Knowledge_Structure

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Unformatted text preview: M08_KNIG7366_02_SE_C08.qxd 5/3/07 9:32 AM Page 236 PART SU M MARY I Newton’s Laws The goal of Part I has been to discover the connection between force and motion. We started with kinematics, which is the mathematical description of motion; then we proceeded to dynamics, which is the explanation of motion in terms of forces. Newton’s three laws of motion form the basis of our explanation. All of the examples we have studied so far are applications of Newton’s laws. The table below is called a knowledge structure for Newton’s laws. A knowledge structure summarizes the essential concepts, the general principles, and the primary applications of a theory. The first section of the table tells us that Newtonian mechanics is concerned with how particles respond to forces. The second section indicates that we have introduced only three general principles, Newton’s three laws of motion. You use this knowledge structure by working your way through it, from top to bottom. Once you recognize a problem KNOWLEDGE STRUCTURE I ESSENTIAL CONCEPTS BASIC GOALS GENERAL PRINCIPLES as a dynamics problem, you immediately know to start with Newton’s laws. You can then determine the category of motion and apply Newton’s second law in the appropriate form. Newton’s third law will help you identify the forces acting on particles as they interact. Finally, the kinematic equations for that category of motion allow you to reach the solution you seek. The knowledge structure provides the procedural knowledge for solving dynamics problems, but it does not represent the total knowledge required. You must add to it knowledge about what position and velocity are, about how forces are identified, about action/reaction pairs, about drawing and using free-body diagrams, and so on. These are specific tools for problem solving. The problem-solving strategies of Chapters 5 through 8 combine the procedures and the tools into a powerful method for thinking about and solving problems. Newton’s Laws Particle, acceleration, force, interaction How does a particle respond to a force? How do objects interact? Newton’s first law Newton’s second law Newton’s third law An object will remain at restror will continue to move with constant velocity r (equilibrium) if and only if Fnet 5 0. r Fnet 5 ma r r FA on B 5 2 FB on A r BASIC PROBLEM-SOLVING STRATEGY Use Newton’s second law for each particle or object. Use Newton’s third law to equate the magnitudes of the two members of an action/reaction pair. Linear motion a Fx 5 max a Fy 5 0 or a Fx 5 0 a Fy 5 may Trajectory motion a Fx 5 max a Fy 5 may Circular motion 2 2 a Fr 5 mv /r 5 mv r a Ft 5 0 or mat a Fz 5 0 Linear and trajectory kinematics Circular kinematics Uniform acceleration: ( as 5 constant) vfs 5 vis 1 as Dt sf 5 si 1 vis Dt 1 1 2 2 as ( Dt ) Uniform circular motion: T 5 2pr/v 5 2p/v uf 5 ui 1 vDt ar 5 v 2/r 5 v2r vt 5 vr Nonuniform circular motion: vf 5 vi 1 aDt uf 5 ui 1 vi Dt 1 1 a ( Dt ) 2 2 vfs2 5 vis2 1 2as Ds Trajectories: The same equations are used for both x and y. Uniform motion: ( a 5 0, vs 5 constant) General case vs 5 ds / dt 5 slope of the position graph sf 5 si 1 vs Dt as 5 dvs /dt 5 slope of the velocity graph tf vfs 5 vis 1 3 as dt 5 vis 1 area under the acceleration curve ti tf sf 5 si 1 3 vs dt 5 si 1 area under the velocity curve ti 236 M11_knig7366_02_SE_C11.qxd 5/7/07 8:02 AM Page 336 PART SU M MARY II Conservation Laws In Part II we have discovered that we don’t need to know all the details of an interaction to relate the properties of a system “before” an interaction to the system’s properties “after” the interaction. Along the way, we found two important quantities, momentum and energy, that characterize a system of particles. Momentum and energy have specific conditions under which r they are conserved. In particular, the total momentum P and the total energy Esys are conserved for an isolated system, one on which the net external force is zero. Further, the system’s mechanical energy is conserved if the system is both isolated and nondissipative (i.e., no friction forces). These ideas are captured in the two most important conservation laws, the law of conservation of momentum and the law of conservation of energy. Of course, not all systems are isolated. For both momentum and energy, it was useful to develop a model of a system interacting with its environment. Interactions between the system and the environment change the system’s momentum and energy. In particular, s Impulse is the transfer of momentum to or from the system: Dps 5 Js. s Work is the transfer of energy to or from the system: DEsys 5 Wext. Interactions within the system do not change P or Esys. The kinetic, potential, and thermal energy within the system can be transformed without changing Esys. The basic energy model is built around the twin ideas of the transfer and the transformation of energy. The table below is a knowledge structure of conservation laws. You should compare this with the knowledge structure of Newtonian mechanics in the Part I Summary. Add the problem-solving strategies, and you now have a very powerful set of tools for understanding motion. r KNOWLEDGE STRUCTURE II ESSENTIAL CONCEPTS BASIC GOALS Conservation Laws Impulse, momentum, work, energy How is the system “after” an interaction related to the system “before”? What quantities are conserved, and under what conditions? Impulse-momentum theorem Work-kinetic energy theorem Energy equation r GENERAL PRINCIPLES Dps 5 Js DK 5 Wnet 5 Wc 1 Wdiss 1 Wext DEsys 5 DK 1 DU 1 DEth 5 Wext r CONSERVATION LAWS For an isolated system, with Fnet 5 0 and Wnet 5 0 r • The total momentum P is conserved. • The total energy Esys 5 Emech 1 Eth is conserved. For an isolated and nondissipative system, with Wdiss 5 0 • The mechanical energy Emech 5 K 1 U is conserved. BASIC PROBLEM-SOLVING STRATEGY Draw a before-and-after pictorial representation, then use the momentum or energy equations to relate “before” to “after.” Where possible, choose a system for which momentum and/or energy are conserved. If necessary, calculate impulse and/or work. Basic model of momentum and energy Impulse and momentum p 5 mv tf r r Environment System r Momentum P Energy K U Work and energy K 5 1 mv 2 2 Impulse Work Energy out sf Js 5 3 Fs ( t ) dt ti Impulse Work Energy in Energy and momentum are transferred to and from the system. W 5 3 Fs ds si r 5 F # Dr r Eth Energy is transformed within the system. Ug 5 mgy Us 5 1 k ( Ds ) 2 2 336 M15_Knig7366_02_SE_C15.qxd 5/30/07 8:04 AM Page 476 PART SU M MARY III Applications of Newtonian Mechanics We have developed two parallel perspectives of motion, each with its own concepts and techniques. We focused on the first of these in Part I, where we dealt with the relationship between force and motion. Newton’s second law is the principle most central to the force/motion perspective. Then, in Part II, we developed a before-and-after perspective based on the idea of conservation laws. Newton’s laws were essential in the development of conservation laws, but they remain hidden in the background when the conservation laws are applied. Together, these two perspectives form the heart of Newtonian mechanics. Our goal in Part III has been to see how Newtonian mechanics is applied to several diverse but important topics. We added only one new law of physics in Part III, Newton’s law of gravity, and we introduced few completely new concepts. Instead, we’ve broadened our understanding of the KNOWLEDGE STRUCTURE III force/motion perspective and the conservation-law perspective through our investigations of rotational motion, gravity, oscillations, and fluids. In reviewing Part III, pay close attention to the interplay between these two perspectives. Recognizing which is the best tool in a particular situation will help you improve your problem-solving ability. Our knowledge of mechanics is now essentially complete. We will add a few additional ideas as we need them, but our journey into physics will be taking us in entirely new directions as we continue on. Hence this is an opportune moment to step back a bit to take a look at the “big picture.” Newtonian mechanics may seem all very factual and straightforward to us today, but keep in mind that these ideas are all human inventions. There was a time when they did not exist and when our concepts of nature were quite different from what they are today. Applications of Newtonian Mechanics Newton’s Theory of Gravity Rotation of a Rigid Body A rigid body is a system of particles. Rotational motion is analogous to linear motion. Rotational motion Angular acceleration a Torque t Moment of inertia I Angular momentum L Linear motion Acceleration a Force F Mass m Momentum p Any two masses exert attractive gravitational forces on each other. Newton’s law of gravity is Fm on M 5 FM on m 5 GMm r2 • Kepler ’s laws describe the elliptical orbits of satellites and planets. • The gravitational potential energy is GMm r • Newton’s second law tnet 5 Ia • Rotational kinetic energy K 5 1 2 2 Iv Ug 5 2 NEWTON’S LAWS 1 CONSERVATION LAWS Oscillations Fluids and Elasticity Systems with a linear restoring force exhibit simple harmonic oscillation. • The kinematic equations of SHM are x ( t ) 5 A cos( vt 1 f0 ) v ( t ) 5 2vmax sin( vt 1 f0 ) where vmax 5 vA and the phase constant f0 describes the initial conditions. • Energy is transformed between kinetic and potential as the system oscillates. In an undamped system, the total mechanical energy E 5 1 mv 2 1 1 kx 2 5 1 m ( vmax ) 2 5 1 kA2 2 2 2 2 is conserved. Fluids are systems that flow. Gases and liquids are fluids. Fluids are better characterized by density and pressure than by mass and force. • Liquids Pressure is primarily gravitational. The hydrostatic pressure is p 5 p0 1 rgd • Gases Pressure is primarily thermal. Pressure in a container is constant. • Archimedes’ principle The buoyant force is equal to the weight of the displaced fluid. For fluid flow, Bernoulli’s equation p1 1 1 rv12 1 rgy1 5 p2 1 1 rv22 1 rgy2 2 2 is really a statement of energy conservation. 476 M19_Knig7366_02_se_c19.qxd 5/23/07 9:02 AM Page 598 PART SU M MARY IV Thermodynamics Part IV had two important goals: first, to learn how energy is transformed; second, to establish a micro/macro connection in which we can understand the macroscopic properties of solids, liquids, and gases in terms of the microscopic motions of atoms and molecules. We have been quite successful. You have learned that: s s Practical devices for turning heat into work, called heat engines, are limited in their efficiency by the second law of thermodynamics. s s s Temperature is a measure of the thermal energy of the molecules in a system, and the average energy per molecule is simply 1 kBT per degree of freedom. 2 The pressure of a gas is due to collisions of the molecules with the walls of the container. Heat is the energy transferred between two systems that have different temperatures. The mechanism of heat transfer is molecular collisions at the boundary between the two systems. Work, heat, and thermal energy can be transformed into each other in accord with the first law of thermodynamics, DEth 5 W 1 Q. This is a statement that energy is conserved. The knowledge structure of thermodynamics below summarizes the basic laws, diagramming our energy model and presenting our model of a heat engine in pictorial form. Thermodynamics, more than most topics in physics, can seem very “equation oriented.” It’s undeniable that there are more equations than we used in earlier parts of this text and more things to remember. But focusing on the equations is seeing only the trees, not the forest. A better strategy is to focus on the ideas embedded in the knowledge structure. You can find the necessary equations if you know how the ideas are connected, but memorizing all the equations won’t help if you don’t know which are relevant to different situations. KNOWLEDGE STRUCTURE IV ESSENTIAL CONCEPTS BASIC GOALS Thermodynamics Work, heat, and thermal energy. How is energy converted from one form to another? How are macroscopic properties related to microscopic behavior? First law of thermodynamics Energy is conserved, DEth 5 W 1 Q. Second law of thermodynamics Heat is not spontaneously transferred from a colder object to a hotter object. GENERAL PRINCIPLES GAS LAWS AND PROCESSES Ideal-gas law pV 5 nRT 5 NkBT V 5 constant and W 5 0 T 5 constant and DEth 5 0 • Isobaric process • Adiabatic process p 5 constant Q50 Heat Engines Environment Work by system W,0 • Isochoric process • Isothermal process Energy Transformation Work on system W.0 Energy in System Thermal energy Eth 1 Other state variables p, V, T, n, M,… First law: DEth 5 W 1 Q Energy out Wout 5 area inside pV curve 5 QH 2 QC Wout h5 QH TC hmax 5 hCarnot 5 1 2 TH Hot reservoir TH QH Heat engine QC Cold reservoir TC Wout Q.0 Heat to system Q,0 Heat out of system Work Requires volume change Gas: W 5 23 p dV 5 2 (area under pV curve) Thermal Energy Eth 5 1 NkBT per 2 degree of freedom Heat Requires temperature difference Q 5 McDT or nCDT Q 5 6ML for phase changes 598 ...
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