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Bilingual Mechanics Chapter 4 Energy and Work 制作 张昆实 制作 张昆实 Yangtze University Yangtze University 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 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 4-1 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. gravitational potential energy elastic mechanic energy kinetic energy ★ In this chapter we learn what is physics through the study of energy and work. 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 be created or destroyed, i.e. the total energy of a system is constant. 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. ★ Kinetic energy K is energy associated with the state of motion of an object. K mv 2 2 SI unit: joule(J), 1J 1kg m / s 1 2 2 (4-1) (4-2) 4-4 Work work W 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. ★ 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; 4-5 Work and Kinetic Energy Finding an Expression for Work ★ A bead can slide along a frictionless horizontal wire ( x axis), A constant force F ，directed at an angle to the wire, accelerates the bead along the wire. Fx max (4-3) v0 ax v v0 2ax d (4-4) F Solve for ax , yields Bead Fx 2 2 1 1 (4-5) 2 mv 2 mv0 Fx d start d W Fx d (4-6) 2 2 v wire x end 4-5 Work and Kinetic Energy ★ To calculate the work done on an object by a force during a displacement, we use only the force component along the object’s displacement. The force component perpendicular to the displacement does zero work. Since General form Fx F cos W Fx d Fd cos Scalar (dot) product W Fd cos F d (4-7) (4-8) Cautions: (1) constant force (magnitude and direction) (2) particle-like object. 4-5 Work and Kinetic Energy ★ Signs for work 0 90 cos 0 W 0 positive work 90 180 cos 0 W 0 negative work 90 cos 0 W 0 does’t do 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. 4-5 Work and Kinetic Energy ★ Units for work The unit for work is the same as the unit for energy 1J kg m / s 1N m 2 2 (4-9) ★ 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. 4-5 Work and Kinetic Energy ★ Work-kinetic Energy Theorem 1 2 mv mv0 Fx d 2 1 2 2 K K f K i W change in the kinetic energy of a particle = (4-10) net work done on the particle K f Ki W kinetic energy after The net work is done Work-kinetic Energy Theorem for particles (4-5) (4-11) = kinetic energy Before the net work + the net Work done 4-6 Work Done by Force A tomato is thrown upward, the work done by the grivatational force : Wg mgd cos (4-12) In the rising process Wg mgd cos180 mgd (4-13) minus sign: the grivatational force transfers energy (mgd) from the object’s kinetic energy, consistant with the slowing of the object. In the falling down process Wg mgd cos0 mgd (4-14) plus sign: the grivatational force transfers energy (mgd) to the object’s kinetic energy, consistant with the speeding up of the object. 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. Lowering an object. Displacement: downward Lifting force: negative work; transfers energy from the object; Gravitational force: positive work; transfes energy to the object. 4-6 Work Done by Force The change K in the kinetic energy due to these two energy transfers is K K f K i Wa Wg(4-15) K f Ki ( 0) Wa Wg 0 Wa Wg Wa mgd cos (4-16) 180 0 (in lifting and lowering ; K f K i ) (4-17) The angle between Fg and d Up: Wa mgd . Down: Wa mgd 4-6 Work Done by Force The spring force Fig(a): A block is attached to a spring in equilibrium (neither compessed nor stretched). X axis along the spring relaxed state 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) The spring force (variable force) K: spring constant stretched compressed F kd (Hooke’s law) (4-18) F kx (Hooke’s law) (4-19) 4-6 Work Done by Force The work done by a spring force assumptions: spring massless; ideal spring(obeys Hooke’s law); contact frictionless. Ws Fxj x x 0 ' F F Calculus metheod (4-20) Sumintegration xf xf xi xi Ws Fx dx kxdx 1 2 1 2 Ws kxi kx f 2 2 ( work by a spring force ) (4-21) (4-22) (4-23) o x P F dW O xf dx xi x x 4-6 Work Done by Force The work done by a spring force Work Ws is positive if the block ends up closer to the relaxed position (x=0) than it was initially. It is negative 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. 1 2 1 2 Ws kxi kx f 2 2 (4-23) If xi 0 and xf x 1 2 Ws kx 2 ' F F o x P then ( work by a spring force ) (4-24) x 4-6 Work Done by Force The work done by a spring force Suppose we keep applying a force Fa on the block, our force does work Wa , the spring force does work Ws on the block. The change K in the kinetic energy of the block F K K f Ki Wa Ws o x (4-25) If the block is stationary before and after the displacement, then K K 0 f i Wa Ws (4-26) Fa P x 4-6 Work Done by Force The work done by a spring force Wa Ws (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. F o Sample problem 4-2 : P93 x Fa P x 4-6 Work Done by Force One-dimensional Analysis the same method as the calculation of the work done by a spring force (calculus). W j Fj ,avg x (4-27) W W j Fj ,avg x (4-28) W lim Fj ,avg x (4-29) x 0 xf W F ( x)dx xi ( Work: variable force ) (4-30) 4-6 Work Done by Force Three-dimensional Analysis A three-dimensional force acts on a body (4-31) F Fx i Fy j Fz k Simplifications: Fx only depends on x and so on Incremental displacement dr dxi dyj dzk The work (4-32) dW done by F during dr dW F dr Fx dx Fy dy Fz dz is (4-33) rf xf yf zf ri xi yi zi W dW Fx dx Fy dy Fz dz (4-34) 4-6 Work Done by Force work-kinetic energy theorem with a variable force Let us prove: xf xf xi xi W F ( x)dx madx (4-35) dv dv dx dv madx m dx m dx m vdx mvdv dt dx dt dx (4-36) (4-37) (4-38) Substituting Eq. 4-38 into Eq. 4-35: xf vf vf xi vi vi W madx mvdv m vdv mv f mvi 1 2 2 1 2 W K f K i K 2 (4-39) work-kinetic energy theorem 2-7 Power Power : The power due to a force is the rate at which that force does work on an object. work down by a force: W in a time interval : t the average power Instantaneous power Unit: 1watt = 1 W = 1 J / s the instantaneous power can be expresscd in terms of the force and the particle’s velocity : W t (4-40) dW P dt (4-41) Pavg P dW dt F cos dx dt P Fv cos F v (4-46) 2-8 Work and Potential Energy W K f K i K work-kinetic energy theorem (4 -10) Discuss the relation: W ~U Exp. A tomato is thrown upward Rising: Fg does W, leads K tomato Energy transferred from the tomato, Where does it go? To Increase the gravitational potential energy of the tomato-earth system! (the seperation is increased ! ) Earth Falling: Fg does W, leads K tomato Energy transferred from the gravitational potential energy of the tomato-earth system to the kinetic energy of the tomato ! 2-8 Work and Potential Energy Work and Potential Energy Discuss the relation: W ~U For either rise or fall, the change U in gravitational potential energy is defined to equal the negative work done on the tomato by the gravitational force. U W Earth (4-47) This equation also applies to a block-spring system (P 98) 4-9 Path Independence of Conservative Forces Conservative and Nonconservative Forces Key elements: 1. A system (two or more objects); 2. A force acts between a object and the rest part of the system; 3. when configuration changes, the force does work W transffering the kinetic energy of the 1 object into some other form of energy. 4. Reversing the configuration changes, the force reverses the energy transfer, doing work W2 . IF W1 W2 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 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 force does negative work Transfer kinetic energy thermal energy So the thermal energy is not a potential energy! The frictional force Nonconservative Forces ! (2) the drag force 4-9 Path Independence 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 not depend on the path taken by the particle. 4-9 Path Independence of Conservative Forces The work done by a conservative force on a particle moving between two points does not depend on the path taken by the particle. Wab ,1 b Wab ,1 b 1 1 2 Wab ,2 a Wab,1 Wba ,2 0 2 a Wba ,2 Wab ,1 Wba ,2 Wab ,2 (4-48) 4-10 Determining Potential Energy Values Find the relation between a conservative force and the associated potential energy : The work done by a variable conservative force on a particle (see Eq. 4-35): xf W F ( x)dx (4-51) xi xf U W F ( x)dx xi (4-52) Eq.4-52 is the general relation we sought. 4-10 Determining Potential Energy Values Gravitational potential energy : A particle is moving vertically along a y axis From xf U W F ( x)dx xi yf yf yi yi U (mg )dy mg (4-52) dy mg y U mg ( y f yi ) mg y (4-53) U U i mg ( y yi ) (4-54) U ( y ) mgy ( yi 0, and Ui 0) Gravitational potential energy (4-55) yf yi 4-10 Determining Potential Energy Values Elastic potential energy : A block-spring system is vibrating, the spring force F kx does work on the block. xf xf U (kx)dx k xdx k x xi xi xi U kx kx 2 f 1 2 1 2 U 0 kx 0 1 2 2 U ( x) kx 1 2 2 2 i 1 2 2 xf (4-56) ( xi 0, and Ui 0) (4-57) 4-11 Conservation of Mechanical Energy Mechanical energy : THe Mechanical energy of a system is the sum of its potential energy and the kinetic energy of the objects within it: Emec K U ( Mechanical energy ) (4-58) A conservative force does work W on the object changing the object’s kinetic energy (4-59) K W The change U in potential energy U W Combining (4-59) , (4-60): Rewriting: (4-60) K U K2 K1 (U 2 U1 ) (4-61) (4-62) 4-11 Conservation of Mechanical Energy Rewriting: Rearranging: K2 K1 (U 2 U1 ) K2 U 2 K1 U1 (4-62) (4-63) ( conservation of Mechanical energy ) The sum of K and U for any state of a system = The sum of K and U for any other state of the system The principle of conservation of mechanical energy ： In a isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system, cannot change. 4-11 Conservation of Mechanical Energy From Eq. 4-61 : K U Emec K U 0 (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 instant without considering the intermediate motion and without finding the work done by the forces involved. The principle of conservation of mechanical energy Newton’s laws of motion 4-11 Conservation of Mechanical Energy A pendulum bob swings back and forth. K U Emec = constant ( the pendulum-Earth system ) 4-11 Conservation of Mechanical Energy ★ Finding the force Analytically Know F ( x ) Know U xf Find U U xi F ( x)dx Find F ( x ) (4-52) U ( x) W F ( x)x From Eq.(4-47) : Solving for F ( x ) and passing to the differential limit : dU ( x) F ( x) dx ( one-dimensional motion ) (4-68) ★ Check Eq.(4-68) : 2 1 U ( x ) kx 1. 2 2. U ( x) mgx F ( x) d ( kx ) dx kx 1 2 2 F ( x) d (mgx) dx mg 4-11 Conservation of Mechanical Energy The Potential Energy Curve U ( x ) x U ( x) K ( x) Emec K ( x) Emec U ( x) x2 x1 , U ( x) K ( x) At x1, K ( x) 0 F ( x) 0 x1 x1 right is a turning point dU ( x) F ( x) dx 4-11 Conservation of Mechanical Energy unstable equilibrium K.E=0 F=0 on both sides deflecting force ! neutral equilibrium K.E=0 F=0 stable equilibrium K.E=0 F=0 on both sides restoring force ! 4-12 Work Done on a System by an External Force Work is energy transferred to or from a system by means of an external force acting on that system. system system Positive W In 4-5 : Negative W K W (only ) The work-kinetic energy theorem In 4-12: Fextetnal E (in other forms) 4-12 Work Done on a System by an External Force No Friction Involved throwing a ball upward, Ball-Earth your applied force system Emec K U does work Positive W Earth kinetic potential W K U (4-71) W Emec (4-72) mechanical energy ( Work done on system, no friction involved ) 4-12 Work Done on a System by an External Force ★ Friction Involved v0 a F fk Emec v W Fd K f d k x d F f k ma v v 2ad 2 2 0 Eth (4-73) BlockFloor system (4-75) Fd K U f k d Fd Emec f k d (4-76) Eth f k d (4-77) Fd Emec Eth (4-78) 2 2 1 1 Fd 2 mv 2 mv0 f k d W Emec Eth (4-79) (4-74) （work done on a system, friction involved） 4-13 Conservation of Energy Countless experiments have proved: The law of conservation of energy The total energy E of a system can change only by amounts of energy that are transferred to or from the system. Total energy Mechanical energy Thermal energy Internal energy of any form The work done on a system = the change in the total emergy W E Emec Eth Eint (4-80) 4-13 Conservation of Energy Isolated System If a system is isolated from its environment No energy transfers to or from it. W=0 ★ The total energy E of an isolated system cannot change. W E Emec Eth Eint 0 Emec Emec，2 Emec，1 Emec，2 Emec.1 Eth Eint 0 Emec，2 Emec，1 Eth Eint (4-81) (4-82) 4-13 Conservation of Energy Isolated System ★ The total energy E of an isolated system cannot change. ★ Emec，2 Emec，1 Eth Eint ★ (4-82) In an isolated system, we can relate the total energy at one instant to the total energy at another instant without considering the energies at intermediate times. This is a powerful tool in solving problems about isolated system 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 F 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 . K K K0 Fd cos K U Fd cos (4-83) (4-84) com F v0 v d x 4-13 Conservation of Energy Fig. 4-26 A vehicle accelerates to the left using fourwheel 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 F acting on the car. However, F 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 F1 F2 4-13 Conservation of Energy Power ★ Power is the rate at which work is done by a force. (P96) ★ ★ Power is the rate at which energy is transferred by a force from one form to another. average power instantaneous power Pavg E t dE P dt (4-85) (4-86)