Conservative Forces Nonconservative Forces
Total Page：16
File Type：pdf, Size：1020Kb
Load more
Recommended publications
-
Classical Mechanics
Classical Mechanics Hyoungsoon Choi Spring, 2014 Contents 1 Introduction4 1.1 Kinematics and Kinetics . .5 1.2 Kinematics: Watching Wallace and Gromit ............6 1.3 Inertia and Inertial Frame . .8 2 Newton's Laws of Motion 10 2.1 The First Law: The Law of Inertia . 10 2.2 The Second Law: The Equation of Motion . 11 2.3 The Third Law: The Law of Action and Reaction . 12 3 Laws of Conservation 14 3.1 Conservation of Momentum . 14 3.2 Conservation of Angular Momentum . 15 3.3 Conservation of Energy . 17 3.3.1 Kinetic energy . 17 3.3.2 Potential energy . 18 3.3.3 Mechanical energy conservation . 19 4 Solving Equation of Motions 20 4.1 Force-Free Motion . 21 4.2 Constant Force Motion . 22 4.2.1 Constant force motion in one dimension . 22 4.2.2 Constant force motion in two dimensions . 23 4.3 Varying Force Motion . 25 4.3.1 Drag force . 25 4.3.2 Harmonic oscillator . 29 5 Lagrangian Mechanics 30 5.1 Configuration Space . 30 5.2 Lagrangian Equations of Motion . 32 5.3 Generalized Coordinates . 34 5.4 Lagrangian Mechanics . 36 5.5 D'Alembert's Principle . 37 5.6 Conjugate Variables . 39 1 CONTENTS 2 6 Hamiltonian Mechanics 40 6.1 Legendre Transformation: From Lagrangian to Hamiltonian . 40 6.2 Hamilton's Equations . 41 6.3 Configuration Space and Phase Space . 43 6.4 Hamiltonian and Energy . 45 7 Central Force Motion 47 7.1 Conservation Laws in Central Force Field . 47 7.2 The Path Equation . -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
Conservative Forces and Potential Energy
Program 7 / Chapter 7 Conservative forces and potential energy In the motion of a mass acted on by a conservative force the total energy in the system, which is the sum of the kinetic and potential energies, is conserved. In this section, this motion is computed numerically using the Euler–Cromer method. Theory In section 7–2 of your textbook the oscillatory motion of a mass attached to a spring is described in the context of energy conservation. Specifically, if the spring is initially compressed then the system has spring potential energy. When the mass is free to move, this potential energy is converted into kinetic energy, K = 1/2mv2. The spring then stretches past its equilibrium position, the potential energy increases again until it equals its initial value. This oscillatory motion is illustrated in Fig. 7–7 of your textbook. Consider the more complicated situation in which the force on the particle is given by F(x) = x - 4qx 3 This is a conservative force and the its potential energy is 1 2 4 U(x) = - 2 x + qx (see Fig. 7–10). From the force, we can calculate the motion using Newton’s second law. The program that you will use in this section calculates this motion and demonstrates that the total energy, E = K + U, is conserved (i.e., E remains constant). Given the force, F, on an object (of mass m), its position and velocity may be found by solving the two ordinary differential equations, dv 1 dx = F ; = v dt m dt If we replace the derivatives with their right derivative approximations, we have v(t + Dt) - v(t) 1 x(t + Dt) - x(t) = F(t) ; = v(t) Dt m Dt or v - v 1 x - x f i = F ; f i = v Dt m i Dt i where the subscripts i and f refer to the initial (time t) and final (time t+Dt) values. -
Conservation of Energy for an Isolated System
POTENTIAL ENERGY Often the work done on a system of two or more objects does not change the kinetic energy of the system but instead it is stored as a new type of energy called POTENTIAL ENERGY. To demonstrate this new type of energy let‟s consider the following situation. Ex. Consider lifting a block of mass „m‟ through a vertical height „h‟ by a force F. M vf=0 h M vi=0 Earth Earth System = Block F m s mg WKnet K0 (since vif v 0) WWWnet F g 0 o WFg W mghcos180 WF mgh 1 System = block + earth F system m s Earth Wnet K K 0 Wnet W F mgh WF mgh Clearly the work done by Fapp is not zero and there is no change in KE of the system. Where has the work gone into? Because recall that positive work means energy transfer into the system. Where did the energy go into? The work done by Fext must show up as an increase in the energy of the system. The work done by Fext ends up stored as POTENTIAL ENERGY (gravitational) in the Earth-Block System. This potential energy has the “potential” to be recovered in the form of kinetic energy if the block is released. 2 Ex. Spring-Mass System system N F’ K M F M F xi xf mg wKnet K0 (Since vi v f 0) w w w w w netF ' N mg F w w w net F s 11 w k22 k s22 i f 11 w w k22 k F s22 f i The work done by Fapp ends up stored as POTENTIAL ENERGY (elastic) in the Spring- Mass System. -
6. Non-Inertial Frames
6. Non-Inertial Frames We stated, long ago, that inertial frames provide the setting for Newtonian mechanics. But what if you, one day, find yourself in a frame that is not inertial? For example, suppose that every 24 hours you happen to spin around an axis which is 2500 miles away. What would you feel? Or what if every year you spin around an axis 36 million miles away? Would that have any e↵ect on your everyday life? In this section we will discuss what Newton’s equations of motion look like in non- inertial frames. Just as there are many ways that an animal can be not a dog, so there are many ways in which a reference frame can be non-inertial. Here we will just consider one type: reference frames that rotate. We’ll start with some basic concepts. 6.1 Rotating Frames Let’s start with the inertial frame S drawn in the figure z=z with coordinate axes x, y and z.Ourgoalistounderstand the motion of particles as seen in a non-inertial frame S0, with axes x , y and z , which is rotating with respect to S. 0 0 0 y y We’ll denote the angle between the x-axis of S and the x0- axis of S as ✓.SinceS is rotating, we clearly have ✓ = ✓(t) x 0 0 θ and ✓˙ =0. 6 x Our first task is to find a way to describe the rotation of Figure 31: the axes. For this, we can use the angular velocity vector ! that we introduced in the last section to describe the motion of particles. -
1 Classical Theory and Atomistics
1 1 Classical Theory and Atomistics Many research workers have pursued the friction law. Behind the fruitful achievements, we found enormous amounts of efforts by workers in every kind of research field. Friction research has crossed more than 500 years from its beginning to establish the law of friction, and the long story of the scientific historyoffrictionresearchisintroducedhere. 1.1 Law of Friction Coulomb’s friction law1 was established at the end of the eighteenth century [1]. Before that, from the end of the seventeenth century to the middle of the eigh- teenth century, the basis or groundwork for research had already been done by Guillaume Amontons2 [2]. The very first results in the science of friction were found in the notes and experimental sketches of Leonardo da Vinci.3 In his exper- imental notes in 1508 [3], da Vinci evaluated the effects of surface roughness on the friction force for stone and wood, and, for the first time, presented the concept of a coefficient of friction. Coulomb’s friction law is simple and sensible, and we can readily obtain it through modern experimentation. This law is easily verified with current exper- imental techniques, but during the Renaissance era in Italy, it was not easy to carry out experiments with sufficient accuracy to clearly demonstrate the uni- versality of the friction law. For that reason, 300 years of history passed after the beginning of the Italian Renaissance in the fifteenth century before the friction law was established as Coulomb’s law. The progress of industrialization in England between 1750 and 1850, which was later called the Industrial Revolution, brought about a major change in the production activities of human beings in Western society and later on a global scale. -
1 Review Before First Test Physical Mechanics Fall 2000 Newton's Laws
Review before first test Physical Mechanics Fall 2000 Newton's Laws – You should be able to state these laws using both words and equations. The 2nd law most important for meteorology. Second law: net force = mass × acceleration Note that it is the net force – all forces acting on the object have to be taken into account – remember the gravity in one of your homework problem? Inertial and non-inertial reference frame – in which reference frame are the Newton's Law valid? Can Newton's Law be used in a non-inertial reference frame? Anything needs to be done to do so? In equation form, the second law is F = m a where a is the acceleration rate. dV a = dt where V is the velocity defined as dx V = . dt x is the coordinate of the objection, and is a function of time t. F = ma can have the following forms: dVdx2 mFmF==and . dtdt 2 Momentum, momentum theorem, impulse For an object with non-constant mass, the more accurate way of expressing the second law is 1 dPd()mVdæödx º==Fand ç÷mF dtdtdtèødt which says that net force = rate of change of momentum where Pº mV is the definition of momentum. The above is the differential form of momentum theorem. The integral form of momentum theorem is t2 P21-P=ºFdt Impulse (6) òt1 which says Impulse from the net force (F × Dt) = change in momentum Note that the change in momentum does not require displacement. The impulse can be applied to an object before an appreciable displacement occurs. -
Mechanics Potential Energy Conservative and Nonconservative Forces
Mechanics Potential Energy Conservative and Nonconservative Forces Lana Sheridan De Anza College Oct 31, 2018 Last time • work done by a varying force • kinetic energy • the work-kinetic energy theorem • power Overview • concept of potential energy • conservative and nonconservative forces • potential energy definition • some kinds of potential energy • potential energy diagrams WALKMC08_0131536311.QXD 12/8/05 17:47 Page 205 8–1 CONSERVATIVE AND NONCONSERVATIVE FORCES 205 8–1 Conservative and Nonconservative Forces In physics, we classify forces according to whether they are conservative or nonconservative. The key distinction is that when a conservative force acts, the work it does is stored in the form of energy that can be released at a later time. In this section, we sharpen this distinction and explore some examples of conserva- tive and nonconservative forces. Perhaps the simplest case of a conservative force is gravity. Imagine lifting a box of mass m from the floor to a height h, as in Figure 8–1. To lift the box with constant speed, the force you must exert against gravity is mg. Since the upward distance is h, the work you do on the box is W = mgh. If you now release the box andWork allow it to Done drop back Lifting to the floor, a gravity Box does the same work, W = mgh, and in the process gives the box an equivalent amount of kinetic energy. Work done by person (applied force) W = Fd cos(0◦) = mgh. Work done by person = mgh Work done by gravity = mgh ▲ FIGURE 8–1 Work against gravity Lifting a box against gravity with con- stant speed takes a work mgh. -
Mechanical Energy Change in Inertial Reference Frames Saeed Ghanbari, University of Zanjan, Zanjan, Iran
Mechanical Energy Change in Inertial Reference Frames Saeed Ghanbari, University of Zanjan, Zanjan, Iran he mechanical energy change of a system in an iner- reference. In an inertial frame of reference, in the presence of tial frame of reference equals work done by the total nonconservative forces such as frictional, normal, and tension nonconservative force in the same frame. This rela- forces, if the work done by nonconservative forces on a system Ttion is covariant under the Galilean transformations from is zero, the mechanical energy is conserved. The work done inertial frame S to S, where S moves with constant velocity by the same nonconservative forces in another inertial refer- relative to S. In the presence of nonconservative forces, such ence frame may be nonzero. Therefore, in the second frame as normal and tension forces, the mechanical energy of a of reference mechanical energy is not conserved. Considering system can be conserved in S and not be conserved in S. In three examples, we discuss the results. In this work, problems this paper we find useful relations between the mechanical involving rotational kinetic and potential energy changes are energy changes in two inertial frames of reference. not studied. Introduction Covariance of DE = Wnc The fact that the work done by an arbitrary force could According to the work-kinetic energy theorem,5 which is have different values in different inertial frames of reference derived from Newton’s second law, the kinetic energy change is not usually covered in the introductory physics textbooks DK in S frame equals work W done by total force F, and is neither well discussed nor explained in the physics classrooms. -
PHYSICS 149: Lecture 15
PHYSICS 149: Lecture 15 • Chapter 6: Conservation of Energy – 6.3 Kinetic Energy – 6.4 Gravitational Potential Energy Lecture 15 Purdue University, Physics 149 1 ILQ 1 Mimas orbits Saturn at a distance D. Enceladus orbits Saturn at a distance 4D. What is the ratio of the periods of their orbits? A) Tm/Te = 1/8 T23∝ r 2 ⎛⎞ 3 B) Tm/Te =1/4= 1/4 Tm ⎛⎞D ⎜⎟= ⎜⎟ ⎝⎠TDE ⎝⎠4 C) Tm/Te = 1/2 ⎛⎞T 11 D) T /T =2= 2 ⎜⎟m = = m e ⎝⎠TE 64 8 Lecture 15 Purdue University, Physics 149 2 ILQ 2 A pendulum bob swings back and forth along a circular path. Does the tension in the string do any work on the bob? Does gravity do work on the bob? A) only tension does work B) both do work C) neither do work D) only gravity does work Lecture 15 Purdue University, Physics 149 3 Energy • Energy is “conserved” meaning it can not be created nor destroyed – Can change form – Can be transferred • Total Energy of an isolated system does not change with time • Forms – Kinetic Energy Motion – Potential Energy Stored – Heat – Mass (E=mc2) • Units: Joules = kg m2 /s/ s2 Lecture 15 Purdue University, Physics 149 4 Definition of “Work” in Physics • Work is a scalar quantity (not a vector quantity). • Units: J (Joule), N⋅m, kg⋅m2/s2, etc. – Unit conversion: 1 J = 1N1 N⋅m = 1kg1 kg⋅m2/s2 • Work is denoted by W (not to be confused by weight W). Lecture 15 Purdue University, Physics 149 5 Total Work • When several forces act on an object, the “total” work is the sum of the work done by each force individually: Lecture 15 Purdue University, Physics 149 6 ILQ 1 You are towing a car up a hill with constant velocity. -
Conservative Forces 11 - 1
PHYS 101 Lecture 11 - Conservative forces 11 - 1 Lecture 11 - Conservative forces What’s important: • conservation of energy; power Demonstrations: none Conservative Forces The concept of potential energy that we introduced in the last lecture has an aspect of reversibility associated with it: • we do work to slide a block up an inclined plane (increase U) • gravity does work as a block slides down an inclined plane (decrease U). What about a force like friction? We can slide a book across a table against a frictional force f vi = 0 vf = 0 We do work on the book, W ≠ 0, but vi = vf = 0 and there is no change in K: ΔK = 0. Further, there is no change in U: after we have stopped pushing the book, it does not move back into its original position (i.e. the potential energy of the book hasn’t changed, so the book can’t reduce its potential energy by moving to its original position)! So, friction does not have a potential energy U associated with it. We say that gravity is a conservative force: it has a potential energy which depends on position. Friction is a non-conservative (or dissipative) force with no potential energy. Are there other differences between conservative and non-conservative forces? Gravity Friction W = Uf - Ui W = f Δx + f Δy + f Δx + f Δy depends only depends on the total path taken on the end-points Δx Δy So, in a conservative force, the work depends only on end-points of the path; in a non- conservative force, the work depends on the path. -
Work Done in Lifting an Object
Work done in lifting an object y = h A force F lifts the mass at constant speed through a height h. F The displacement is h a = 0 The applied force in the direction of the y = 0 displacement is: F = mg (no acceleration) mg The work done by the force F is: W = Fh = mgh But the kinetic energy has not changed – the gravity force mg has done an equal amount of negative work so that the net work done on the mass is zero 14 Work done in lifting an object y = h Alternative view: define a different form of energy – Gravitational potential energy, PE = mgy F a = 0 Define: Mechanical energy = kinetic energy + potential energy y = 0 Mechanical energy, E = mv2/2 + mgy mg Then: Work done by applied force, F, is (change in KE) + (change in PE) So W = Fh = ∆KE + ∆PE 15 Check, using forces and acceleration y = h v Net upward force on the mass is F – mg F Apply Newton’s second law to find the acceleration: a F – mg = ma, y = 0 vo so, a = (F – mg)/m mg One of famous four equations – 2 2 v = v0 + 2ah So, v2 = v2 + 2(F mg)h/m 0 − ( m/2) × mv2/2 mv2/2 = Fh mgh That is, DKE = Fh DPE − 0 − − Or, W = Fh = DKE + DPE 16 Or, W = Fh = DKE + DPE If there is no external force, W = 0 and 0 = DKE + DPE so that DKE = DPE − As the mass falls and loses potential energy, it gains an equal amount of kinetic energy.