Chapter 12 – – Simple Machines

Chapter 12 – Simple MachinesA PowerPoint Presentation byPaul E. Tippens, Professor of PhysicsSouthern Polytechnic State University 2007

SIMPLE MACHINES are used to perform a varietyof tasks with considerable efficiency. In thisexample, a system of gears, pulleys, and leversfunction to produce accurate time measurements.Photo Vol. 1 PhotoDisk/GettyPhotoDisk/Getty

Objectives: After completing thismodule, you should be able to: Describe a simple machine in general terms andapply the concepts of efficiency, energyconservation, work, and power. Distinguish by definition and example between theconcepts of the ideal and actual mechanicaladvantages. Describe and apply formulas for the mechanicaladvantage and efficiency of the following devices:(a) levers, (b) inclined planes, (c) wedges, (d)gears, (e) pulley systems, (f) wheel and axel, (g)screw jacks, and (h) the belt drive.

A Simple MachineIn a simple machine,input work is done bythe application of asingle force, and themachine does outputwork by means of asingle force.A simplemachineFoutFinsinsoutWWin FinsinWout FoutsoutConservation of energy demands that the workinput be equal to the sum of the work outputand the heat lost to friction.

A Simple Machine (Cont.)InputInput workwork outputoutput workwork workwork againstagainst frictionfrictionEfficiency e is definedas the ratio of workoutput to work input.Work outputWork inputFout soute Fin sinA simplemachineFinininsinWin Finsine FFoutoutWWsoutWout Foutsout

Example 1. The efficiencyof a simple machine is80% and a 400-N weightis lifted a vertical height of2 m. If an input force of20 N is required, whatdistance must be coveredby the input force?A simplemachineFin ?sinEfficiencyFout soute Fin sinsoutWWTheTheefficiencyis 80%e 0.80,thereforeadvantageis a orreducedinputforce,but it is Fat theexpense ofFdistance.Thessoutout outor sina greater outmustinput eforcemovedistance.Fin sineFin(400 N)(2 m)sin (0.80)(20 N)sin 5.0 m

Power and EfficiencySince power is work perunit time, we may writeA simplemachineWorkP or Work PttWout P0te WinPtiP0e PiEfficiency is the ratioof the power outputto the power input.Fin ?sinsoutWWEfficiencyPoute PinPower out P0e Power inPi

Example 2. A12-hp winch motor lifts a 900-lb loadto a height of 8 ft. What isthe output power in ft lb/s if A simple Fin ?the winch is 95% efficient? machinesinFirst we mustfind the poweroutput, Po:P0e PiP0 ePisoutWWEfficiencyPoute PinPo (0.95)(12 hp) 11.4 hp 550 ft lb/s (1 hp 550 ft/s): Po (11.4 hp) 6600 ft lb/s 1 hp Po 6270 ftlb/s

Ex. 2 (cont.) A12-hp winch motor lifts a 900 lb loadto a height of 8 ft. Howmuch time is required ifthe winch is 95% efficient?We just found thatPo 6270 WWork out Fo soPo ttNow we solve for t :Fin ?A simplemachinesinsoutWWEfficiencyPoute PinFo so (900 lb)(8 ft)t Po6270TimeTime required:required: tt 1.151.15 ss

Actual MechanicalAdvantageThe actual mechanicaladvantage, MA, is theratio of Fo to Fi.MA 80 Noutput force Fo Fiinput forceFin ?A simplemachineFoutsinsoutWWFor example, if an40 N input force of 40 Nlifts an 80 N weight,the actual mechanicaladvantage is:ActualMechanicalAdvantageMA80 NMA 40 NM A 2.0

An Ideal MachineConservation of energy demands that:InputInput workwork outputoutput workwork workwork againstagainst frictionfrictionFi si Fo so (Work ) fAn ideal or perfect machine is 100%efficient and (Work)f 0, so thatFo siFi si Fo so or si soThe ratio si/so is the ideal mechanical advantage.

Ideal MechanicalAdvantageThe ideal mechanicaladvantage, MI, is theratio of sin to sout.MA A simplemachineFin ?2mFoutsin distance iout distance soFor example, if an input forcemoves a distance of 6 m whilethe output force moves 2 m, theideal mechanical advantage is:soutWW6msinIdealMechanicalAdvantageMI6mMI 2mM I 3.0

Efficiency for an Ideal EngineFor 100% efficiency MA MI. In other words,in the absence of friction, the machine IS anideal machine and e 1.Fo 80 NMA 4IDEAL EXAMPLE:Fi 20 NAsimplemachineFout 400 NWWFin 80 NSin 8 mSout 2 me 100%si 8 mMI 4so 2 mMAe 1.0Mi

Efficiency for an Actual EngineThe actual efficiency is always less than theideal efficiency because friction always exits.The efficiency is still equal to the ratio MA/MI.The efficiency of anyengine is given by:MAe MiInIn ourour previousprevious example,example, thethe idealideal mechanicalmechanicaladvantage. IfIf theadvantage waswas equalequal toto 44.the engineengine waswas onlyonly50%, the50% efficientefficient,the actualactual mechanicalmechanical advantageadvantagewould. Thenwould bebe 0.5(4)0.5(4) oror 22.Then 160160 NN (instead(instead ofof 8080N)-N weight.N) wouldwould bebe neededneeded toto liftlift thethe 400400-Nweight.

The LeverA lever shown hereconsists of input andoutput forces at differentdistances from a fulcrum.FoutroutThe input torque Firi is equalto the output torque Foro.The actual mechanicaladvantage is, therefore:rinFulcrumFinFi ri Fo roFo ri MA Fi ro

The LeverFriction is negligibleso that Wout Win:FoutFo siFi si Fso or Fi sosoutrout rinNote from figure that angles are the same andarc length s is proportional to r. Thus, the idealmechanical advantage is the same as actual.The ideal MI is: M I Fo ri Fi roand M I M AsinFin

Example 3. A 1-m metal lever is used to lift a800-N rock. What force is required at the left endif the fulcrum is placed 20 cm from the rock?1. Draw and label sketch:2. List given info:Fo 700 N; r2 20 cm800 Nr2riF ?r1 100 cm - 20 cm 80 cm3. To find Fi we recall the definition of MI :riMI roThus,and80 cmMI 4;20 cmFoMA 4FiandFor lever: MA MI800 NFi 200 N4

Other Examples of Levers

Wheel and Axel:Application of Lever Principle:With no friction MI MA andFor Wheeland Axel:Fo ri MA Fi roFiRrFoWheel and AxelFor example, if R 30 cm and r 10 cm, aninput force of only 100 N will lift a 300-N weight!IfIf thethe smallersmaller radiusradius isis 1/31/3 ofof thethe largerlarger radius,radius,youryour outputoutput forceforce isis 33 timestimes thethe inputinput force.force.

Single Fixed PulleysSingle fixed pulleys serve only to change thedirection of the input force. See examples:Fin FoutFoutWFinFoutFin

Single Moveable PulleyFin2m1m80 NFinFinFout 80 NFin Fin Fout40 N 40 N 80 NA free-body diagram shows an actual mechanicaladvantage of MA 2 for a single moveable pulley.sinNote that the rope moves a distance Fofo 22 Fin Fout or M A M2 I 2 m while the weight is lifted only 1 m.soutFi

Block and TackleArrangementFiWe draw a free-body diagram:FiFi Fi FiFoWFo4 Fin FoutFoMA 4FiThe lifter must pull 4 m of ropein order to lift the weight 1 m

The Belt DriveA belt drive is a device used to transmit torquefrom one place to another. The actual mechanicaladvantage is the ratio of the torques.MA ForoBeltDriveoutput torque o input torque iSince torque is defined as Fr,the ideal advantage is:Fo roMI MA Fi ririFiro DoBelt Drive: M I ri Di

Angular Speed RatioThe mechanical advantageof a belt drive can also beexpressed in terms of thediameters D or in terms ofthe angular speeds .Belt Drive:MI Do i Di oNote that the smaller pulleydiameter always has thegreater rotational speed.Do BeltDrive Di iSpeed ratio: o

Example 4. A 200 N m torqueis applied to an input pulley12 cm in diameter. (a) Whatshould be the diameter of theoutput pulley to give an idealmechanical advantage of 4?(b) What is the belt tension?To find Do we use the fact thatDoMI 4; Do 4 DiDiDo 4(12 cm) 48 cmNow, i Firi and ri Di/2.Belt tension is Fi and ri isequal to ½Di 0.06 m.ForoDoMI DiMI 4riF i Fi ri 200 N m200 N mFi 3330 N0.06 m

GearsMechanical advantageof gears is similar tothat for belt drive:Gears:Do N oMI Di N iIn this case, Do is thediameter of the drivinggear and Di is diameterof the driven gear. N isthe number of teeth.NiNoIf 200 teeth are in theinput (driving) gear, and100 teeth in the output(driven) gear, the mechanical advantage is ½.

Example 5. The driving gear on a bicycle has 40teeth and the wheel gear has only 20 teeth. Whatis the mechanical advantage? If the driving gearmakes 60 rev/min, what is the rotational speed ofthe rear wheel?N o 22MI ; M I 0.5N i 44Remember that theangular speed ratio isNo 20Ni 40opposite to the gear ratio.N o i i 1; MI OutputOutput angularangular speed:speed:N i o o 2 o 2 2(60 rpm) 120120 rpmrpm

The Inclined PlaneThe Inclined Planesi Fo WIdeal MechanicalAdvantageFisoslope siMI height soWActual Advantage: M A FiBecauseBecause ofof friction,friction, thethe actualactual mechanicalmechanicaladvantageadvantage MMAA ofof anan inclinedinclined planeplane isis usuallyusually muchmuchlessless thanthan thethe idealideal mechanicalmechanical advantageadvantage MMII.

Example 6. An inclined plane has a slope of 8 mand a height of 2 m. What is the ideal mechanical advantage and what is the necessary inputforce needed to push a 400-N weight up theincline? The efficiency is 60 percent.Si 8 mFi Fo 400 NFoM A 2.4 Fi2msi 8 mMI ; MI 4so 2 mMAe ; M A eMI (0.60)(4)MIFo 400 NFi 2.42.4FFi i 167167 NN

The Screw JackFoRFip2 RMI pScrewJackAn application of theinclined plane:Input distance: si 2 ROutput distance: so pScrew Jacksi 2 RMI sopDue to friction, the screw jack is an inefficientmachine with an actual mechanical advantagesignificantly less than the ideal advantage.

Summary for Simple MachinesEfficiency e is definedas the ratio of workoutput to work input.Efficiency is the ratioof the power outputto the power input.Work outpute Work inputFout soute Fin sinPower out P0e Power inPi

SummaryThe actual mechanicaladvantage, MA, is theratio of Fo to Fi.MA A simplemachineoutput force Fo Fiinput forcesinsoutWWThe ideal mechanicaladvantage, MI, is theratio of sin to sout.Fin ?MA EfficiencyPoute Pinsin distance iout distance so

Summary (Cont.)The actual mechanicaladvantage for a lever:Fo ri MA Fi roApplication of lever principle:With no friction MI MAFor Wheeland axel:Fo ri MA Fi ro

Summary (Cont.)Belt Drive:MI Do i Di oForoMI 4output torque oMA input torque iro DoBelt Drive: M I ri DiBeltDriveriF

SummaryDo NoGears: M I Di NiThe InclinedPlanesiFi Fo WsoNiIdeal MechanicalAdvantageslope siMI height soNoWActual Advantage: M A Fi

Summary (Cont.)FoRFip2 RMI pScrewJackAn application of theinclined plane:Input distance: si 2 ROutput distance: so pScrew Jacksi 2 RMI sop

CONCLUSION: Chapter 12Simple Machines

A Simple Machine In a simple machine, input work . input work is done by the application of a single force, and the machine does output work . work by means of a single force. Conservation of energy demands that the work input be equal to the sum of the work output and the heat lost to friction. A simple machine s . s. in s . s. out. W. inin .