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Energy and Work in Physics: Understanding Kinetic and Potential Energy

This chapter explores the fundamental concepts of energy and work in physics, including kinetic and potential energy, work done by force, and conservation of mechanical energy.

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Energy and Work in Physics: Understanding Kinetic and Potential Energy

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  1. BilingualMechanics Chapter 4 Energy and Work 制作 张昆实 Yangtze University 制作 张昆实 Yangtze University

  2. Chapter 4 Energy and Work 4-1 What Is Physics? 4-2 What Is Energy? 4-3 Kinetic Energy 4-4 Work 4-5 Work and Kinetic Energy 4-6 Work Done by Force 4-7 Power

  3. Chapter 4 Energy and Work 4-8 Work and Potential Energy 4-9 Path Independence of Conservative Forces 4-10 Determining Potential Energy Values 4-11 Conservation of Mechanical Energy 4-12 Work Done on a System by an External Force 4-13 Conservation of Energy

  4. gravitational elastic mechanic energy 4-1 What Is Physics ★ One of the fundamental goals of physics is to investigate the topic “energy”, which is obviously important. ★ One job of physics is to identify the different types of energy in the world. potential energy kinetic energy ★ In this chapter we learnwhat is physics through the study of energy and work.

  5. 4-2 What Is Energy? . ★ potential energy is the energy 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 becreated or destroyed, i.e. the total energy of a system is constant.

  6. ★Kinetic energyis energy associated with the state of motion of an object. (4-1) (4-2) SI unit: joule(J), 4-3 Kinetic Energy ★Newton’s 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.

  7. ★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 your force is negative; 4-4 Work work is energy transferred to or from an object by means of a force acting on the object. Energy transferredto the object is positive work, and energy transferredfrom the object is negative work. work is energy transferredto or from an object by means of a force acting on the object. Energy transferredto the object is positive work, and energy transferredfrom the object is negative work.

  8. (4-3) (4-4) Bead Solve for , yields wire (4-5) start end (4-6) 4-5 Work and Kinetic Energy Finding an Expression for Work ★ A bead can slide along a frictionless horizontal wire ( axis), A constantforce ，directed at an angle to the wire, accelerates the bead along the wire.

  9. Since (4-7) General form Scalar (dot) product (4-8) 4-5 Work and Kinetic Energy To calculate the workdone on an object by a force during a displacement, we use only the forcecomponentalong the object’s displacement. The force component perpen- dicular to the displacement does zero work. ★ Cautions: (1) constant force (magnitude and direction) (2) particle-like object.

  10. positive work negative work does’t do work 4-5 Work and Kinetic Energy ★Signs for work A 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 has no such vector component.

  11. (4-9) 4-5 Work and Kinetic Energy The unit for workis the same as the unit for energy ★ Units for work ★net work done by several forces When two or more forces act on an object, their net work done on the object is the sum of the works done by the individual forces.

  12. (4-11) (4-5) (4-10) change in the kinetic energy of a particle net work done on the particle = kinetic energy Before the net work = kinetic energy after The net work is done the net Work done 4-5 Work and Kinetic Energy ★Work-kinetic Energy Theorem Work-kinetic Energy Theorem for particles +

  13. A tomato is thrown upward, the work done by the grivatational force : (4-12) In the rising process (4-13) d F F g g In the falling down process (4-14) d • 4-6 Work Done by Force minus sign: the grivatational force transfers energy (mgd) from the object’s kinetic energy, consistant with the slowing of the object. plus sign: the grivatational force transfers energy (mgd) to the object’s kinetic energy, consistant with thespeeding up of the object.

  14. F F d d F F g g Lowering an object.Displacement: downward • 4-6 Work Done by Force ★Work done in lifting and lowering an object Lifting an object. Displacement: upward Lifting force: positive work; transfers energy to the object; Gravitational force: negative work; transfes energy from the object. Lifting force: negativework;transfers energy from the object; Gravitational force: positive work; transfes energy to the object.

  15. F F (4-15) d d F F g g (4-16) (4-17) (in lifting and lowering ; ) Up: Down: The angle between and . • 4-6 Work Done by Force The change in the kinetic energy due to these two energy transfers is

  16. The spring force (Hooke’s law) (4-18) (variable force) K: spring constant (Hooke’s law) (4-19) • 4-6 Work Done by Force X axis along the spring The spring force relaxed state Fig(a): A block is attached to a spring in equilibrium (neither compessed nor stretched). stretched Fig(b):when stretched to right The spring pulls the block to the Left (restoring force) Fig(c): when compressed to left The spring pulls the block to the right (restoring force) compressed

  17. F dW x xf xi O dx Sumintegration (4-21) (4-22) (4-23) • 4-6 Work Done by Force The work done by a spring force assumptions: spring massless; ideal spring(obeys Hooke’s law); contact frictionless. Calculus metheod (4-20) ( work by a spring force )

  18. (4-23) If and then ( work by a spring force ) (4-24) • 4-6 Work Done by Force The work done by a spring force Work is positive if the block ends upcloser to the relaxed position (x=0) than it was initially. It is negative if the block ends upfather away fromx=0. It is zero if the block ends up at the same distance fromx=0.

  19. 4-6 Work Done by Force The work done by a spring force Suppose we keep applying a force on the block, our force does work , the spring force does work on the block. The changein thekinetic energy of the block (4-25) If the block is stationarybefore and after the displacement, then (4-26)

  20. 4-6 Work Done by Force The work done by a spring force (4-26) If a block that is attached to a spring is stationary before and after a displacement, then the work done on it by the applied force displacing it is the negative of the work done on it by the spring force. Sample problem 4-2 : P93

  21. ( Work: variable force ) (4-30) • 4-6 Work Done by Force One-dimensional Analysis the same method as the calculation of the work done by a spring force (calculus). (4-27) (4-28) (4-29)

  22. 4-6 Work Done by Force Three-dimensional Analysis A three-dimensional force acts on a body (4-31) Simplifications: only depends on and so on Incremental displacement (4-32) The work done by during is (4-33) (4-34)

  23. (4-35) (4-39) work-kinetic energy theorem • 4-6 Work Done by Force work-kinetic energy theorem with a variable forceLet us prove: (4-36) (4-37) (4-38) Substituting Eq. 4-38 into Eq. 4-35:

  24. work down by a force: in a time interval : the average power (4-40) Instantaneous power (4-41) Unit:1watt =1= 1 (4-46) 2-7 Power Power :The power due to a force is the rate at which that force does work on an object. the instantaneous powercan be expresscd in terms of the force and the parti-cle’s velocity :

  25. (4 -10) Discuss the relation: ~ Exp.A tomato is thrown upward Rising: does , leads Energy transferred from the tomato, Where doesit go? To Increase the gravitational potential energy of the tomato-earth system! (the seperation is increased ! ) Earth Falling: does , leads Energy transferred from thegravitational potential energy of the tomato-earth system to the kinetic energy of the tomato ! work-kinetic energy theorem • 2-8Work and Potential Energy

  26. Discuss the relation: ~ (4-47) Earth • 2-8Work and Potential Energy Work and Potential Energy For either rise orfall, the change in gravitational potential energy is defined to equal the negative work done on the tomato bythe gravitational force. This equation also applies to a block-spring system (P 98)

  27. IF is always true, The other form of energy is a potential energy The force is a conservative force! • 4-9 Path Independence of Conservative Forces • Key elements: • A system (two or more objects); • 2. Aforce acts between a object and the rest part • of the system; • when configuration changes, the forcedoes • worktransfferingthe kinetic energy of the • object into some other form of energy. • Reversing the configuration changes, the force • reverses the energy transfer, doing work . Conservative and Nonconservative Forces

  28. 4-9 Path Independence of Conservative Forces Conservative and Nonconservative Forces Nonconservative Forces: a force that is not conservative Exp. : (1) the kinetic frictional force : A block is sliding on a rough surface, the kinetic frictional forcedoes negative work Transfer kinetic energythermal energy So the thermal energy is not a potentialenergy! The frictional forceNonconservative Forces! (2) the drag force

  29. 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 not depend on the path taken by the particle. • 4-9 Path Independence of Conservative Forces The closed-path test : to determine whether a force is conservative or nonconservative.

  30. b b 1 1 The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle. 2 2 a a • 4-9 Path Independence of Conservative Forces (4-48)

  31. 4-10 Determining Potential Energy Values Find the relation betweena conservative forceand the associated potential energy: The work done by a variable conservative force on a particle (see Eq. 4-35): (4-51) (4-52) Eq.4-52 is the general relation we sought.

  32. (4-53) (4-54) Gravitational potential energy (4-55) • 4-10 Determining Potential Energy Values Gravitational potential energy: A particle is moving vertically along a y axis From (4-52)

  33. (4-56) (4-57) • 4-10 Determining Potential Energy Values Elastic potential energy: A block-spring system is vibrating, the spring forcedoes work on the block.

  34. ( Mechanical energy ) (4-58) A conservative force does work on the object changing the object’s kinetic energy (4-59) The change in potential energy (4-60) Combining(4-59) , (4-60): (4-61) Rewriting: (4-62) • 4-11 Conservation of Mechanical Energy Mechanical energy: THeMechanical energy of a system is the sum of its potential energy and the kinetic energy of the objects within it:

  35. Rearranging: (4-63) The sum of andfor any state of a system The sum of andfor any other state of the system = Rewriting: (4-62) • 4-11 Conservation of Mechanical Energy ( conservation of Mechanical energy ) The principle of conservation of mechanical energy ： In a isolated system where only conservative forces cause energychanges, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system, cannot change.

  36. The principle of conservation of mechanical energy Newton’s laws of motion • 4-11 Conservation of Mechanical Energy From Eq. 4-61 : (4-64) When the mechanical energy of a system is conserved we can relate the sum of kinetic energy and potential energy at one instant to that at another instantwithoutconsidering the intermediate motion and withoutfinding the work done by the forces involved.

  37. = constant • 4-11 Conservation of Mechanical Energy A pendulum bobswingsback and forth. ( the pendulum-Earth system )

  38. (4-52) KnowFind KnowFind From Eq.(4-47) : Solving for and passing to the differential limit : (4-68) ( one-dimensional motion ) ★ Check Eq.(4-68) : 1. 2. • 4-11 Conservation of Mechanical Energy ★ Finding the force Analytically

  39. 4-11 Conservation of Mechanical Energy (4-68) The Potential Energy Curve At , right is a turning point

  40. K.E=0 F=0 on both sides deflecting force ! K.E=0 F=0 on both sides restoring force ! • 4-11 Conservation of Mechanical Energy unstable equilibrium neutral equilibrium neutral equilibrium K.E=0 F=0 stable equilibrium

  41. system system Negative Positive In 4-5 : (only ) In 4-12: (in other forms) The work-kinetic energy theorem • 4-12 Work Done on a System by an External Force Work is energy transferredto or froma system by means of an external force acting on that system.

  42. Ball-Earth system Positive • 4-12 Work Done on a System by an External Force NoFriction Involved throwing a ball upward, your applied force does work Earth kinetic potential (4-71) (4-72) mechanical energy ( Work done on system, no friction involved )

  43. (4-75) (4-76) (4-73) (4-77) (4-78) (4-79) (4-74) （work done on a system, friction involved） • 4-12 Work Done on a System by an External Force ★ Friction Involved Block- Floor system

  44. The law of conservation of energy Mechanical energy Thermal energy Internal energy of any form Total energy The work done on a system = the change in the total emergy (4-80) • 4-13 Conservation of Energy Countless experiments have proved: The total energyE of a system can change only by amounts of energy that are transferredto or from the system.

  45. (4-81) (4-82) • 4-13 Conservation of Energy If a system is isolated from its environment No energy transfers to or from it. W=0 Isolated System The total energy E ofan isolated systemcannotchange. ★

  46. (4-82) ★ In an isolated system, we can relate the total energy atone instant to the total energy at another instant without considering the energies at intermediate times. • 4-13 Conservation of Energy The total energy E ofan isolated systemcannotchange. ★ ★Isolated System This is a powerful tool in solving problems about isolated system

  47. 4-13 Conservation of Energy ★External Forces and Internal Energy Transfers An initially stationary ice skater pushes away from a railing and then slides over the ice. Her kinetic energy increases because of an external force on her from the rail. However, that force does not thansfer energy from the rail to her. Thus, the force does no work on her. Rather, her kinetic energy increases as a result of internal thansfers from the biochemical energy in her muscles. (4-83) (4-84)

  48. 4-13 Conservation of Energy Fig. 4-26 A vehicle accelerates to the left using four-wheel drive. The road exerts four frictional forces (two of them shown) on the bottom surfaces of the tires. Taken together, these four forces make up the net external force acting on the car. However, does not thansfer energy from the road to the car and does no work on the car. Rather, the car’s kinetic energy increases as a result of internal thansfers from the energy stored in the fuel

  49. (4-85) (4-86) • 4-13 Conservation of Energy ★ ★ Power is the rate at which work is done by a force. (P96) Power is the rate at which energy is transferredby a forcefromone formtoanother. Power ★ averagepower instantaneouspower

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