Newtonian Mechanics E-Book

Derek Raine

• Herausgeber: Packt Publishing
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 332

Veröffentlichungsjahr: 2024

Beschreibung

Newtonian mechanics is fundamental in physics education due to its intellectual significance, diverse applications, and its role in teaching modeling and problem-solving. This text covers both introductory and advanced topics, making it suitable for extended study. Emphasizing problem-solving, it guides readers through the process of constructing models and finding solutions, thus enhancing their analytical skills.
Starting with mechanical models and forces, the course progresses through kinematics, energy, and motion, providing a solid foundation. Further chapters delve into momentum, orbital motion, and oscillations, offering insights into dynamic systems. Advanced topics like rigid bodies, stability of motion, and Lagrangian and Hamiltonian mechanics ensure a comprehensive understanding.
The journey through this course equips learners with the skills to approach complex problems, construct effective models, and develop robust solutions, making it invaluable for students aiming to excel in physics and related fields.

Leseprobe

NEWTONIANMECHANICS

Second Edition

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NEWTONIANMECHANICS

A Modelling Approach

Second Edition

DEREK RAINE

MERCURY LEARNING AND INFORMATIONDulles, VirginiaBoston, MassachusettsNew Delhi

Reprint and Revision Copyright ©2021 by MERCURY LEARNINGAND INFORMATION LLC. All rights reserved.

Original title and copyright: Newtonian Mechanics: A Modelling Approach 2/E. Copyright ©2020 by D.J. Raine. All rights reserved. Published by The Pantaneto Press.

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Publisher: David PallaiMERCURY LEARNINGAND INFORMATION22841 Quicksilver DriveDulles, VA [email protected]

Derek Raine. Newtonian Mechanics: A Modelling Approach, 2/E.ISBN: 978-1-68392-682-5

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CONTENTS

Preface

Chapter 1Mechanical Models

1.1 Introduction

1.2 Models

1.3 Estimates

1.4 Units and Dimensions

1.5 Equations

1.6 Chapter Summary

Chapter 2Forces

2.1 Action and Reaction

2.2 Forces in Equilibrium

2.3 Horse Before Cart

2.4 Static Friction

2.5 Sliding Friction

2.6 A Friction Paradox

2.7 Rolling Friction

2.8 Contact Area

2.9 Torque: The Moment or Couple of a Force

2.10 Condition for Static Equilibrium

2.11 Center of Gravity

2.12 An Example

2.13 Problem Summary

2.14 Inclined Planes

2.15 Pulling at an Angle on a Flat Plane

2.16 Pulling at an Angle on an Inclined Plane

2.17 Solution of Problem 2

2.18 Tipping Point

2.19 Tipping on an Inclined Plane

2.20 Levers

2.21 Stress and Strain

2.22 Chapter Summary

2.23 Exercises

Chapter 3Kinematics

3.1 Constant Speed

3.2 Constant Acceleration

3.3 A Body Projected Vertically under Gravity

3.4 Motion in Two Dimensions

3.5 Addition of Velocities

3.6 Projectile Motion

3.7 Approximate Solutions

3.8 Air Resistance

3.9 Addition of Accelerations

3.10 Other Forms of Acceleration

3.11 Chapter Summary

3.12 Exercises

Chapter 4Energy

4.1 Work

4.2 Kinetic Energy and Work

4.3 Definition of Mass

4.4 Work and Potential Energy

4.5 Conservative Forces

4.6 Nonconservative Forces

4.7 Friction and “Zero Work Forces”

4.8 Conservation of Energy

4.9 Units for Energy

4.10 Example

4.11 Bound Systems

4.12 Virtual Work

4.13 Elastic Energy

4.14 Example – Bungee Jumping

4.15 Solution to the Problem

4.16 Chapter Summary

4.17 Exercises

Chapter 5Motion

5.1 Newtonian Dynamics

5.2 Equations of Motion

5.3 An Example

5.4 Motion in Higher Dimensions

5.5 Rate of Doing Work

5.6 Inertial Forces

5.7 Systems of Particles

5.8 Example: Motion under Air Resistance

5.9 Sky Dive

5.10 Tower Problem

5.11 Model 1

5.12 Model 2: Terminal Speed

5.13 Model 3

5.14 The Shape of the Shot

5.15 Upthrust

5.16 Simple Harmonic Motion

5.17 Why SHM Is Important

5.18 Energy of a Harmonic Oscillator

5.19 Chapter Summary

5.20 Exercises

Chapter 6Momentum

6.1 Conservation

6.2 Conservation and Invariance

6.3 Impulse

6.4 Collisions in One Dimension

6.5 Center of Momentum Frame

6.6 Inelastic Collisions

6.7 The Problem

6.8 Collisions in Two Dimensions

6.9 Collision Timescales

6.10 Rocket Equation

6.11 Chapter Summary

6.12 Exercises

Chapter 7Orbital Motion

7.1 Angular Speed: Geometric Approach

7.2 Angular Speed: Algebraic Approach

7.3 Angular Velocity as a Vector

7.4 Angular Acceleration: Geometric Approach

7.5 Angular Acceleration: Algebraic Approach

7.6 Angular Momentum

7.7 Circular Motion: Dynamics

7.8 Particle in a Magnetic Field

7.9 Centrifugal Force

7.10 Rotating Frames

7.11 Gravity

7.12 Extended Bodies

7.13 Gravitational Potential and Potential Energy

7.14 Escape Speed

7.15 Radial Infall

7.16 Circular Orbits

7.17 Virial Theorem

7.18 Changing Orbits

7.19 Elliptical Orbits

7.20 Properties of the Ellipse

7.21 Kepler’s Laws

7.22 Derivation of Kepler’s Laws for Elliptical Orbits

7.23 Extended Bodies: Multipole Expansion

7.24 The Poisson Equation

7.25 Motion Inside Matter: Falling Through the Earth

7.26 Tidal Forces

7.27 Solution of the Problem: Roche Limit

7.28 What Is Gravity?

7.29 Chapter Summary

7.30 Exercises

Chapter 8Oscillations

8.1 Resonance

8.2 Damping

8.3 Quality Factor

8.4 Forced Oscillations

8.5 Impedance

8.6 Energy and Phase

8.7 Power Curve

8.8 Complex Exponentials

8.9 Fourier Analysis

8.10 Coupled Oscillators

8.11 Coupled Oscillators with Dissipation

8.12 Forced Coupled Oscillators

8.13 Chapter Summary

8.14 Exercises

Chapter 9Rigid Bodies

9.1 Rotational Energy

9.2 Moments of Inertia

9.3 Angular Momentum

9.4 The Receding Moon

9.5 Space Tether

9.6 Equation of Motion

9.7 Compound Pendulum

9.8 A Model of Running

9.9 Rolling and Slipping

9.10 Galileo’s Inclined Plane

9.11 Spin and Precession

9.12 Euler Equations

9.13 Chapter Summary

9.14 Exercises

Chapter 10Stability of Motion

10.1 Perturbations

10.2 Cubic Potential

10.3 Motion of the Planet Mercury

10.4 Stability: General Formulation

10.5 An Example of Stability: Non-Newtonian Orbits

10.6 A Warning

10.7 Solution to Problem

10.8 Phase Portraits: Harmonic Oscillator

10.9 Phase Portraits: Damped Oscillator

10.10 Chaos

10.11 Chapter Summary

10.12 Exercises

Chapter 11Lagrangian and Hamiltonian Mechanics

11.1 Principle of Least Action

11.2 Euler–Lagrange Equations

11.3 Newton’s Laws

11.4 Simple Harmonic Oscillator

11.5 Acceleration in Polar Coordinates

11.6 Rotating Coordinate System

11.7 Bead on a Wire

11.8 Cycloidal Pendulum

11.9 Spherical Pendulum

11.10 Compound Pendulum

11.11 Small Oscillations Revisited

11.12 An Example

11.13 Hamiltonian Mechanics

11.14 Conservation Laws and Noether’s Theorem

11.15 Energy and the Hamiltonian

11.16 Action Angle Variables and Integrable Systems

11.17 Quantum Theory

11.18 Chapter Summary

11.19 Exercises

Index

PREFACE

Newtonian mechanics is taught as part of every physics program for several reasons. It is a towering intellectual achievement; it has diverse applications; and it provides a context for teaching modelling and problem solving. I have tried to give equal prominence to all three missions in this text. To do this I have included some advanced material as well as the customary introductory topics. The book therefore is designed to be studied over an extended time-frame somewhat beyond the first year of a university physics program. This enables me to develop the problem-solving aspects more fully than in many other texts, as well as including some more advanced content. In particular I have tried to show how problems are approached in order to bring out the way one goes about constructing a solution or model. Tidy solutions and appropriate models rarely come fully formed, yet many texts present them as such, assuming that students will learn through their own trial and error. I think the trial-and-error process needs to be taught.

Each chapter begins with a problem, for which the following text provides the background to a solution. I hope that knowing what the question is makes the following material more digestible. I have included some end-of-chapter questions, but not drill exercises. These are so readily available (and constructible) that it seemed an extravagant use of paper to write yet more. The text itself contains some solved drill exercises which help to illustrate a particular concept.

The level of mathematics varies through some of the chapters. The more difficult sections can be omitted on first reading. I have assumed that students will be taking a parallel course in mathematical methods, but the early parts of chapters use plug-and-chug verification to avoid overburdening the student. On the other hand, if we avoid mathematical sophistication entirely, it is not possible to reach the required level of skill to build up the modelling expertise that Newtonian mechanics is supposed to teach.

It will be clear to readers that my approach to many subjects is not entirely conventional. For example, Newton’s third law is treated first, weight is introduced before mass, energy is introduced before the equations of motion. This last I do for the particular reason of making contact with contemporary physics: the physics of elementary particles is encapsulated (roughly speaking) in an expression for the (quantum) energy of the Universe, and their dynamics follows from this. It also makes direct contact with Hamiltonian mechanics, an understanding of which makes quantum mechanics a little less impenetrable.

Much of the material for this book was developed in collaboration. I am particularly grateful to Dr. Edwin Thomas who not only originated some of the problems but read an initial version of the text and helped in proofreading for accuracy. It goes without saying that any remaining errors are mine alone. Sarah Symons and Naomi Banks also made helpful suggestions.

Derek RaineLeicesterMarch 2021

CHAPTER 1

MECHANICAL MODELS

1.1 INTRODUCTION

We observe that the world changes. At first most of these changes appear random, but then we begin to observe the regularity of day and night, the periodicities of the seasons, the flow of water, and the transforming effect of fire. We wonder if we can perhaps control some of these changes. Gradually, we learn that to exploit nature, we must first understand changes. Progress in understanding change means describing it and isolating regularities, it means that understanding a surface complexity in terms of deep simplicity. We might link the start of this endeavor to Plato’s challenge to the academicians of Athens to understand the complex movement of the stars and the planets in terms of motion on interlinked circles. We might highlight the development of kinematics in Oxford and Paris in the thirteenth century, isolating the features of motion under constant acceleration and describing it graphically. We could note the complexity of Ptolemy’s epicycles brought to order by Kepler’s discovery of the elliptical motion of the planet Mars, and Galileo’s experimental finding that bodies fall with constant acceleration. Or, we could begin with the laws of motion as synthesized by Newton from the work of Huygens and Descartes into a code that can unravel the motion of all bodies – unless they are moving near the speed of light or inhabit the micro-world of the atom. Wherever we start, this is above all a story of progress in stripping away the inessentials for the given purpose; in short, a story of how to make models of the world in the language of mathematics.

This book is about that story: as a history, it is one of the greatest narratives of human endeavor; and as current science, it is one of the most significant underpinnings of modern technology.

Let us begin, amusingly and totally unfairly, with a speech to the British Association for the Advancement of Science given by Dionysius Lardner in 1838. Lardner said “Men might as well project a voyage to the Moon as attempt to employ steam navigation against the stormy North Atlantic Ocean.” One hundred and fifty years separated the accomplishment of the two events but neither was as impossible as he had predicted. We do not have any record of why Lardner thought we could not travel to the Moon, but we do know why he thought that steamships could not cross the Atlantic. He believed that the resistance of a ship increases with its size; so more coal is required to feed the boilers of the larger ship that produce the power to overcome the resistance. But the size of the ship then has to be increased to carry the extra coal, which in turn increases the resistance requiring ever more coal. Eventually, Lardner believed that the maximum range would be reached using (presumably) an infinite amount of coal in an infinitely large ship.

A little mathematics, and some knowledge of ship design, enables us to see how the problem is, in fact, overcome. In order to proceed, it is easier to imagine ourselves in the frame of reference of the ship (or, stated more simply, just imagine ourselves on the ship). Then, the resistance force on the ship is proportional to the rate at which it destroys the momentum of the sea that tries to flow past it. This is proportional to the transverse cross-sectional area of the ship. Let us take some length scale L to characterize the size of the ship, the width or length, for example. Then, we imagine the ship to grow proportionately as we increase L (e.g., multiply all lengths by factor 2). The area will increase as ∝L2 with the scale, L (so by a factor of 4 if we double L). However, the amount of coal carried increases as the volume, which increases more rapidly (∝L3). Thus, larger ships are, in fact, more suited to long distances than small ones. We can do even better if we make the ship long and thin (which is why ships are long and thin): we can increase the volume, while the transverse area remains almost constant.

What do we learn from this? First, that we need mathematics, or at least mathematical ideas, to clinch an argument, not mere words. Second, that to apply mathematics we need to simplify the situation to retain only what is relevant: here, it does not matter what the ship is made of, or even how it differs from a cuboid; the properties of the sea are unimportant, other than that it flows. And, we can adopt a convenient point of view (from the ship or from the land) in assessing the problem: the outcome cannot depend on which frame of reference we choose.

1.2 MODELS

We are going to look at a systematic way of thinking about models in physics. Let us introduce this through another example. Consider the orbit of the Earth around the Sun. There are two agents involved here: the Earth and the Sun; they are, if you like, the players on the stage. The Sun is going to be an external agent: that is to say, its properties are going to be fixed and unaffected by the presence of the Earth. Its only role will be to exert a gravitational pull on the Earth. Our second agent, the Earth, will treat it as a point mass with the properties that it has a position and a velocity. The two agents interact through the gravity of the Sun, which falls off with the inverse square of the distance between the Sun and the Earth. With this set-up, we look for possible orbits of the Earth around the Sun which repeat – that is to say which the Earth – in this model will track year after year. The outcome, as you probably know, is that the Earth must move in an elliptical orbit with the Sun at a special point called the focus of the ellipse, which particularly depends on how the system was formed (i.e., on the initial conditions).

Is this what really happens? No. The Sun is not at rest – it too moves under the influence of the gravity of the Earth, the Earth is not a point, it is not spherical, also it spins and the pair interacts through solar radiation and the solar wind as well as through gravity. And, that is before; we have taken account of the influence of the other planets of the solar system. Some of these differences do not affect the orbit, but some do. The point of the model is that it allows us to investigate the effect of the hypothesis that gravity follows an inverse square law. No other law would provide us with the gross features of the orbits. The model can then be extended under the same hypothesis to see if we can account for the detailed departures of the orbit from a perfect ellipse by adding in the previously omitted details to a more comprehensive model. Once we have used these models to establish our hypothesis about the nature of gravity, this will become part of our knowledge of physics that will be used in any other situations where we need to model gravitational interactions, for example in other planetary systems: our models should be consistent and we develop a body of knowledge of the laws of physics to ensure this.

To complete the story, you may know that things work out pretty well for the inverse square law, but not exactly once Einstein comes on the scene. Einstein’s general theory of relativity enables us to say that the hypothesis of the inverse square law is not exactly true – no model based on it will agree exactly with all observations of the motion of the planets. In Einstein’s theory of relativity, there are, in effect, forces of gravity on the Earth (and on the other planets) that modify the inverse square law, and which do enable us to account for planetary motion precisely.1

We use Einstein’s theory to calculate departures from Newtonian gravity in any model of bodies orbiting under gravity. So let us think about the orbit of a GPS satellite around the Earth. To calculate this, we would need a model of the Earth. There is a number to choose from:

1. The Earth is a uniform sphere.

2. The Earth is a non-uniform sphere with density varying with radius.

3. The Earth is an ellipsoid.

4. The Earth is a body the mass distribution (and shape) of which has been mapped (to some level of accuracy).

5. The Earth is exactly the shape and density of the Earth (the real world “experiment”).

All (except the last) are approximations. Whether they are useful depends on what we want to do. Which model is the most appropriate to study the following? We will leave you to decide.

a) Satellite orbits

b) Earthquake determinations of the structure of the core

c) Tidal forces

d) Weather prediction

e) Solar system models.

1.3 ESTIMATES

Before we tackle a problem in detail, it is important to build an approximate model to get a rough idea of what to expect. Here is a historical example.

Newton used a much better estimate for the acceleration of the Moon, but a rather worse estimate of the distance to the Moon, with the result that for several years, he did not believe the inverse square law to be exact. With a better knowledge of the distance to the Moon, the numbers worked out and Newton went on to write the Principia.

Two important points to remember in making estimates: quantities raised to high powers need to be known fairly accurately to get a good estimate; on the other hand only rough values are needed for quantities raised to fractional powers. Also, if a quantity is bounded by a large range, then the geometric mean is the best estimate for that quantity. For example, a useful estimate of a quantity that varies between 1 and 100 is usually not 50.5 (the arithmetic mean) but (the geometric mean).

It is useful to practice using approximate models and approximate values to obtain the order of magnitude estimates. Here are some examples: which of the following are true?

b) ½ degree ~ the angle subtended by a penny coin at arm’s length.

c) A piece of paper folded 25 times could stretch to the Moon.

1.4 UNITS AND DIMENSIONS

In the SI system, the standard base units in mechanics are the meter, kilogram, and second, corresponding to the dimensions of mass [M], length [L], and time [T]. Apart from the need to attach units to physical quantities, the dimensions of derived quantities are useful in several ways.

Dimensions have to balance in an equation, a fact which often allows one to check an equation – provided the equation is written with all the physical quantities in symbols and not subsumed in numerical values.

For example, the pressure at the center of the Sun supports the Sun against its own gravity, so the energy per unit volume must be roughly equal to the gravitational energy. In Chapter 6, we shall see that the gravitational potential energy can be estimated as GM2/R4, where M is the mass of the Sun, R its radius, and G Newton’s gravitational constant. Putting in values for the solar mass and radius, we find that the pressure at the center of the Sun must be of order 1014 N m−2. This is a remarkable result: we have used a little mathematical physics to construct a “device” that “measures” the pressure at the center of the Sun. (Actually, we could go further: this pressure must also be roughly the energy density of the solar plasma, from which we could estimate the temperature of the solar interior.)

Finally, one can sometimes use dimensional analysis to extract the dependence of one physical quantity on others. For example, the drag of a body in a fluid must have the dimensions of a force and must depend on the area of the body, A, its speed v (a body at rest experiences no drag), and the density of the medium ρ (at low enough density the medium may as well not be there). The only combination of A, v, and ρ that has the dimension of a force (MLT−2) is Aρv2. Of course, the shape of the body will add a numerical factor. In addition, there would be a viscous drag on the body, which can also be estimated by dimensional considerations, up to a numerical factor.

The disadvantage of units is that there are many different ones in use for the same quantity. This is partly historical and partly, sometimes, for the convenience of using numerical values as close as possible to order unity. This being so, it is often necessary to convert between units. There are various algorithmic ways of doing this. For example,

because there are 8/5 × 1000 m in a mile and 3600 s in an h. Note that meters (m) and seconds (s) cancel from the intermediate formula. Your speed in miles per second will be less than that in meters per second by a factor of the number of meters in a mile (divide by 8/5 × 1000) and your speed in meters per hour will be greater than that in meters per second by the number of seconds in an hour (multiply by 3600).

1.5 EQUATIONS

Estimates inform mathematics as well as numerical calculations. The most important aspect, once one has learned to work with symbols and not numerical values, is to learn to neglect small quantities. Let us look at some examples.

so

Thus, adding a meter to the length gives a height of about 15 cm – on whatever planet you choose!

Why has it worked out like this? Dividing through by 2πR, we can write the equation another way:

In other words, a 1% change in the circumference (δ/R) produces a roughly % (proportionate) increase in the radius (h/R) because the radius and circumference are linearly related. Put this way, the answer is entirely reasonable.

Consider next a completely different problem. What rise in sea level would result from a 1 degree rise in sea temperature? What model shall we choose? The simplest one, which we shall take as our starting point, is a sphere of radius R covered to a uniform depth h in a thin layer of water. Suppose that the coefficient of expansion of water is α. Then, the change in volume of the sea on expansion is α times the original sea volume:

(5.1)

This looks like a lot of work; however, our model has both h << R and δ << R. So, expanding the brackets and canceling, we can approximate:

(5.2)

neglecting the extra terms with powers of δ and h higher than the first, or

(5.3)

since h << R.

Equation (5.1) makes it look as if the final answer should involve the radius of the Earth, R. The result (5.3) shows that the relative rise (δ/h) is independent of the radius of the planet. Is this reasonable? We cannot make a dimensional argument here, because there are too many lengths involved: the final answer could have been multiplied by any number of factors of h/R. The easiest way to see that the result is reasonable is to imagine a strip of water from around the circumference laid out (approximately) on a flat surface. Then, it does not matter how long the strip is: the rise in height will always be the same when it expands. Another way of seeing this is to compare it to putting a girdle around the Earth: the extra height (radius) is accommodated by a proportionate increase in length (circumference) without reference to the radius of the planet.

Note that we could have written down Equation (5.2) immediately by approximating the volume of a thin covering of the ocean on a sphere as area × depth. So, this is another check on the model.

For our final example, we look at the fall-off of pressure with height in the Earth’s atmosphere. Suppose that a student, asked to estimate the height of the atmosphere, claims that the inverse square law of gravity means that gravity gets weaker as you get to greater heights in the atmosphere, and hence, that the top of the atmosphere is where gravity is so much weaker than it cannot stop the air molecules escaping. What do we make of this?

Of course, to be fair it all depends on what you mean by the top of the atmosphere, but we can agree that what most people mean by a significant atmosphere does not extend as far as low Earth orbit at a few hundred kilometers. (It is actually much less the FAI3 defines the boundary between the atmosphere and outer space as the Karman line at 100 km.) We can see that the student’s answer must be wrong with just a little appreciation of mathematics. The acceleration due to gravity, g, falls off with radius from the center of the Earth as an inverse square: The only length scale in the gravitational model is the radius R. (The presence of the atmosphere does not alter this: gravity is essentially unaffected by the atmosphere.) So R is the length scale on which gravity gets significantly weaker, a scale very much greater than the height of the atmosphere. Thus, as far as the atmosphere is concerned we can treat g as approximately constant. The explanation for the thinness of the layer of atmosphere around the Earth must lie elsewhere.

Another way of looking at this is to work out how much g changes by over a height h << R. We do not do this by tapping numbers into a calculator. Instead, we derive a feeling for the way g falls off by expanding the inverse square law for h << R:

using the binomial theorem and neglecting terms in higher powers of h/R. So close to the surface, g falls off linearly with height.

The Earth is 6400 km in radius, so if the atmosphere were to extend this by as much as 200 km, it would amount to no more than 3%. Gravity is an inverse square law, so a 3% increase in radius means roughly a 6% decrease (double 3%) in gravity: scarcely noticeable. The atmosphere would extend by several Earth radii if the explanation given were really true. In fact, the height of the atmosphere is governed by the amount of air, and the way pressure falls off with height in an approximately constant gravitational field and has a true scale height (height to fall by a factor 1/e) of around 8 km.

1.6 CHAPTER SUMMARY

• Physics in general, and mechanics in particular, involves making mathematical models of the world.

• A model seeks to simplify reality as much as possible for the purpose to which it is being put.

• A model is defined in terms of the agents and their interactions. Simplification, therefore, means identifying the significant agents and their essential interactions.

• Models in mechanics should be described in terms of mathematical symbols for dimensional quantities with the entry of numerical values reserved for the final step. This allows dimensions to be checked for consistency.

• The mathematics should be approximated appropriately to the model, especially in the neglect of small quantities where justified. This enables the results to be interpreted more readily.

• The result of a model should be expressed and explained in words (and/or graphically) and examined to check that it is reasonable.

1 Even Einstein’s theory may not be the final word: string theories, for example, suggest that there may be higher-order corrections to the equations of general relativity, although these would have a negligible effect on the solar system.

2 The philosopher Ludwig Wittgenstein liked to quote this as an example where a mental picture of the relation between big and small leads us astray: pouring a glass of water into the ocean does not have much effect on sea level. Quoted in, for example, Wittgenstein and the Philosophy of Mind, Ed Jonathan Ellis & Daniel Guevara (2012) OUP.

3 Fédération Aéronautique Internationale.

CHAPTER 2

FORCES

In this chapter, we are going to address the following problem:

Problem 1: The figure shows a horse and cart. In due course, the farmers will have had enough of being photographed and will want to transport their harvest to market or storage. How does the horse pull the cart?

What are the forces in and on the system of horse and cart; why do these forces move the cart in some circumstances but not others (e.g., if the cart is too laden)?

Picture credit: http://www.flickr.com/photos/hartlepool_museum/5933914248/sizes/z/in/photostream/

2.1ACTION AND REACTION

We start by considering various cases where the forces on a body are in equilibrium, hence where the forces do not change the state of motion of the body.

Figure 2.1: Horizontal forces on a block at rest on a horizontal plane

Consider a block at rest on a flat plane as in Figure 2.1. We imagine that the block is subject to equal and opposite horizontal forces acting through a common point, as indicated by the arrows. By symmetry, the block cannot move. If the forces on a body do not change its state of motion, we say that the forces are in equilibrium. This suggests that a body that does not move must be acted on by equal and opposite forces in both magnitude and direction, hence must be subjected to no net force (or no forces at all).

If forces of the same magnitude in Figure 2.1 were not to act through a common point, we should have a more complicated situation in which the block could tip over. We shall deal with this later: for the moment, all forces on an extended body are assumed to act through a common point. Alternatively, we can consider the body to be a point particle with no extension, so that all forces on it act through the same point by construction.

Figure 2.2: Vertical forces on a block at rest on a horizontal plane. The reaction force R is equal and opposite to the weight W

Consider now a block at rest on a flat plane as in Figure 2.2. If we were to imagine ourselves in the role of the plane, for example, by holding the block in our hand, we would feel the block pushing down. We attribute this to the weight of the block. Let us call this force W.

Thus, the block has a weight W, which is the force acting down on the plane. Experience shows that an unbalanced force causes an object to move. So we expect that the plane must act back on the block with a force equal and opposite to the weight of the block. In fact, if we imagine ourselves now in the role of the block, we feel this reaction as our weight. This is shown in Figure 2.2, where each force is represented by an arrow that points in the direction of the force and has a length proportional to the magnitude of the force.

Actually, in general, everyday experience alone does not always show that an unbalanced force causes an object to move. In one of the earliest systematic considerations of the issue, Aristotle pointed out that a man cannot move a ship.1 It was Newton’s insight to argue that the reason for this was not, directly, the weight of the ship, but the resistance offered by the water. Thus, even in this case, the ship does not move perceptibly because the forces on it are balanced. More than that, Newton proposed that in all cases, an action is balanced by an equal and opposite reaction – even when the reaction is not obviously visible. Thus, we have

Newton’s Third law:

To every Action there is an equal and opposite Reaction.

Note that the law refers to the action and reaction between two agents (the block and the plane above): the action of agent A on B is equal and opposite to that of B on A. Each agent is acted on by the respective reaction.

There is a lot of confusion on the issue of action–reaction pairs and you may well have been told that what you have just read is wrong. The reason offered is that action–reaction pairs have to be of