Simplifications

1、 制作 張昆實Yangtze University制作 張昆實Yangtze UniversityBilingual Mechanics Chapter 4 Energy and Work Chapter 4 Energy and Work4-1 What Is Physics?4-2 What Is Energy?4-3 Kinetic Energy4-4 Work4-5 Work and Kinetic Energy4-6 Work Done by Force4-7 Power Chapter 4 Energy and Work4-8 Work and Potential Energy4-

2、9 Path Independence of Conservative Forces4-10 Determining Potential Energy Values4-11 Conservation of Mechanical Energy4-12 Work Done on a System by an External Force4-13 Conservation of Energy4-1 What Is Physics One of the fundamental goals of physics is to investigate the topic “energy”, which is

3、 obviously important . In this chapter we learn what is physics through the study of energy and work. One job of physics is to identify the different types of energy in the world. potential energy kinetic energymechanic energy gravitational elastic. 4-2 What Is Energy? potential energy is the energy

4、 detemined by the configuration of a system of objects that exert forces on one another. kinetic energy is the energy a body has because of its motion. energy may be transformed from one form to another, but it cannot be created or destroyed, i.e. the total energy of a system is constant.4-3 Kinetic

5、 EnergyNewtons laws of motion allow us to analyze many kinds of motion. However, the analysis is often complicated, requiring details about the motion.There is another technique for analyzing motion , which involves energy. Kinetic energy is energy associated with the state of motion of an object. (

6、4-1)SI unit: joule(J),(4-2)4-4 Work work is energy transferred to or from an object by means of a force acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work. work is energy transferred to or from an object by means of a forc

7、e acting on the object. Energy transferred to the object is positive work, and energy transferred from the object is negative work. You acceletate an object its kinatic energy increaced the work done by your force is positive; You deceletate an object its kinatic energy decreaced the work done by yo

8、ur force is negative;4-5 Work and Kinetic Energy Finding an Expression for Work A bead can slide along a frictionless horizontal wire ( axis), A constant force ，directed at an angle to the wire, accelerates the bead along the wire.BeadwirestartendSolve for , yields(4-4)(4-5)(4-3)(4-6)4-5 Work and Ki

9、netic Energy To calculate the work done on an objectby a force during a displacement, we use only the force component along the objectsdisplacement. The force component perpen-dicular to the displacement does zero work.Since(4-7)General form(4-8)Scalar (dot) productCautions: (1) constant force (magn

10、itude and direction) (2) particle-like object. 4-5 Work and Kinetic Energy Signs for workA force does positive work when it has a vector component in the same direction as the displacement, and it does negative work when it has a vector component in the opposite direction. It does zero work when it

11、has no such vector component. positive work negative workdoest do work4-5 Work and Kinetic Energy Units for work The unit for work is the same as the unit for energy(4-9) net work done by several forcesWhen two or more forces act on an object, their net work done on the object is the sum of the work

12、s done by the individual forces. 4-5 Work and Kinetic Energy Work-kinetic Energy Theorem(4-11)+Work-kinetic Energy Theoremfor particleschange in the kinetic energy of a particle net work done on the particle=(4-5)(4-10)kinetic energy afterThe net work is done kinetic energy Before the net work = the

13、 netWork done4-6 Work Done by ForceA tomato is thrown upward, the work done by thegrivatational force :(4-12)dFgIn the rising process(4-13)dFgminus sign: the grivatational force transfers energy (mgd) from the objects kinetic energy, consistant with the slowing of the object.plus sign: the grivatati

14、onal force transfers energy (mgd) to the objects kinetic energy, consistant with the speeding up of the object.In the falling down process(4-14)4-6 Work Done by Force Work done in lifting and lowering an objectdFgFLifting force: positive work;transfers energy to the object;Gravitational force: negat

15、ive work; transfes energy from the object.dFgFLowering an object. Displacement: downwardLifting force: negative work; transfers energy from the object;Gravitational force: positive work; transfes energy to the object.Lifting an object. Displacement: upward4-6 Work Done by ForcedFgFdFgFThe change in

16、the kinetic energy due to these two energy transfers isThe angle between and .(4-15)(in lifting and lowering ; )(4-17)Up: (4-16)Down:4-6 Work Done by Force The spring forceFig(a): A block is attached to a spring in equilibrium (neithercompessed nor stretched).Fig(b): when stretched to rightThe sprin

17、g pulls the block to theLeft (restoring force)relaxed statestretchedcompressedFig(c): when compressed to leftThe spring pulls the block to theright (restoring force)X axis along the spring(Hookes law) (4-18)The spring force(variable force)(Hookes law) (4-19)K: spring constant4-6 Work Done by Force T

18、he work done by a spring forceassumptions: spring massless; ideal spring(obeys Hookes law); contact frictionless.xFdxdWxixfO(4-20)(4-21)Calculus metheod( work by a spring force )Sumintegration(4-22)(4-23)4-6 Work Done by ForceWork is positive if the block ends up closer to the relaxed position (x=0)

19、 than it was initially. It isnegative if the block ends up father away from x=0.It is zero if the block ends up at the same distance from x=0.If and then(4-23)(4-24)( work by a spring force ) The work done by a spring force4-6 Work Done by ForceIf the block is stationary before and after thedisplace

20、ment, thenSuppose we keep applying a force on theblock, our force does work , the spring forcedoes work on the block.(4-25)(4-26)The change inthe kinetic energy of the block The work done by a spring force4-6 Work Done by Force(4-26)If a block that is attached to a spring is stationary before and af

21、ter a displacement, then the work done on it by the applied force displacing it is thenegative of the work done on it by the spring force. Sample problem 4-2 : P93 The work done by a spring force4-6 Work Done by Force One-dimensional Analysisthe same method as the calculation of the work done by a s

22、pring force (calculus). (4-29)(4-28)(4-27)(4-30)( Work:variable force )4-6 Work Done by ForceThe work done by during isSimplifications: only depends on and so on Three-dimensional Analysis(4-34)A three-dimensional force acts on a body(4-33)(4-32)(4-31)Incremental displacement4-6 Work Done by Force w

23、ork-kinetic energy theorem with a variable force Let us prove:(4-35)Substituting Eq. 4-38 into Eq. 4-35:(4-36)(4-39)work-kinetic energy theorem(4-37)(4-38)2-7 Power the instantaneous power can be expresscd in termsof the force and the parti-cles velocity : Power : The power due to a force is the rat

24、e at which that force does work on an object.(4-46)Instantaneous power(4-41)(4-40)work down by a force: in a time interval :the average powerUnit: 1watt = 1 = 12-8 Work and Potential Energy work-kinetic energy theorem(4 -10)Exp. A tomato is thrown upwardRising: does , leadsEnergy transferred from th

25、e tomato, Where does it go? To Increase the gravitational potential energy of the tomato-earth system!(the seperation is increased ! )EarthDiscuss the relation: Falling: does , leads Energy transferred from the gravitational potential energy of the tomato-earth system to the kinetic energy of the to

26、mato !2-8 Work and Potential Energy Work and Potential Energy Discuss the relation: For either rise or fall, the change in gravitational potential energyis defined to equal the negative work done on the tomato by thegravitational force.Earth(4-47)This equation also applies to a block-spring system (

27、P 98)IFis always true,The other form of energy is a potential energy Conservative and Nonconservative Forces Key elements: A system (two or more objects);2. A force acts between a object and the rest part of the system; when configuration changes, the force does work transffering the kinetic energy

28、of the object into some other form of energy.Reversing the configuration changes, the force reverses the energy transfer, doing work .The force is a conservative force!4-9 Path Independence of Conservative Forces Conservative and Nonconservative ForcesNonconservative Forces: a force that is not cons

29、ervativeExp. : (1) the kinetic frictional force : (2) the drag forceA block is sliding on a rough surface, the kinetic frictional force does negative work Transfer kinetic energy thermal energySo the thermal energy is not a potential energy!The frictional force Nonconservative Forces !4-9 Path Indep

30、endence of Conservative Forces The closed-path test : to determine whether a force is conservative or nonconservative. The net work done by a conservative force on a particle moving around every closed path is zero.The work done by a conservative force on a particle moving between two points does no

31、t depend on the path taken by the particle.4-9 Path Independence of Conservative Forces4-9 Path Independence of Conservative ForcesThe work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle.(4-48)ab121ab2 Find the relation between

32、a conservative force and the associated potential energy :The work done by a variable conservative force on a particle (see Eq. 4-35):(4-51)Eq.4-52 is the general relation we sought.4-10 Determining Potential Energy Values(4-52) Gravitational potential energy :A particle is moving vertically along a

33、 y axisFrom4-10 Determining Potential Energy Values(4-52)(4-54)(4-53)(4-55)Gravitational potential energy4-10 Determining Potential Energy Values Elastic potential energy :A block-spring system is vibrating, the spring force does work on the block. (4-57)(4-56)4-11 Conservation of Mechanical Energy

34、Mechanical energy : THe Mechanical energy of a system is the sum of its potential energy and the kinetic energy of the objects within it: ( Mechanical energy )(4-58)Combining (4-59) , (4-60):(4-61)(4-62)Rewriting:A conservative force does work on the object changing the objects kinetic energy (4-59)

35、(4-60)The change in potential energy4-11 Conservation of Mechanical EnergyThe sum of and for any other state of the systemThe sum of and for any state of a system= In a isolated system where only conservativeforces cause energy changes, the kinetic energy and potential energy can change, but their s

36、um, the mechanical energy of the system, cannot change. ( conservation of Mechanical energy )(4-62)Rewriting:(4-63)Rearranging:The principle of conservation of mechanical energy ：4-11 Conservation of Mechanical EnergyWhen the mechanical energy of a system is conserved we can relate the sum of kineti

37、c energy and potential energy at one instant to that at another instant without consideringthe intermediate motion and without finding the work done by the forces involved. (4-64)From Eq. 4-61 :The principle of conservation of mechanical energyNewtons laws of motion4-11 Conservation of Mechanical En

38、ergyA pendulum bob swings back and forth.= constant( the pendulum-Earth system ) Finding the force AnalyticallySolving for and passing to the differential limit :(4-68)( one-dimensional motion )Know Find(4-52)Know FindFrom Eq.(4-47) :4-11 Conservation of Mechanical Energy1. Check Eq.(4-68) :2. At ,T

39、he Potential Energy Curve rightis a turning point4-11 Conservation of Mechanical Energy(4-68)K.E=0F=0 on both sidesrestoring force !F=0 on both sidesK.E=0deflecting force !neutral equilibriumneutral equilibriumstable equilibriumunstable equilibriumK.E=0F=04-11 Conservation of Mechanical Energy Work

40、is energy transferred to or from a system by means of an external forceacting on that system.systemNegativesystemPositive In 4-12:(in other forms)In 4-5 :(only )The work-kinetic energy theorem 4-12 Work Done on a System by an External Force No Friction Involved( Work done on system, no friction invo

41、lved )Ball-Earth systemPositive (4-71)(4-72)throwing a ball upward,your applied force does work Earthkineticpotentialmechanical energy 4-12 Work Done on a System by an External ForceBlock-Floorsystem Friction Involved(4-73)(4-74)(4-75)(4-76)(4-77)(4-78)(4-79)（work done on a system, friction involved

42、） 4-12 Work Done on a System by an External Force The total energy E of a system can change only by amounts of energy that are transferred to or from the system. Countless experiments have proved:The work done on a system = the change in the total emergy (4-80)The law of conservation of energy Total

43、 energyMechanical energyThermal energyInternal energy of any form 4-13 Conservation of EnergyThe total energy E of an isolated system cannot change.(4-81) If a system is isolated from its environment No energy transfers to or from it. W=0 Isolated System(4-82) 4-13 Conservation of EnergyThe total energy E of an isolated system cannot change. Isolated S