今日課程內容 CH8 位能與能量守恆 保守力與非保守力 保守力做功與路徑積分 位能與保守力做功的關係 力學能與力學能守恆 外力做功 能量守恆

8.1 Potential energy( 位能 ) 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(U): 1.Gravitational Potential Energy, 2.Elastic Potential Energy

8.2 Work and potential energy The change  U in potential energy (gravitational, elastic, etc) is defined as being equal to the negative of the work done on the object by the force (gravitational, elastic, etc)

8-2 Conservative( 保守 ) and Nonconservative( 非保守 ) Forces A force is conservative if: the work done by the force on an object moving from one point to another depends only on the initial and final positions of the object, and is independent of the particular path taken. Example: gravity.

8-2 Conservative and Nonconservative Forces If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

8-2 Conservative and Nonconservative Forces Potential energy( 位能 ) can only be defined for conservative forces.

8.3 Path Independence ( 路徑積分 )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 W ab,1 and the work done from b back to a along path 2 as W ba,2. If the force is conservative, then the net work done during the round trip must be zero If the force is conservative,

Sample problem, slippery cheese

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:

8.4: Determining Potential Energy values: Gravitational Potential Energy( 重力位能 ) A particle with mass m moving vertically along a y axis (the positive direction is upward). As the particle moves from point y i to point y f, the gravitational force does work on it. The corresponding change in the gravitational potential energy of the particle–Earth system is: The gravitational potential energy associated with a particle–Earth system depends only on the vertical position y (or height) of the particle relative to the reference position y =0, not on the horizontal position.

8.4: Determining Potential Energy values: Elastic Potential Energy( 彈力位能 ) 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 x i to point x f, the spring force F x =- 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 x i = 0.

Sample problem: gravitational potential energy

A 1000-kg roller-coaster car moves from point 1 to point 2 and then to point 3. (a) What is the gravitational potential energy at points 2 and 3 relative to point 1? That is, take y = 0 at point 1. (b) What is the change in potential energy when the car goes from point 2 to point 3? (c) Repeat parts (a) and (b), but take the reference point ( y = 0) to be at point 3.

8.5: Conservation of Mechanical Energy Principle of conservation of energy: In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy E mec of the system, cannot change. The mechanical energy E mec 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: 沒有外力時，力學能守恆 ????

A pendulum swings back and forth. During one full cycle the values of the potential and kinetic energies of the pendulum– Earth system vary as the bob rises and falls, but the mechanical energy E mec of the system remains constant. The energy E mec can be described as continuously shifting between the kinetic and potential forms. In stages (a) and (e), all the energy is kinetic energy. In stages (c) and (g), all the energy is potential energy. In stages (b), (d), ( f ), and (h), half the energy is kinetic energy and half is potential energy. If the swinging involved a frictional force then E mec would not be conserved, and eventually the pendulum would stop. 8.5: Conservation of Mechanical Energy

Sample problem: water slide

Two water slides at a pool are shaped differently, but start at the same height h. Two riders, Paul and Kathleen, start from rest at the same time on different slides. (a) Which rider, Paul or Kathleen, is traveling faster at the bottom? (b) Which rider makes it to the bottom first? Ignore friction and assume both slides have the same path length.

Estimate the kinetic energy and the speed required for a 70-kg pole vaulter to just pass over a bar 5.0 m high. Assume the vaulter’s center of mass is initially 0.90 m off the ground and reaches its maximum height at the level of the bar itself.

A dart of mass kg is pressed against the spring of a toy dart gun. The spring (with spring stiffness constant k = 250 N/m and ignorable mass) is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length ( x = 0), what speed does the dart acquire?

A ball of mass m = 2.60 kg, starting from rest, falls a vertical distance h = 55.0 cm before striking a vertical coiled spring, which it compresses an amount Y = 15.0 cm. Determine the spring stiffness constant of the spring. Assume the spring has negligible mass, and ignore air resistance. Measure all distances from the point where the ball first touches the uncompressed spring ( y = 0 at this point).

A swinging pendulum( 擺 ). This simple pendulum consists of a small bob of mass m suspended by a massless cord of length l. The bob is released (without a push) at t = 0, where the cord makes an angle θ = θ 0 to the vertical. (a) Describe the motion of the bob in terms of kinetic energy and potential energy. Then determine the speed of the bob (b) as a function of position θ as it swings back and forth, and (c) at the lowest point of the swing. (d) Find the tension in the cord, T. Ignore friction and air resistance.

8.6: Reading a Potential Energy Curve A plot of U(x), the potential energy function of a system containing a particle confined to move along an x axis. There is no friction, so mechanical energy is conserved. A plot of the force F(x) acting on the particle, derived from the potential energy plot by taking its slope at various points.

8.6: Reading a Potential Energy Curve The U(x) plot with three possible values of E mec shown.

8.6: Potential Energy Curve, Equilibrium Points If we place the object at any point to the right of x 5, the system’s mechanical energy is equal to its potential energy, and so it must be stationary. A particle at such a position is said to be in neutral equilibrium( 中 性平衡，任意平衡 ). x 3 is a point at which K =0. If the particle is located exactly there, the force on it is also zero, and the particle remains stationary. However, if it is displaced even slightly in either direction, a nonzero force pushes it farther in the same direction, and the particle continues to move. A particle at such a position is said to be in unstable equilibrium( 不穩定平衡 ). If we place the object at x 4, it is stuck there. It cannot move left or right on its own because to do so would require a negative kinetic energy. If we push it slightly left or right, a restoring force appears that moves it back to x 4. A particle at such a position is said to be in stable equilibrium( 穩定平衡 ).

Sample problem: reading a potential energy graph

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.

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

Sample problem: change in thermal energy

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 E mec is any change in the mechanical energy of the system, E th is any change in the thermal energy of the system, and E int is any change in any other type of internal energy of the system. The total energy E of an isolated system cannot change.

8.8: Conservation of Energy External Forces and Internal Energy Transfers 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.

8.8: Conservation of Energy: Power In general, power P is the rate at which energy is transferred by a force from one type to another. If an amount of energy E is transferred in an amount of time t, the average power due to the force is and the instantaneous power due to the force is

Sample problem: energy, friction, spring, and tamales

33 長庚大學畢業生應具備之核心能力 a. 服務社會、肩負責任的使命感與主流價值的道德觀 b. 本國語文及英文的溝通能力 c. 資訊應用之基本能力 d. 人文、社會及自然科學之基礎知識 e. 蒐集資料、分析數據、書面及口頭報告的能力 f. 終身自我學習的能力 g. 協調、溝通及團隊合作之能力 h. 國際觀及國際競爭力

醫學院畢業生應具備的核心能力 a. 人文素養人道關懷 (Humanities) b. 團隊合作及溝通協調能力 (Teamwork 、 Communication and Coordination skill) c. 基本電腦資訊應用能力 (Basic computer skill) d. 醫學專業素養及倫理 (Medical professionalism and Ethics) e. 生物醫學知識及研究能力 (Biomedical knowledge and Research ability) f. 國際觀及競爭力 (Global perspective and Competitiveness) g. 終身學習能力 (Lifelong learning)

a. 人文素養 (Humanities) b. 行為及社會科學知識 (Behavior and social sciences knowledge) c. 基本資訊應用 (Basic computer skill) d. 物理治療專業風範及倫理 (Professionalism & Ethics in Physical Therapy) e. 醫學知識與物理治療專業知識 (Professional knowledge in medical science & Physical Therapy) f. 生物醫學研究能力 (Biomedical Research ability) g. 物理治療專業臨床技能 (Clinical professional skill in Physical Therapy) h. 終身學習能力 (Lifelong learning)

a. 人文素養與專業倫理 Humanities & Professional Ethics b. 行為及社會科學知識 Behavioral and Social Science c. 基本資訊應用技能 Basic Information Processing Skill d. 職能治療相關專業 Occupational Therapy Related Profession e. 研究方法概念 Reseach Ability f. 臨床專業技能 Clinical professional Techniques g. 終身學習能力 Lifelong Learning

a. 基礎科學知識 Basic Scientific knowledge. b. 基本資訊應用技能 Basic computer and information application ability. c. 生物醫學知識 Basic Biomedicine knowledge. d. 生醫實驗及研究技巧 Biomedical experiments and research techniques. e. 專業臨床醫檢技能 Professional Clinical Diagnostics Skills. f. 人文關懷及社會服務 Humanities. g. 掌握問題導向學習技巧，以達終身學習 To achieve a life long learning attitude based on problem-based learning training.

習題 Ch8: 6, 7, 15, 22, 27, 28, 31, 38, 53, 57,