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Unformatted text preview: CHAPTER NEWTON’S LAWS Then from these forces, by other propositions which are also
mathematical,  deduce the motions of the planets, the comets, the moon, and
the sea. I wish we could derive the rest of the phenomena of Nature by the
same kind of reasoning from mechanical principles, for I am induced by many
reasons to suspect that they may all depend upon certain forces by which the
particles of bodies, by some cause hitherto unknown, are either mutually
impelled towards one another, and cohere in regular figures, or are repelled
and recede from one another. These forces being unknown, philosophers have
hitherto attempted the search of Nature in vain, but i hope the principles here
laid down will afford some light either to this or some truer method of philosophy.
Isaac Newton, Principia, 1687 6.1 THE END OF THE CONFUSION In 1543, Copernicus published his book, and a tremor rocked the foundations of the
Aristotelian world. A century later the Aristotelian world lay in ruins, but nothing had
arisen to replace it. Galileo and Kepler had made mighty discoveries, but there was no
central principle that could organize the world. The unified harmony of the Aristotelian view had been replaced by buzzing confusion.
1 1 1 112 NEWTON’S LAWS Galileo was concerned not with the causes of motion, but instead with its description.
The branch of mechanics he reared is known as kinematics; it is a mathematically de
scriptive account of motion without concern for the causes. Central to Galileo’s arguments
was the law of inertia, which we discussed in Chapter 4. Armed with this principle,
Galileo could neutralize Aristotelian arguments against a moving earth, but the recon
struction of a new mechanics that he promised in his final book had scarcely begun. The law of inertia created confusion in the minds of seventeenthcentury scholars
because it overturned a centuries01d teaching that a body needs a force to keep it in
motion. Not fully grasping the ideas of the new mechanics, these scholars failed to define
accurately the numerous scientific terms they employed. The confusion was aggravated
even further when the term force was used in different ways. Originally force was
considered chieﬂy as the cause of movement. But some thinkers considered it to be a
result of motion. The term centrifugal force, for example, was invented for the tendency
of a revolving body to ﬂy away from the center because of its motion. Traditionally
gravity had been considered to be a property of a body, its heaviness, which causes it
to fall. Some physicists now considered it to be the result of the motion of a swirling
ﬂuid of matter. In 1642, Galileo — nearly 80 years old, blind, and imprisoned — died. In the same
year, Isaac Newton was born. Newton’s work was the culmination of the Copernican
Revolution. It was his task to create order in the chaos of terms, notions, and miscon
ceptions that permeated seventeenthcentury physics. Newton was not a man of half—hearted pursuits. When he thought on something, he
thought on it continually, to the neglect of food, sleep, and human society. The period
from the autumn of 1684 to the spring of 1686 is a virtual blank in his life except for
his Principia. Once he adopted the principle of inertia and developed the central concept
of force, his dynamics quickly fell into place. He had seized on the essence of his second
law of motion 20 years before and had never altered it as he wrestled with the first law,
which was a statement of the law of inertia. He realized that an impressed force alters a
body’s motion and that the corresponding change in motion is proportional to it. By this
understanding, he capped and completed Galileo’s kinematics with a dynamics — a theory
of the causes of motion. Few periods of history have been so intense or held greater
consequences for science than the six months in the autumn and winter of 1684—5, when
Isaac Newton created the modern science of dynamics. 6.2 NEWTON’S LAWS OF MOTION The Principia was published in 1687, when Newton was 44, and details all his work on
the motion of bodies. The style of the Principia is reﬂective of its author: cold and rigid;
its pages are laden with diagrams and geometric proofs. Although Newton undeniably
arrived at his results by using his newly developed calculus, in the Principia he presented
geometric proofs — the language of physics in the seventeenth century. At Newton’s own
Cambridge University, a stately institution not given to undue haste, the Principia was
used as a textbook right into the twentieth century. To remove the confusion in terminology, Newton began by saying carefully what
he meant by terms such as inertia, mass, and force. He used these terms in his axioms
of motion, which he presented as statements that need no proof. These axioms are the
basis from which the nature of all types of motion can be deduced. 6.2 NEWTON'S LAWS OF MOTION 113 Newton inherited from Galileo and Descartes the essential idea that motion along a
straight line with a constant speed was the natural state of any body, needing no further
explanation. This is Newton’s first law, the law of inertia. Stated in his own words, First Law: Every body continues in its state of rest, or of uniform motion in a
straight line, unless it is compelled to change that state by forces
impressed upon it. Example 1
What would be the path of the planets if there were no force acting on them? The first law tells us that in the absence of any force, a body will continue to move
in a straight line. Therefore, if there were no force on the planets, they would travel in
straight lines, not in nearly circular orbits about the sun. Newton, like Galileo before him, realized that an object’s inertia was somehow
connected to its mass. He defined mass as the quantity of matter that arises conjointly
from an object’s density and size. The greater the mass of an object, the more difficult
it is to prevent it from continuing in motion with a constant velocity. This idea led to
his second law. In the Principia it is modestly stated as Second Law: The change of motion is proportional to the force impressed;
and is made in the direction of the straight line in which the
force is impressed. What Newton meant by “motion” involved not only a body’s velocity, but also its
mass. It is the quantity we call momentum, the product of mass m and velocity v. Stated
as an equation, the second law is d F — dt(mv) . (6.1)
The d/dt is a mathematical symbol meaning “the instantaneous rate of change of .” When
we understand that, the equation expresses Newton’s second law almost as we would in
an English sentence in which F is the subject and “ = ” is the verb. The mathematical
sentence says, “Force is equal to the rate of change of momentum of an object.” If the second law is applied to a body for which the mass m is a constant, then by
a rule of differentiation this means that d(mv)/dt = m dv/dt. But we know that the
instantaneous rate of change of velocity is acceleration, a = dv/dt. So for an object (or
collection of objects) whose mass doesn’t change, Newton’s second law tells us that acceleration is caused by forces. It is usually written Second Law: F = ma. (62) 114 NEWTON’S LAWS This form was first presented by the Swiss mathematician Leonhard Euler 65 years after
the publication of the Principia. It is probably the most useful equation in all of physics. To understand Newton’s second law, we need to understand the concept of force.
In everyday language, force is associated with a push or a pull. When you push on
something, you can feel yourself exerting a force. Once armed with that sensation, you
look around and find countless examples of things exerting forces on other things. Pushes,
pulls, gravity, tension in a string, and friction are all examples of forces that enter Newton’s
second law. But these forces must originate outside the object whose motion we’re trying
to describe. In other words, only external forces acting on an object can change its motion. The force F need not be just one force acting on the body. It is the vector sum of
all external forces acting on the object. Even though vectors hadn’t been invented yet,
Newton knew that forces have both a magnitude and a direction. Whenever we write
F = ma, F symbolizes the vector sum of external forces acting on a body. The acceleration
of an object is the result of the total force acting on it. Some physicists use Fnet to remind
them that it is the net or total force that enters the equation. Others write 2 F to symbolize
the vector sum. The symbol 2 is the Greek letter sigma and means sum. The second law, being a vector equation, is shorthand for three equations involving
Cartesian components: 2 F, = max, (6.3a)
2 Fy = may, (6.3b)
2 F, = maz. (6.30) It is this form that is more useful for solving problems. Once the respective components
of the external forces acting on a body are known, the components of the acceleration
are determined by the equations, and the motion of the body can be deduced. Example 2
A Rolls Royce Silver Shadow and a Volkswagen Rabbit are traveling at the same speed
along a level road. How do the forces required to stop them compare? A Rolls Royce is not only more expensive, but also has a greater mass than a
Volkswagen. By Newton’s second law, if the acceleration of each car is the same, as
they are here because they are to be brought to rest from the same speed, then the more
massive car requires a greater force. The ratio of the forces is equal to the ratio of the
masses, because from the relations FRR = mRRa and FVW = mvwa, we get F RR mRR Fvw mvw Newton needed one additional law to express what happens when several bodies
interact with each other. His third law is 6.2 NEWTON'S LAWS OF MOTION 115 Third Law: To every action there is always opposed an equal reaction: or, the
mutual actions of two bodies upon each other are always equal,
and directed to contrary parts. When you push on anything — a door, a pencil — it pushes back on you with a force
equal in magnitude but in the opposite direction. In other words, you can’t touch without
being touched. That’s the essence of the third law — a law of interactions. As illustrated
in Figure 6.1, if Body 1 exerts a force F12 on Body 2, then Body 2 exerts a force F2,
on Body 1 such that F12 = —F21. Body 2 Figure 6.1 An illustration of Newton’s third law. Sometimes it is difficult to isolate the action—reaction pairs of forces in Newton’s
third law. As a guide, remember that they always act on diﬁ’erent bodies, never on the
same body. If you know one force — for example, you pull on a rope — you can find the
reaction force by turning around the sentence: the rope pulls on you. As with the second
law, the third law is best understood through applications. Example 3
Identify the action—reaction pair of forces for each of the following cases: (a) a child pulling a dog’s leash,
(b) a bird ﬂapping its wings,
(c) raindrops hitting a roof. Let’s form two categories: Action Force and Reaction Force. Action Force Reaction Force
(a) child pulls on leash (a) leash pulls on child
(b) wings push down on air (b) air pushes up on wings
(c) raindrops push down on roof (c) roof pushes up on raindrops If we wanted to understand the motion of the Child, the bird, or the raindrops, we would 116 NEWTON'S LAWS use the forces exerted on these objects as listed in the second column. Those are the
external forces that enter the second law. Questions 1. 10. Suppose you have two identical cans, one filled with lead and the other empty, in
an orbiting spacecraft where everything is weightless. How can you tell which can
is empty without looking inside? What physical principles are behind the reasoning for making wrecking cranes
with massive weights at the end of a cable? Without seatbelts, you would hit the windshield during a quick stop. Why? Why
would you be in danger of whiplash if your car had no headrests and you suffered
a collision from behind? What kind of motion does a constant force produce? . A train consisting of an engine and three boxcars moves down the track with a constant acceleration. Between what two cars is the tension in the coupling the
greatest? the least? Why? . Often when parents spank a child they say, “This hurts me as much as it does you.” Is there any physical basis for this statement? While you are driving along the freeway, a bug splatters on your windshield.
Which experiences the greater force, the bug or the windshield? . Discuss whether the following pairs of forces are action~reaction forces: (a) An athlete standing on a scale pushes down on it; the scale pushes up on the
athlete. (b) The earth attracts a stone; the stone attracts the earth. (c) The tires of a car push on the road; the earth pulls down on the tires. (d) A chair pushes down on the floor; gravity pulls down on the chair. A farmer urges an Aristotelian horse to pull his wagon, but the horse refuses to
try. In his defense, the horse cites Newton’s third law and claims, “If I pull on
the wagon, the wagon pulls equally back on me. I can never exert a greater force
on the wagon than it exerts on me, so I could never start it moving.” What advice
would you give the farmer to counter this argument? A fan is mounted on a cart as shown below. If the fan is turned on, does the cart
move? If so. in which direction? Suppose a sail were added to the cart. What would be the motion of the cart
if the fan were now turned on? 6.3 UNITS OF MASS, MOMENTUM, AND FORCE 117 6.3 UNITS OF MASS, MOMENTUM, AND FORCE Mass, length, and time form the basic physical quantities used in mechanics. The unit
of mass in the metric system (SI) is the kilogram, abbreviated kg. The standard kilogram
is a platinum—iridium cylinder kept in a vault at the International Bureau of Weights and
Measures in Sévres, France. It is the only SI unit still defined by such an artifact.
Secondary standards are housed all over the world. With an equalarm balance, these
standards can determine the mass of objects to a precision of two parts in 100,000. The
kilogram is also equal to 1000 grams (1000 g); the gram is the unit of mass in the cgs
(centimeter, gram, second) system. Other SI prefixes can be used with the gram, such
as milligram (1 mg = 10‘3 g) and microgram (1 pg = 10‘6 g). The unit of mass in
the British engineering system is the slug; one slug is equal to 14.58 kg. Momentum, being a product of mass and velocity, is a derived physical quantity. In
the metric system, the basic unit is kg m/s. It has been suggested that this unit be called
a descartes, after the French mathematicianphilosopher, or a clout, which is perhaps
more descriptive. In the British system, momentum comes in units of slug ft/s. Through Newton’s second law, force should have the same units as mass times
acceleration. Therefore, the SI unit of force is the kg m/sz, called a newton and abbreviated
N. One newton is the force required to accelerate a lkg mass at 1 m/szz 1 N = 1 kg 92 .
s The unit of force in the cgs system is called a dyne and is the force that will accelerate
a lg mass at an acceleration of l cm/sz. Using 1 kg = 103 g and 1 INS2 = 102 cm/sz,
you can show that l N = 105 dyn. In the SI and cgs systems, mass, length, and time are the fundamental quantities.
Force is a derived unit. In the British system, however, the standard quantities are force,
length, and time. The unit of force in the British system is the pound, abbreviated 1b,
which is 1 slug ft/sz. By working out the conversion of units, you can show that 1 lb =
4.45 N. When an object is in free fall, gravity accelerates it downward with a constant
acceleration g. Newton’s second law tells us that the force must be F=mg, where the direction of the force is vertically downward, toward the center of the earth.
This is what we mean by the weight of an object; weight is the force of gravity acting
on an object (whether it is falling or not). Being a force, weight is a vector. If we call
the magnitude of the vector W, then W = mg near the surface of the earth. In countries that still use the British system, people are often confused between
kilograms and pounds. These units refer to different physical quantities. Yet labels list
the weight of an item in pounds along with its mass in kilograms and do not specify that
one is weight and the other is mass. Unlike the mass of a body, which is an intrinsic
property of a body, the weight of a body depends on its location. If you know the mass
of an object, you can find its weight if you also know the acceleration of gravity at that
location. Moving an object around on the surface of the earth doesn’t change its weight
very much, but moving it to the moon does change its weight considerably, without changing its mass. In Chapter 8 we’ll find out why weight varies with location, when
we discuss Newton’s universal law of gravity. 118 NEWTON’S LAWS Table 6.1 summarizes the units and conversions between the three systems of units. Table 6.1 Units of Mass and Force in the SI, cgs, and British Systems
Units of Mass
kilogram gram slug
1 kg 1 103 0.0685
1g 10‘3 1 6.85 x 10’5
1 slug 14.58 0.0146 1
Units of Force
newton dyne pound
1 N 1 105 0.2248
ldyn 10‘5 1 2.248 X 10‘6
11b 4.448 4.448 x 105 1
Questions 11. Determine whether the following combinations of units are units of mass, momen
tum, or force: (a) N S, (b) dyn sz/cm,
(c) slug ft/s, (d) 1b s. 12. How many newtons does a typical l60lb man weigh?
13. What is the mass of a 0.75lb can of beans?
14. What does it mean for an object to be weightless? 15. A rock weighs 60 N on the moon, where the acceleration due to gravity is one
sixth that on earth. What is the mass of this rock on the earth? 16. A jar of lightning bugs is tightly capped. Does it weigh more, less, or the same
when the bugs are ﬂying around compared to when they are at rest? 17. Suppose you hand an object to each of two people and ask them to guess its
weight. One holds it still and guesses; the other hoists it up and down before
guessing. One is estimating the mass, and the other the weight. Which is which?
Explain. 6.4 PROJECTILE MOTION: AN APPLICATION OF NEWTON'S SECOND LAW 119 6.4 PROJECTILE MOTION: AN APPLICATION OF NEWTON’S
SECOND LAW Galileo was the first to describe the motion of a projectile correctly. Using his laws of
inertia and of falling bodies, he showed that projectiles follow parabolic trajectories and
was able to deduce many other properties of their motion. Let’s reexamine how a projectile
moves through space by starting with Newton’s second law. If a cannonball has mass m, and you know the net force F acting on it, you can
predict the cannonball’s motion using F = ma. What force acts on a cannonball? Initially
there is the force from the explosion of gunpowder. That force acts on the cannonball
only momentarily and determines the initial speed of the cannonball. What we need to
know is the force acting on the cannonball after it has left the cannon, while it is in the
air. And that’s simple; the only force acting on it is gravity (neglecting air resistance). We already deduced what that force is for any body near the surface of the earth.
Since force equals mass times acceleration, the force on the cannonball must be mg, its
weight. The direction of this force is vertically downward. If we choose a coordinate
system like that in Fig. 6.2, then gravity is directed in the negative 2 direction. Using
k as the unit vector in the positive 2 direction, we have F = —mgl:1 (6.4) Figure 6.2 Coordinate system used to describe a projectile’s motion. Substituting this force into the second law, we get —mgk = ma. The mass is the same on both sides of the equation and cancels out, so we are left with
—g12 = a. This tells us that the ﬂight of a projectile doesn’t depend on its mass. Galileo knew that!
He told us about it in Chapter 2. Our equation looks quite simple, but remember it is a vector equation — it stands
for three equations, not ju...
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 Fall '07
 Goodstein

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