Chapter 7

1. Chapter 7 Kinetic energy and work

2. 7.3 Kinetic energy Kinetic energy K is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy. For an object of mass m whose speed v is well below the speed of light, The SI unit of kinetic energy (and every other type of energy) is the joule (J), 1 joule = 1 J = 1 kgm2/s2.

3. 7.4: 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.

4. 7.5: Work and kinetic energy To calculate the work a force F does on an object as the object moves through some displacement d, we use only the force component along the object’s displacement. The force component perpendicular to the displacement direction does zero work. For a constant force F, the work done W is: A constant force directed at angle f to the displacement (in the x-direction) of a bead does work on the bead. The only component of force taken into account here is the x-component. When two or more forces act on an object, the net work done on the object is the sum of the works done by the individual forces.

5. 7.6: Work done by gravitational force (b) An applied force lowers an object. The displacement of the object makes an angle with the gravitational force .The applied force does negative work on the object. (a) An applied force lifts an object. The object’s displacement makes an angle f =180° with the gravitational force on the object. The applied force does positive work on the object.

6. 7.5: Work and kinetic energy Work-kinetic energy theorem ΣW = Kf – Ki= ΔK It holds for both positive and negative work: If the net work done on a particle is positive, then the particle’s kinetic energy increases by the amount of the work, and the converse is also true. The theorem says that the change in kinetic energy of a particle is the net work done on the particle.

7. Workdone onanaccelerating elevator cab

8. 7.8: Work done by a general variable force A. One-dimensional force, graphical analysis: • We can divide the area under the curve of F(x) into a number of narrow strips of width x. • We choose x small enough to permit us to take the force F(x) as being reasonably constant over that interval. • We let Fj,avg be the average value of F(x) within the jth interval. • The work done by the force in the jth interval is approximately • I • Wj is then equal to the area of the jth rectangular, shaded strip.

9. 7.8: Work done by a general variable force A. One-dimensional force, calculus analysis: We can make the approximation better by reducing the strip width Dx and using more strips (Fig. c). In the limit, the strip width approaches zero, the number of strips then becomes infinitely large and we have, as an exact result,

10. 7.8: Work done by a general variable force B. Three dimensional force: If where Fx is the x-components of F and so on, and where dx is the x-component of the displacement vector drand so on, then Finally,

11. Hooke’s Law When x is positive (spring is stretched), F is negative When x is 0 (at the equilibrium position), F is 0 When x is negative (spring is compressed), F is positive 7.7: Work done by a spring force Fs = - kx Slide 11

12. Work Done by a Spring Identify the block as the system Calculate the work as the block moves from xi = -xmax to xf = 0 The total work done as the block moves from –xmax to xmax is zero xf xi Slide 12

13. 7.7: Work done by a spring force Hooke’s Law: To a good approximation for many springs, the force from a spring is proportional to the displacement of the free end from its position when the spring is in the relaxed state. The spring force is given by The minus sign indicates that the direction of the spring force is always opposite the direction of the displacement of the spring’s free end. The constant k is called the spring constant (or force constant) and is a measure of the stiffness of the spring. The net work Ws done by a spring, when it has a distortion from xi to xf , is: 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 farther away from x =0. It is zero if the block ends up at the same distance from x= 0.

14. Work Done by Gravitational Force Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation: Calculate the work as the objectmoves from ri to rf Slide 14

15. 7.8: Work kinetic energy theorem with a variable force A particle of mass m is moving along an x axis and acted on by a net force F(x) that is directed along that axis. The work done on the particle by this force as the particle moves from position xi to position xfis : But, Therefore,

16. Power The time rate of energy transfer is called power The average power is given by Slide 16

17. Instantaneous Power The instantaneous power is the limiting value of the average power as Dt approaches zero This can also be written as The SI unit of power is the joule per second, or Watt (W). In the British system, the unit of power is the footpound per second. Often the horsepower is used. 1N=1 kg-m/s2 = 0.225 lb 1 m = 3.281 ft Slide 17

18. 7.9: Power

19. Chapter 8 Potential energy and conservation of energy

20. 8.1 Potential energy • Technically, potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another. • Some forms of potential energy: • Gravitational Potential Energy, • Elastic Potential Energy

21. Conservative Forces The work done by a conservative force on a particle moving between any two points isindependent of the path taken by the particle The work done by a conservative force on a particle moving through any closed path is zero Slide 21

22. 8.3 Path Independence of Conservative Forces The net work done by a conservative force on a particle moving around any closed path is zero. If the work done from a to b along path 1 as Wab,1 and the work done from b back to a along path 2 as Wba,2. If the force is conservative, then the net work done during the round trip must be zero If the force is conservative,

23. Nonconservative Force, Example Friction is an example of a nonconservative force The work done depends on the path The red path will take more work than the blue path Slide 23

24. This system consists of Earth and a book Do work on the system by lifting the book through Dy The work done is mgyb - mgya Determining Potential Energy Gravitational Potential Energy Ub F外力 Ua = mgyb- mgya Slide 24

25. 8.4: Determining Potential Energy values: For the most general case, in which the force may vary with position, we may write the work W:

26. 8.4: Determining Potential Energy values: Elastic Potential Energy In a block–spring system, the block is moving on the end of a spring of spring constant k. As the block moves from point xi to point xf , the spring force Fx =- kx does work on the block. The corresponding change in the elastic potential energy of the block–spring system is If the reference configuration is when the spring is at its relaxed length, and the block is at xi = 0.

27. Find W for each one of the three paths.

28. 8.5: Conservation of Mechanical Energy Principle of conservation of energy: In an isolated system where onlyconservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change. The mechanical energy Emec of a system is the sum of its potential energy U and the kinetic energy K of the objects within it: With and We have:

29. 8.7: 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.

30. 8.7: Work done on a System by an External Force FRICTION INVOLVED FRICTION NOT INVOLVED

31. (b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing ? (a) With what minimum speed must Jane begin her swing to just make it to the other side ? mJ mT Fig. P7-57, p.218

32. 8.8: Conservation of Energy 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. where Emec is any change in the mechanical energy of the system, Eth is any change in the thermal energy of the system, and Eint is any change in any other type of internal energy of the system. The total energy E of an isolated system cannot change.

33. 8.8: Conservation of Energy External Forces and Internal Energy Transfers internal energy=biochemical in the muscles An external force can change the kinetic energy or potential energy of an object without doing work on the object—that is, without transferring energy to the object. Instead, the force is responsible for transfers of energy from one type to another inside the object.

34. Fnet=Σfriction ( f ) The net external force Fnet from the road change the kinetic energy of the car. However, wiFnetdoes not transfer energyfrom the roadto the carand so out does no work on the car.Instead, the force is responsible for transfers of energy from the energy stored in the fuel.

35. Conservative Forces and Potential Energy Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system The work done by such a force, F, is DU is negative when F and x are in the same direction Slide 35

36. Conservative Forces and Potential Energy The conservative force is related to the potential energy function through The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system This can be extended to three dimensions Slide 36

37. Conservative Forces and Potential Energy – Check Look at the case of an object located some distance y above some reference point: This is the expression for the vertical component of the gravitational force Slide 37

38. Energy Diagrams and Stable Equilibrium The x = 0 position is one of stable equilibrium Configurations of stable equilibrium correspond to those for which U(x) is a minimum x=xmax and x=-xmax are called the turning points Slide 38

39. Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the particle is in equilibrium For any other value of x, the particle moves away from the equilibrium position This is an example of unstable equilibrium Configurations of unstable equilibrium correspond to those for which U(x) is a maximum Slide 39

41. Solution • Stable equilibrium exists for a separation distance at which the potential energy of the system of two atoms (the molecule) is a minimum. Take the derivative of the function U(x): Slide 41

42. Solution • Minimize the function U(x) by setting its derivative equal to zero: • Evaluate xeq the equilibrium separation of the two atoms in the molecule: Slide 42

43. Solution • We graph the Lennard-Jones function on both sides of this critical value to create our energy diagram as shown in Figure. Slide 43

44. Solution • Notice that U(x) is extremely large when the atoms are very close together, is a minimum when the atoms are at their critical separation, and then increases again as the atoms move apart. When U(x) is a minimum, the atoms are in stable equilibrium. indicating that the most likely separation between them occurs at this point. Slide 44

45. Neutral Equilibrium Neutral equilibrium occurs in a configuration when U is constant over some region A small displacement from a position in this region will produce neither restoring nor disrupting forces Slide 45