Therefore we give some suggestions below for minimum coverage of chapters. Sometimes it is not desirable to include any electrical or magnetic problems in the beginning course. We believe that the text can be used in this fashion, but it is true that many students find the electrical problems very interesting.
Many instructors find it difficult to be ruthless in cutting material. Our own experience is that it is better to cover some material well than to cover more material less well. The advanced sections and the Advanced Topics should give the talented students something with which to stretch their abilities and the students who go on in physics a reference work that can be used in connection with later studies. With these comments we proceed to the details of the several chapters. Chapter 1.
As in the first edition, this chapter is not an essential part of the study of mechanics, but it may provide interesting reading for those with broader interests. For instructors who wish to assign the reading, it may provide a good place to illustrate the concept of order of magnitude. Chapter 2. Vectors introduce the student to the language that is very useful in physics.
As pointed out in the text, the vector product can be omitted here along with the examples of magnetic forces in which v and Bare not perpendicular.
One can proceed to Chap. The scalar product is used often in finding magnitudes and in Chap. In addition it provides a tool for solving numbers of interesting problems. The section on vector derivatives is "also useful, but the parts treating the unit vectors rand jj can be omitted and I Teaching Notes introduced much later.
Hopefully, circular motion is a good introduction of the dynamics to come. Chapter 3. This is a long chapter with a good many applications. Newton's laws are introduced in conventional form and we proceed to applications of the Second Law. For a shortened course or one that does not include electrical and magnetic applications, the section on them can be omitted entirely or the magnetic field can be treated only for the case of velocity and magnetic field perpendicular.
Conservation of momentum is then introduced through Newton's Third Law. Kinetic energy is referred to in collision problems even though it is not introduced until Chap. Most students have heard of it in high school and do not find difficulty with it; but it can be omitted if desired. Chapter 4. As pointed out in the text, this chapter is not of the conventional type.
Many physicists find appeal in the introduction of galilean transformations, and for those planning to go on to special relativity, it does provide a nice introduction to transformations of coordinates. However, to nonphysics students and to those with limited time, it may be too much "frosting on the cake" and should be omitted. Some reference to accelerated frames of reference and fictitious forces should probably be included, but these can be taken from the first few pages.
Chapter 5. Work and kinetic energy are introduced, first in one dimension and then in three dimensions. The scalar product is really necessary here, but certainly the use of the line integral can be skirted. Potential energy is treated in detail. In a shorter course, the discussion of conservative fields could well be omitted as could the discussion of electrical potential.
However, this is an important chapter and should not be hurried through. Chapter 6. This chapter treats collisions again and introduces the centerof-mass system of reference. Center of mass is an important concept for rigid bodies, and although the center-of-mass system is widely used, a shortened version of a mechanics course could well omit this.
The introduction of angular momentum and torque requires the use of the vector product. By this time, students have achieved a level where they can grasp and use the vector product, and if it has been omitted earlier, it can be taken up here. The conservation of angular momentum is an appealing topic to many students. Chapter 7. Here the Mathematical Notes should be studied first if the students have had difficulty with differential equations. The mass on the spring and the pendulum provide straightforward examples of this important subject of oscillatory motion.
In a shortened version, the sections on xiii xiv Teaching Notes average values of kinetic and potential energy, damped motion, and forced oscillations can be omitted completely. The laboratory can provide excellent examples of this type of motion.
The Advanced Topics on the Anharmonic Oscillator and the Driven Oscillator will be interesting to the more advanced student. Chapter 8. The present authors believe that an introductory treatment of rigid bodies is valuable to all students. The ideas of torque and angular acceleration about a fixed axis are not difficult, and they provide the student connections with the real, visible world.
The simple treatment of the gyro is also valuable; but the introduction of principal axes, products of inertia, and rotating coordinate systems should probably be omitted in most courses. Chapter 9. Central-force problems are very important. Some instructors may not wish to spend so much time on evaluating the potential inside and outside spherical masses, and this of course can be omitted. They may also find the labor of integrating the r equation of motion too much, in which case they can omit it.
They should enjoy the Advanced Topic. There is a good deal that can be cut from this chapter if necessary, but the work of mastering it is very rewarding. The two-body problem and the concept of reduced mass are also useful but again can be omitted in a shortened course. Chapter This chapter reviews a number of methods of determining the speed of light. For a course in mechanics, this material is not essential.
We believe that students will be interested in it, but it could be assigned as outside reading. Then comes the Michelson-Morley experiment, which in a course like this is the most convincing evidence of the need for a change from the galilean transformation. The doppler effect is introduced because of the evidence that the recessional doppler effect provides for high speeds of distant stars, and the chapter closes with a section on the speed of light as the ultimate speed for material objects and the failure of the newtonian formula for kinetic energy.
For those with limited time for the study of special relativity, a cursory reading of the chapter might be sufficient. In this chapter the Lorentz transformation equations are derived and applied to the most common characteristics of special relativity, length contraction, and time dilation. The velocity transformations are introduced and some examples given. This chapter is the basis for the following chapters, and consequently ample time should be alloweq for the study of it.
Teaching Notes Chapter The results of Chap. The relation to experiments with high-energy particles and to high-energy nuclear physics needs to be emphasized.
At this stage students may be only vaguely aware of, for example, nuclear physics; but the examples are so pertinent to the public today that it should be easy to teach.
Finally the subject of particles with zero rest mass will answer the questions of many alert students. A number of examples of the subjects developed in the previous chapter are treated here. The center-of-mass system is brought in and its advantages pointed out. In a shortened course all this can be omitted. Good students will be interested in it, and it can be referred to as outside reading in other physics courses treating special relativity. In recent years the study of general relativity has become quite popular, and this chapter could provide a bridge to reading in general relativity.
It is, of course, not central to the subject of special relativity in the usual sense, but many students may be interested in the difference between gravitational and inertial mass, and almost all will have heard about the tests of general relativity. In the first year many more new ideas, concepts, and methods are developed than in advanced undergraduate or graduate courses. A student who understands clearly the basic physics developed in this first volume, even if he may not yet be able to apply it easily to complex situations, has put behind him many of the real difficulties in learning physics.
What should a student do who has difficulty in understanding parts of the course and in working problems, even after reading and rereading the text?
First he should go back and reread the relevant parts of a high school physics book. Then he should consult and study one of the many physics books at the introductory college level.
Many of these are noncalculus texts and so the difficulties introduced by the mathematics will be minimized. The exercises, particularly worked-out exercises, will probably be very helpful.
Finally, when he understands these more elementary books, he can go to some of the other books at this level that are referred to in the Appendix. Of course, he should remember that his instructors are the best source for answering his questions and clearing up his misunderstandings.
Many students have difficulty with mathematics. In addition to your regular calculus book, many paperbacks may be helpful. Note to the Student Units Every mature field of science and engineering has its own special units for quantities which occur frequently. The acre-foot is a natural unit of volume to an irrigation engineer, a rancher, or an attorney in the western United States.
The MeV or million electron volts is a natural unit of energy to a nuclear physicist; the kilocalorie is the chemist's unit of energy, and the kilowatt-hour is the power engineer's unit of energy. The theoretical physicist will often simply say: Choose units such that the speed of light is equal to unity.
A working scientist does not spend much of his time converting from one system of units to another; he spends more time in keeping track of factors of 2 and of plus or minus signs in his calculations. Nor will he spend much time arguing about units, because no good science has ever come out of such an argument. Physics is carried out and published chiefly in the gaussian cgs and the SI or mks units.
Every scientist and engineer who wishes to have easy access to the literature of physics will need to be familiar with these systems. The text is written in the gaussian cgs system; but a number of references are made to the SI units Systeme Internationale , which until recently were more commonly called mks or mksa units.
The transformation from cgs to SI units in mechanical problems is easy, as will be explained in the text. However, when one comes to problems in electricity and magnetism there is difficulty. In the text, explanation is given of both systems, and some examples are worked in both systems. It is not clear whether the change to the SI units that began more than twenty years ago will continue.
In the current physics literature there still seem to be more papers in the cgs system, which is the reason for retaining it in this volume. In a course such as this, we want to make it as easy as possible for both sceintists and engineers to read the journals, particularly physics journals.
Notation xviii Notation Physical Constants Approximate values of physical constants and useful numerical quantities are printed inside the front and back covers of this volume.
More precise values of physical constants are tabulated in E. Cohen and J. DuMond, Rev. Taylor, W. Parker, and D. Langenberg, Rev. Signs and Symbols In general we have tried to adhere to the symbols and unit abbreviations that are used in the physics literature-that are, for the most part, agreed upon by international convention. We summarize here several signs which are used freely throughout the book. The American Institute of Physics encourages use of the sign where others might write either or -.
Style Manual, American Institute of Physics, rev. Order of Magnitude By this phrase we usually mean "within a factor of 10 or so. It is an exceptionally valuable professional habit, although it often troubles beginning students enormously.
We say, for example, that is the order of magnitude of the numbers and 25, In cgs units the order of magnitude of the mass of the electron is 27 g; the accurate value is 0.
Notation We say sometimes that a solution includes is accurate to terms of order x 2 or E, whatever the quantity may be. This is also written as O x 2 or O E. The language implies that terms in the exact solution which involve higher powers such as x 3 or E2 of the quantity may be neglected for certain purposes in comparison with the terms retained in the approximate solution.
The same argument is valid within our three-dimensional space. Values less than can be observed out to about 3 X 10 20 cm, t the limit of angle measurement with present telescopes. By a similar argument we have d - A'B dt. Heath and Company, Boston, Ernst Mach, "The Science of Mechanics," chap.
An excellent elementary discussion of the Foucault pendulum is given under Foucault. This is a quite complete list of works. A conservation law is usually the consequence of some underlying symmetry in the universe. There are conservation laws relating to energy, linear momentum, angular momentum, charge, number of baryons protons, neutrons, and heavier elementary particles , strangeness, and various other quantities.
In Chaps. In this chapter we discuss the conservation of energy. In Chap. The entire discussion at present will be phrased for the nonrelativistic regime, which means a restriction to galilean transformations, speeds very much less than that of light, and independence of mass and energy. If all the forces in a problem are known, and if we are clever enough and have computers of adequate speed and capacity to solve for the trajectories of all the particles, then the conservation laws give us no additional information.
But since we do not have all this information and these abilities and facilities, the conservation laws are very powerful tools. Why are conservation laws powerful tools?
Conservation laws are independent of the details of the trajectory and, often, of the details of the particular force. The laws are therefore a way of stating very general and significant consequences of the equations of motion.
A conservation law can sometimes assure us that something is impossible. Thus we do not waste time analyzing an alleged perpetual motion device if it is merely a closed system of mechanical and electrical components, or a satellite propulsion scheme that purportedly works by moving internal weights. In the exploration of new and not yet understood phenomena the conservation laws are often the most striking phYSical facts we know.
They may suggest appropriate invariance concepts. Many physicists have a regular routine for solving unknown problems: First they use the relevant conservation laws one by one; only after this, if there is anything left to the problem, will they get down to real work with differential equations, variational and perturbation methods, computers, intuition, and other tools at their disposal.
These concepts, which can be understood from a simple example, arise very naturally from Newton's Second Law and will be treated in detail later. To start with, we discuss forces and motions in only one dimension, which simplifies the notation.
The development is repeated for three dimensions; the student may find the repetition helpful. To develop the concepts of work and kinetic energy we consider a particle of mass M drifting in intergalactic space and initially free of all external interactions. We observe the particle from an inertial reference frame.
A force F is applied to the particle at time t O. The force thereafter is kept constant in magnitude and direction; the direction is taken to he the y direction. The particle will accelerate under the action of the applied force. Notice that Eq. The change is caused by the force l' acting for a distance y - Yo ' It is a useful definition of work to call l' X y - Yo the work done on the particle by the applied force.
With these definitions Eq. Conservation of Energy ter of definition, but the definitions are useful and they follow from Newton's Second Law. To convert joules to ergs, multiply by 10 7 the value of the work expressed in joules since we have seen in Chap. In talking about work, one must always specify work done by what. In the case above, the work is done by the force that accelerates the particle. Such forces are often integral parts of the system that we are investigating; for example, they may be gravitational, electric, or magnetic forces.
Later, when we talk about potential energy, we shall call these forces of the field, or forces of the system; but we shall also consider forces applied by an external agent perhaps by us , and it will be important to distinguish work done by field forces from that done by the agent. For example, if the agent applies a force always equal and opposite to the field force, then the particle will not be accelerated and no change in kinetic energy will be produced.
It is important to note that we are excluding effects of friction forces in the present discussion; we are using ideal situations to establish our definitions and concepts. As the body falls toward the surface of the earth, the work done by gravity is equal to the gain in kinetic energy of the body see Fig.
Mv 2 FIG. Equation 5. What happens to the potential energy when a particle at rest on the earth's surface is raised to a height h? Note that we call the force that we exert Fag; in other words, we and the external agent are identical. Of course, it is easy to talk about "we" and "us," and the terms are used below; but the important point to remember is that here an external agent is conceptually brought into the problem only for the purpose of evaluating the potential energy.
In the absence of friction forces a specific definition of the potential energy of a body particle at a point of interest can now be formulated: Potential energy is the work we do in moving the body without acceleration from an initial location, arbitrarily assigned to be a zero of potential energy, to the point of interest. A few comments may aid our understanding of this definition. We are free to arbitrarily assign the location of zero potential energy according to convenience, and so the value at the point of interest will always be relative to this assignment.
Presumably there are field forces acting upon the body, and to move it without acceleration we must exert a force equal and opposite to their resultant force. Under this condition we move the body without acceleration from the zero position to the point where we wish to evaluate the potential energy. The work we have done is equal to the potential energy. Since, in the absence of friction, the force we apply is always equal and opposite to the field forces present in the Conservation of Energy problem, the work we do is equal and opposite to the work done by those forces.
Therefore, we can equally well define potential energy as the work done by the forces of the problem, the field forces, in moving the system in the other direction from the point under consideration to the arbitrary zero. For example, the work done by gravity [Eq. Equally valid is the definition of positive potential energy at a point as the kinetic energy generated by the forces in the free motion of the body to the arbitrary zero, as in Fig..
This definition, as stated, does not apply to cases in which the potential energy is negative relative to the zero; but an obvious modification of the definition is valid. An example is given on pages Two further points are worth emphasizing. First, the potential energy is purely a function of position, Le.
Second, the zero point must always be specified. It is only the change in potential energy that is meaningful; for example, it may be converted into kinetic energy or, conversely, created from it.
The absolute value of the potential energy is meaningless. Since this is true, the choice of the location of zero is arbitrary. In many cases a certain zero is particularly convenient, e. We denote the potential energy by U or PE. If in Eq. This is illustrated in Fig. Because E is a constant we have in Eq.
At height y. The potential energy IS converted Into kinetic energy. After the car makes contact with the ground, the kinetic energy IS converted to heat In the shock absorbers, springs. The symbol E denotes the total energy, which is constant in time for an isolated system. Two illustrations are given in Figs. U defines a potential energy field; it is a scalar function of y. The forces are derivable from this field function. Note here that the zero will appear in U as a constant term so when the force is derived in Eq.
Assume the level of projection is the position of zero potential energy. The lotal energy. The law of conservation of energy states that for a system of particles with interactions not explicitlyl dependent on the time, the total energy of the system is constant. We accept this result as a very well established experimental fact.
More specifically, the law tells us there exists some scalar function [such as the function! For example, the mass m or the elementary charge e must not change with time. Besides the energy function, there are other functions that are constant in the conditions specified here. We treat other functions in Chap. The energy is a scalar constant of motion. We interpret the phrase external interaction to include any change in the laws of physics or in the values of the fundamental physical constants such as g or e or m during the relevant time interval as well as any change in external conditions such as gravitational, electric, or magnetic fields.
In our present treatment we do not consider changes of energy from mechanical form kinetic and potential into heat. For example, we omit forces of friction; they are not what we define later as conservative forces. By consistent we mean that, for example, -.! You can check this for Eq. The hamiltonian formulation of mechanics, in particular, is one way that is very well suited to reinterpretation in the language of quantum mechanics.
But here at the beginning of our course we need a simple direct formulation more than we need the generality of the hamiltonian or lagrangian formulations, which are the subject of later courses.!
Work We begin by generalizing the definition of work. Suppose F is not constant but is a function of the position r.
Equation ,5. The limit lim", F r. The integral is called the line integral of F from A to B. We want to generalize Eg. But we can rearrange the integrand.! This is a generalization of Eg. We recognize' 5. We see from Eg. Here we have let the gravitational force play the role of the force F.
The initial value K A of the kinetic energy is! We can understand that if the potential energy increases in going from A to B, the kinetic energy of a free particle moving in that direction will decrease of course, Fag is not acting , whereas if the potential energy decreases, the kinetic energy will increase. A linear restoring force is one that is directly proportional to the displacement measured from some fixed point and in a direction tending to reduce the displacement see Fig.
This is called Hooke's law, For sufficiently small displacements such a force may be produced by a stretched or compressed spring.
For large elastic displacements we must add terms in higher powers of x to Eq. FIG,5,7 In order to stretch or compress the spring, we must exert a force In opposition to the restoring force In displaCing the spnng an amount.. Here the force on the particle is a function of position. To calculate the work we do, we use the definition [Eq. We obtain the answer directly from Eqs. Alternatively, we can use the conservation of energy. The water at the top of the waterfall has gravitational potential energy, which in falling is converted into kinetic energy.
The velocity v is determined by this equation if the initial velocity V o of the water is known. The kinetic energy of the falling water can be converted in a powerhouse into the rotational kinetic energy of a turbine; otherwise the kinetic energy of the falling water is converted at the foot of the falls into thermal energy or heat.
At a high temperature the random molecular motion is more vigorous than at a low temperature. EXAMPLE Energy Transformations in the Pole Vault A rather amusing example of the interconversion of energy among various formskinetic energy, potential energy of the bent elastic pole, and potential energy due to elevation-is afforded by the sequence of pictures in Fig.
We can easily see that a central force is conservative. A central force exerted by one particle on another is a force whose magnitude depends only on the separation of the particles and whose direction lies along the line joining the particles. In Fig. Two paths labeled 1 and 2 connect points A and B as shown. The dashed curves are sectors of circles centered at O. We may regard F.
Now the magnitudes F j and F2 are equal on the two segments because they lie at equal distances from the point 0; the projections dr cos of the path segments on the respective vectors F are equal because, as we can see, the separation of the circles measured along the direction of F j is equal to the separation measured along F 2' Therefore on the path segments considered.
D Mgy. For conservative forces the work done around a closed path is zero. Suppose the force depends on the velocity with which the path is traversed. The force on a charged particle in a magnetic field depends on velocity. Can such a force be conservative? It turns out that the important fundamental velocity-dependent forces are conservative because their direction is perpendicular to the direction of motion of the particle, so that F.
You can see this for the Lorentz force Chap. Frictional forces are not really fundamental forces, but they are velocity-dependent and not conservative. All of our discussion presupposes two-body forces. This is an important assumption; it is likely that some of the students in this course will be called upon in their research careers to do battle with many-body forces.
A discussion of what is involved in the two-body assumption is given in Volume 2 Sec. This result for interactions between elementary particles is inferred from scattering experiments; for gravitational forces the result is inferred from the accuracy of the prediction of planetary and lunar motions, as discussed in the Historical Notes at the end of this chapter.
We also know that the earth has made about 4 X 10 9 complete orbits around the sun without any important change in distance to the sun, as judged from geologic evidence on the surface temperature of the earth. The relevant geologic evidence extends back perhaps yr and cannot be taken as entirely conclusive because of the numerous factors, including solar output, that affect the temperature, but the observation is suggestive. Further examples are discussed in the Historical Notes.
We need to say more about central vs noncentral forces. In consideration of the force between two particles there are two possibilities: the particles have no coordinates other than their positions; one or both particles have a physically distinguished axis. In the first possibility there can only be a central force, while in the second the specification that the particle Conservation of Energy be moved from A to B is incomplete-we have also to specify that the axis be kept in the same direction relative to something.
A bar magnet has a physically distinguished axis; if we move the magnet bodily around a closed path in a uniform magnetic field, we mayor may not do a net amount of work on the magnet. If the magnet ends up at the same location and in the same orientation as it started out, no work is done.
If the location is the same but the orientation is different, work will have been done. The work may have a positive or negative sign. It is easy to see that friction is not a conservative force. It is always opposed to the direction of motion, and so the work done by a constant frictional force in a motion from A to B, a distance d, will be Ffric d; if the motion is from B to A, it will also be Ffric d.
But if friction is a manifestation of fundamental forces and they are conservative, how can friction be nonconservative? This is a matter of the detail of our analysis. If we analyze all motion on the atomic level, that of fundamental forces, we shall find the "motion" conservative; but if we see some of the motion as heat, which is useless in the mechanical sense, we shall consider that friction has acted. The identity of heat and random kinetic energy is treated in Volume 5.
In the discussion of conservation of momentum in Chap. Kinetic energy was not conserved; but the sum of the kinetic and internal excitation energy for the two particles was called the total energy and was assumed to be conserved, in agreement with all known experiments. We return now to our discussion of potential energy. The discussion of conservative forces emphasizes the remark on page that the potential energy at a point can be uniquely and hence usefully defined only in the case of conservative forces.
We have seen how to calculate the potential energy from a knowledge of the forces acting in a problem; we choose a zero and then calculate the work we do or the agent does in moving the system slowly, without changing the kinetic energy, from the zero to the desired position. POSitions of equilibrium.
Thus E must be zero if the total energy of the particle is constant between launching and escape; whence the escape velocity De is given by FIG. For bodies launched from earth, escape from the solar system is more difficult than escape from the earth. If R e is the radius of the earth and y is the height above the surface of the earth, we wish to show that f Still later. It is shown as follows.
Divide numerator and denominator by Re : We can now use the expansion equation 2. Using Newton's already solved the problem. Here we must use the fact that V x does not change. Potential What is the potential at this point? From Eqs. The difference 7. What is the kinetic energy when the protons have moved infinitely far apart?
The proton moves through a potential drop of V. What is its final kinetic energy'? But we know from Eq. Rewriting Eq. Let us write down the generalization of Eq.
Notice that U r 1 - r 2 is put in only once: If two particles interact, the interaction energy is mutual! If particles 1 and 2 are protons in the earth's gravitational field, the energy E in Eq. The last term is the coulomb energy of the two protons; the next-to-last term is their Conservation of Energy mutual gravitational energy. The ratio of the last two terms is GM2 ;; 7 X 48 - 36 1O- HJ - showing, since the forces depend on the distance in the same way, that the gravitational force between protons is extremely weak in comparison with the electrostatic force.
To obtain the power in watts from the value expressed in horsepower, multiply by , approximately. Express the answer in ergs and in joules, and refer the potential energy to the surface of the earth. Neglect friction. The sum of c and el should equal a or b. Potential energy above earth What is the potential energy U R e of a mass of I kg on the surface of the earth referred to zero potential energy at infinite distance?
If charge is expressed in esu, the result will be in ergs. Pay special attention to the sign of the answer. The velocity of the satellite relative to the center of the earth is v, and the mass is M. What is the kinetic energy of the moon relative to the earth? The relevant data are given in the table of constants inside the cover of this volume.
Anharmonic spring. A peculiar spring has the force law -Dx:J. Does the angle of projection affect the answer? Atwood's machine. An Atwood's machine was described in Chap.. Compare with the result of Eq. Electron in bound orbit about proton. Suppose that an electron moves in a circular orbit about a proton at a distance of 2 X lO-8 em. Consider the proton to be at rest. Potential energy?
Give values both in ergs and in electron volts. Pay careful attention to the various signs. Spring paradox. What is wrong with the following argument? The mass is now attached to the spring, which will be stretched because of the gravitational force mg on the mass.
Escape velocity from the moon. Potential energy of pair of springs. In the following assume that either may expand or contract in length without buckling see Fig.. Find F. Sketch a graph of potential energy as a function of r in the xy plane, and find the equilibrium position.
Loop the loop. A mass m slides down a frictionless track and from the bottom rises up to travel in a vertical circle of radius R. Find the height from which it must be started from rest in order just to traverse the complete circle without falling off under the force of gravity. Time-oI-flight mass spectrometer. The operation of a time-of-Hight mass spectrometer is based on the fact that the angular frequency of helical motion in a uniform magnetic field is independent of the initial velocity of the ion.
In practice, the device produces a short pulse of ions and measures electronically the time of Hight for one or more revolutions of the ions in the pulse. Show that the gyroradius is approximately R; yVM B em where V is the ion energy in electron volts. Given a magnetic field of G, calculate the time of Hight for 6 revolutions of singly ionized potassium K3! Electron heam in oscilloscope. The plates, which have a length I and a separation d, sustain a potential difference b with respect to each other.
The screen of the tube is located at a distance L from the center c 1 70 Chapter 5 of the plates. For a two-particle system R c.
We differentiate Eq. In the absence of external forces the total momentum is constant, so that FIG. For two masses M 1 and ;'H 2 at arbitrary positions R c. IS about to decay. This is true, for example, for a radioactive nucleus that decays in flight see Fig. It is a simple matter to show from Eg. If Fn is the force on particle n, then, on differentiating Eg.
This is another significant result: In the presence of external forces the vector acceleration of the center of mass is equal to the vector sum of the external forces divided by the total mass of the system. In other words, we can use the methods we have developed in Chaps. This principle is particularly important in the rigid-body problems treated in Chap. As anot. We are now going to illustrate the usefulness of the center of mass by working out some important collision problems.
We have already worked out several problems in Chaps. Figure 6. Regardless of whether the collision is elastic or inelastic, the total momentum is unchanged in a collision.
We will be happy if you revert us anew. Mechanics: berkeley physics course. Mechanics: Berkeley Physics Course. Charles Kittel, Walter D.
Knight, and Malvin A. McGraw-Hill, New York, Berkeley physics course 1. Mechanics Charles Kittel, Walter D. Knight, Malvin A. Ruderman, A. Carl Helmholz, Burton J. Moyer digital library Bookfi BookFi.
Carl; Moyer, Burton J. Berkeley physics course. Ruderman and R. Bruce Lindsay, American Institute of Physics. Mechanics berkeley physics course, vol. Carl Helmholz, Burton. Mechanics book Berkeley physics course 1. Moyer, Charles Kittel, Malvin A. Ruderman, Walter D. Mechanics by A. Download eBook. Special topics in classical mechanics physics - Course Details. I Charles Kittel, Walter D.
Mechanics: berkeley physics course, vol. Ruderman, Clark Kerr] on Amazon. Knight, Malvin a. Moyer-Solutions Manual to Mechanics. Knight, Malvin A Theoretical mechanics 1. Ruderman, Mechanics, Berkeley Physics. Berkeley physics course mechanics: amazon.
Ruderman Walter D. Knight Charles. Kittel c, knight w d , helmholz a c, moyer b j Jun 04, knight w d, helmholz a c, moyer b j berkeley physics course Charles Kittel, Walter D. Knight,Philip I. Mechanics Kittel, Knight, Ruderman. One hundred sixty four. Ruderman and a great selection of. Moyer] on Amazon.
0コメント