Newton’s Incredible Laws
Introduction. This book discusses Newton’s laws of mechanics and gravitation in a non-mathematical way. A rudimentary familiarity with calculus will enhance the reading experience, but is not necessary. Any student contemplating an introductory college level course in mechanics will benefit from reading it. And those who have completed such a course may gain new insight into the three laws that profoundly changed the world. The first chapter, on coordinate systems and frames of reference, sets the stage for Newton’s three laws. In Chapter 2 Newton’s first law is discussed, and the second and third laws are taken up in subsequent chapters. Historically, Newton’s laws proved to be the well spring for several related developments (e.g. statistical mechanics, rocketry, and architecture to name but a few). Other areas of inquiry (e.g. thermodynamics and fluid dynamics), originally thought to be independent and self-contained, were in time seen also to be related to the three laws of mechanics (particularly after the discovery of atoms). Isaac Newton announced his three laws
way back in the 17 All of the statements made in this book are well-grounded in theory and experiment. Some of the field effects (discussed herein in a predominantly qualitative way) can and have been derived analytically and/or on computers. Those interested in such details are invited to visit www.grdmax.net.
Chapter 1. Coordinate Systems and Frames of Reference. When the author was young, one of his favorite places to visit was a park where there were "monkey bars." These were a set of steel pipes forming a three-dimensional cubic grid. Many pleasant hours were spent climbing around in this grid. Although we didn't know it then, the monkey bars were our first experience with a rectangular coordinate system. A good deal of that branch of physics
called "mechanics" has to do with the study of In order to One of the easiest coordinate systems to
work with is the We have constructed our coordinate system so that the distance between any two line-intersections in the x- and y-directions is one meter. But we could as well have used feet, inches, miles, light years, etc.
Figure 1.1
One Plane in a Rectangular Coordinate System If we place a clock at each point where the lines in Fig. 1.1 intersect, we have a frame of reference. As already mentioned, all of the clocks must be synchronized to be of any practical use … say they all read t = 0 at the same instant. Table 1.1 shows data about some
particular particle moving in the xy-plane. It provides partial
information about
Table 1.1
doesn’t tell us when the particle was at x = -2.5 meters, etc. We can
refine our data by adding more lines in Fig. 1.1, say at Now it may have occurred to you that our "where" and "when" data is somewhat subjective. We could as well have used some other instant for t = 0. And of course we could have used some other intersection point for the origin. Indeed we could have oriented our grid some other way in space (for example, with the x-axis vertical and the y-axis horizontal). The transformations from one such coordinate system to another are straightforward. Life gets more interesting when two
frames of reference
Figure 1.2
Particle Path Relative to Frame B Now one of the main questions answered
by Newtonian mechanics is this: Given a force acting on a particle, what
will the particle’s motion be? In Frame A the particle is at rest, and
in Frame B it most decidedly is
Chapter 2. Newton’s First Law. Newton’s first law answers the
following question: Of the infinite set of possible reference frames,
which one (or ones) should be used to correlate force with particle
motion? In modern language, Newton’s first law can be stated as follows:
Let us consider the particle in Chapter
1, viewed from frames A and B. If no force acted on the particle then,
according to Newton, frame A would qualify as an inertial frame (since the
particle remains at rest in frame A). Since the particle is In order to do the physics, Newton
admonishes us to
It should be noted that Newton does not
specify any negative x-direction of frame C with constant
velocity –. Newton’s first law says "at rest"
vor "moves with a constant velocity." So frame C is also
an inertial frame. We could as well apply Newton’s second law
(next chapter) in frame C, and get results in agreement with experiment.
Chapter 3. Newton’s Second Law. Newton’s second law can be stated as
follows: .
In more fundamental units, it is the product of a particle’s mass times
its velocity: p=mp. Thus according to Newton, vF=d.
(Newton called momentum the "quantity of motion.")/dt=d(mp)/dtvLet us suppose that a particle’s mass
remains constant in time. In this case Newton’s second law can
alternately be stated as follows: is the
acceleration of a particle relative to one inertial frame of reference,
and if all inertial observers agree on the force acting on the particle,
then a is its acceleration relative to aevery inertial
frame of reference. Case in point: Frames A and C discussed above.
Presumably observers in both frames agree that zero force acts on
the particle. In frame A the particle is at rest, and in frame C it moves
with a constant velocity. In both cases the acceleration is zero.
Indeed in any inertial frame the acceleration would be zero.The idea, that Newton’s second law ( )
works in aevery inertial frame is referred to as co-variance.
Although a particle’s velocity may differ from inertial frame to
inertial frame, its acceleration is the same in every inertial frame. Thus
this one law, =mF, works in every inertial
frame. It’s an amazingly beautiful and useful result.aFor example, we might consider the
motion of a planet (say the Earth) around the much more massive Sun (which
is practically at rest in an inertial frame). Under the influence of the
Sun’s gravitational force, the Earth travels nearly in a circle around
the Sun once every year. Given a knowledge of what the gravitational force
is, this orbit can be derived using .aBut what is the situation viewed from
the inertial rest frame of a space traveler, traveling past the Sun at a
constant velocity? Relative to the space traveler’s frame the Sun moves
in the produces
the right curve! Newton’s second law is indeed co-variant!a
Chapter 4. Newton’s Third Law. In modern terms Newton’s third law can
be stated as follows: We can =mF.
We can tell how great the force acting on the particle is by reading the
scale. But we could as easily say that the scale indicates the magnitude
of the force acting on us! And that magnitude is, of course, the same as
the magnitude of the force acting on the particle, the difference being
that these two forces act in opposite directions.aAccording to Newton, things
Chapter 5. Momentum and Energy Conservation. If we abbreviate a particle’s momentum
(m , then Newton’s second law can be
stated as: p=dF/dt. (dp is the pinfinitesimal
change in momentum that occurs in the infinitesimal time interval
dt. The ratio of two infinitesimal quantities can be a perfectly finite
quantity. It can even be a very large quantity.) Multiplying
through by dt, we find that (dt)=dF. In
words, the (infinitesimal) change of a particle’s momentum equals the
product of the force acting on the particle times the (infinitesimal) time
interval during which said force acts.pNow according to Newton’s third law,
for all of the time a driving agent exerts the force – on the agent.
Thus for every change of particle momentum, dF, there is a
change of –dp in the agent’s momentum. pThe total
change in momentum, of both agent and particle, is zero. In other
words, the total momentum of both agent and particle is constant in time.
This result … a necessary consequence of Newton’s third law … is
called The Conservation of Momentum Law. It is enormously useful in
solving many problems. No violation of this law has ever been observed,
even when the modifications suggested by Einstein (and by experiment) are
factored into Newtonian mechanics.When we multiply the force on a particle
by the
acting through the infinitesimal distance dx. Here again, Newton’s third
law implies something. For the work done on the Fagent, by the equal
and oppositely directed force exerted by the particle, is –F(dx)=-dE.
Thus the total work done on the particle and on the agent is zero.Now work is one form of
Chapter 6. Force Laws. Given knowledge of some particle’s
initial position and velocity at time t=0, Newton’s second law ( )
can be used to calculate or compute the particle’s future motion,
provided we know what a is at times greater than t=0. The
question is: How do we know what F is? It is the Fforce
laws that hypothetically answer this question.In general the force experienced by a
particle can be attributed to its environment (other particles, etc.) One
of the first force laws was deduced by Newton. It is usually referred to
as the Law of Universal Gravitation. According to this law, every pair of
particles attract one another. The force of attraction … the
gravitational force … is proportional to the product of the particle
masses, and inversely proportional to the square of the distance between
them: F=Gm Knowing the force law for gravity, we
can plug it into Newton’s second law to get the equation of motion for
mass m Another important force law was
announced by Newton’s contemporary, Hooke. It states that when an agent
stretches or compresses a spring, then the force exerted by the spring (on
the agent In the 19 Lorentz was the first to specify the
force law for the electromagnetic force. According to Lorentz, the _{B}, where
E is the electric field of qE_{A} at q_{B}’s
position. Maxwell’s equations prescribe how to calculate the fields of q_{A}
throughout space. Thus Maxwell and Lorentz together define one of the most
important force laws in physics.
Chapter 7. Maxwell and Newton’s Third Law. In Newtonian mechanics it is axiomatic that a particle cannot exert a force on itself. It can only exert a force on other entities (i.e. on other particles, on an agent, etc.) For many years the equal and oppositely directed reaction force, described by Newton’s third law, was something of a mystery. Now according to Maxwell’s equations,
whenever a
Figure 7.1
Induced Electric Field, Accelerated Charge It turns out that the induced electric
field illustrated in Fig. 7.1 , thus indicates that
the charge experiences an electric force oppositely directed to E
whenever a is unequal to zero. At first glance we might
conclude that the Newtonian axiom … that no particle can exert a force
on itself … is violated by electric charge. But it should be noted that
the acceleration-opposing Lorentz force is present aonly when
is nonzero, and a is nonzero aonly when the electric
charge experiences an external force (exerted by one agent or another).Without getting into the details, it
turns out that a charge can be assigned a so-called electromagnetic mass,
say m _{em}.
The equality is not precise in the case of charge, since the Lorentz force
implied by Fig. 7.1 has two parts: an "inertial" part and a
"radiation reaction" part. The inertial part is the part that
equals –ma_{em}. The second part of the Lorentz
force, implied by Fig. 7.1, is called the radiation reaction force. When
forced to move in certain ways (e.g. when forced to oscillate back and
forth on the x-axis), charges emit radiant energy in the form of
electromagnetic waves. Part of the work done by the driving agent force
during any given cycle is radiated away into infinite space. But here
again the fields exert an equal and oppositely directed force on the
driving agent. In brief, Newton’s third law (like the conservation laws)
appears always to be true.aIn a sense the charge is the point of
interaction between the agent causing the charge to accelerate and the
charge’s electromagnetic field. In cases where an agent causes a charge
to accelerate, we might rephrase Newton’s third law to say:
Chapter 8. Charge and Neutral Matter. Knowledge of electric charge was non-existent (or at best only very sketchy) during Newton’s time. The world view in those days was that space is populated with quantities of "neutral matter." The connection between matter and light (or electromagnetic radiation) was unknown (although Newton performed several experiments on light). The reason why charge lay undiscovered at first is of course because atoms contain equal amounts of positive and negative charge. Today we know that "neutral matter" … even elementary particles such as neutrons … ultimately contain charged constituents (called quarks). But in Newton’s time such internal details were unknown. When scientists discovered the existence
of charge and began to experiment with it, a startling parallel between
mass-mass and charge-charge interactions was noted. According to
Newton’s Law of Universal Gravitation, the (attractive) force between
masses m Since charge satisfies _{em}
for many practical purposes, aand (thanks to the
acceleration-induced electric field) since charge always exerts an equal
and oppositely directed force to any external agent force, it makes one
wonder whether "neutral matter" really exists. All material
particles … even uncharged ones like neutrons … contain electrically
charged constituents (or quarks). As with atoms, the fields
"outside" such neutral particles may cancel out more or less
completely. But "inside", the individual quarks may
experience the acceleration-induced electric fields discussed in Chapter
7.There is certainly plenty to think
about. The reason for bringing Maxwell and Lorentz into the Newtonian
paradigm has been two-fold. First it is interesting to see that the
mysterious reaction force of Newton’s third law is, at least in the case
of charge, not so mysterious after all; it is an
Chapter 9. Kinetic and Potential Energy. Any particle, in motion relative to an
inertial frame of reference, can do work if it is brought to a state of
rest. To the extent energy is defined to be the ability to do work, the
particle has energy. This energy is called In the pre-relativity era the formula
for kinetic energy was mv Sometimes it may seem that the
Work-Energy Theorem fails. For example, let us suppose that we lift some
particle (of mass m) to a height h above the ground. We know that we must
exert a force of at least GMm/R The answer lies in the concept of
Like kinetic energy, potential energy
can do work. For example, we need only return the particle, discussed
above, back to ground level. In this case, if the speed is again kept
constant, the particle does work on There are other forms of potential
energy. For example, the potential energy of two particles, attached to
the ends of a spring which is compressed or stretched an amount x, is kx When one of the particles, in a system possessing potential energy, remains at rest in an inertial frame (for example, if one end of the spring discussed above is anchored to an immovable wall), then it is customary to attribute the potential energy to the particle that moves.
Chapter 10. Conservative Systems. Let us suppose that, instead of lifting the particle (in Chap. 9) to a height h above the ground, we shoot it upward. If we give it the right initial speed, the particle will rise to a height h, and will then fall back to the ground. On the way up, the particle’s speed (and thus its kinetic energy) steadily decreases. And of course its potential energy steadily increases. An important fact (and a very useful
one) is that the In going from point A to point B the particle may, in one instance, travel in a straight line (the shortest distance). Another time the particle may travel a more circuitous, longer route. But in the particle/Earth system the difference in potential energy between any two points depends only upon the relative positions of the two points, and not on the path to go from one to the other. Such systems are said to be
conservative. The hallmark is that the total energy is a Recalling the case (in Chap. 9), where
the particle’s speed remained constant as we lifted it to height h, that
system was
Chapter 11. Least Action. There is a principle in optics (called Fermat’s Principle) which deals with the fact that light travels at different speeds through different media. For example, the speed of light in air is greater than it is in water. Consequently the light reflected from a fish in the water may not travel in a straight line to the fisherman’s eye (assumed to be in the air). At the air/water interface the light may abruptly change its direction. Fermat’s Principle, or the Principle of Least Time, provides information about the light’s path. According to this principle, the path taken by the light, in going from point A to point B, is the path that requires the least time. In mechanics there is a similar
principle, dealing with a conservative system particle’s path from point
A to point B, in some time interval t=(t Now since Newton’s second law provides
a basis for computing the exact particle path in any given case, it might
be wondered why we should concern ourselves with paths that never occur.
The relevant fact in this matter is that computers were not available to
our forebears. The abiding idea before computers was to In both Newtonian and Lagrangian
mechanics the particle path in any given case is perfectly precise. But in
quantum mechanics it might be said that In the microscopic world, the apparent
exactness and repeatability of Newton, Lagrange and Fermat opens out into
a more blurred paradigm. And it is
Chapter 12. Fictitious Forces. In Chapter 2 it was emphasized that
Newton’s laws work only in Having emphasized the importance of
inertial frames, we now change our minds and say that physics Imagine that we are standing on a large, round floor that can rotate about a vertical axis. (An example would be a merry-go-round without the horses.) When the rotation rate is zero, the only forces we experience are gravity (pulling downward) and the floor (pushing upward). These two forces sum to zero, and our acceleration is accordingly zero. If the floor rotates, however, then
relative to the inertial frame in which the axis of rotation is fixed, we
go in a circle. At any given rotation rate our One of the convenient things about
Newton’s second law is that it works for both linear and radial
acceleration. Thus the floor must exert an inward force upon us in order
for the radial acceleration to occur. Indeed we can Now quite often our inclination is to
assume that our "personal" rest frame is an inertial frame. And
viewed from the rotating floor’s (non-inertial) rest frame, it might
seem that there are now The new force … the one that seems to
pull us outward … is called a The centrifugal force resembles a gravitational force, in that its magnitude is proportional to our mass. It is as though a gravitational field points radially outward at all points in the non-inertial, rotating frame of reference, said field’s strength being proportional to the distance from the frame’s rotation axis. There is a second fictitious force which
we must contend with if we Relative to an inertial frame our speed depends on our distance from the rotating floor’s center. The farther out we are from the center, the faster we go. To be precise, our speed is wr, where w is the floor’s rotation rate and r is our distance from the center. What happens if we move away from the
center? Viewed from the inertial frame, the
floor must exert a atangential force to speed us up. If we are
walking upright, we must not only lean backwards (toward the rotating
floor’s center) to counterpoise the centrifugal force, but we must also
lean to one side to counterpoise the new Coriolis force.Like the centrifugal force, the Coriolis force depends on our mass. But it also depends on our distance from the center and the rate at which we walk away from the center. The faster we walk, the greater the force seems to be. These fictitious forces … these forces
that must be invoked if we wish to apply Newton’s second law in
non-inertial frames … are of course nothing new. They aren’t It is of historic interest that Mach was
dissatisfied with the concept of fictitious forces. He suggested that the
centrifugal and Coriolis forces can be thought of as
Chapter 13. Spaces. Fig. 13.1 depicts a particle, A, in the
xy-plane of a rectangular coordinate system which we’ll call
"K." An arrow has been drawn from the origin to A’s location.
It is customary to call this arrow the
Figure 13.1
Particle A in Rectangular Coordinate System K If we drop perpendiculars from the tip
of With regard to the displacement of A
from the origin, there is nothing sacred about our coordinate system’s
axes. For example, someone else might prefer to work in coordinate system
K’, whose origin might coincide with the origin of K, but whose
x’-axis makes an angle q
with the x-axis of K, etc. In system K’ the x’-component, R Quantities that transform from one
coordinate system to another using this transformation are said to be The Pythagoras and other ancient Greeks
(notably Euclid) worked out many important formulas in Now here is a thought provoking idea:
there are as many "spaces" as there are vectors! We seem
intuitively inclined to think of "displacement space" as being
the one true space. But there is no reason why we shouldn’t, for
example, define a Sometimes changing from "regular" (or displacement) space to some other space will simplify a problem’s solution. A first course in mechanics may not require that students solve problems in spaces other than displacement space. Nevertheless it is a good idea to be aware that there are (mathematically speaking) as many spaces as there are vectors. Deeper insight can often be gained by thinking about things in the context of a space other than displacement space.
Chapter 14. An Example in Velocity Space.
Following is a derivation of g(v)dv, the Maxwell/Boltzmann distribution of molecular speeds (i.e. velocity magnitudes) in a gas containing N molecules, where N is arbitrarily large and where each molecule’s velocity is unique (so that there are N distinct velocities). As a matter of definition, g(v)dv is the fraction of the N molecules with speeds in the range v to v+dv. Maxwell derived the result in 1859. (Unlike the other chapters in this book, this chapter is necessarily more mathematical. It is offered as an example of how useful spaces other than displacement space can sometimes be.) ~
In the following discussion, "(v,dv)" is shorthand for "the range v to v+dv." Let Ng(v)dv be the number of molecular velocities with magnitudes in (v,dv). Then . (1) From thermodynamics, the mean kinetic energy equals 3kT/2, where k is Boltzmann’s constant and T is the absolute temperature. Thus . (2) In postulating a form for g(v)dv, we expect few (or no) molecules to be standing still, and none to be moving with infinite speed. And of course g(v)dv must integrate (normalize) to unity. Integrals of the form have the potential of satisfying both of these requirements, provided n>0. Thus we postulate that , (3) where a, b and n are constants to be determined. Let the fraction of velocities with
x-components in (v In The number of velocities in the
spherical shell, of volume 4pv (4) In particular, for v , (5) or (6) where the constant C equals 4pf . (7) From Eq. 1, . (8) Or, since , (9) we find that (10) and . (11) From Eq. 2 we have . (12) Or, since , (13) we find that (14) and thus . (15) Substituting in Eq. 10, , (16) and thus . (17) Or, , (18) which is the Maxwell-Boltzmann distribution. ~ Maxwell’s formula for g(v)dv was experimentally corroborated by Stern in 1926. Several other experiments provided further corroboration. In brief, Maxwell was right. Other derivations by Boltzmann and Gibbs arrived at the same result.
Chapter 15. Speed-Dependent Mass. For a long time after Newton announced
his three laws of mechanics, it was believed that the mass of a particle
is invariant (or constant). The Conservation Laws for momentum and energy
were augmented by a Conservation of Mass Law. Given some total mass
contained in a volume of space, it was believed that the mass remained
constant (so long as none came into the volume from the outside, and none
left the volume). This law appeared to be well-corroborated by experiment.
One could burn the material, put it through diverse chemical reactions,
beat it out of (or into) shape, etc. Nothing seemed to change the total
mass within an experimental volume. As previously mentioned, this meant
that Newton’s second law … originally written )/dt
… could alternately be stated as v=m(dF/dt)=mv.aWhen physicists began thinking about the
inertia of The suggested dependence of mass on
speed was soon corroborated by experiments with high-speed particles. The
reason why this dependence remained undiscovered for so long owes to the
very great speed of light. In the early days experiments were limited to
cases where v was much less than c. And the adjustment term (1-v
Now one of the truly wondrous things
about Newton’s second law is this: despite the dependence of mass on
speed, Newton’s second law is still true in the form he originally wrote
it down. That is, )/dt is true in
relativity! Newton of course couldn’t have vknown that mass
increases without bound as v approaches c. He couldn’t have known
that =d(mF)/dt is generally true, whereas v=mF
is not. (Could he???)a
Chapter 16. Mass-Energy Equivalence Albert Einstein, the author of Special
(and later of General) Relativity, reasoned that mass was not only
speed-dependent but, under certain circumstances, mass could be changed
completely into energy (notably radiant energy). Before Einstein there
were (in the minds of men) energy and matter (or mass). Each of these
existed in its own world, so to speak, and the manner whereby matter
emitted radiant energy was something of a mystery. Certainly, even with
the dependence of mass on speed factored in, it wasn’t at first clear
that mass could be completely transformed into energy (and vice versa).
When Einstein suggested that this could be the case, there was
understandably skepticism. Part of this skepticism no doubt stemmed from
the now-famous equation, E=mc In modern times we live with the knowledge (and dread) that Einstein was right. This became common knowledge with the detonation of the first atomic bombs. Today the Interestingly enough, what is true for
the splitting (fission) of heavy nuclei is also true for the joining
(fusion) of certain Technically, the problem with fusion is to get two Deuterium nuclei close enough to fuse. The repulsive forces become relatively huge when the distance between the nuclei approaches fusing range. On the Sun, this is accomplished by the immense gravitational field at the Sun’s surface (and the great speeds of particles in such a hot environment). Here in earthbound labs other techniques must be found. The supply of Deuterium fuel in the oceans is inexhaustible. Thus solving the controlled fusion problem is one of mounting interest, as our fossil fuel reserves dwindle. Thanks to E=mc
Chapter 17. Relativistic Mechanics. Newton’s second law, in the form )/dt,
is relativistically correct. Many of the axiomatic beliefs and assumptions
from the pre-relativity era fall, but Newton’s fabulous second law
persists. (Indeed all three of his laws are relativistically correct.)vThe new feature in relativity is that
mass depends on particle speed. Thus (as you may know from calculus),
Newton’s second law expands to /dt) + vdm/dt).
In general the algebra is more involved than in the good old v(=mF
days. But two important cases warrant mention.aWhen (for example, when a particle is made to go in a circle
at constant speed), then particle mass remains constant and Newton’s
second law simplifies to v=gmF_{o},
where g is a popular
shorthand for 1/(1-va^{2}/c^{2})^{1/2}. When
acts in the same (or opposite) direction as F, then
Newton’s second law simplifies to v=gF^{3}m_{o}.aThere are some surprising (and even
entertaining) ramifications of the dependence of mass on speed. Let us
imagine that a particle accelerates at a constant rate down the y-axis of
inertial frame K. A force of ^{3}m_{o}
must be applied downward. (Note that F increases as v increases.)aViewed from some second frame, K’, moving in the positive x-direction of frame K at constant speed u, the particle moves along a curved path. And, like the K observer, the K’ observer would agree that particle speed (and hence particle mass) is increasing with time. But here is an important difference
between K and K’: for the K observer the particle’s x-component of
momentum is constantly zero. Thus there is no x-component of force. For
the K’ observer, the particle’s x-component of
To cut to the chase, in relativity
theory all inertial observers do .
But in relativity the transformation is more complicated. Without getting
into the details, FF_{x}’ might depend on F_{x}, F_{y},
and F_{z}. Similar remarks apply to the transformation of
acceleration from one inertial frame to another.The fact that the old, so-called
Galilean transformation for force was (regrettably) incorrect was almost a
breath of fresh air for students of electromagnetic theory. For in
electromagnetism a charge might or might not have a magnetic field,
depending on whether or not it was moving. And a second charge might or
might not experience a But regarding the first charge’s magnetic field, we might wonder "moving relative to which frame?" If the charge moves with a constant velocity relative to frame K (and accordingly has a magnetic field), then it will be at rest in some other frame, K’ (and will have no magnetic field). The answer to this conundrum is that a
second charge may feel a magnetic force according to frame K, and no
magnetic force according to frame K’. Classically it’s something of a
paradox. But the To the extent )/dt
is relativistically correct in all inertial frames, vall of the
forces of physics must transform from one frame to another quite as d(m)/dt
does.v
Chapter 18. The Principle of Superposition. In chapter 12 we discussed the case
where we stood on a floor that rotated around a vertical axis. Viewed from
the rotation axis’ rest frame (presumed to be an inertial frame), we
went in a circle. That is, we constantly accelerated radially inward
toward the axis of rotation. In addition, we experienced the force of
gravity. Thus the floor had to exert Now when two or more forces are exerted
on a particle, they can be replaced by a single force (namely their vector
sum). Replacing two or more vectors by their vector sum is generally
referred to as the Principle of Superposition. For example, Fig. 18.1
depicts the
Figure 18.1
F, the Combined Gravity-Counteracting and Radial Acceleration-Causing Forces. Many problems, where a particle
experiences several forces attributable to other entities in its
environment, can be simplified by invoking the Principle of Superposition
before applying .aIt should be borne in mind that, in the case of forces, the Principle of Superposition generally applies only to particles. It may not suffice in the case of macroscopic objects, composed of many bound particles. For example, consider the case of the macroscopic bar depicted in Fig. 18.2. Equal but oppositely directed forces act on its two ends. The forces sum to zero. Yet the bar doesn’t remain motionless. Parts of it to the left of center accelerate upward, and parts to the right of center accelerate downward.
Figure 18.2
When Superposition Doesn’t Suffice As it turns out, there How is it that we could use
superposition in Fig. 18.1? Leroy is, after all, scarcely a particle. The
reason we could use superposition in that case is because Leroy is
composed of bound atoms, and the line of passes through Leroy’s cm.
Leroy has learned to lean inward at just the right angle to make this be
so. Were he to lean at any other angle, he would topple.Had Without belaboring the point, the
Principle of Superposition, in the case of multiple forces, can be useful
in macroscopic, "rigid" cases as well as in particle cases,
provided the lines of action of all of the forces act through the center
of mass. In such cases works for all
parts of the macroscopic object. That is, all parts of the object have the
same motion as the object’s center of mass. (But even in these cases the
individual forces may cause something to happen. For example, if equal and
oppositely directed forces act to the left and right, on the ends of the
bar depicted in Fig. 18.2, then the bar is astretched … the amount
depending upon the bar’s elasticity. Strictly speaking, particles
don’t stretch, bend, etc.)
Chapter 19. The Center of Mass. Let us consider a "non-rigid" system of "particles," say our solar system. Relative to an inertial frame, all of the particles are accelerating. (In our solar system, even the massive Sun is not at rest in any inertial frame; it wobbles under the influence of the planets.) The Moon pulls on the Earth, the Earth pulls on Mars, etc. Every force can be accurately determined. What total force do we obtain when all of the interactive forces are summed? You can probably see, by applying Newton’s third action/reaction law, that the forces sum to zero. Now the forces acting on a given planet
are not completely specified by the attractions toward other planets and
the Sun. Every "particle" in our solar system is also attracted
to each and every star in our galaxy (and beyond). We can sum the forces,
attracting the particles in our solar system to the nearest star, and this
sum is of course When we do the math, however, we find a
beautifully simplifying result. There is a Rigorously speaking, then, , where
a is the sum of all the forces, on all our stars, and m is
the sum of their masses.FAs an example of the utility of this
center of mass concept, we might consider a pair of particles, of masses m It is worth mentioning that the motions
of the constituent particles may be difficult (or impossible) to calculate
analytically. But if we know the forces then we can always total force acting on all of a system’s member particles,
and if m is their combined mass, then the system’s center of mass has
the relatively simple motion prescribed by Newton’s second law: F_{total}=m_{total}a_{cm}.
Chapter 20. Symmetries and Simplifications. Isaac Newton derived his Law of
Gravitation by thinking of the Sun, the Moon and the planets as particles,
despite the fact that they obviously are not. How did he get away with
that? Part of the answer owes to the great distance between the Sun and
the planets (compared to their diameters). But Newton himself discovered a
great, mathematical simplification in the case of Consider a particle at rest on the Earth’s surface. Rigorously speaking, every atom in the Earth attracts the particle according to Newton’s Gravity Law. Fig. 20.1 illustrates the case for two diametrically opposed Earth atoms.
Figure 20.1
Earth Atoms Attracting Surface Particle It’s clear in Fig. 20.1 that the
x-components of At first glance, calculating the magnitude of this force seems to be considerably complicated by the fact that the distances between the particle and pairs of Earth atoms varies from zero to the Earth’s diameter. That is a mind-boggling lot of forces to add up! But Newton derived a wonderfully simple result: when the force is inversely proportional to the square of the distance, then the sum of all the forces is the same as what the force would be if all the Earth’s atoms were replaced by a "super" particle at the Earth’s center (said "super" particle’s mass being that of the entire Earth). According to Newton’s third law, we can turn the situation around: the summed forces on all the Earth’s atoms, exerted by the particle, is the same as the force that the particle would exert on a "super" particle at the Earth’s center. Now picture Small wonder the ancient Greeks called the sphere the most perfect of all geometric forms. Newton himself discovered that a sphere can be adequately represented as a particle in many cases. Here is another interesting fact: given N gravitationally attracting atoms (each exclusively occupying some small volume of space), the arrangement that lets the atoms get as mutually close to one another as possible is a sphere. Given a volume of matter, the smallest surface area that can enclose that volume is a spherical surface. That is why all the stars and all the planets (which were fluid at their births) are spheres!
Chapter 21. Force Couples and Torques Fig. 21.1 depicts a box, subjected to
two equal and oppositely directed forces. The total force exerted on the
box is zero. Thus according to the
box’s center of mass does not accelerate. But since the two forces do
not act along a common line, something happens. The box begins to spin.
The combined effect of a and F_{1}
is called a torque (for ‘twisting’).F_{2}
Figure 21.1
A Box Subjected to a Torque Since There are important parallels between
and the equations relating torques to the rates at which the spins of
objects change. Indeed just as a macroscopic body can have alinear
momentum, as a consequence of the motion of its center of mass along some
line, so it can have angular momentum as a consequence of rotation
of its parts about some axis. The direction of the angular momentum can be
found by curling the fingers of the right hand around in the direction of
spin. The thumb will then point in the direction of the angular momentum.
Similarly, this ‘right hand rule’ can be used to determine the
direction of the applied torque. (In Fig. 21.1 the torque points into the
page.)It is customary to denote the angular
momentum as .
And just as tF_{resultant}=dp_{cm}/dt,
it can also be said that =dt/dtLOne of the world’s most fascinating
toys is a gyroscope. Fig. 21.2 schematically depicts a typical gyro.
Application of the right hand rule indicates that its angular momentum,
Figure 21.2
A Gyroscope Now here is an interesting question:
what effect will the indicated force couple have? Intuitively it may seem
that the gyroscope should rotate counterclockwise around the z-axis. But
the torque, created by forces
points F_{2}out of the page (in the positive z-direction). Thus the
torque and angular momentum vector are at right angles. According to =dt/dt,
this torque should cause L to start rotating around the Lx-axis!Let us see if we can understand this weird behavior. Consider the particle indicated by a dot near the gyroscope’s rim. In the absence of any torque, this particle travels a circular path in the xz-plane. It does this because the material between the particle and the gyro axis exerts a radial force inward on the particle. And of course the particle exerts an outward force on that material. The magnitude of these action and reaction forces increases with the spin rate of the gyro. Indeed if the spin rate becomes too great, the gyro will fly apart. Now owing to the gyro’s rigidity, the
effect of the force couplet in the figure is to cause a Note that since each act at right angles to the motions of
the axle’s ends, they do zero work for as long as they cause the gyro to
precess. Thus there’s no change in the gyro’s spin rate or rotational
kinetic energy. As soon as the torque is removed, the precession halts.F_{2}Many devices depend upon gyroscopic
behavior. A child’s spinning top does not fall over as a result of it.
And we are able to keep ourselves upright on a moving bicycle by turning
the front wheel to the right or left. In general gyros maintain their
orientation in inertial space when subjected to no torques. Thus a good
variation on Newton’s first law would be: Just as there are parallels between
and rotational mechanics, so does rotation have its own conservation laws.
That is, angular momentum and rotational kinetic energy are conserved,
quite as linear momentum and energy are. These laws stem from the
corollary to Newton’s third law, that afor
every torque applied by some agent to an object (or system), an equal and
oppositely directed torque is applied by the object on the agent.
Chapter 22. Heat. As scientists explored the many
ramifications of Newton’s laws, a second movement was underway
investigating the phenomena of thermodynamics (notably heat). In general
any object (like the gyroscope in Chap. 21) or any volume of gas contains
heat energy. Heat is now known to be a form of The answer became evident with the discovery of atoms. Every macroscopic (big enough to see) object is composed of many atoms. And at temperatures above absolute zero these atoms are not at rest. In a solid, each atom vibrates around some nominal center point (as if connected to its neighbors by several microscopic springs). And in the case of a gas, the atoms dart about, rebounding elastically off other gas atoms (or molecules). Of course the motions (or momenta) are
in all different directions, and the Historically it wasn’t at first
obvious that macroscopic objects are composed of myriad atoms. With the discovery of atoms, however, it
soon became evident that heat energy is in fact internal kinetic energy,
and temperature is proportional to the Of course scientists couldn’t To the eye, a solid’s surface may appear to be placid. But microscopically the surface atoms are darting out and in. Newton’s laws of mechanics can be used to show that heat energy flows from a hotter to a cooler object when their surfaces are brought in contact. Other phenomena and laws (e.g. Boyle’s law) were seen to be natural consequences of Newton’s laws once atoms had been discovered. It is safe to say that these three laws, coupled with the knowledge that atoms are the fundamental building blocks of nature, paved the way for one of the greatest theoretical syntheses in human history.
Chapter 23. Fluid Dynamics. Another area of study that initially paralleled Newtonian mechanics was the study of fluid behaviors (moving air, water, etc.) A matter of particular interest was the pressure (or force per unit area) exerted by a fluid on a material surface. Before the discovery of atoms it was known that the greater the flow rate of a fluid over a material surface, the less pressure the fluid exerts on the surface. But the underlying reasons for this phenomenon were unclear. Fig. 23.1 depicts the flow of a fluid over the top and bottom surfaces of an object. Let us assume for openers that the fluid is incompressible. Then two elements of fluid starting from the leading edge (one going over the top surfaces and the other going over the bottom surfaces) must nominally meet up again at the trailing edge.
Figure 23.1
Fluid Flowing over Top and Bottom Surfaces Now clearly the distance over the top
surfaces, from leading to trailing edge, is greater than the distance over
the bottom surfaces. Thus the fluid must travel faster over the top than
over the bottom if everything is to meet up again at the trailing edge.
The pressure on the top is accordingly less than the pressure on the
bottom. Part of this difference is offset by the greater surface area of
the top surface. But the offset is not complete. Of course we can as well view things from the fluid’s rest frame. In this case the object moves through the fluid. The net force upward is still there. Let us now replace the incompressible
fluid in Fig. 23.1 with Air molecules encountering the
object’s leading surfaces will rebound from them. But the net force on
these two surfaces is zero. In effect the air molecules will be
"swept" upward and downward. Since the object’s trailing
bottom edge is parallel to the air flow, this sweeping action has
negligible effect on the number of molecules rebounding from it every
second. But the The lift on the trailing surfaces of the wing can ultimately be attributed to the rarefaction (or partial vacuum) created above the wing’s upper trailing surface. It is contingent upon the air not backfilling (due to thermal motion) before the trailing part of the wing has passed out of the rarefaction zone. If the temperature is too high … if the mean air molecular speed is too great … partial backfilling will occur before the trailing edge of the wing passes out of the rarefaction zone … the wing will start losing its lift. Thus the lift effect works best in cool air. Indeed in places where it gets really hot in the summertime (e.g. Arizona), airport authorities may sometimes deem it advisable to prohibit takeoffs and landings. There won’t be enough lift on airplane wings, at normal takeoff and landing speeds, to provide adequate climb and controlled descent.
Chapter 24. Collisions. We can define a gas (and a liquid) to be
a collection of particles that aren’t bound to one another. (A solid
would correspondingly be a collection of particles that A collision between two particles is
defined to be elastic when the total kinetic energy after the collision
equals the total kinetic energy before the collision. "Total kinetic
energy" in this case refers to mv A macroscopic example of near-perfect elastic collisions is found in billiards or pool. Pool balls are by design very hard, and they rebound elastically off one another. A microscopic example of elastic collisions is found in gases.
What happens to the lost kinetic energy
when a collision is inelastic? For the most part it may be converted to Now it is a useful fact that light
interacts with material particles in a particulate fashion. And the
"collision rules" when a "light particle" (or a
photon) collides with a material particle (such as an electron) are
essentially the same as when two material particles collide with one
another. Interestingly enough, however, photons don’t seem to collide
with one another (as electrons do).
Chapter 25. Periodic Motions. One of the more interesting toys that were in vogue for a time were "Super Balls." These are hard rubber balls which rebound almost elastically from a hard floor. That is, you can drop a super ball onto a hard floor, and the ball will almost bounce all the way back up to your hand. The collisions of ball and floor are nearly elastic. Of course there are never truly Motions that are repeated in time are said to be periodic. The time from the arbitrarily defined start of one cycle to the start of the next is called the motion’s period, and is often symbolized as t. The motion is periodic if the velocity at time (t+t) is the same as the velocity at time t. The An (t).vAnother important example of periodic
motion is that of a particle oscillating on an ideal, anchored spring. In
this case the motion is "w"
is called the
Figure 25.1
Particle in Circular Orbit For this type of motion, x(t)=Rcos(q)
and y(t)=Rsin(q). Let
us say that the particle is at x=R, y=0 at time t=0. Then q=wt,
where q is expressed
in radians and w is
expressed in radians per second. (2p
radians equals 360 degrees. Like cycles, radians are dimensionless … the
units of w are second Sinusoidal motions are of great interest in physics, since they occur (or are closely approximated) in many situations. For example, the motions of a solid’s atoms, about their nominal center points, are practically sinusoidal over a broad range of temperatures. Mathematically it is a useful fact that
the x- and y-position components of a particle, circling the origin at
constant speed, are the same as those of two particles, one oscillating on
the x-axis and the other on the y-axis. But
Chapter 26. Waves. Imagine that we have a very long wire, anchored at one end to a rigid wall and with Leroy’s hand holding the other end. Fig. 26.1 illustrates.
Figure 26.1
Leroy and the Wire At time t=0 Leroy begins shaking his
hand up and down such that the position of his hand is y = A sin(wt).
What happens? As you might know from experience, the wire begins to
undulate up and down, not only at Leroy’s hand, but progressively
outward toward the wall. The speed at which these undulations propagate
away from Leroy will depend, in part, on how tightly Leroy pulls the wire
in the negative x-direction (i.e., the speed will depend, in part, on the If we observe the wire at some point x>0, between Leroy and the wall (and before the undulations have reached the wall), we find that y(x,t) = A sin(wt-vt), where v is the speed at which the undulations move away from Leroy’s hand and toward the wall. In other words, the vertical acceleration that Leroy imparts at x=0 is reproduced at point x>0 and at time t>0 according to x = vt. Although any small section of the wire,
at any given x Waves of all kinds are encountered in the physical world. The space we occupy is constantly flooded with a staggering mix of electromagnetic waves. We need only turn on a radio or television to sample them. (Or, where there is light, we need only open our eyes.) Waves like those in the wire, where the
oscillations occur perpendicular to the direction of wave propagation, are
referred to as In order to understand electromagnetic waves (light, radio, TV, …) we must use Maxwell’s equations. But Newton’s laws can be used to analyze the waves in taut wires, sound waves, and a host of other wave types. In the case of a taut wire (Fig. 26.1)
Leroy
Chapter 27. Interference One of the author’s more memorable
experiences occurred one day when a physics professor turned a low
frequency sound generator on in a large lecture hall. The acoustic wave
from the sound generator quickly spread out to the walls, floor and
ceiling, and reflections generated a host of additional waves. In a
fraction of a second the hall was filled with sound waves traveling in
every imaginable direction. Like all waves, The fascinating thing about this experiment was the nature of the waves. We were instructed by the professor to slowly move around in the lecture hall. And when we did so, we made an astounding discovery. In one place we could hear a loud, low frequency hum. It was so loud that it was all but impossible to make out what other students were saying. Yet only a step or two away we heard no hum at all, and could easily hear what other students were saying. Some years later the author saw a similar experiment performed with an electromagnetic wave generator and a large (reflecting) metal sheet. The professor turned the generator on and held a rod antenna between generator and sheet. The ends of the antenna were wired to a small bulb, and the electromagnetic signal was powerful enough to make the bulb glow. Yet when the antenna was moved farther away from the generator, the bulb went out. It was as if no wave existed at the second location. Further displacement toward the reflecting sheet caused the bulb to glow again! What lay at the base of these two
phenomena were standing waves (acoustic and electromagnetic). Standing
waves owe their existence to the ability of two or more waves to pass
through a common point in space. When this occurs, the undulations of the
thing(s) that oscillate (air molecules, electric field vectors, etc.) This adding behavior of waves is known as interference. Where the undulations reinforce each other, the interference is said to be constructive. (And where the undulations oppose each other the interference is said to be destructive.) In mechanics, In Fig. 26.1 Leroy can (with the
wall’s help) set up standing waves. The propagating waves, initially
created by Leroy, will eventually reach the wall and a One of the requirements for creating a
standing wave is that Leroy wiggle his end of the wire at the right
frequency. A standing wave presumes that an integral number of half
wavelengths exist when both ends of an oscillating wire (for example) are
anchored. Frequencies meeting this requirement are said to be
"resonant" frequencies. They play an important role not only in
"classical" phenomena (those explainable by Newtonian and
Maxwellian theory), but also in At first (after Newton but before the discovery of atoms) many waves were not understood to be a consequence of Newton’s laws. But with the discovery of atoms it became possible to analyze such waves in terms of Newtonian mechanics. The notable exceptions were electromagnetic waves. One must use Maxwell’s equations to understand them. Indeed Maxwell was the first to realize that light is an electromagnetic wave! His announcement sent shock waves throughout the scientific community.
Concluding Remarks. If ever there were two inspired moments in human history, it was when Newton wrote his three laws of mechanics, and when Maxwell and Lorentz worked out the electromagnetic force law. For all of these laws work in every inertial frame of reference and are relativistically correct. Equally as marvelous, Newton’s laws
and the Maxwell-Lorentz force law are connected in some very fundamental
ways. Part of the force exerted on a naked charge … usually the dominant
part … equates to d(m electric force when the fields of an accelerated charge are solved
for.Einstein’s Special Theory of Relativity wrought profound changes to Newtonian mechanics. Many of the cherished, "axiomatic" assumptions of the pre-relativity era fell. Mass, force and acceleration were no longer matters of agreement among inertial frames. Indeed inertial observers even disagree about the lengths of solid objects and the rates at which clocks run. It is a fascinating subject, fully worthy of its own book. Despite such revolutionary (and
sometimes counter-intuitive) changes, Newton’s three laws come shining
through (as do Maxwell’s equations and Lorentz’s force law). )/dt
is relativistically correct in every inertial frame of reference. But,
owing to the dependence of mass on particle speed (which of course varies
from inertial frame to inertial frame), v=mF
is not valid in relativity.aHow did Newton know to write his second law in the relativistically correct form? Was it simply fortuitous? Or did he have something going for him that we can only speculate about? And did he understand that his third law contained the seeds of the great conservation laws? With regard to his Universal Law of Gravitation, no mention of time is made. But today it is believed that gravitational interactions, like electromagnetic interactions, pass between material objects at a finite speed (namely the speed of light). Does an "uncharged" body generate some sort of gravitational waves when forced to oscillate, as a charged body does? General Relativity theory suggests that it does. The search is on to detect such waves! It is an intriguing fact that Maxwell’s field equations (and the Lorentz force law), coupled with Newton’s second law, can be used to demonstrate that a moving "atom" is actually length-contracted, and that temporal processes occur more slowly in moving systems (a phenomenon referred to as "time dilation"). Thus the seeds of some of the most counter-intuitive aspects of Special Relativity lay waiting to be derived by applying Newton’s laws and the theory of Maxwell/Lorentz. As these words are written, it has been discovered that even uncharged elementary particles (e.g. neutrons) are composed of charged particles (quarks). The possibility that "neutral matter" may, in the final analysis, be only a figment of our imaginations raises some intriguing questions. Might we find that the "inertial," equal and oppositely directed reaction forces of Newton’s third law, in the case of "uncharged" atoms, neutrons, etc., are ultimately electric forces when we "peel back the veil," and peer within? All in all, mechanics in one form or another continues to be one of the most interesting branches of physics. Observing things moving around in our personal spaces, and wanting to predict those motions, seems to be rooted in our very genes. Many ingenious recipes for making such predictions have been suggested through the years. But none have enjoyed greater staying power than Isaac Newton’s incredible laws! |