero. What timewill it takethebody totravel: (1) thefirst metre, (2) thelastmetre? Disregard the resistanceoftheair.1.14. Duringthelast second ofitsfree fall abody covers half ofthe total distance traveled. Find: (1) the height h fromwhich the bodyfalls, (2) thedurationof fall ing.22PROBLEMS(1.15-1.231.15. A body Ais thrown vertically upward with the initial velocityVI; abodyBfalls from the height h withtheinitial velocity V2=0.Findhow thedistancexbetween thebodies Aand Bdepends onthetimet if thebodies began to move simultaneously.1.16. Thedistance betweentwoundergroundstations is 1.5kilo-metres. Thefirst half of thedistanceis covered bya train withauni-formly accelerated speed, andthe second half withauniformlyretar-ded speed. The maximumspeed of thetrain Is 50 km/h. Find: (1) theacceleration, taking it to be numerically equal to the retardation,(2) the time the train travels betweenthe stations.1.17. A trainis funning ata speed of 36 km/h. If the supplyof cur-rent to thetraction motorsis stopped, thetrain, moving withauni-formly retardedspeed. will stopin 20 seconds. Find: (1) thenegativeaccelerationofthetrain, (2) thedistancefromthestationat whichthe current should be switched off.1.18. Brakinguniformly reduces the speed of atrainfrom 40 km/hto 28 km/hduringone minute. Find: (1) thenegativeaccelerationofthetrain, (2) thedistancetraveled bythetrainduringthetimethebrakes areapplied.1.19. A car runs ata uniformly retarded speed with a negative acce-leration of -0.5 m/s>, The initial velocity of the car is 54 km/h.Inhowmuchtime and howfar fromtheinitial point will thecarstop?1.20. A body Abegins to move withtheinitialvelocity v; and con..tinuesto move withtheconstant accelerationat. A body Bbegins tomove at the same time as thebodyAwiththeinitial velocity v ~ andcontinues to movewith the constant negative acceleration az Whattime isrequiredforthe twobodies to acquire the same velocity aftermotionhasbegun?1.21. A body Abegins to move withtheinitial velocity v ~ = 2 m/sand continues to move at a constantacceleration a. In L\t=10secondsafter the body Abegins to move, a body B departs from the same pointwiththeinitial velocityv;=12 m/s andmoves withthe same accele-rationa. What is themaximumaccelerationaat whichthe bodyBcan overtake the body A?1.22. The relationshipbetween the distance s traveled by a body andthetime t is given by the equation s=At-Bt'+Ct3, where A==2 mis, B=3 rn/s! and C=4 m/s'. Find: (I) how thevelocity vand accelerationadepend on thetimet, (2) thedistance traveledbythebody. thevelocityandacceleration of thebody in 2 seconds aftermotionhas begun. Plotadiagram showing thedistance, velocityandacceleration for O ~ t ~ 3 s after every 0.5 s.1.23. The relationshipbetween the distance s traveled by a body andthe time t is expressed by the equation s=A-Bt+Ct2, whereA=6 m, B=3 mls and C=2rn/ss. Detesmine theaveragevelo..1.24-1.32JPHYSICAL FUNDAMENTALSOFMECHANICS23cityand theaverage accelerationof thebody within thetime intervalfrom 1to 4 seconds. Plot the diagram of the distance, velocity and ac-celeration for O ~ t ~ 5 secondsafter every second.1.24. The relationship between the distance s traveled by a body andthe timet is described by the equations=A+Bt+Ct', where A==3m, 8=2 mls and C=1 miss. Determine theaveragevelocityandtheaverage accelerationof thebodyduringthefirst, second andthirdseconds of motion.1.25. The relationshipbetweenthe distancestraveledbyabodyandthe timet is described by the equation s=AfBt+Ctl+Dt3,whereC=O.14 m/s! andD=O.OI misS. (1) In what time after motionbegins will theacceleration of thebodybe equal to1 m/s'?(2) Whatisthe averageaccelerationthe bodyacquires during this time?1.26. A stoneis thrownhorizontallywiththevelocity Vo= 15 mlsfrom a tower with a height of H=25 metres.Find: (1) the time duringwhich the stone is in motion, (2) the distanceSx from the tower base towherethe stone will drop onto the ground, (3) the velocity v with whichit will touch the ground, (4) the angleq> formed by the trajectory of thestone withthe horizontal at the point whereit reachestheground.Disregard theresistanceoftheair.1.27. A stonethrownhorizontallyfell ontotheground after0.5 se-condat a distanceof5metresfromwhereit wasthrown. (1) Fromwhat height h was thestonethrown? (2) What was the initial velocityVo ofthestone? (3) What velocityvdid thestonetouchthegroundwith? (4) What angleq> was formed by thetrajectory of the stone withthehorizontal at the point where itreached the ground? Disregard theresistance of the air.1.28. A ball thrownhorizontallystrikes a wall5 metres away. Theheight of thepointstruck by theball is 1 metrelower than the heightwhichit was thrownfrom. (1) What velocityVo was theball thrownwith? (2) At what angleq>didtheball reachthe wall?Disregardtheresistanceoftheair.1.29. A stoneis thrownhorizontally. In0.5 second after thestonebegan tomove, thenumerical valueofits velocitywas1.5 timesitsinitial velocity. Findtheinitial velocityofthestone. Disregard theresistance oftheair.1.30. A stoneis thrownhorizontally withthevelocity Vx=15 m/s.Determine the normal andtangential accelerations of the stone in 1se-condafter it begins tomove. Disregardtheresistanceof theair.1.31. Astone is thrown horizontally with the velocity 10m/s.Findthe radiusQf curvature of its trajectoryin 3 seconds after the mo-tionbegan. Disregardtheresistanceof theair.1.32. Aball is thrown withthevelocityVo= 10 m/sat anangle ofa=40 to thehorizon. Find: (1) the height Sywhich the ball willriseto, (2) the distanceSx fromthepoint ofthrowingtowhere the ball24PROBLEMS(1.33-1.41will dropontotheground, (3) thetimeduring whichthe ball will beinmotion. Disregard the resistance of the air.1.33. An athlete puts a shot 16 m 20 em iu Leningrad. What distancewillbe coveredbyanidentical throwinTashkent,assuming that theinitial velocity and angle to the horizon arethe same? The accelerationofgravityis 981.9cm/s! inLeningradand980.1em/slinTashkent.1.34. A bodyis thrown withthe velocityVo at anangle tothe hori-zon, The duration of motion t==2.2 seconds. Find the maximumheight reached bythe body. Disregard the resistance of the air.1.35. Astonethrown withthevelocityVo= 12 m/sat anangleof(%=45 tothe horizon droppedtotheground at thedistance sfromthe point whereit was thrown. From what height h should the stone bethrowninahorizontal directionwiththesame initial velocityVoforit tofall at the samespot?1.38. A bodyis thrown withthe velocityVo= 14.7 m/s at an angleof (%=300tothehorizon. Find thenormal and tangential accelerati-ons of the body in t=1.25 s after it began tomove. Disregard the re-sistanceof theair.1.37. A bodyis thrown with the velocity00=10m/s at an angleof a=450 to the horizon. Findthe radius of curvature of its trajectoryint=1 s after the body began to move. Disregardthe resistanceoftheair.1.38. Abodyis thrown withthe velocityVo at anangle of extothehorizon. Determine Vo ande if the maximum height which the bodyrea-ches ish=3mand the radius ofcurvature at theupper point of itstrajectoryR=3m. Disregardtheresistanceoftheair.t.39. Astone is thrown fromatower with aheight of H=25 matVo= 15 mls andan angle cx=3O tothe horizon. Find: (1) thetimeduringwhich the stone willbe in motion, (2) the distance fromthe to-wer baseto wherethe stone will droponto the ground. (3) the velocitywith whichthe stone will fall tothe ground, (4) theangleq> formedbythe trajectory of the stone withthe horizon at the pointof fall. Disre-gard the resistance of the air.1.40. A boythrows aballwith the velocity Vo= 10 m/s at anangleof cx=45 tothe horizon. The ball strikes awall at adistance of s==3 mfromtheboy. (1) When willthe ball strike the wall (whentheball ascends or descends)? (2) Find the height y at which theballwill strike the wall (counting from the height whichthe ball was thrownfrom). (3) Determine the velocity of the ball at the moment ofimpact.Disregard the resistance of the air.1.41. Find the angular velocities of: (1) daily rotation of theEarth) (2) a watchhourhand, (3) a watchminute hand, (4) anartifi-cialsatellite of the Earth rotating along a circular orbit with the period~ f .revolution T=88 min, (5) the linear velocity of this satelliteIf Itsorbit isat adistanceof200kmfrom"theEarth'ssurface.1.42-1.53]PHYSICAL FUNDAMENTALSOFMECHANICS251.42. Determinethelinear velocityof revolutionofpointsontheEarth's surface ata latitude of 60.1.43. What shouldthevelocityof anairplane flying from east towest be ontheequator forthepassengers tosee theSunmotionlessinthesky?1.44. An axle with two disks mounted at a distanceof 1=0.5 m fromeach otherrotateswithanangular velocitywhlch corresponds tothefrequency , , ~ 1,600rpm. A bullet flyingalong theaxle pierces bothdisks. Thehole inthe seconddiskisdisplaced withrespect to thatin thefirst one by theangle q>= 12. Findthe velocityof the bullet.1.45. Findtheradiusofarotatingwheelifthelinear velocityVIof apointon therimis 2.5 times greater thanthelinearvelocityV2of apoint 5 centimetrescloser tothe wheel axle.t .46. Auniformlyacceleratedwheel reachesthe angular velocity00=20 rad/s in N=10 revolutionsafterrotationbegins. Determinethe angular acceleration of the wheel.1.47. In t=1 minute after it begins torotateaflywheel acquiresavelocitycorresponding to v=720 rpm. Findtheangular accelera-tionof thewheel andthenumberof itsrevolutionsper minute. Themotion isuniformly accelerated.1.48. When braked, a uniformly retarded wheel reducesits velocityfrom 300 rpm to180 rpm during one minute. Findthe angularaccele-rationof the wheel andthenumber of revolutionsit completes in thistime.1.49. A fan rotateswitha .velocity corresponding to8frequency of900 revImin. When its motoris switched off, thefan uniformly slowsdown and performs 75 revolutions before it comes to a stop. Howmuchtime elapsed fromthe moment the fan was switched offto the moment itstopped?1.50. Ashaft rotates at aconstant velocitycorrespondingtothefrequency180revImin. At a certainmoment theshaft is brakedandbeginstoslow down uniformlywithanangular accelerationnumeri-callyequal to3rad/s>. (1)Inhow muchtimewill theshaft stop?(2) What number ofrevolutionswill it performbefore stopping?1.51. Apoint moves along a circle having a radius of r=2O cmwithaconstant tangential acceleration of Gt=5 cm/s-, How much timeisneeded after motionbegins for the normal acceleration an of the pointto be: (1) equal to the tangential acceleration, (2) double the tangentialacceleration?1.52. Apoint moves along acirclehavingaradius of r= 10 cmwitha constant tangentialaccelerationCIt. Findthe tangentialaccele..ration atof the point if its velocity is v=79.2 cm/s atthe end of thefifth revolutionaftermotionhas begun.1.53. A point moves along a circle having a radius of r= 10 emwitha constant tangentialacceleration at. Findthe normal acceleration a"26PROBLEMS(1.54-1.61of thepoint in t=20 seconds aftermotionbegins if thelinear velo-cityof thepoint is V= 10cm/s at the end of thefifth revolutionaftermotionhasbegun.1.54. It maybe assumed toafirst approximationthat anelectronmoves in an atom of hydrogen along a circular orbitwiththe constantvelocityv. Findtheangularvelocityof electronrotationaroundthenucleus andits normal acceleration. The radiusof the orbit r=O.5xX 10-10mandthe velocity of the electron alongthis ' orbit v=2.2 Xx 108m/s,1.55. Awheel havinga radius r=10 emrotateswith aconstantangular accelerationa=3.14rad/s>. Findfor points on the wheel rimat theendof the first second after motionhasbegun:(I) theangularvelocity, (2) thelinear velocity, (3) thetangential acceleration, (4)thenormal acceleration, (5) thetotal acceleration, and (6) theangleformed by the direction of the total acceleration withthe wheel radius.1.56. Apoint moves alonga circle with a radius of r=2 cm.Therelationshipbetweenthe distance andthetime isgivenbytheequation x=Ct8, where C=O.l em/st. Find the normalandtangen-tial accelerationsof thepoint at themoment when itslinear velocityv=O.3 rn/s.1.57. A point moves along a circle with the relationship between thedistance and the time conformingto the equations=A+Bt+Ct2,where B= -2m/s and C=1 mist. Find the linear velocityof thepoint, anditstangential, normal and total accelerationsint=3se-conds after motion begins if the normal accelerationof the pointwhen/'=2secondsism/ss,1.58. Findtheangularaccelerationof awheelif thevectorof thetotal accelerationof a point on the rimforms an angleof 600with thedirection of the linear velocity of this pointin 2 seconds after uniformlyacceleratedmotion begins.1.59. Awheel rotates with a constant angular acceleration=2 rad/s". Int=0.5 second after motionbegins, thetotal acceleration .of the wheel becomes a=13.6 em/51Determine the radius of the wheel.1.60. Awheel witha radiusof r=O.l m sorotatesthattherelati-onshipbetween theangle of rotation of the wheel radiusandthetimeis described by the equation =---2 2 2 2The peri-odofsmall oscillations of a physical pendulumT=2n y ~ I lwhere J=moment of inertiaof thependulumrelative toitsaxis ofrevolutionm=mass of the pendulumd=distance from the axis of revolution to the centre of gravityg=acceleration of gravity.3. t , Findthe moment of inertiaandthe angular momentum of theEarth relative to its axis of revolution.3.2. Two balls withthe radii r1=r,=5em are attachedto the endsof athinrod witha weight much smallerthanthatof theballs. Thedistance between the centres of the balls R=O.5 m. The mass of eachball m=1 kg. Find: (1) the moment of inertiaJ 1 of thissystem withrespect to an axis passing throughthe middle of the rod perpendicularto itslength, (2) the moment of inertiaJt of this system relativeto thesame axis assuming the balls to be materialparticles whose masses areconcentrated attheir centres, (3) the relative error 6= 117/1madeincalculatingthemoment ofinertia ofthis systemwhen weuseItinsteadof J1.3.3. Aconstant tangential forceF=98.1 N is applied to the rim ofa homogeneous disk with a radius of r=0.2 m. When the disk rotates,it is actedupon bythemoment of frictionforces M,,=O.5 kgf -m.48PROBLEMS(3.4-3.13Findtheweight 0of thedisk when itrotateswitha constantangularaccelerationof cx=100 rad/s>.3.4. A homogeneous rod with alength of 1 m and a weight of 0.5 kgfrotates in a vertical planeabout a horizontal axispassing through themiddle of therod.What angular accelerationwill therod rotate withifthe rotational moment is9.81X 10-2N -m?3.5. A homogeneousdisk witharadiusof r=0.2 m andaweightof 0=5 kgf rotates aroundanaxis passingthrough itscentre. Therelationbetween theangular velocityof disk rotationandthetimeisdescribed by the equation ro=A+Bt, where8=8 rad/st, Find thetangential force appliedtotherimof thedisk. Disregardfriction.3.6. A flywheel withthe moment of inertia 1=63.6 kg -rn! rotateswith a constant angular velocity 00=31.4 rad/s. Findthe braking mo-mentM wnich stops theflywheel in t=2Oseconds.3.7. A tangential force of 10 kgfis applied totherimof awheelhaving the form of a disk witha radius of 0.5 m and amass m=50kg.Find: (I)the angular accelerationof the wheel, (2) in what timeaftertheforce is appliedwill thewheel rotateat 100 rev/5.3.8. A flywheelwith a radius of r=0.2 m and a mass of m=10kgis connected toamotorbymeans of adrivebelt. Thetensionof thebelt whichruns withoutslippingis constantand equals T=14.7N.What number of revolutions willbe developed by theflywheel per se-cond in ~ t = 10seconds after motion begins? Consider theflywheel asahomogeneous disk. Disregardfriction.3.9. Aflywheel withamoment ofinertiaof245 kg m' rotatesat20rev/so The wheel stops in one minute after thetorque stops acting onit. Find: (I)the moment of the forces of friction, (2) the number of re..volutions completedbythe wheel fromthe moment the forces stopactingonit until it stops.3.10. Twoweights 01=2kgfandQs=1kgfare linked by a threadandthrownover apulley weighing G=1 kgf. Find: (I) the accelera-tion a withwhich the weights move, (2) thetensions T1 andT, of thethreads whichtheweightsareattachedto. Consider the pulleyasahomogeneous disk. Disregardfriction.3.11. A loadwith a mass ofm=2kg is attached to the end of a cordwrappedaround a drumhaving a mass ofM=9 kg. Findthe accelera-tion of the load. Consider the drum to be a homogeneous cylinder. Dis-regardfriction.3.12. A load of01=10 kgf is attached to the end of a cordwrappedaround adrumwith a radius of r=0.5 m. Find the moment of inertiaof the drum if the load is lowered withan accelerationof a=2.04 m/s>,3.13. A load of01=0.5 kgf is attachedto the end of a cord wrappedaroundadrumwitha radiusof r=20 em. The moment of inertiaofthe drumI =0.1 kg -rnt. Before thedrum begins to rotate, the heightof theload 01abovethe floor is hI = 1 rih Find: (1) thetimeneeded3.14-3.25]PHYSICAL FUNDAMENTALSOFMECHANICS49by theload to reach thefloor t (2) the kinet ie energy of theload at themoment ofimpact against thefloor, (3) thetensionof thecord. Dis-regardfriction.3.14. Two different weights are connectedbya threadpassing overa pulleywhose moment of inertia1=50 kg-rn- and radius r=20 em.Thepulleyrotateswithfrictionandthemoment of thefrictionforcesMIr=98.1N m.Find thedifference in the tensions of the threadT1-- T2 on bothsides of thepulleyif itrotateswitha constant angularacceleration a=2.36 rad/s-.3.15. A pulley weighingG= 1kgf is secured to the edge of a table(see Fig. 1 and Problem 2.31). The equal weights A andB(01=02==I kgf) are linked by a threadthrown over the pulley. The coefficientof friction of the weight B against thetable f=O.I. Considerthe pulleyto bea homogeneous diskand disregardthe friction inthe pulley.Find: (1) the accelerationwhich the weights move with, (2) thetensi-onsT1 andT2of thethreads.3.16. A disk weighing 2 kgf rolls without slippingover a horizontalplanewithavelocityof4m/s. Findthekineticenergyofthe disk.3.17. Aball 6 em in diameterrolls without slipping over a horizon-tal planewithavelocityof 4 revIs. Themass of theball is 0.25kg.Findthekineticenergy of therollingball.3.18. Ahoop andadiskhavethesame weight G androllwithoutslipping withthe same linear velocity v. The kinetic energy of the hoop1=4kgf-rn. Findthekineticenergy ofthedisk E23.'19. A ball with a massof m=I kgrolls without slipping, strikesa wallandrollsback. Before theimpact thevelocityof theball v1:=:= 10em/s and after the impact vz=8 cm/s.Find theamount of heat Qevolvedduringthe impact.3.20. Determinetherelative error obtainedin calculating thekine-tic energy of a rollingball if rotation of theball is neglected.3.21. A disk witha weight of I kgf and adiameter of 60 em rotatesabout anaxispassing throughits centreperpendicular to-itsplaneat20 tests. What work shouldbe performed to stopthedisk?3.22. Thekineticenergy of ashaft rotatingat aconstant velocityof5revIsis60 J. Find theangular momentumoftheshaft.3.23. Findthekinetic energy of a cyclist ridingat aspeed of V==9km/h. Thecyclist withhisbicycle weighs 0=78 kgf, andthewheels 01=3kgf. Considerthebicycle wheels as hoops.3.24. Aboydrivesahoop overahorizontal pathwithaspeed of7.2 km/h. Over what distance can the hoop run uphillatthe expense ofitskineticenergy? The slope of thehill is1in 10.3.25. What is theminimumheighth from which a cyclistcan startto travel by inertia (withoutfriction) over a pathin the formof a loopwitha radius of R=3 mso asnot to leave the pathat thetop of theloop? Themass ofthecyclist together withthe bicyclem =75 kg,4-357450PROBLEMS(3.28-3.33the mass of the wheels beingnZI =3 kg. Consider thebicycle wheels ashoops.3.26. A copperball witha radius of r=10 em rotates witha vela..citycorrespondingtov =2revls about anaxis passingthrough itscentre. What work should be performed to increase theangular velocityofrotationofthe ball twofold?3.27. Find the linear accelerations of the centres of gravity of:(1) a ball, (2) a disk, and(3) a hoop, whichroll without slipping downaninclinedplane. Theangle of inclinationis 30, andtheinitial ve-locityofall thebodiesiszero. (4) Comparetheseaccelerationswiththat of abodywhichslides oft' the inclinedplane without friction.3.28. Findthe linearvelocitiesofthe centres ofgravityof: (1) aball) (2) adisk, and (3) ahoop, whichroll without slippingdown aninclinedplane. Theheight ofthe inclinedplane h=O.5 m, andtheinitial velocityof all thebodiesis zero. (4) Comparethesevelocitieswiththat of a body which slides off theinclinedplane without friction.3.29. The surfacesof twocylinders-aluminium(solid) and lead(hollow)-having the same radius r=6 ernandthe same weight G==0.5 kgf arepaintedthesame colour. (1) How canthecylindersbedistinguished by observingtheir translational velocitiesat thebase oftheinclinedplane? (2) Findthe moments of inertia of these cylinders.(3) How much timedoes ittake each cylinder to rolldown theinclinedplane without slipping? The height of the inclined plane h=O.5 m anditsangleofinclination a= 30. Theinitial velocityof eachcylinderis zero.3.30. A wheel is uniformlyretardedbybraking anditsvelocityofrotationdrops from 300 to180 rev/minin one minute. The moment ofinertiaof thewheel is 2 kg -mi. Find: (1) theangularaccelerationofthewheel, (2) thebrakingmoment, (3) theworkofbraking, (4) thenumber of revolutions completed by the wheel during this minute.3.31. A fanrotates witha velocity of 900 rev/min. Whenitsmotorisswitchedoff, the fanhas uniformlyretarded rotation and makes75 revolutions before it stops. The work of thebraking forces is 44.4 J.Find: (1) the momentof inertia of thefan, (2) themoment of the fric-tionforce.3.32. A flywheel witha moment of inertia of 1=245 kg -m> rotatesat 20revIs. After theaction ofthetorqueisdiscontinued, thewheelstopsupon completing1,000 revolutions. Find: (1) themomentof thefriction forces, (2) thetimewhich elapses from themoment theactionofthetorquediscontinuestothemoment whenthewheel stops.3.33. A load of 1kgf is fixed to theendofathread passing aroundthe rimof- a pulleyfitted on the same axleas aflywheel. Over whatdi-stanceshouldthe loadlower forthewheelandthe pulleytoacquirea velocityof 60 rev/min?Themoment ofinertiaof thewheel and thepulleyis 0.42kg -rn>, andthepulleyradiusis10 em.3.34-3.44)PHYSICAL FUNDAMENTALSOFMECHANICS513.34. A flywheel begins torotate witha constant angular accelera-tion of a=0.5 rad/s> and acquires an angular momentum10>==73.5 kg m'ls int1= 15 secondsafter motionbegins. Findthekine-ticenergyofthewheel int2= 20 seconds after rotationbegins.3.35. Aflywheel rotates withaconstant velocitycorrespondingto\'=10 revls anditskinetic energy Ek=800 kgf-m. In whattime willthetorque M= 50N mappl iedtothe flywheel doubleitsangularvelocity?3.36. A constant tangential forceF=2 kgf is appliedto the rim ofa diskwith a mass of m=5 kg. What kinetic energy will be impartetlto thediskin L\t=5 seconds after theforce begins to act?3.37. Through what angle should a homogeneous rod suspendedfrom a horizontal axispassing throughtheupper end of therod devi-ate for the lower end of the rod to move at 5 mls when it passes throughthe position of equilibrium? The rod is I m long. .3.38. A homogeneous rod 85 cm long is suspended from a horizontalaxis passingthrough its upperend. What minimumvelocityshouldbe impartedtothelower end of therodtomakeitcomplete one fullrevolution about the axis?3.39. A pencil placedverticallyonatablefallsdown. What willtheangularandlinear velocitiesbe at theendof thefall of: (1) themiddleof thepencil, (2) itsupper end?Thepencilis15 cmlong.3.40. Ahorizontal platformwith a mass of 100kg rotates at10 revIminaroundavertical axispassing throughitscentre. A manweighing 60 kgf is standing onits edge. What velocitywill theplat-form begin torotate withif the man moves from the edge of theplate.form to its centre?Regard the platform as a circular homogeneous diskandthemanas apoint mass.3.41. Whatwork willbe performed by a man moving from the edgeof the platformto its centrein the conditions of the previous problem?Theradius oftheplatformis1.5 m.3.42. A horizontal platformwitha weight of 80 kgf anda radiusof1 m rotates atan angular velocity corresponding to 20 rev/min. A manstandsin the centre of the platformand holds weights in his out-stretchedhands. Howmany revolutions will theplatformmakeperminute if the man lowers his hands. thus reducinghis moment ofinertia from 2.94kg-rn! to0.98kgm'? Considerthe platformasacircular homogeneous disk.3.43. How many times willthekineticenergy of theplatformwiththemanincreaseinthe previousproblem?3.44. A man weighing 60 kgf standson animmobile platformwithamass of 100 kg. What numberofrevolutions will be madebytheplatform a minute if the man moves along a circle with a radius of 5 maroundtheaxisof rotation?The manmoves relativetotheplatformwitha velocity of 4 km/h. The radius of the platform is 10rn. Consider4*52PROBLEMS13.45- 3.50the platform as a homogeneous disk and the man as a pointmass.3.45. A homogeneous rod oscillates inaverticalplane about a hori-zontal axis passing through its top. Thelengthof therod1=0.5rn.Findtheperiodof oscillations of therod.3.46. Findtheperiod of oscillations of therodin thepreviouspro-blemif the axis of rotation passesthrough a point 10centimetresfromits topend.3.47. Two weightsareattachedtotheendsofavertical rod. Thecentre of gravityoftheseweights isbelow the middleoftherodbyd==5 cm. Findthe length of therod if .the period of small oscillationsof therod withtheweightsaroundahorizontal axispassingthroughits centre T=2 sec. Neglect the weight of the rod with respect tothatofthe weights.3.48. A hoop56.5 emindiameterhangsonanail hammeredintoa wall andperforms small oscillationsin aplaneparallel tothewall.Findthe periodoftheoscillations.3.49. What shouldbetheminimum length1of a threadonwhicha homogeneous ball witha diameter of D=4 ern is suspended to regardthis ball asamathematical pendulumindeterminingthe periodofsmall oscillations? The error made when assumingthisshouldnot ex-ceed1 percent.3.50. A homogeneous ball is suspended from a thread withalengthequal to the radius of theball. How manytimesis theperiod of smalloscillations of thispendulumgreater than that of a mathematicalpen-dulumsuspendedat thesamedistancefrom thecentreof gravity?4. Mechanics of FluidsLiquidsand gases are also known underthecommon name of fluids.The steadymotionof anideal incompressiblefluid is described bytheBernoulli equationpv2P+T +pgh= constwhere p=densityof the fluidv=velocityof the fluidinthegivencross sectionof thepipeh=height ofthis cross sectionaboveacertainlevelp=pressure.It followsfromthe Bernoulli equationthat a fluidflows out fromasmall orifice with the velocity V=V2gh, where h is the height ofthesurface of thefluid abovetheorifice. Since thesame quantities offluid passthroughany crosssection of a pipe, thenAtVl =A2Va, whereVI andV2 are thevelocities of thefluidin two sections of thepipe withthe areasA1 andAI. "4.1-4.3) PHYSICAL FUNDAMENTALSOFMECHANICS53Theforce of resistanceacting onaball fallingin aviscousfluidisdetermined by the Stokes formulaF=6 nfJrvwhere f)=coefficient of internal friction ofthe fluid (dynamicvis-cosity)r=radius of theballof the ball.The Stokes law is true only for laminar motion, when the volume of afluid passing during the timet through a capillary tube withthe radiusr andthe length1canbedeterminedfrom thePoiseuilleformulaV=nr4tl1pBIl]where 11=dynamic viscosity of the fluidAP=difference ofpressures at thetubeends.Thenatureof motionof afluidis determinedbythedimensionlessReynolds numberDvp DvRe=-=-T) 'Ywhere D=quantity characterizing thelinear dimensions of thebodyaround which the fluid flowsv=velocity of flowp=densityll=dynamic viscosity.The ratio v=fJ/pis known as thekinematic viscosity. The criticalvalueof the Reynolds number whichdetermines thetransitionfromlaminartoturbulent motionis different for bodies of differentshape.4.1. Find the flowvelocity ofcarbon dioxidegasalongapireif0.51 kgof gas flows inhalf anhour throughthecross section0 thepipe. The densityof the gas is 7.5kg/m3and the pipe diameteris 2 cm(inProblems4.1through4.9thefluids are considered asideal andin-compressible).4.2. The bottomof a cylindrical vessel nas a circular hole d=lemin diameter. The diameter of thevessel D=O.5 m. Findtherelation-shipbetweenthevelocity vwithwhichthewater level inthevesseldrops and the height h of this level. Also determinethe numerical valueof this velocityfortheheight h=O.2m.4.3. A vessel filled withwaterstands on a table. In its side the ves- .sel has a small orifice arranged atthe distance hi from the bottom of thevessel and at thedistance h" from thelevel of the water, which is con-stant. At what distance fromthe orifice (ina horizontal direction)will the jet ofwater fall ontothe table?Solvethe problemfor thePROBLEMS(4.4-4.9aFig. 5following cases: (I) h1=25emandh,=16 ern, (2) h1=16emandh2=25em.4.4. AvesselAfilled with water (Mariottevessel) communicateswiththe atmosphere through a glass tube a passing throughthethroatof thevessel (Fig. 5) Afaucet Fis h.=2 cm from thebottomof thevessel. Find the velocitywith which thewater flows out ofthe. faucet F whenthedistance between the endof thetubeandthebottomof thevessel is: (I) hi =2em,(2) h1=7.5cm,and (3) ht=IOcm.4.5. Acylindrical tank with a height ofh=1 mis filled with water up toits brim.(I) What time is required toempty thetankthrough an orifice initsbottom?Thecross-sectional area ofthe ori fleeis 1/400of that of the tank. (2) Comparethis timewith that required for thesame amountofwater toflow outof thetankif the waterlevel in thetank is maintained constantat aheight of h=1 mfromthe orifice.4.6. Water is poured intoavessel atarateof 0.2litreasecond. What shouldthe diameter d of anorificeinthebottomof thevessel befor thewater to remain at a constant le-vel of h=8.3 cm?4.7. What pressure will be builtupby a compressor in apaint gunif astreamofliquidpaint flowsout of it withavelocityof 25 m/s?Thedensityof thepaint is 0.8 g/cms.8..Fig. 84.8. Aliquidflows alongahorizontal pipeAB(Fig. 6). Thedif-ference between thelevels of theliquidin tubes a andbis10 em. Thediameters of tubes a and b are thesame. Determinethe velocity of theliquidBowing alongpipeAB.4.9. Airis blown throughapipeAB(Fig. 7) at arateof15 litresper minute. Thecross-sectional areaof the broadportionof pipeAB4.10-4.17)PHYSICALPUNDAMENTALSOF MECHANICSB6556PROBLEMS[4.18-4.20=900kg/rn" and adynamicviscosity of T)=0.5 N -s/rn>. Thelevelof the oil in the vessel is kept at a height of ht=50em above the capil-larytube. Findthedistance from the endC'f the capillarytube(alongahorizontal line) totheplace where the streamof oil dropsontothetable.4.18. A steel ball fallsin a broad vessel filled withtransformeroilhavingadensityof p=900kg/rn" andadynamicviscosityof 1]==0.8 N -s/m", Assumingthat the Stokes law is true when R e ~ O . 5 (ifin calculatingRethe ball diameter is taken tobethequantityD),find themaximumdiameter oftheball.4.19. Assuming that laminarmotionof afluidis retained in acy-lindrical pipe when R e ~ 3 , O O O (if when calculating Re thepipe dia-meteris takento be thequantity D), show that the conditions of Prob-lem 4.1 correspond tolaminar motion. Thekinematicviscosity of thefluid is tobe taken equal to v=I.33x 10-8m2/s.4.20. Water flows along a pipe ata rate of 200 em" per second. Thedynamic viscosity of thewater in the conditionsof theexperiment is0.001N -s/rn>. At what maximumpipediameter will thewater flowremainlaminar? (See the conditions of the previous problem.)Chapter 2MOLECULARPHYSICSANDTHERMODYNAMICSTHERMALUNITSThe International Systemof Units (Sf) incorporatesthe MKSDsystem designed for measuring thermal units (GOST8550-61). Table 7gives the basic and the most important derived unitsused to measurethermal quantities inthis system.QuantityandsymbolLength lMass mTimetTemperatureTAmount of heatHeat capacity of asystemEntropy ofasystemSpecific heatSpecific entropySpecific heat of phasetransitionTemperature gradientThermal fluxSurfaceradiation den-sity. density of ther-mal fluxTABLE 7Formula Unitlsymbol of IDimensionolunit quantityBasic Unitsmetre m 1kilogramme kg msecond s tdegree deg eDerivedUnitsQ=W=E joule J l2mt-"I.C=!L [ouIeper degree J/deg1'Jmt-S8- 1ATs= L\QjouIe per degree J/degl'Jmt-S6- 1Tc=.JLjoule per kilogram- J/kgdeg['Jt-38' - 1m ~ Tme-degreeS joule per kilogram- J/kgdegl't-tO-ls=-me-degreemQ joule per kilogram- J/kgl 2 t - ~q=-memgradT = degree per metre deg/rn 1- 16L\T=/f{ =AQwatt Wl2mt-SAt(J>watt per square W/m'mt-aq=A metre58Quantityandsymbol I FormulaPROBLEMSUnitTable7, concludedIs ymbOI of IDimensionofunit quantl tyThermal conductivity A= __Q_watt per metre- W/m-deglmt-38- 1coefficient At AAT degreeAlA square metre per mt/s Itt-1Thermal ditJusi vity a =cp secondCoefficient of heat , respectively. Findthe number of hydrogen moleculesin1 rnain these conditions.5.146. Thecoefficients ofdiffusion andinternal frictionof oxygenare equal to 1.22xIO' rn2/sand1)=1.95xIO-r. kg/rn-s respecti-vely. Findin conditions: (1) theof theoxygen, (2) themean free pathof Its molecules, and(3) thearithmetic mean velocityof itsmolecules.5.147. What maximum velocity can a rain drop 0.3 mm in diameterreach?Thediameter ofanair moleculeis 3x 10-10mandtheairtemperatureis 00C. Assumethat theStokeslawis truefor theraindrop.5.148. Anairplanefliesat avelocityof 360 km/h. Assumingthatthe layer of air at the airplane wing carried along owing to viscosity is4 ern, findthe tangential force acting on each square metre of the wingsurface. The diameter of an air molecule isax10-8cm. The tempera-ture of theair is 0 C.5.149. Thespace between two coaxial cylinders isfilled withgas.The radiiof the cylinders are equal to r=5 cm and R=5.2 em, res-pectively.The height of the internalcylinder h=25 em, The externalcylinder rotateswith avelocity corresponding tov=360rev/min.For the internal cylinder toremainimmobile, atangential force ofF= 1.38X 10-8Nshouldbe appliedto it. Considering thiscase to thefirst approximation as a plane problem, use the data of this experimenttodetermine the viscosity coefficient of the gas between the cylinders.5.150. Find the thermal conductivity coefficient of hydrogen ifits coefficient of internal friction is 8.6x 10-8Ns/mlin these condi-tions.5.15 t. Findthethermal conductivity coefficient of airat atempe-rature of 10 Cand a pressure of 10' Nzcrn>. The diameter of an air mo-lecule is 3x 10-8cm.5.152. Plot a diagram showing the thermal conductivity coefficientof hydrogen versus the temperaturewithin the rangeof 100 Kfor intervals of 100.5.153. Avessel with a volume of V=2 litres contains N=4XX10.22molecules ofabiatomicgas. Thethermal conductivitycoef-ficient of the gas "'=0.014 W/m -deg. Findthe diffusion coefficient ofthe gas intheseconditions.5.154. Carbondioxide gas andnitrogenareunder the same tempe-rature and pressure. Findfor these gases the ratio of: (I) the diffusioncoefficients, (2) thecoefficients ofinternal friction, and (3) thether-mal conductivitycoefficients. Assume thediameters of the moleculesof thesegasesto beidentical. .5.155. The distance between the walls of a Dewar flaskis 8 mm. Atwhat pressure will the thermal conductivityoftheair between thewalls of theflaskbegin to diminish during its evacuation? The tempe-78PROBLEMS(5.156-5.162rature of theair is17 C andthe diameterof anair molecule is 3xX10-"1mm.5.156. A cylindrical vacuumbottle withan externalradius of ',,==10 ern, an internal radius of '1=9 ernand a height ofh=20 ernisfilled withice. The temperature of theice is 0 C and the ambient tem-perature of the air is 200C. (I) Findthe maximum air pressure betweenthewalls of the vacuumbottle at whichthe thermal conductivitycoefficient will still dependonthepressure. Thediameterof theairmolecules is 3x 10-8cmandthetemperatureof theair betweenthebottle walls isequal tothe arithmetical meanof the temperaturesof the ice and the ambient medium. (2) Find the thermal conductivitycoefficient of the air between the bottle walls at pressures of: (a) 760mmHg, (b) 10- mmHg (J.t=29 kg/mole). (3) What amount of heat passesin one minute through the side surface of the vacuum bottle with a meanradiusof 9.5 cmdue tothermal conductivity? Solvetheproblemforpressures of: (a) 760 mm Hg, and(b) 10-&mm Hg.5.157. What amount of heat is lost everyhour through adoublewindowowing to the thermal conductivity of the air enclosed betweenthe panes? The area of each pane is 4 mtand the distance between themis30em. Thetemperature intheroomis 180C, andoutside it is-20 C. Thediameter of theair molecules is Sx 10-8em, andthetemperatureoftheair betweenthe panesis equal tothearithmeticmeanbetween thetemperaturesintheroom andoutdoors. Thepres-sureis 760 mm Hg.5.158. There isair betweentwoplates arranged at a distance ofI mmfromeach other andatemperature difference of ~ T = lOismaintained between them.The area of each plateA=100 ems, Whatamount of heat is transferred by conductivity fromone plateto the oth-er during10 minutes? The airis in standard conditions and the diame-terof an airmolecule is equal to 3x 10-10m.5.159. Tengrammes of oxygenare under a pressure of 3X 10' N1m2at a temperature of 10 C. The gas heatedat a constant pressure occu-pies a volume of 10 Iitres, Find: (1) the amount of heat received bythegas, (2) the change in theinternal energy of the gas, (3)thework per-formedbythegasduringexpansion.S.160. Hydrogenat a temperatureof270Cand inanamount of6.5 grammesexpands twofoldat p=const owing to theinflux of heatfromoutside. Find: (1) thework of expansion, (2) thechangeintheinternal energy of the gas, (3) theamountof heatreceived by the gas.5.161. A closed vessel contains 20grammes of nitrogen and 32gram-mes of oxygen. Findthe change intheinternal energy of the gas mix-turewhen it cools by28 C.5.162. Twokilomolesofcarbon dioxidegas areheated by50 Cat constant pressure. Find: (I)the change intheinternal energy of thegas, (2) the workof expansion, (3)the amount o1Jleat received by thegas,5.163-5.176) MOLECULAR PHYSICS ANDTHERMODYNAMICS795.163. Five hundred calories of heat are imparted to a blatornicgas. The gas expandsat constant pressure. Determinethework of ex-pansionofthegas.5.164. Workequal to 16 kgf-mwasperformedduring isobaricex-pansionof abiatomic gas. What amount of heat didthegasreceive?5.165. Agas occupying a volume of 5 litres underapressure of 2 XXIOCN/m2and a temperature of 17 C is heated andexpandsisobari-cally. The workofexpansionis 20 kgf-m. What temperaturewas thegas heated to?5.166. Sevengrammes of carbon dioxide gasareheatedby 10 Cin conditions of free expansion. Findthework of expansion of thegasandthe change inits internal energy.5.167. One kilornoleof a multiatomicgas isheated by 100 Cinconditions of free expansion: Find: (1) theamount of heat receivedbythe gas, (2) the changein itsinternal energy, (3) the work of expansion.5.168. One gramme of nitrogenis present in a vessel undera piston.(1) What amount of heatshould be spentto heat the nitrogen by10 C?(2)Howmuch will the pistonrise?The pistonweighs I kgfand itscross-sectional area is 10 em>. Thepressureabovethe piston isI at.5.169. Oxyhydrogen gasis in a vessel undera piston. Determine theamount of heat evolvedupon explosionof the gas if itsinternal energychanges by 80.2 caloriesandthepistonrises 20 ern. Thepistonweighs2kgf anditscross-sectional area is10 ern". Theair abovethepistonis in standard conditions.5.170. Nitrogenamountingto 10.5 grammes expands isothermallyat a temperature of -230C from a pressure of Pi=2.5 at to Pl=1at.Findtheworkperformedbythegasduringexpansion.5.171. Upontheisothermal expansion of 10 grammes of nitrogen ata tempetature of 17 C, work was performed equal to 860 J. How manytimes didthe pressureof the nitrogenchange uponexpansion?5.172. The workof isothermal expansionof 10 grammes ofagasfromthe volume VI to V2=2V1is 575 J. Determinethemeansquarevelocityofthegasmoleculesat thesametemperatureasabove.5.173. One litre of heliumin standard conditions expands isothermal-ly to a volume of twoIitres at the expenseof heat receivedfromahotsource. Find: (1) the work performed by the gas during expansion,(2)theamount ofheat receivedbythegas.5.174. Uponthe isothermal expansionof 2 m3of agasitspressurechanges fromPI =5at toP2=4at. Determinetheworkperformed.5.175. What temperature will air at 0C be cooled toif itexpandsadiabaticallyfromavolumeof VI to V2=2V1?5.176. Oxygenamountingto7.5litresis compressedadiabaticallytoavolumeof 1 litre, andthepressureat theendof compression is1.6x 108N/m3 Under what pressure was the gas beforecompres-sion?80PROBLEMS18.177-5.1885.177. Air is compressedadiabaticallyin the cylinders of anInter-nal-combustionengine and its pressure changes from PI=1 atto p,,==35 at. Theinitial temperature of the airis 40 C. Findits tempera-tureat theendof compression.5.178. A gas expands adiabatically, anditsvolume doubles, whileits absolute temperaturedrops 1.32 times. What number of degrees offreedomdo the gas molecules have?5.179. A biatomic gas at atemperatureof 27 C andunder apres-sure of 2x 10" N/mlis compressed adiabaticallyfromthe volumeVI to V2= 0.5VI.Findthetemperatureandpressure of thegasaftercompression.5.1BO. Oxyhydrogengas in a vessel under a piston occupies a volumeof10- rna in standardconditions. When compressedrapidly thegasignites. Find the temperature of ignition of the gas if the work of com-pressionisequal to4.73kgf-m.5.181. A gasis in avessel under apistonin standard conditions.Thedistancebetween thebottomof thevessel andthecrown of thepiston is 25em. A load of 20 kgf is placed on the piston and it lowersby13.4 em. Assuming thecompression tobeadiabatic, findthe ratio~ / c " forthis gas. Thecross-sectional areaofthe piston is 10 em".Disregardtheweight ofthepiston.5.182. A biatomicgas occupies avolume of VI = O.5litreunderapressure of PI=0.5 at. The gas is compressed adiabatically to a certainvolume V. and a pressure p. andis then cooled to theinitialtempera..tureat aconstant volume of VI. Hereitspressure becomesequal topo=1 at. (1) Plot the diagram of thisprocess. (2) Determine the vol-ume VI and the pressurePI.5.183. A gas so expands adiabaticallythat itspressure drops from2 atto 1 at. The gas is then heated at a constant volume to the initialtemperatureanditspressure increases to1.22 at. (1) FindtheratioepIc" for this gas. (2)Plot the diagramof this process.5.184. One kilomoleof nitrogen in standard conditions expands adia-baticallyfrom the volumeVI to V,=5V t Find: (1)the change in theinternal energy of the gas, (2) thework performedduringexpansion.6.185. It is necessary to compress 1X 10-1rna of air to a volume of2x 10-8mi. What is thebest compression process-adiabaticoriso-thermal?5.186. Thework of 146kJ is spent tocompressadiabaticallyonekilomole of a biatomic gas. Howmuch will the temperature of the gasincrease duringcompression?5.187. How many times will the mean square velocity of themole-cules of abiatomic gas decreaseupon8 twofold adiabaticincrease inthevolume of thegas?5.188. Ten grammesof oxygen in standard conditions are com-pressed to a volume of 1.4x 10-8mI. Find 1he pressureand tempera3.189-5.197J MOLECULAR PHYSICS AND THERMODYNAMICS81v,Fig. 8PADp, ----0pz---c, ,BI II Itureof theoxygen after compression if: (1) theoxygen is compressedisothermally, (2) the oxygeniscompressedadiabatically Findtheworkofcompressionin eachcase. 5.189. Nitrogeninanamount of28 grammesat atemperatureof40 Cand a pressure of 750 mm Hg is compressed to a volume of 13 lit-res. Findthetemperature andpressure of thenitrogenaftercompres-sion if: (1) the nitrogenis compressed isothermally, (2) the nitrogeniscompressed adiabatically. Fi!1d thework of compression ineach case.5.190. How manytimes willthe free path of the molecules of a bia-tornic gas increase if its pressure ishalved? Considerthe caseswhen:(1) thegas expands isothermally, (2)thegasexpands adiabatically.5.191. Twodifferent gases-onemonoatomic andtheother biato-mic-are at the same temperature and occupy the same volume.The gases are so compressed-adiabatically that their volume is halved.Whichof thegases will be heatedmore andhow manytimes?5.192. One kilogramme of airat a temperature of 300C andapres-sure of 1.5 atmexpands adiabatically and thepressure drops to1 atm.Find: (1) theexpansionratio, (2) the final temperature, (3) the workperformedbythegasduringexpansion.5.193. The volumeof 1kmoleof oxygen instandardconditionsincreasestoV=5Vo Plot a diagramshowingthe relationship p==f(V) if: (1) expansionoccurs isothermally,and(2) expansion occurs adiabatically. Findthevalueof pforthevolumes: Vo, 2Vo, avo,4Voand 5Vo5.194. A certainquantityof oxygen occu-pies a volume of VI =3litres at a tempera-tureof t1=27 C anda pressure of Pl=8.2xX10& N/m' (Fig. 8). In the secondstatethe gas parametersare: Vt = 4.5 litres andP2=6x 10' N/ml Find the amount of heatreceived by the gas, the work performedby thegas during expansion, and the change in theinternal energyofthe gas.Solve the problemon conditionthat the transitionof thegas fromthe first to the second state is 'via: (1)ACB, and (2) ADB.5.195. Anideal heat engine operatesaccordingtothe Carnot cycleandreceives 600 calfrom a hotsource each cycle. The temperature ofthehot sourceis 4000Kandthat of thecoldsink3000K. Find thework performed by the engine per cycle and the amount of heat rejectedtothecoldsinkpercycle.5.196. An ideal heat engine operatesaccording tothe Carnotcycle. "Determinetheefficiency of thecycleif work equal to300 kgf-rnwasperformedduringone cycle andthecooler received 3.2kcal.5.197. Anidealheat engine operatesaccordingtothe Carnot cycleand performs work equalto 7.35x 10 Jduring one cycle. The tempe-6-357482PROBLEMS(5.198-5.202rature of thehotsource is1000C andthatof thecoldsinkDOC. Find:(1) the engine efficiency, (2) theamount of heat received by theengineper cyclefrom thehot source, (3) theamount of heat rejectedtothecoldsink duringonecycle.5.198. Anidealheat engine operates accordingtothe Carnot cycle.Eightyper cent of theheat received from thehotsourceis rejectedtothe coldsink. The amount of heat received fromthe hot source is1.5 kcaI. Find: (1) thecycle efficiency, (2) thework performedduringa complete cycle.5.199. Anideal heat engineoperates accordingtotheCarnot cycleusing heatedair takenat aninitial pressure of 7 atmandatempera-tureof 127 C. The initial volume of theair is 2x 10-3rn-. Afterthefirst isothermal expansionthe air occupiesavolumeof 5litres, andafter adiabatic expansion8litres. Find: (1) thecoordinates of the in-tersectionof theisothermal andadiabaticlines, (2) thework oneachsectionofthecycle, (3) thetotal workperformedduring the entirecycle, (4) the cycle efficiency, (5) the amount of heat received from thehot source per cycle, (6) theamount of heat rejectedtothecoldsinkduringonecycle.5.200. One kilornole of anidealgas completesa cycle consisting oftwoisochoricandtwoisobaric lines. ThevolumeofthegaschangesfromVI=25m8toV,=50mSand the pressure fromPI = 1 atmtoP2=2atm. How many timesis theworkperformedin this cyclelessthan the work inaCarnot cycle whoseisothermal lines correspondtothemaximum andminimumtemperatures of thecycle underconsi-derationif thevolumedoublesinisothermalexpansion?5.201. An ideal refrigerator operates according to the reverse Carnotcycleand performs work equal to3.7x lOt J percycle. It receivesheat from a body witha temperature of -10 C and transfers this heatto a body witha temperature of +17 C. Find: (1) the cycle efficiency,(2) theamount of heat rejectedfromthecoldbodypercycle, (3) theamount of heat impartedtothehot bodyper cycle.5.202. Anideal refrigerator operatesasaheat pumpaccording tothereverse Carnot cycle. It receivesheat from water witha tempera-ture of 2 Cand transfers it to air with atemperatureof 27C. Find:(1)thequantity TIl-the ratio between the amount of heat impartedto theair duringacertaintimeandtheamount rejected during thesame timefrom thewater, (2) thequantity T)z-the ratiobetweentheamount of heat rejectedfrom thewaterduringa certaintimeandtheenergyspent during thesametime tooperatethe refrigerator (112 isknown as the refrigeration coefficient), (3) the quantity 'tl3-theratiobetween theenergy spent to operate therefrigerator during a cer-taintimeandtheamount of heat transferredduringthesametimetotheair (113is known as the cycle efficiency). Findthe relationshipbet-weenthe quantities 111t l } ~ and '13. "5.203-5.207J MOLECULAR PHYSICS AND THERMODYNAMICS83Dv, v,Fig. 95.203. An ideal refrigerator operates accordingto the reverse Carnotcycle and transmitsheat froma cold.source with water at a tempera-ture of 0 C to aboiler with water at atemperature of100 CWhatamount of water must be frozenin the cooler to convert 1 kg of waterintovapour intheboiler?5.204. A roomis heatedby a refrigerator operating accordingto thereverse Carnot cycle. Howmany times is the amount of heat Qo re-ceivedby theroombyburningfirewood ina furnace lower than theheat Ql impartedtothe roomby. arefrigerator actuatedbyaheatengine consumingthesame amountoffirewood. The engine operates with- P 8 Cin thetemperature range T1 = 100 C p, --,.--.....--'and T,=oo C. Atemperature of T; == 16Cshould bemaintained intheroom. Theambient temperature T ~ ==-lOC.5.205. The working cycle of an idealsteamengineisshownin Fig. 9: (a)whenadmission of the steam fromthe boilerinto thecylinderis begun, the pressure po - A + - - - ~ - - - - ....... Ein the latterincreases at a constant vo- Ilume Vo frompo to PI (line AB); (b)uponfurther admissionofthe steam,the piston movesfrom left to right(line Be) ata constant pressurePI; (c)whenthe pistoncontinues to moveto the right, the supply of steamfromthe boiler into the cylinder is shut off and the steam expands adiaba-tically(line CD); (d) when the piston is in its extreme right-hand po-sitionthe steam emergesfromthe cylinderinto a cold sink, andthepressure drops ata constant volumeV, to po (line DE); (e) during thereverse stroke the piston expels the remaining steam at a constant pres-sure po and the volume drops fromVI to Vo (line EA).Determine thework done bytheengine percyclejf Vo= O.5 litre, V1=1.5 litres,V2=3.0 litres, po= Iat, PI=12 at and theadiabaticexponent is1.33.5.206. A steam engine ratedat 14.7kW consumes8.1kg of coalwltha calorific value of 3.3x 10' J/kg during one hour of operation.The temperature of the boiler is 200Cand that of the cold sink 58 C.Determine theactual efficiency of the engine '11and compare itwiththeefficiencyT)2 of an ideal heat engine operatingaccording to theCarnot cyclewithin the sametemperature range.5.207. The piston area in a steam engine rated at20 hp is 200 em'andthe pistonstroke1=45em, The "isobaric processBe (Fig. 9)takes place when the piston travels one-third of its stroke. The volumeV. as compared to the volumes VI and V.may be neglected. The steam84P DPi -P, -Fig. 10EPROBLEMS(5.208-5.2115.212-5.221) ,\\OLECULAR PHYSICS ANDTHERMODYNAMICS85V,chamber beafter compression? (3) Findtheworkperformedduringcompression. Thepolytropic exponent is l.3.5.212. Determinethe efficiency ofan internal-combustion carbu-rettor engine if theexponentis 1.33 and the compression ra-tiois: (1) =4, (2) v:=6, (3) =8.5.213. The carburettor engineof a "Volga"car consumes a minimumof 265grammes of petrol per hp/h. Determine the losses due to frictionthermal conductivity, etc. Thecomp- Pc'ression ratiois 6.2 and the calorific va- P, -- .Dlue of the petrol 4.6x 101J /kg. Thepolytropic exponent isequal to1.2.5.214. The cycleof afour-strokeDiesel engine is depictedin Fig. 11: (a)line AB-airis suckedinto a cylinder(Po= 1at); (b) line Be-the air is comp-ressed adiabatically to thepressurePI;(c) atthe end of the compression stroke,fuel is injected into thecylinder, it ig- PDnitesinthehot air and burns; the pis-ton moves to the right first isobarically(line CD)and thenadiabatically (lineDE); (d) atthe end of adiabatic expan-sion, theexhaust valveopens and the pressure drops to p (line 8);(e) when thepistonmoves totheleft. themixture isexpelledfromthecylinder (line BA). Findtheefficiencyof theengine.5.215. Theadiabaticcompressionratioofaninternal-combustionDiesel engine is 16and the adiabaticexpansion ratio 6.4. Whatis theminimumamountof fuel oil consumedbytheengine per hourifitspower is 50 hp, thepolytropic exponent 1.3 andthe calorific value ofthe fuel 4.6X 101J /kg?5.216. Find the change in entropy upon the conversionof 10gram-mes of ice at -200C into steam at 100 C.5.217. Find the increment in entropy upon the conversion of 1gram-meof waterat 0 C into steam at 100C.5.218. Findthe change inentropyupon the melting of 1 kg of iceat 0 C.5.219. Liquidlead at the meltingpoint is poured onto ice at 0 C.Findthe change in entropy during thisprocess if the amount of lead is640 grammes.5.220. Findthe change in entropyduring thetransitionof 8 gram-mes of oxygen from avolume of10 litresat atemperatureofBO Cto a volume of 40 litresata temperature of 300 C.5.221. Find the change in entropy during the transition of6grammes of hydrogen from a volume of 20 litres at a pres-86PROBLEMS(5.222-5.231sure of 1.5xI06N/m2to a volume of 60 litres at a pressureof1x10&N/m25.222. Hydrogen amounting to6.6 grammes expands isobaricallyuntil its volume doubles.Findthe change in entropy during expansion.5.223. Findthechangein entropyduringtheisobaric expansionof8 grammes ofheliumfromthevolume VI= 10 litres to V2=25litres.5.224. Find thechangeinentropyuponthe isothermal expansionof 6 grarnmes of hydrogenfrom10' toO.5x 10' N/m25.225. Nitrogen amountingto 10.5 grammesexpands isothermallyfrom a volume of VI =2 Iitres to V2=5litres, Findthe entropyincre-ment during this process.5.226. Oxygen (10grammes) is heated from/1=50 Cto /2=== 150 C. Find thechangein entropyif the oxygenis heated: (1) iso-choricallyt (2) isobarically.5.227. Theheating of1 kmole of abiatornicgasincreases itsabso-lutetemperature 1.5times. Find thechangeinentropyifthe gas isheated: (1) isochorically, (2) isobaricallyt5.228. Theheating of 22 grammes of nitrogenincreases itsabsolutetemperature1.2 times andits entropyby 4.19J/deg. Under what con-ditions wasthe nitrogenheated (at aconstant volumeoraconstantpressure)?5.229. Findthe changein entropyupon thetransition of a gas fromstate Ato state B for the conditions of Problem5.194if thetransitionoccurs: (1)alongACB, (2)alongADB(see Fig. 8).5.230. Onecubicmetreof air at atemperatureof 0C andapres-sureof 2kgf /cm! expands isothermallyfromthevolume VI toV2==2V1o Findthechange inentropy duringthis process.5.231. The change inentropy betweenthe twoadiabatic linesofa Carnot cycleis1 kcal/deg. Thetemperaturedifference betweenthetwoisothermal linesis equal to1000C. What amount of heat will beconverted intoworkduringthiscycle?6. Real GasesTheequationof stateforreal gases (Van der Waals equation) forone kilornole has the form:where Vo= volumeof onekilornoleofagasaandb =constants different for different gasesp= pressureT= absolute temperatureR= gas constant. '-8.1-6.6) MOLECULAR PHYSICS AND THERMODYNAMICS81TheVanderWaals equationreferredto any mass m of a gas may bewritten as( P+;,';,) (V-: b) = ~ RTwhereV =volumeof all thegasIt =mass of one kilomole.In this equation ;,';, =Pi isthepressure duetotheforcesof inter-action of the molecules, and~ =Viisthevolumeconnectedwith....the own volumeof the molecules.The constants a and b of a given gas arerelated toits critical tempe-rature Ttor, critical pressure Pel' and critical volume VOL''' as follows:a 8aVocr =3b; PCT = 2761; TCT =27bR'Theseequationscan besolvedwithrespect totheconstants aandb:a =27TJrR' and b =~ c r R64Pcr PCI'Ifweintroducethefollowingnotation:'t=...!..- n=..E...., w=.!!..Tel' ' Per VocrtheVander Waalsequationwill takethetransformedform(for onekilomole)(n+;,) (3(1)-1) =816.1. Expressin 51 units the constants a and b containedinthe VanderWaals equation.6.2. By usingdataonthecritical valuesof Tel'andPCI'forcertaingases (see Table V), find forthemthe constants a and b inthe VanderWaals equation.6.3. What isthetemperatureof2grammesof nitrogenoccupyingavolume of 820 ern! at a pressure of 2atm? Consider the gas to be:(1) ideal, (2) real.6.4. What isthetemperature of 3.5grammes of oxygenoccupyinga volume of 90 ern! at a pressure of 28 atm? Consider thegas tobe: (1)ideal, (2)real.6.5. 10 grammes of helium occupy a volume of100 cmSat a pressureof 108N/m2 Find the temperature of the gas, considering it as:(1) ideal, (2) real.6.6. One kilomole of carbon dioxide gas has a temperature of 1000c.Find the pressure of the gas, consideringit as: (1) real, (2) ideal.Solve the problem for the volumes: (a) V1=1rna, and(b) Vz=O.05m388PROBLEMS(6.7-6.186.7. A closed vessel with a volume of V=0.5 mScontains 0.6 kmoleof carbondioxide gas at a pressure of .3x 108N/m2 By using the Vander Waalsequation, find thenumberof tinesthetemperature of thisgasis toberaisedtodoublethepressure.6.8. Onekilomoleofoxygenhas a temperature of t=27 CandapressureofP= 107N/m2 Find the volume of the gas, assumingthat it behavesintheseconditionsasareal gas.6.9. Onekilomoleofnitrogenhasatemperatureof t=27 C andapressure of 5x 108N/m2 Find thevolume of the gas assumingthatit behaves intheseconditionsasareal gas.6.tO. Findtheeffectivediameterof anoxygenmolecule, assumingthat thecritical values Tcr andPer for theoxygenareknown.6.11. Find theeffectivediameterofanitrogenmolecule: (I) fromthe given value of the mean freepathof the molecules in standardconditions l=9.5x 10-8ern, (2) from the knownvalue ofthecon-stant binthe Van derWaals equation.6.12. Findthe mean free pathof a carbondioxide molecule in stand-ard conditions. Calculate the effective diameter of the molecule, assum-ing thecritical temperatureTel' andpressure p.;tobe known for car-bondioxide gas.6.13. Findthe diffusion coefficient ofheliumat atemperatureoft=17C and a pressure of p=I.5X10' N/m2 Calculate theeffectivediameter of a helium atom, assuming Terand PCI' to be known for helium.6.14. Plot theisothermal lines p=f(V)for one kilomole of carbondioxidegasat atemperatureof0 C.Consider thegasas: (1)ideal,and (2) real. Takethefollowingvaluesof Vinmt/kmolefortherealgas: 0.07, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, 0.20, 0.25, 0.30, 0.35and0.40, andforthe ideal gaswithin0.2~ V ~ 0.4 m-/kmole.6.15. Findthepressure due to theforces of interaction of themole..cules containedinonekilomoleof agas instandardconditions. Thecritical temperature and pressure of this gas are Te,.=417Kand Pcr== 76 atm, respectively.6.16. Theforces ofinteractionbetweenhydrogenmolecules arene-gligible, the most important factor being the sizeof the molecules.(1) Write the equationof state for such a semi-ideal gas. (2) Find theerror if thedimensions of themolecules aredisregarded in calculatingthe number of kilomoles of hydrogenin a certain volume at a tempera-tureoft=O C andapressureof p=2.8x 107N/m'.6.17. A vessel with a volume of 10 litres contains 0.25 kg of nitrogenat atemperatureof 21 C. (1) What partof thegaspressure will thepressure duetotheforces of molecular interactionbe?(2) What partof thevolume of the vessel willbe occupiedby theown volume of themolecules?6.18. Acertaingasamountingto0.5krnole occupies avolumeofV1=lm3 When the gas expandstoVs=1.2m3, workisdoneagainst6.19-6.26)MOLECULAR PHYSICS ANDTHERMODYNAMICS89the forcesof interactionof the moleculesequal to W=580 kgf-m.Findtheconstant aintheVanderWaalsequationforthisgas.6.19. Twenty kilogrammes of nitrogen expand adiabatically intoa vacuumfromVl=l mStoV2=2rn", Find the dropof the tern.perature duringexpansion ifthe constant a in the Vander Waalsequation isknownfor nitrogen. (Seethe answer to Problem6.2.)6.20. Half a kilomole of a triatomicgas expands adiabatically intoavacuumfromVl =0.5rnato V2=3rna. Thetemperature of thegasdrops by12.2. By using these data, find the constant ain the Van derWaals equation.6.21. (1) What pressureis required toconvert carbondioxidegasinto liquid carbon dioxide at a temperature of: (a) 31 Cand (b)50 C?(2) What maximumvolume canbe occupiedbyI ~ g of liquidcarbondioxide?(3) What is the highest pressureof the saturatedva-pours of liquidcarbon dioxide?6.22. Findthedensityof watervapoursin the critical state if theirconstant bintheVander Waalsequationis known. (See theanswerto Problem 6.2.)6.23. Findthedensityof heliumin the critical state, assuming thecritical valuesof TcrandPcrtobe known for it.6.24. One kilomole of oxygen occupies a volume of 0.056 rna at a pres-sureof 920 at. Findthetemperature of thegasusing thetransformedVanderWaalsequation.6.25. One kilomole of helium occupies a volume of V=0.237rn3atatemperature of t=-200 C. Findthepressure of the gas using thetransformedVander Waalsequation.6.26. Determinehowmany times the pressureofagas is greaterthanits critical pressure if thevolume andtemperature of the gas aredouble their critical values.7. SaturatedVapoursandLiquidsTheabsolutehumidityis thepartial pressure of thewater vapourspresent in theair. The relative humidity wis the ratio of theabsolutehumidity to thepartialpressure of watervapours saturating the spaceat the given temperature.Thespecific heat ofvaporization (evaporation) r istheamount ofheat requiredto convert aunit mass of aliquidintovapour at a con..stant temperature.Themolecular heat ofvaporization'0is equal to'0= t-L'wheref.1 is the mass of one kilomole.The relationship betweenthe pressure of a saturated vapour P,and the temperature is describedby the Clausius-Clapeyron equa-90tlonPROBLEMSdps '0dT=T(Vo-V,)where Vo=volume of one kilomole of vapourV,=volume of onekilomole ofliquid.The relativechange in the volume of aliquid when heatedis deter-minedbytheformulaAVy=' V ~ twhere'V is thecoefficient of volume expansion.The relative change in the volume of a liquid when the pressure chan-gesiswhere k is the coefficient of compression.The coefficient of surface tension ais numerically equal to the forceapplied to a unit length of the edge of the surface film of a liquid, i.e.Fa=TWhen thefilmarea changes by L\Athefollowing work isperformedAW= a ~ AThe additional pressure caused by curvature of the surface of theli-quidis determinedbytheLaplace formulaL\p = a(-.!-. +-!-)RtR2where R1 andR are the radiiof curvature of two mutuallyperpendi-cular cross sections of the surface of the liquid.The radius R is positiveif thecentre of curvatureis inside theliquid(a convex meniscus) andnegative if it is outside the liquid(a concave meniscus).Theheight of theliquidin a capillarytubeh=2(% cos 8'pgwherer=radius of the tubep=density of the liquid8=wettingangle.Withcomplete wetting6=0andwithno wetting8=n.The pressureof saturatedvapOUTS PI above a concave surface of ali-quidisless.andaboveaconvexsurfacerrtorethan thepressurepo7.1-7.2) MOLECULAR PHYSICS AND91aboveaflatsurface. Theadditional pressureisA 2apOUP=Pt --PO=+ pRwhere p=densityofthe liquidpo== densityof the saturatedvapoursof the liquidR=radiusofcurvatureofthe surface ofthe liquid.Theosmoticpressurepof asolutionisrelatedtoits absolutetem-perature Tbythe Van't HoffformulaP= CRTwhere Risthegas constant andC=::visthenumber of kilomolesof the substance being dissolvedin a unit volumeof the solution(molarconcentration of the solution).For solutionsof undissociatedmolecules ofasubstancem NC=f1V=NAwhere NA;=:Avogadro's numberN=number of moleculesof thedissolvedsubstance inaunitvolume.Whendissociationoccurs, thenumberofparticlesinaunit volumewill be greater, and theosmoticpressurewill increase.Thepressureof thesaturatedvapoursabovethesolutionis smallerthanabovethe puresolvent. Withasufficientlylow concentrationofthesolution, therelativedropinthepressureof thesaturatedvapourabovethesolutioncanbedeterminedfromtheRaoult lawpo-p z'--,;;- =z+z'where po=pressure of the saturated vapour above the pure solventp=pressure of the saturated vapour above the solutionz' =numberof kilomoles ofthe dissolvedsubstancez=number of kilomoles ofthe liquid.Problems relating totheviscosityofliquidsaregiveninSection 4 ofChapter 1.7.1. Table VI givesthepressure of water vapours saturating a spaceat various temperatures. How can these databe used to compile a tableshowing theamount of water vapoursin1 rna of airsaturated withthevapours at various temperatures?Bywayof example, calculate theamountof saturated water vapours in I rn3of air at atemperature of50C. 7.2. Find thedensityof saturatedwatervapours at atemperatureof 50 C.92PROBLEMS(7.3-7.137.3. How manytimesis thedensityof saturatedwater vapoursata temperature of 16Clower than that ofwater?7.4. Howmany times is the density ofsaturated water vapoursat a temperature of 200 C greaterthanat a temperatureof100 C?7.5. Whatis the weight of the water vapours in 1rnBof air on a warmdayat a temperature of 30 Cand a relative humidity of 75percent?7.6. The relative humidity of the air in a closed space with a volumeof V= 1 rna is equal to60 per cent at atemperatureof 20 C. Howmuch more water should be evaporatedinto this space for the vapourstobesaturated?7.7. Thetemperatureinaroomis18C and the relative humidityis 50 per cent. A metal tea-kettle is filled with cold water. Whatis thetemperature of the wateratwhich thekettlestops being covered withmist?7.8. Find the number of molecules of saturated water vapour contai-ned in 1 em' at a temperature of 30 C.7.9. Halfa gramme of water vapouroccupies avolume of 10 litresat a temperature of 50 C. (1) Determine the relative humidity.(2) What amount of vapour will be condensed if the volume is halvedisothermally?7.10. A Wilson cloud chamber witha volume of 1 litre contains airsaturated withwatervapours. Theinitialtemperatureof thechamberis 20 C. As the pistonmoves, thevolume of thechamber increases1.25 times. The expansion is adiabatic,x=CplCfj being equal to1.4.Find: (1) thepressure of thewatervapoursbefore expansion, (2) theamount ofwater vapours inthe chamber before expansion, (3) thedensityof thewatervapours before expansion, (4) thetemperature ofthevapour after expansion (disregardthechange inthetemperaturedue toheatevolutionduring vapour condensation), (5) theamount ofwater vapours condensed into water, (6) thedensity ofthe water vapo-urs after condensation, (7) the degreeof supersaturation,l.e., the ratioof thedensity of the water vapour after expansion(but before conden..sation)tothedensityof thewatervapour saturating thespace at thetemperaturewhichset inafter condensation.7.11. Findthe specific volume of water in theliquidandvaporousstates instandardconditions.7.12. Using the first lawof thermodynamics and the data in Tables VandVI, findthespecific heat ofwater vaporizationat 200 C. Thecritical temperaturefor water Tcr=647 K andthecritical pressurePcr=217 atm. Checktheresult obtainedwiththeaidof thedata inTable VII.7.13. What part of the specific heat of watervaporizationata tem-peratureof100C is spent toincrease the i ~ e r n a J energy of thesys-tem?7.14-7.23) MOLECULAR PHYSICS AND THERMODYNAMICS937.14. The s ~ e c i ~ c heat of vaporizatioyn of .benzene (CeHe) at atem-perature of 77 CIS equal to 95 cal/g. What IS the change in theinter.nal energy upon the evaporation of 20 grammes of benzene at thistem-perature?7.15. UsingtheClausius-Clapeyronequationandthedata inTab-le VI, find thespecific heat of watervaporizationat a temperature of5 C. Checkthe result obtained with the aid of the data in Tab-le VII.7. 16. The pressureof saturatedmercuryvapours at temperaturesoftl= 100 Candt2=1200Cis equal toPI= 0.28mm HgandP2==0.76mmHg, respectively. Determine the meanspecificheat ofmercuryvaporizationwithinthis temperaturerange.7.17. Theboiling point of benzene(CaHe) at p=1 atrnis equal to80.2 C. Determinethepressure of its saturated vapours at a tempera-ture of75.6 Cifthemeanspecific heat of vaporizationisequal to4X 106J/kgwithinthegiventemperaturerange.7.18. The pressure of the saturated vapours of ethyl alcohol(C2H,.OH)is equal to133 mm Hgat atemperatureof40 C, andto509 mm Hg at a temperature of 68 C. Find the change in entropy uponthe vaporizationof 1 grammeofethyl alcohol at a temperatureof50C.7.19. Thechange in entropyuponthevaporizationof 1 kmoleofa certainliquidat atemperatureof 50C is133 J/deg. Thepressureofthesaturatedvapours ofthis liquid at a temperatureof 50Cisequal to 92.5mm Hg. By how much will thepressure of the saturatedvapours of this liquidchangewhenthetemperaturechanges from50to 51 C?7.20. Findthelimitpressure to which a vessel can be evacuated bymeans of amercury-diffusionpumpoperating withoutamercurytrapIf thetemperature of thepump water jacket is15 C. Thepressure ofthesaturatedmercuryvapours at atemperatureof0Cisequal to1.6x 10-- mmHg. The specific heat of vaporization of mercuryshouldbetakenequal to75.6 cal/gwithinthetemperaturerangeof0-15 C.7.21. Knowing that the densit y of mercury at a temperature of0C is 13.6 g/cm-, find its densityat3000C. The coefficient of volumeexpansionof mercuryis tobe considered constantandits meanvalueequal to1.85x 10-- deg-1withinthegiventemperaturerange.7.22. Thedensityof.mercuryis equal to13.4 g/cm"at a tempera-tureof1000C. At what temperatureisthemercurydensityequal to13.1 g/cm3? The coefficient of volume expansion of mercuryis equal to1.8x 10-' deg-17.23. Taking the mean value of the coefficient of water compressionequal to 4.8 X 10-&atrn, find thedensityof seawater at adepthof5 krn if its density at the surface is 1,030kg/rn". In calculating the hyd-94PROBLEMS(7.24-7.33rostatic pressureof the seawater, assume its densityto beapproxi-matelyequal tothedensityofthewaterat thesurface.7.24. At 0 C andatmosphericpressure, the coefficient of compres-sion of benzene is equal to 9x 10-1atm-1 and its coefficient of volumeexpansion to1.24x 10-3deg- 1 What externalpressure must be appli-ed for thevolumeof the benzenetoremainunchangedwhen heatedby I 0 c.7.25. Thecoefficient of volumeexpansionof mercury isequal toy=1.82xIO-' deg-1 Find the coefficient of compression iftheex-ternal pressure must be increased by 47atmfor the volumeof themercurytoremainunchangedwhen it isheatedby 10c.7.26. Findthedifference inthelevels ofmercury int\VOidenticalcommunicating glass tubes if the left-hand tube is maintained ata tern..perature of 0 C andthe right-hand oneis heated to100 C. Theheightofthe left-handtubeis 9Ocm. The coefficientof volume expansionofmercury is 1.82x 10-' deg-1 Disregard theexpansionof theglass.7.27. Mercury is poured intoaglass vessel withaheight of H== 10 ern. At a temperature of t=2O Cthe level ofthe mercuryish= 1 mmbelow theupper edgeofthevessel. By howmuchcanthemercurybe heated so that it does not flow outof thevessel?Thecoef-ficient of volume expansion of mercury is y=I.82x 10-4deg-1Disregardtheexpansionof theglass.7.28. A glassvessel filledwithmercuryat atemperature of 00Cuptoitsedges weighs 1 kgf. The emptyvessel weighsO.1kgf. Neglect-ing the expansion of the glass, find the amount of mercury which can becontainedinthevessel at atemperature of t()()O C. Thecoefficient ofvolumeexpansionofmercury is 1.8 x10-' deg- t.7.29. Solve the previous problemtakinginto account the expansionof theglass. Assume that thecoefficient of volumeexpansionof theglass is equal to 3X10-&deg-17.30. A glassvessel is filleduptoitsedges withliquid oil at a tern..perature of 0 C. Whenthevessel withtheoil washeatedto 1000C,sixper cent of the oil flowed out. Find the coefficient of volume expan-sionof the oilYoil' assuming the coefficient of volume expansion of theglass to beequal toy =3 x10-1deg-17.31. What will therelative error indeterminingthecoefficientofvolume expansion of the oil inthe conditions of theprevious problembe if the expansionof the glass is neglected?7.32. The temperaturein aroomis 37 C andthe atmospheric pres-sure760 mmHg. What pressure(inmmHg)will be shownbyamer-curybarometer hangingintheroom? Consider theexpansionof theglass tobe small as compared with that of the mercury. The coefficientof volumeexpansionof mercury is 1.82X10- deg-l.. 7.33. (I) What force must be applied to a horizontal aluminiumring with aheight of h== lOmm, anInternal-diameter of d1==50 mm7.34-7.41)MOLECULAR PHYSICS AND THERMODYNAMICS95DFig. 12andanexternal diameterofd2=52 mmtotear theringawayfromthesurface of water? (2) What part of theforce determinedis duetotheforce ofsurfacetension?7.34. Aringwithaninternal diameter of25mmandanexternaldiameter of 26 mm is suspended on a spring withadeformationcoef-ficient of 10-4kgf/mmandtouchesthesurface of aliquid. Whenthesurface of theliquid lowered, the ringbroke away from itupon expan-sionof the springby 5.3 mm. Find the coefficient ofsurface tension of the liquid.7.35. Frame ABeD (Fig. 12) with a movablebar KLis coveredwitha soapfilm. (1) What shouldthe diameter of the copper bar KL be for it to remaininequilibrium? (2) What is the length 1of thebar ifisothermal workequal fo4.5X 10-&J is performedwhen the bar movesover a distance of 1 ern. Forsoapywater ex = 0.045 N/m.7.36. Alcohol flowsout drop by drop from a vessel Athrough a vertical tube with an internal diameterof2 mm. Find the timeduring which10grammes ofthealcohol will flowout if the intervalbetween dropsis 1 second. Assume thatthe diameter of the neckof a drop at themoment it breaks away is equal to the internal diameter of thetube.7.37. Water flows out drop by drop from a vessel througha verticaltubewithaninternal diameterofd =3mm. Whenthewater coolsfromt1=1000Cto t2= 200C, the weight ofeachdropchanges byAG =13.5X 10-0kgf. Knowingthecoefficient of surfacetensionofwaterat 20 C, find this coefficient at100 C. Assumethat the diameterof theneck of adrop at themoment it breaksawayis equal tothein-ternal diameter ofthe tube.7.38. Twenty drops of lead were formed when the lower end of a ver-t ically suspended lead wire 1mm in diameter was melted. By howmuchdidthewirebecome shorter?Thecoefficient of surfacetensionof li ...quidlead is 0.47 N/m. Assume that the diameter of the neck of a dropat themoment it breaksawayis equal tothediameter of thewire.7.39. Drops of water fall from a vertical tube withan internal radiusof r= I rom. Find the radius of a drop at the moment when it breaksaway, considering ittobe spherical. Assume that thediameter of theneck of a drop at themoment it breaksawayis equal totheinternaldiameter of the tube.7.40. By how much will a mercurydrop obtained from themergingof twodrops each witha radius of 1 mm be heated?.7.41. What workshouldbeperformed against theforces of surfacetension to split a spherical mercury drop with a radius of 3 mm into twoidentical drops?96PROBLEMS(7.42-7.537.42. What work shouldbe performed against theforces of surfacetension todouble the volume of a soap bubble witha radius of 1em.Thecoefficient of surfacetension of a soap solutionis 43x 10-3N/m.7.43. What work shouldbe performed against theforces of surfacetension to blowa soap bubble (a=O.043N/m) 4 ern in diame-ter?7.44. Determine the pressure of the air (in mm Hg) in an airbubblewithadiameter of d=O.OI mmat adepthof h=20 cmbelowthesurface of water. The external pressure Pl=765 mmHg,7.45. The pressureof the air inside a soapbubble is I mmHggreater thantheatmosphericpressure. What is thediameter ofthebubble? The coefficient of surface tension of the soapsolution is0.043 N/m.7.46. Findthedepth of anairbubbleunder waterif thedensity ofthe air in the bubble is 2 kg/m", The diameterof the bubbleis0.015rnrn, the temperature 20 Cand the atmospheric pressure760 mm Hg.7.47. How manytimesis thedensity of air in a bubblein waterat8 depth of 5 m belowthe surface greater thanthe density of the air atatmospheric pressure (at the sametemperature)? "The radius of thebubble is5x 10- mm.7.48. Anopencapillary tube with an internal diameter of d==3 mmisloweredinto a vessel with mercury. The differencebetweenthelevels of themercuryin the vessel andin the capillary tube ~ h =E::3.7 mm. What is theradiusof curvature of themercurymeniscusin the capillarytube?7.49. Anopencapillary tube with an internal diameter of d=== 1mm is loweredinto a vessel with water. The difference between thelevelsofthewater inthevessel and inthecapillarytube is6.h==2.8ern, (1) What istheradius of curvature of themeniscus inthecapillarytube? (2) What wouldthe differencebetween the levelsinthevessel andthecapillarytubebe if wetting were complete?7.50. To whatheight will benzene rise in a capillary tube whose in-ternal diameter is d=1 mm?Consider wetting tobecomplete.7.51. What shouldtheinternal diameterof a capillarytubebe forthe water to rise in it by 2cmwith complete wetting? Solve the problemfor the cases when the capillarytubeis: (1) on theEarth, (2) on theMoon.7.52. Findthedifferencein thelevels of mercuryin two communi-cating capillary tubes withthe diameters d1=1 mm andd2=2 mm,respectively. Consider that thereis absolutelyno wetting.7.53. What should the maximumdiameter of thepores in the wickof an oil stove be for the oil to rise from the bottom of the stove to theburner(height h=10cm)? Consider the p o r ~ as cylindrical tubes andwettingtobe complete.7.54-7.65JMOLECULAR PHYSICS AND THERMODYNAl\\lCS917.54. Acapillarytubewithan internal radiusof2mmisloweredinto a liquid. Find the coefficientof surface tension of the liquid if theweight of the liquid that has risenin the capillary tube is9x 10-1 kgf.7.55. Acapillarytubewithan internal radiusof r=O.16 mmisloweredverticallyinto avessel withwater. Whatshouldtheairpres-sureabovethe liquidin the capillary tube be for the waterlevel in thecapillary tube andin a broad vessel tobe the same?The external pres-surePo=760mm Hg, and wettingiscomplete.7.56. A capillary tube is lowered verticallyintoa vessel with water.Theupper endof''thetubeis soldered. Forthelevelof the waterin thetube andin a broad vessel to be the same, the tube has to be submergedinto the water by 1.5 per cent of itslength. What isthe internal radius of thetube?Theexternal pressure Ais 750 mmHg andwettingis complete.7.57. Theinternal diameter d of barometric tube Afilled with mercury(Fig. 13) is: (a) 5 mm, (b) 1.5 em.Can the atmospheric pressurebedetermined directlyfromthe height of the mercury column? Find the __._ _ __height of the mercurycolumnin eachcaseif theat- -=----=.:.:mospheric pressure Po=758 mmHg. Consider that -=-=-::---=there isabsolutelynowetting.7.58. The internal diameterofabarometric tube Fig. 13is 0.75cm. What correctionshould be made whenmeasuring the atmospheric pressure. according tothe height of themercury column? Consider that thereis no wetting.7.59. What will therelative error bein calculating theatmosphericpressureequal to760 mmHgaccording tothe height of a mercury co-lumnif the internal diameter of thebarometrictubeis: (1) 5 mmand(2)10 mm?Considerthat thereisnowetting.7.60. A greasedsteel needlewhichis unwettable by wateris placedontothesurfaceof water. What will themaximumdiameter of theneedlebeat which it will still remainonthe surface?7.61. Will a greased (unwettablebywater) platinum wire1 mmindiameter float onthesurfaceofwater?7.62. The bottomof a vessel withmercuryhas 8 hole. What canthe maximumdiameter oftheholebeat whichno mercurywill flowout fromthevessel when themercurycolumn is3cmhigh?7.63. Thebottom ofaglassvesselwithanareaof A=30 em> hasaround holewithadiameter of d=O.5mm. Thevessel isfilledwithmercury. Howmuch mercury will remain in the vessel?7.64. Find theweight of a water skater running overwater if undereachof the sixlegs of theinsecta hemisphere witha radius of 0.1mmis formed.7.65. What force must be applied todetach two wetted photographicplates 9x 12 em insizefromeachotherwithout shifting them? The7-357498PROBLEMS(7.66-7.72hathicknessof the water layer betweentheplates is0.05mmand thewetting is complete.7.66. Aliquid is poured between two vertical flat and parallelglassplates at a distance of 0.25 mm from each other. Find thedensityof theliquid if the height whichit risestobetween the plates is 3.1em(a;:::30dyne/em). Consider wettingtobecomplete.7.67. There are fivegrammes of mercury between two horizontalflat and parallel glass plates. Aloadof5kgf isplacedontheupperplate andthedistance between the plates becomes equal to0.087mm.Neglectingtheweightof theplateas comparedwiththat of theload,find the coefficient of surface tension of the mercury. Consider that thereis no wetting.7.68. Anopencapillarytubecontainsa dropofwater. When thetubeisinitsvertical positionthedropformsacolumn withalengthof: (1) 2 em, (2) 4 em, (3) 2.98 cm. Theinternal diameter of the capil-lary tube is 1mm. Determine the radii of curvature oftheupperand lower meniscuses in each case. Consider wetting to becom-plete.7.69. Wateris pumpedinto ahorizontal capillarytubewithanin-ternal diameter of d=2 mmso thata column h=10 em longis for-med. How many grammes of the water wUI flowout of the tubeif it isplacedvertically?Consider wettingto becomplete.Note. Bear inmindthat themaximumlengthofthewatercolumnleft inthe capillarytube should correspondtotheradius of curvatureof the lower meniscus, equaltotheradius of the tube(see the solutionofthepreviousproblem).7.70. A column of alcohol is containedin an openverticalcapillarytube with aninternalradius of ,=0.6 mm. The lower meniscus of thecolumnhangs fromthe bottom endof thetube.Find the height h of the alcohol column at whichtheradius of curvature R of thelower meniscusis equal to: (1) 3 r (2) 2 r and (3) r, Considerwetting tobecomplete.7.71. The tube shown in Fig. 14is openatboth endsandfilledwithkerosene. Theinternalradii a and b areequal "tor1=0.5 mmand '2==0.9 rnm, respectively. At what difference ~ hbetween thelevelswill the meniscus at the endof tube abe: (1) concave witharadius of cur ..vatureof RX= rl' (2) flat, (3) convex withaFig. 14 radiusofcurvatureofRX=r2 t(4) convexandequaltor1? Consider wetting tobe complete. .7.72. Acapillary tube is sosubmerged intoa broad vessel withwater that theupper endof the tubeis above thelevel of the waterinthe vesselby h=2 em, The internalradius'of the capillary tube' =7.73-7.83)MOLECULAR PHYSICSAND THERMODYNAMICS99=0.5 mm. Find the radius of curvature R of the meniscus in the tubeConsider wettingtobe complete. 7.73. Anaerometer floatsinwater whichwets its walls comple-tely. Thediameter ofthevertical cylindrical tubeoftheaerometerd=9 mm. Howmuch will thedepth of submergence of the aerometerchangeif several drops of alcoholarepoured ontothesurface of thewater?7.74. Anaerometer floats in a liquid having a density of p==800 kg/rn"anda coefficient of surface tensionof a,=30 dyne/em.Theliquidcompletelywets thewalls of theaerometer. Thediameterof the verticalcylindrical tube of the aerometer d=9 mm. Howmuchwill the depth ofsubmergenceof the aerometer change if greasingmakes it completelyunwettable?7.75. When 10grammes of sugar(C12H22011) are dissolved in 0.5 lit ...reofwater theosmoticpressure ofthesolution is equal to 1.52XX 10' Nzm>. What is the temperature of the solution? The sugar mole-culesarenot dissociated.7.76. Theosmoticpressure of a solutionat atemperature of 87 Cis equal to 1.65x'l0' N/m2 Whatnumber of water molecules is thereper molecule of thesubstancedissolvedinthesolution? Thereis nodissociation.7.77. Twogrammes of table salt aredissolvedin 0.5litreof wa-ter. The degreeof dissociation of thesalt molecules is 75 per cent.Determine the osmotic pressureofthesolutionat atemperatureof17 C.7.78. When tablesalt is dissolved in water, thedegree ofdlssocia-tion of its molecules is 40 per cent. The osmotic pressure of the solutionis equal to1.21 kgf/cm! at a temperature of 27C. How muchof thesalt is dissolved in1 litreof water?7.79. Table salt in an amount of 2.5 grammes is dissolvedIn 1 litreof water at a temperature of 18C. The osmotic pressure of thesolutionis1.6x 10' N/m2. (1) What is the degreeof dissociationof thesaltmolecules? (2) Howmany particles of the dissolved substance are therein1 ern"of thesolution?7.80. Fortygrammes of sugar (C12H2S011)are dissolved in 0.5litreof water. Thetemperature of the solutionis 50 C. What is thepres-sure ofthesaturatedwater vapoursabovethesolution?7.81. The pressure of saturated vapours above a solution is 31.5 mmHg at a temperature of 30 C. Findtheir pressure at atemperatureof60 C.7.82. Thepressureofsaturatedvapours 2bovea solution Is1.02times smallerthanthat of pure water. Howmanymolecules of waterarethereper molecule of thedissolved substance?7.83. One hundred grammes of an unvolatile substance are dissolv-ed in 1 litre of water. The temperatureof the solutionis 90 C and the7*100PROBLEMS (7_84pressure ofthesaturatedvapours above thesolution515.9 nunHg.Determine the mass of one kilomole of thedissolvedsubstance.7.84. An unvolatile substancewitha mass of one kilomole of J.L=z=60 kg/kmole is dissolved in water. The temperature of the solutionis BO C andthe pressure of the saturated vapours above the solution353 mmHg. Find the osmotic pressure of the solution.8. SolidsThe change in the melting point dT upon a change in the pressurebydp isdescribedby theClausius-ClapeyronequationdT=Tv,-Vsdpqowhere qo=molecular heat offusionV,=volume of one kilomole of liquIdV,,==volume of one kilomole of solidT=melting point.When the temperatures are not too low, solids obey Dulong and Pe-tit's law, according to which the atomic heat of all chemically simplesolids is approximately equal to 3 R=25x 108J /kg-atorn-deg==6 cal/g-atom -deg.The amountof heat transferred byconduction duringthetimecanbe determined from theformula AAAtwhere =temperature gradient in a direction perpendicular to theareaAAA== coefficient of thermal conductivity.When thetemperature rises, thelength of solids increases to afirstapproximationlinearlywiththetemperature, i.e.,l,=lo{ I +a,t)whereIt=length of asolidbodyat thetemperaturet10= length of thebody at thetemperature 0 Cex, = coefficientof linearthermal expansion.For isotropic solids a ='V' where 'v is thecoefficientof volumethermal expansion.. Indeformation due tolongitudinal (or unilateral compres-sion) of a rod,the relativechange initslength according toHooke's8.1-8.7]lawisMOL-ECULARPHYSICSANDTHERMODYNAMICS101Ai PI7=7where Pl=specific load, i.e., Pl=~ (here Fis the tensile or compres-siveforce, andAis thecross.. sectional area)E =modulus of elasticity(Young's modulus).The relative change in the thickness of a rod in longitudinal tension is~ dd = ~ P Iwhere p isthecoefficient oflateral compression. ThequantityJA.=pEIs knownasPoisson's ratio.To twist a rod(a wire) throughacertainangle q> themoment ofacoupleof forces shouldbe appliedM _nG'&q>- 2lwhere l=lengthof thewirer=radius of the wirea= shearmodulusof thewire material.8.1. WhenI kmole oficeis melted.. thechangeintheentropy is22.. 2 kJ /deg. Findthe change in the melting pointof theice when theexternal pressure is increased by1X 105N/m'.8.2. The melting point of tinis 231.90Cat a pressure of 106Nzrn",and232.2 C at apressure of107N/m2 Thedensityofliquidtinis7.0 g/crn", Findtheincrease in entropywhen 1 kmole of tin is melted.8.3. Themelting point of iron changes by 0.012 C when thepres-sure changes by 1 kgf/cm'.Findthe change in thevolume of one kilo-moleofiron.8.4. Byusing Dulongand Petit's law, find the specificheat of:(I)copper, (2) iron, (3) aluminium.8.5. By using Dulong and Petit's law, find the material which a me-tallicball 0.025 kgf in weight is made of if 117J of heat are requiredto heat it from10 C to 30 C.8.6. By using Dulong and Petit's law, find howmanytimes the spe-cific heat of aluminiumis greater thanthat of platinum.8.7. A lead bullet strikes a wall wit