Mechanics can predict forces and motions. The three pillars of mechanics are: I. models of physical behavior, II. geometry, and III. the basic mechanics balance laws. The laws of mechanics are informally summarized in this introductory chapter. The extreme accuracy of Newtonian mechanics is emphasized. Despite relativity and quantum mechanics supposedly having ‘overthrown’ seventeenth-century physics, classical mechanics is still incredibly accurate for almost all engineering purposes. Various uses of the word ‘model’ are described.
Mechanics is the study of force, deformation and motion. And also of the relations between force, deformation and motion. We care about forces because we want to know how hard to push something to make it move or whether it will break when we push. We care about deformation and motion because we want things to move or not move in certain ways. Towards these ends, our goals are to solve special versions of this general mechanics problem:
The general mechanics problem: Given some (possibly idealized) information about the properties, forces, deformations, and motions of a mechanical system, make useful predictions about other aspects of its properties, forces, deformations, and motions.
By system, we mean a tangible thing such as a wheel, a gear, a car, a bridge, a human finger, a butterfly, a skateboard and rider, a quartz-watch timing crystal, a building in an earthquake, a rocket, or the piston in an engine. Will a wheel slip? a gear tooth break? a car tip over? What is the biggest truck that can cross a given bridge? Which muscles are used when you hit a key on your computer? How do people balance on skateboards? How does size affect the frequency of crystal vibration? Which buildings are more likely to fall in what kinds of earthquakes? What is the relation between gas-ejection rate and thrust in a rocket? What forces are on the connecting rod in an engine?
For each special case of the general mechanics problem we need to identify the system(s) of interest, idealize the system(s), use classical (high school, Euclidean) geometry to describe the layout, deformation and motion, and finally use the laws of Newtonian mechanics. Those who want to know how machines, structures, plants, animals and planets hold together and move about need to know Newtonian mechanics. As best we can extrapolate, in another two or three hundred years people who want to design robots, buildings, airplanes, boats, prosthetic devices, and large or microscopic machines will likely still use the equations and principles we now call Newtonian mechanics
.
Any mechanics problem can be divided into three parts, which we think of as the three major pillars that hold up the subject:
Constitutive laws: the mechanical behavior of objects and materials;
Kinematics: the geometry of motion and distortion; and
Kinetics: the laws of mechanics (, etc.).
Let’s discuss each of these ideas (these pillars) a little more. This overview should shape your thinking when working out details in later chapters.
The first pillar of mechanics is mechanical behavior. The mechanical behavior of something is the description of how loads cause deformation (or vice versa). When something carries a force, it stretches, shortens, shears, bends, or breaks. Your finger tip squishes when you poke something. Too large a force on a gear in an engine causes it to break. The force of air on an insect wing makes it bend. Various geologic forces bend, compress and break rock. This relation between force and deformation can be viewed in a few ways.
Definition of force. First, the relation between force and deformation gives us a definition of force. Force can be defined by the amount of spring stretch it causes on a particular known spring. Thus, most modern devices for measuring force do so indirectly. They measure the deformation that the force causes in a calibrated spring of some kind. One justification for calling ‘mechanical behavior’ the first pillar is that force can be defined in terms of deformation of a known elastic object. The first pillar gives us a notion of force, even before we introduce the laws of mechanics.
Steel vs chewing gum. Second, a piece of steel distorts under a given load differently than a same-sized piece of chewing gum. This observation, that different objects deform differently with the same loads, implies that an object’s properties affect its mechanics. The relations of an object’s deformations to the forces that are applied are called the mechanical properties of the object. Mechanical properties are sometimes called constitutive laws because the mechanical properties describe how an object is constituted (meaning ‘what it is made from’), at least from a mechanics point of view. The classic example of a constitutive law is that of a linear spring which you remember from your elementary physics classes:
(spring tension is proportional to stretch). To do mechanics, we have to make assumptions and idealizations about the constitutive laws applicable to the parts of a system. How stretchy (elastic) or gooey (viscous) or otherwise deformable is an object? The set of assumptions about the mechanical behavior of the system is sometimes called the constitutive model.
Deformation is often hard to see. Distortion in the presence of forces is easy to see or imagine in the flesh of squeezed fingertips, in chewing gum between teeth or when a piece of paper bends. But pieces of rock or metal have deformation that is essentially invisible and sometimes hard to imagine. With the exceptions of things like rubber, flesh, or objects that are very small in one or two directions (thin sheets and wires), solid objects that are not in the process of breaking typically change their sizes much less than 1% when loaded. Most structural materials deform less than one part per thousand with working loads. These small deformations, even though essentially invisible, are important. These deformations are enough to break bones and collapse bridges.
Rigid-object mechanics. Part of good engineering is to idealize-away things that are not important. Unimportant features unnecessarily clutter the mind and also make calculations harder. When deformations are not of much consequence, engineers usually wish them away.
Mechanics calculations in which deformation has been neglected are called rigid-object (or rigid-body††margin:
‘Rigid body’ means ‘rigid object’. The old-fashioned word for ‘Rigid-object’ was ‘rigid-body’ because in the old days physical things were abstractly called ‘bodies’.
Think of a guy with a robe and beard squinting through a brass telescope and deeply pondering ‘celestial bodies’.
Nowadays, Newton’s laws are applied to biological things like people, so the
word ‘body’ can be confusing. For example, one could mis-infer that ‘rigid-body’ biomechanics is the study of people with rigid rigor mortis bodies, so called
‘stiffs’. Then, there is the problem of distinguishing a whole organism from its parts. In modern biomechanics modeling of human motion people usually model each of the limb segments as a rigid object. It is confusing to say that the human body is a collection of rigid bodies. It is clearer to say the human body is modeled as a collection of rigid objects.
Here, we will usually adopt the ordinary English language, rather than the classical mechanics language. Things are objects (not bodies), and things whose deformation we neglect are rigid objects (not rigid bodies).
)
mechanics
because a rigid (infinitely stiff) solid would not deform at all.
Rigidity, the assumption of infinite stiffness, is an extreme constitutive assumption.
However, assuming rigidity greatly
simplifies many calculations while still generating adequate predictions for many
practical problems. The assumption of rigidity also simplifies the introduction of
more general mechanics concepts.
Thus, for understanding the steering dynamics of a car we might
treat the car as a rigid object, whereas for crash analysis where
rigidity is clearly a poor approximation, we might treat a car as
highly deformable.
Contact behavior. Most constitutive models describe the material inside an object. But to solve a mechanics problem involving friction or collisions one also has to have a constitutive model for the contact interactions. The standard friction model (or idealization) ‘’ is an example of a contact constitutive model, as is the elementary ‘restitution’ model for collisions ‘’.
In summary, we need a model of a system’s mechanical behavior before we can make useful predictions. Useful constitutive models can sound absurdly extreme, as in the assumption that a piece of a human body is rigid.
In classical mechanics we use classical Greek (Euclidean) geometry to describe the layout, deformation and large-motions of objects. Deformation is defined by changes of lengths and angles between various pairs and triplets of points. Motion is defined by the changes of the position of points in time. Length, angle, similar triangles, the curves that particles follow and so on can be studied and understood without Newton’s laws and thus make up the second independent pillar: geometry and kinematics.
Large motions. Many machines and machine parts are designed to move something relatively far. Bicycles, planes, elevators, and hearses are designed to move people; a clockwork, to move clock hands; insect wings, to move insect bodies; and forks, to move potatoes. A connecting rod is designed to move a crankshaft; a crankshaft, to move a transmission; and a transmission, to move a wheel. And wheels are designed to move skateboards, bicycles and cars of various kinds.
The description of the motion of these things, of how the positions of the pieces change with time, of how the connections between pieces restrict the motions, of the curves traversed by the parts of a machine, and of the relations of these curves to each other is called kinematics. Kinematics is the study of the geometry of motion (or, of geometry in motion).
Motion versus deformation. The idea behind the word deformation is correctly conveyed by the misspelling, ‘deform-motion’. Deformations usually involve small changes of distance between points on one object, whereas net motion (see the paragraph above) involves large changes of distance between points on different objects. We often need to understand deformation of individual parts to predict when they will break. Sometimes the motion associated with deformation is important in itself, like when designing a building to not sway too much in the wind. And sometimes the larger, net transport kind of motion is of interest; for example we would like all points on a plane to travel about the same large distance from New York to Bangalore. Really, deformation and motion are not distinct topics; both involve keeping track of the positions of points. The distinction we make is for simplicity. Trying to simultaneously describe deformations and large motions is just too complicated for beginners to understand, and also too complicated for most engineering practice. So the ideas are kept (somewhat artificially) separate in elementary mechanics courses such as this one. As separate topics, the geometry needed to understand small deformations (called ‘strains’) and the geometry needed to understand large motions of rigid objects (called ‘particle and rigid-object kinematics’) are both basic parts of mechanics. (This book, however, limits discussion of deformation to that of linear springs.)
The same intuitive ‘force’ that causes deformation also causes motion, or more precisely, acceleration of mass. The relation between force and acceleration of mass makes up the third pillar holding up the subject of mechanics. We loosely call this Newton’s laws; synonyms include the laws of mechanics, momentum and energy balance and kinetics.
Force is related to deformation by material properties (elasticity, viscosity, etc.) and force is related to motion by the laws of mechanics summarized in the front cover. In words and informally, these are:
The laws of mechanics apply to any system (rigid or not):
Force and moment are the measures of
mechanical interaction;
and
Action minus reaction applies to all interactions,
( ‘every action has an equal and opposite reaction’);
The net force on a system causes a net
linear acceleration
(linear momentum balance),
The net turning effect of forces on a system causes it
to
rotationally accelerate (angular momentum balance), and
The change of energy of a system is due to the
energy flow
into the system (energy balance).
A non-minimal set of assumptions. The principles of action and reaction, linear momentum balance, angular momentum balance, and energy balance, are actually redundant in various ways. Linear momentum balance can be derived from angular momentum balance and sometimes vice-versa (as discussed in the Statics and Dynamics books).
Energy balance equations can often be derived from the momentum balance equations. And the principle of action and reaction can be derived from the momentum balance equations. In engineering practice, however, we worry little about which idea could be logically derived from the others for the problem under consideration. Rather, we take them all as operationally true. The four assumptions in O-III above are not a mathematically minimal set, but they are all accepted truths by practitioners of mechanics.
A lot follows from the laws of Newtonian mechanics, including the contents of this book. When these ideas are supplemented with idealizations of the mechanical behavior of particular systems (e.g., of machines, buildings or human bodies), they lead to predictions about motions and forces. There is an endless stream of results about the mechanics of one or another special system. Some of these results are classified into entire fields of research such as ‘fluid mechanics,’ ‘vibrations,’ ‘seismology,’ ‘granular flow,’ ‘biomechanics,’ or ‘celestial mechanics.’
The four basic ideas also lead to mathematically advanced formulations of mechanics with names like ‘Lagrange’s equations,’ ‘Hamilton’s equations,’ ‘virtual work’, and ‘variational principles.’ If you go on in mechanics, you may learn some of these things in more advanced courses.
Elementary mechanics is sometimes partitioned into three courses named ‘statics’, ‘dynamics’, and ‘strength of materials’. These subjects all use, but vary in how much they emphasize, material properties, geometry, and Newton’s laws.
Statics is mechanics with the idealization that the acceleration of mass is negligible in Newton’s laws. The second book in this series provides a thorough introduction to statics. Things need not be standing exactly still to be well idealized with statics. Actually, nothing is exactly still anyway. But, as the name implies, statics is generally about things that don’t move much. In statics, the first pillar of mechanics, constitutive laws, is generally introduced without fanfare by the (implicit) assumption of rigidity. Other constitutive assumptions used in statics include inextensible ropes, linear springs, and frictional contact. The material properties used as examples in elementary statics are generally simple. Also, because things don’t move or deform much in statics, the geometry of deformation and motion are all but ignored. Despite these commonly applied vast simplifications, statics is useful for the analysis of natural and engineered structures, of slow machines or the light parts of fast machines, and of other things (say, the stability of boats).
Dynamics concerns the non-negligible acceleration of mass. Chapters 9 and on of this book introduce dynamics. As with statics, the first pillar of mechanics, constitutive laws, is given a relatively minor role in the elementary dynamics presented here. For the most part, the same library of elementary properties are used with little fanfare (rigidity, in-extensibility, linear elasticity, and friction). Dynamics thus primarily concerns kinematics and kinetics. Once one has mastered statics, the hard part of dynamics is the kinematics. Dynamics is useful for the analysis of, for example, fast machines, vibrations, and ballistics.
Strength of Materials expands statics to include material properties and also pays more attention to distributed forces (e.g., ‘traction’ and ‘stress’). This book only occasionally touches lightly on strength-of-materials topics like stress (loosely, force per unit area), strain (a way to measure deformation), and linear elasticity (a commonly used constitutive idealization of solids that generalizes the concept of a spring). Strength-of-Materials gives equal emphasis to all three pillars of mechanics. Strength-of-Materials is useful for predicting the amount of deformation in a structure or machine, where it is most likely to break with a given load, and whether or not it is likely to break with that load.
0.1.1 What are the three pillars of Mechanics?
We are repeatedly reminded that Newtonian ideas have been replaced by relativity and quantum mechanics. So why, in the 21st century, should you read this book and learn ideas, remnants of the nineteenth century, which are known to be wrong?
First off, this criticism is maybe a bit off base: general relativity and quantum mechanics are inconsistent with each other, not yet††margin: (as of this writing) united by a universally-accepted deeper theory of everything. So, strict consistency with modern physics, as we know it, isn’t possible.
How accurate is Newtonian mechanics? In practice, how big are the errors we make when we do classical mechanics, neglecting various more modern physics discoveries?
The errors from neglecting the effects of special relativity are on the order of where is a typical speed in your problem and is the speed of light. The biggest errors are associated with the fastest objects. For, say, calculating space shuttle trajectories with speeds of about 5 miles/s ()
this leads to an error of about
errors having to do with the non-flatness of space are so small that Albert Einstein††margin: Time Magazine’s man of the century, 1900-2000 had trouble finding a place where the deviations from Newtonian mechanics could be observed at all. Finally, he predicted a small, barely measurable effect on the predicted motion of the planet Mercury. Newtonian mechanics predicts a fixed elliptical orbit. Einstein’s equations correctly predicted that the elliptical path itself rotates (precesses) once every 3 million years, that’s about 45 arc-sec (an 80th of a degree) per century. So the Newtonian ‘error’ is about one part in (like a one cent error in a millionaire’s bank balance). One engineering example where this error could matter is with satellite-based Global Position Systems (GPS). GPS location calculations do take general relativity into account to prevent errors of about one part in a billion (a millimeter error over a thousand kilometers).
In Newtonian classical mechanics, we assume we can know exactly where something is and how fast it is going. But according to quantum mechanics this is impossible. The product of the uncertainty in position of an object, and the uncertainty of its momentum must be greater than Planck’s constant . Planck’s constant is small; . The fractional error in position is biggest for small objects moving slowly. So if one measures the location of a computer chip with mass ††margin: A tenth of a gram, about a 300th of an ounce to within a twenty fifth of a thousands of an inch, the uncertainty in its velocity is only
In classical mechanics we usually (although not always) neglect fluctuations associated with the thermal vibrations of atoms. But any object in thermal equilibrium with its surroundings constantly undergoes changes in size, pressure, and energy, as it interacts with the environment. For example, the internal energy per particle of a sample at temperature fluctuates with amplitude
where is Boltzmann’s constant, is the absolute temperature, is the number of particles in the sample, and is the specific heat. Water has a specific heat of 1 calK, or around 4 Joule/K. At room temperature of 300 K, for molecules of water, these values lead to an uncertainty of only Joule in the internal energy of the water. Thermal fluctuations are big enough to visibly move pieces of dust in an optical microscope (Brownian motion), and to generate variations in electric currents that are easily measured, but for most engineering mechanics purposes they are negligible. However, if thermal fluctuations are of interest, they can be modeled reasonably accurately using Newtonian mechanics at the atomic scale.
Physics errors vs modeling errors. As described above,
classical Newtonian physics is an accurate approximation of Nature for engineers, with errors typically on the order of parts per billion.
On the other hand, the errors within mechanics, due to imperfect modeling or inaccurate measurement, are, except in extreme situations (like GPS), far greater than the errors due to the imperfection of Newtonian mechanics theory. For example, mechanical force measurements are typically off by a percent or so, distance measurements by a part in a thousand, and material properties are rarely known to one part in a hundred and often not even one part in 10. That is, even in the most accurate of circumstances, your mechanics calculations will typically be off by at least 100,000 times more than the laws of mechanics themselves are off. The errors in your ability to measure, or in your understanding of the properties of your system, are far bigger than the errors in the laws of classical Newtonian mechanics.
If your engineering mechanics calculations make inaccurate predictions this is surely be because of errors in modeling or measurement (let’s assume no math mistakes), not inaccuracies in the laws of mechanics.
Only in special circumstances are classical mechanics predictions off because of neglect of relativity, quantum mechanics, or statistical mechanics. And the chances are high that if you don’t know all about these things, then you are not dealing with one of these circumstances.
You can trust Newtonian mechanics. In summary, Newtonian mechanics is accurate enough, and also much simpler to use than the theories which have ‘overthrown’ it. You have trusted your life many times to engineers who treated classical mechanics as ‘truth’. In turn, your engineering mechanics work will justly be based on the laws of classical mechanics. Although perhaps philosophically objectionable, it is reasonable, common, and accepted engineering practice to
Think of the laws of mechanics as absolute truth.
0.2.1 About how big are your engineering calculation errors due to the fact that Classical Mechanics neglects Quantum Mechanics and Relativity?
A plastic toy car guided by a child’s hand crashes into another toy car (fig. 1). The toys are models of cars. In Engineering and Science, however, the word model has a broader meaning. This broader meaning is well described with a ”commuting diagram”. Actually, a commuting diagram is itself a model of the word model. Very meta. If you spend a few minutes on this short section we think it will help your thinking about mechanics and other things
.
Toy cars and Moomins For example, in this broader sense the toy crash is a model of a real car crash. The model of the crash event is that two plastic things are guided together by human hands. It’s as if there are two parallel universes, the ‘real’ one and the ‘model’ one. And the real process of car collision is ‘modeled by’ the crashing of toy cars. The word model then means that cars are replaced by plastic toys and the laws of mechanics replaced by the guiding of the child’s hands. And the results of the collision are replaced by whatever damage occurs to the plastic toys.
The commuting diagram. A model, in this broader sense, is represented abstractly by a commuting diagram, as shown in Fig. 2. The top row is the system to be modeled, say the real cars. The real car collision is the workings of the system w, as dictated by nature’s laws in their full subtlety and complexity, taking into account all known and as-yet unknown physics. And the way the cars move and deform and end up damaged is the system behavior SB. Parallel to this in the bottom row of the figure is the model universe. A plastic car R represents a real car by having about the same shape. The laws of nature w are ‘modeled by’ the manipulation rules in the model m, in this case the guidance of the child’s hands. And the result of the real crash SB is ‘modeled by’ the result of the play crash. The model is compared to reality by making an association between bent car metal with scratched toy plastic.
We will rate this as a ‘good model’ if the damage to the plastic mimics the damage to a real car. This is expressed by the success at ‘commuting’, in the mathematical sense of the word commuting. Is the result of making a model and then carrying out the model process (down then right) the same as the result of the process then modeled (right then down)? In the language of the commuting diagram the question is,
Does
System Model Model Behavior
give the same result as
SystemReal System BehaviorModel Behavior ?
For example, we compare the prediction of damaged plastic to what the real car damage would translate to as cracks and scratches on the plastic? If they agree well then the model ‘commutes’. That is, starting with the real system you get the same answer these two ways: 1) by applying the real workings and then the translate to what you would expect to see in the model and 2) by modeling the car in plastic and applying the model workings. Using the toy car example we can see aspects that commute and aspects that don’t. That both the real cars and the toy cars end up with a crooked orientation is a sign of the model “commuting”. That’s a good feature of the model. That the toy car passengers have no scratches and that the people in the real cars have aching necks is a lack of commuting. The lack of commuting is a defect in ‘the model’.
Mathematical vs physical models. In the toy crash example above, the ‘model’ included a physical object, the toy car. More commonly in science and engineering, the model is a constellation of ideas with no physical object involved. For example, if a solid ‘is modeled as’ a rigid object that means the motion of the object will be calculated by assuming that the solid does not deform. No piece of plastic representing the object is needed.
Models in engineering. In engineering, we use models to make predictions about reality. So the ‘commuting’ ability is usually expressed by comparing the model’s predictions to reality, wrapping three-quarters of the way counter-clockwise around the diagram from the system (at the upper left) down to its model representation through the model manipulations to the model behavior and back up to the prediction for reality (at the upper right). But, instead of plastic cars and children’s hands moving them, in an engineering model an object is represented by its physical parameters, and the object’s behavior is modeled by governing equations.
Models are pervasive. All this abstraction about modeling is confusing partly because we are surrounded by it all the time. Explaining modeling to you is like explaining water to a fish††margin: A model of your understanding of modeling is a fishes understanding of water. . For example, language and thought are themselves, in a sense, models of reality.
What makes a good model? A good model:
Applies to a broad range of systems,
Predicts of a broad range of phenomena,
Makes accurate predictions (i.e., ‘commuting’ (see Figure 2)
so that
the route S RRB
gives the same result as SSBRB),
Is simple, and
Lacks ambiguity in the rules , and ; that is, a good model makes clear (definitive) predictions.
Usually when setting up or choosing a model, you need to make tradeoffs. For example, accurate (good) models are often complicated (bad). And, simple (good) models are often ambiguous (bad).
In a course like this we are concerned with a hierarchy of models.
Space and time. Most basically, we model space, time and matter as having all the common-sense features that we are used to. For example we assume that the location of any point in space can be described by its , , and coordinates relative to some origin.
The laws of mechanics. Second, we model all of Nature’s rules for motion with the basic laws of (classical, Newtonian) mechanics. As stated in the previous section with reference to modern physics concepts, Newtonian mechanics is a high-quality model whose errors (or lack of ability to commute) will likely be of no significance to you in your use of mechanics.
Physical properties. Third, we have models of objects and forces. In this book, as opposed to a book about structural mechanics, we generally ‘model’ solid things as particles or as non-deforming rigid objects. This non-deformation model gives an error that typically ranges from a small fraction of a percent up to a few percent. Models of forces can be very accurate. For example, you can know gravity forces, if you know where you are on the earth (see page 4.1), to about one part in . Some other force models are also reasonably accurate, like the description of linear springs (typically 1% accurate or so). And some force models are basically poor, like for friction and collisions (with typical errors of 20-50%). We don’t know good models for friction and collision forces††margin: ‘We’ being us members of the human race, not just the textbook authors . So, in engineering analysis we do the best we can with our bad models for these things (and try to estimate, and take account of, our modeling errors).
The modeling process. Given this hierarchical set of mechanics models how, given a real machine, do we ‘model’ it in detail using the models from the paragraphs above? Which parts do we approximate as rigid objects, which as massless linear springs, etc? This modeling task is an important part of engineering practice.
However, before one can develop the art of engineering modeling one needs to know how to work with the range of common engineering models. In terms of the diagram in Fig. 2 you need to know how to do the manipulations for a given candidate model. After that, you can develop the art of making up particular models applicable to your system. Much more specifically, for elementary mechanics you need to know how particles and rigid objects interact and move, assuming they, in turn, are governed by the common accepted models for their interactions. Understanding how particles and rigid objects interact and move, as ruled by Newton’s laws, is the core of this book.
Models in homework problems. Most often the problem statement implicitly tells you what model to use, although sometimes in a mildly disguised language (disguised so as to start training your modeling skills). Judging whether or not a given model is good (i.e., commutes, corresponds well with reality) is an important part of engineering practice. So, we will point out deficiencies in various models here and there. Further, because some of these models are pretty good, you can use your intuition (another model!) to guide your learning of mechanics models and, conversely, you can use your new understanding of mechanics models to improve your intuition about reality.
Utility of rigid-object-mechanics models. Many things of engineering interest are well-modeled, for many purposes, as particles or rigid objects. So, the bottom line of this section is this:
If you understand how particles and rigid bodies interact and move according to the ‘rigid-object-mechanics’ model, basically the contents of these books, you will understand a lot about how many real things hold together, fall apart, stay in place and move.
0.3.1 Give an example of a model in mechanics.