This preview shows pages 1–11. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Efrain Lopez shows that
when the forces balance to
zero, no acceleration
occurs. liliﬁ“ Newton’s Second Law Newton’s Second
Law of Motion n Chapter 2, we discussed the concept of mechanical equilibrium, 2F: 0, where
Forces are balanced. In this chapter, we consider what happens when Forces aren’t
balanced—when net Forces do not equal zero. The net Force on a kicked soccer ball, For
example, is greater than zero, and the motion of the ball changes abruptly. Its path
through the air is not a straight line but curves downward due to gravityiagain, a
change in motion. Most of the motion we see undergoes change. This chapter covers
changes in motion—accelerated motion.
In Chapter 3, we learned that acceleration describes how quickly motion changes.
Speciﬁcally, it is the change in velocity during a certain time interval. Recall the deﬁni
tion of acceleration: change in velocity acceleration I . .
time Interval We will now Focus on the cause or” acceleration: force. Force Causes FIGURE 4.1
Kick the ball and it
accelerates. 58 Acceleration Consider a hockey puck at rest on ice. Apply a force, and it starts to move—
it accelerates. When the hockey stick is no longer pushing it, the puck moves
at constant velocity. Apply another force by striking the puck again, and again
the motion changes. Applied force produces acceleration. Most often, the applied force is not the only force acting on an object.
Other forces may act as well. Recall, from Chapter 2, that the combination of
forces acting on an object is the net force. Acceleration depends on the net force.
To increase the acceleration of an object, you must increase the net force acts
ing on it. If you double the net force on an object, its acceleration doubles; if
you triple the net force, its acceleration triples; and so on. This makes good
sense. We say an object’s acceleration is directly proportional to the net force
acting on it. We write acceleration ~ net force Chapter 4 Newton ’5 Second Law of Motion 59 The symbol ~ stands for “is directly proportional to.” That means any change
(:3 in one is the same amount of change in the other. Force Causes Acceleration CHECK YOURSELF 1. You push on a crate that sits on a smooth floor, and it accelerates. If you apply
four times the net force, how much greater will be the acceleration? 2. If you push with the same increased force on the same crate, but it slides on a
very rough ﬂoor, how will the acceleration compare with pushing the crate on a
smooth Floor? (Think before you read the answer below!) I '
Friction
When surfaces slide or tend to slide over one another, a force of friction aets.
When you apply a force to an object, friction usually reduces the net force
and the resulting acceleration. Friction is caused by the irregularities in the
surfaces in mutual contacr, and it depends on the kinds of material and how
much they are pressed together. Even surfaces that appear to be very smooth
have microscopic irregularities that obstruct motion. Atoms cling together at
many points of contact. When one object slides against another, it must either
rise over the irregular bumps or else scrape atoms off. Either way requires
force.
Force 0f hand The direction of the friction force is always in a direction opposing
OCCEleres motion. An object sliding down an incline experiences friction directed up
lh_e_ bl.le the incline; an object that slides to the right experiences friction toward the
m left. Thus, if an object is to move at constant velocity, a force equal to the
opposing force of friction must be applied so that the two forces exactly cancel
Twice as much force each other. The zero net force then results in zero acceleration and constant
produces twice as velocity. much acceleration FIGURE 4.3 Friction results from the mutual contact of
irregularities in the surfaces of sliding objects.
Even surfaces that appear to be smooth have
irregular surfaces when viewed at the
microscopic level. Twice the force on
twrce the mass gives
the same acceleration FIGURE 4.2
Acceleration is directly
proportional to force. _—________.__——_—————— CHECK YOUR ANSWERS glecs 1. It will have four times as much acceleration. Friction 2. it will have less acceleration because friction will reduce the net force. W 60 Part One Mechanics FIGURE 4.4 The direction ofthe force of
friction always opposes the
direction ofmotion. (Left)
Push the crate to the right,
and friction acts toward the
left. (Right) The sack falls
downward, and air Friction
(air resistance) acts upward.
(What is the acceleration of
the sack when air resistance
equals the sack’s weight?) Air Resistance No friction exists on a crate that sits at rest on a level ﬂoor. But, if you
push the crate horizontally, you’ll disturb the contact surfaces and friction is
produced. How much? If the crate is still at test, then the friction that opposes
motion is just enough to cancel your push. If you push horizontally with, say,
70 newtons, friction builds up to become 70 newtons. If you push harder—
say, 100 newtonsiand the crate is on the verge of sliding, the friction between
the crate and floor opposes your push with 100 newtons. If 100 newtons is the
most the surfaces can muster, then, when you push a bit harder, the clinging
gives way and the crate slides.‘ Interestingly, the friction of sliding is somewhat less than the friction that
builds up before sliding takes place. Physicists and engineers distinguish
between static friction and sliding friction. For given surfaces, static friction
is somewhat greater than sliding friction. If you push on a crate, it takes
more force to get it going than it takes to keep it sliding. Before the time of
antilock brake systems, slamming on the brakes of a car was quite prob—
lematic. When tires lock, they slide, providing less friction than if they are
made to roll to a stop. A rolling tire does not slide along the road surface,
and friction is static friction, with more grab than sliding friction. But once
the tires start to slide, the frictional force is reduced—not a good thing. An
antilock brake system keeps the tires below the threshold of breaking loose
into a slide. It’s also interesting that the force of fricrion does not depend on speed. A
car skidding at low speed has approximately the same friction as the same car
skidding at high speed. If the friction force of a crate that slides against a floor
is 90 newtons at low speed, to a close approximation it is 90 newtons at a
greater speed. It may be more when the crate is at rest and on the verge of
sliding, but, once the crate is sliding, the friction force remains approximately
the same. More interesting still, friction does not depend on the area of contact. If
you slide the crate on its smallest surface, all you do is concentrate the same
weight on a smaller area with the result that the friction is the same. So those
extra wide tires you see on some cars provide no more friction than narrower
tires. The wider tire simply spreads the weight of the car over more surface LEven though 1{ may not seem so yet. most of the COHCEFN in physics are not really complicated. But friction
is different. Unlike most concepts in physics. it is a very complicated phenomenon. ‘l‘he ﬁndings are empirical
[gained from a wide range of experiments) and the predictions are approxnnatc (also based on experimenti. FIGURE 4.5 Friction between the tire and
the ground is nearly the same whether the tire is wide
or narrow. The purpose of
the greater contact area is to
reduce heating and wear. Chapter 4 Newton’s Second Law ofMotion 61 area to reduce heating and wear. Similarly, the friction between a truck and the
ground is the same whether the truck has four tires or eighteen! More tires spread
the load over more ground area and reduces the pressure per tire. Interestingly,
stopping distance when brakes are applied is not affected by the number of tires.
But the wear that tires experience very much depends on the number of tires.
Friction is not restricted to solids sliding over one another. Friction occurs
also in liquids and gases, both of which are called ﬂuids (because they flow).
Fluid friction occurs as an object pushes aside the fluid it is moving through.
Have you ever attempted a IOU—m dash through waist—deep water? The friction
of fluids is appreciable, even at low speeds. A very common form of fluid
fricrion for something moving through air is air resistance, also called air drag.
You usually aren‘t aware of air resistance when walking or jogging, but you
notice it at the higher speeds when riding a bicycle or when skiing downhill. Air
resistance increases with increasing speed. The falling sack shown in Figure 4.4
wiil reach a conStant velocity when air resistance balances the sack’s weight. CHECK YOURSELF What net force does a sliding crate experience when you exert a force of 110 N and
friction between the crate and the floor is 100 N? Mass and Weight FIGURE 4.6 An anvil in outer space—
between the Earth and the
Moon, for example—may
be weightless, but it is not
massless. The acceleration imparted to an object depends not only on applied forces and
friction forces, but on the inertia of the object. How much inertia an object
possesses depends on the amount of matter in the object—the more matter, the
more inertia. 1n speaking of how much matter something has, we use the term
mass. The greater the mass of an object, the greater its inertia. Mass is a mea
sure of the inertia of a material object. Mass corresponds to our intuitive notion of weight. We casually say that
something has a lot of matter if it weighs a lot. But there is a difference between
mass and weight. We can deﬁne each as follows: Mass: The quantity of matter in an object. It is also the measure of the
inertia or sluggisbness that an object exhibits in response to any effort
made to start it, stop it, or change its state of motion in any way. Weight: The force upon an object due to gravity. In the absence of acceleration, mass and weight are directly proportional
to each other.2 If the mass of an object is doubled, its weight is also doubled; CHECK YOUR ANSWER 10 N in the direction ofyour push (110 N  100 N). ___________.__—_.—————— 2Weight and mass are directly proportional; weight = mg. where g is the constant of proportionality and has the
value 9.8 leg, Fquii'alently, g is the acceleration due to gravity. 9.8 I'ni's2 [the units kag are equivalent to Ii‘lfszl.
In Chapter 9. we'!l extend the deﬁnition of weight as the force that an object exerts on a supporting surface. 62 Part One Mechanics FIGURE 4.7 The astronaut in space ﬁnds
that it isjust as difﬁcult to
shake the “weightless” anvil
as it would be on Earth. if
the anvil were more massive
than the astronaut, which
would shake morerthe
anvil or the astronaut? FIGURE 4.8 Why will a slow, continuous
increase in downward Force
break the string above the
massive ball, while a sudden
increase would break the
lower string? if the mass is halved, the weight is halved. Because of this, mass and weight
are often interchanged. Also, mass and weight are sometimes confused
because it is customary to measure the quantity of matter in things (mass)
by their gravitational attraction to the Earth (weight). But mass is more fun—
damental than weight; it is a fundamental quantity that completely escapes
the notice of most people. There are times when weight corresponds to our unconscious notion of inertia.
For example, if you are trying to determine which of two small objects is the heav
ier one, you might shake them back and forth in your hands or move them in
some way instead of lifting them. In doing so, you are judging which of the two
is more difﬁcult to get moving, feeling which of the two is most resistant to a
change in motion. You are really comparing the inertias of the objects. In the United States, the quantity of matter in an object is commonly
described by the gravitational pull between it and the Earth, or its weight, usu—
ally expressed in pounds. In most of the world, however, the measure of matter
is commonly expressed in a mass unit, the kilogram. At the surface of the Earth,
a brick with a mass of 1 kilogram weighs 2.2 pounds. In metric units, the unit
of force is the newton, which is equal to a little less than a quarterrpound (like
the weight of a quarterpound hamburger after it is cooked). A 1kilogram brick
weighs about 10 newtons (more precisely, 9.8 NI).3 Away from the Earth’s sur
face, where the influence of gravity is less, a 1kilogram brick weighs less. It would
also weigh less on the surface of planets with less gravity than Earth. 0n the
Moon’s surface, for example, where the gravitational force on things is only one
sixth as strong as on Earth, a 1kilogram brick weighs about 1.6 newtons (or
0.36 pounds). On planets with stronger gravity, it would weigh more, but the
mass of the brick is the same everywhere. The brick offers the same resistance to
speeding up or slowing down regardless of whether it’s on Earth, on the Moon,
or on any other body attracting it. In a drifting spaceship, where a scale with a
brick on it reads zero, the brick still has mass. Even though it doesn’t press down
on the scale, the brick has the same resistance to a change in motion as it has
on Earth. just as much force would have to be exerted by an astronaut in the
spaceship to shake it back and forth as would be required to shake it back and
forth while on Earth. You’d .have to provide the same amount of push to accel
erate a huge truck to a given speed on a level surface on the Moon as on Earth.
The difﬁculty of liftng it against gravity (weight), however, is something else.
Mass and weight are different from each other (Figures 4.6 and 4.7). A nice demonstration that distinguishes mass and weight is the massive ball
suspended on the string, shown in Figure 4.8. The top string breaks when the
lower string is pulled with a gradual increase in force, but the bottom string
breaks when the lower string is jerked. Which of these cases illustrates the
weight of the ball, and which illustrates the mass of the ball? Note that only
the top string bears the weight of the ball. 50, when the lower string is grad
ually pulled, the tension supplied by the pull is transmitted to the top string.
The total tension in the top string is caused by the pull plus the weight of the
ball. The top string breaks when the breaking point is reached. But, when the ASo 2.2 lb equals 9.8 N, or 'i N is approximately equal to 0.22 Ibrabour the weight of an apple. In the metric
system, it is customary to specify quantities of matter in units of mass [in grams or kilograms) and rarely in
units of weight (in newtons). In the United States and countries that use the British system of units, however,
quantities of matter are customarily specified in units of weight (in pounds). (The British unit of mass, the
slug, is not well known.) See Appendix I for more about systems of measurement. When two things are
directly proportional
to each other, as one
increases, the other
increases also. How
ever, when two things
are inversely propor
tional to each other, as one increases, the
other decreases. Chapter 4 Newton’s Second Law of Motion 63 bottom string is jerked, the mass of the ball—its tendency to remain at rest—
is responsible for the bottom string breaking. It is also easy to confuse mass and volume. When we think of a massive
object, we often think of a big object. An object’s size (volume), however, is
not necessarily a good way to judge its mass. Which is easier to get moving:
a car battery or an empty cardboard box of the same size? So, we ﬁnd that
mass is neither weight nor volume. CHECK YOURSELF 1. Does a 2kg iron brick have twice as much inertia as a 'lkg iron brick? Twice as
much mass? Twice as much volume? Twice as much weight? 2. Would it be easier to lift a cement truck on the Earth’s surface or to lift it on
the Moon’s surface? 3. Ask a friend to drive a small nail into a piece of wood placed on top ofa pile of
books on your head. Why doesn’t this hurt you? Acceleration Push your friend on a skateboard and your friend accelerates. Now push equally
hard on an elephant on a skateboard and the acceleration is much less. You’ll see
that the amount of acceleration depends not only on the force but on the mass
being pushed. The same force applied to twice the mass produces half the acceler
ation; for three times the mass, one—third the acceleration. We say that, for a given
force, the acceleration produced is inversely proportional to the mass. That is, 1
mass Acceleration ~ CHECK YOUR ANSWERS 1. The answers to all parts areyes. A 2—kg iron brick has twice as many iron atoms
and therefore twice the amount of matter and mass. In the same location, it
also has twice the weight. And, since both bricks have the same density (the
same ratio of mass to volume), the 2kg brick also has twice the volume. 2. A cement truck would be easier to lift on the Moon because the gravitational
force is less on the Moon. When you lift an object, you are contending with
the force of gravity (its weight). Although its mass is the same on the Earth,
on the Moon, or anywhere, its weight is only 1/6 as much on the Moon, so
only U6 as much effort is required to lift it there. To move it horizontally,
however, you are not pushing against gravity. When mass is the only factor,
equal forces will produce equal accelerations, whether the object is on the
Earth or the Moon. 3. The relatively large mass of the books and block atop your head resists being
moved. The force that is successful in driving the nail will not be as successful
in accelerating the massive books and block, which don’t move very much when
the nail is struck. Can you see the similarity between this and the suspended
massive ball demonstration, where the supporting string doesn’t break when
the bottom string isjerkecl? —___—_———————— 64 Part One Mechanics FIGURE 4.1 0 An enormous Force is
required to accelerate this
threestory—high earth mover
when it carries a typical BSDton loads FIGURE 4.9 Interactive Figure * The greater the mass, the
greater the Force must be For a given acceleration, By inversely We mean that the two values change in opposite directions. As the denominator increases, the whole quantity decreases. For example, the quantity
1/100 is less than 1/10. Newton’s Second Law of Motion conceptsfacceleration, force, and mass. He proposed one of the most impor—
tant rules of nature, his second law of motion. Newton’s second law states sics Newton was the ﬁrst to discover the relationship among three basic physical
CB Newton’s Second Law The acceleration of an object is directly proportional to the net force
acting on the object, is in the direction of the net force, and is
inversely proportional to the mass of the object. In summarized form, this is , net force
Force of hand Acceleration ~ ———
OﬁCEIcrates mass
1' I .  I cs' ' 3‘
e brlci We use the Wiggly line m as a symbol meaning is proportional to. We say i that acceleration a is directly proportional to the overall net force F and
inversely proportional to the mass in. By this we mean that, if F increases,
The same force a increases by the same factor (it P doubles, a doubles); but if in increases, accelerates Z bricks it decreases by the same factor (if m doubles, a is cut in half).
V2 ‘35 mUCh By using consistent units, such as newtons (N) for force, kilograms (kg) for mass, and meters per second squared (m/szi for acceleration, the proportional
ity may be expressed as an exact equation: .' . net force
3 bricks, 1/3 as acceleration = 1 much acceleration mass In its bricfest form, where a is acceleration, PM is net force, and m is mass, it becomes
4 FUE[
a _
in
FIGURE 4.11
Acceleration is inversely An object is accelerated in the direction or the force acting on it. Applied in proportional to mass. the direction of the object’s motion, a force will increase the object’s speed. Chapter 4 Newton’s Second Law ofMation 55 Applied in the opposite direction, it will decrease the speed of the object. Applied
at right angles, it will deflect the object. Any other direction of application will
result in a combination of speed change and deﬂection. The acceleration ofari
object is always in the direction of the net force. CHECK YOURSELF 1. in the previous chapter, acceleration was deﬁned to be the time rate of change
in velocity; that is, a : (change in vly‘time. in this chapter, are we saying that
acceleration is instead the ratio of force to mass—that is, a : F/m? Which is it? 2. Ajumbo jet cruises at a constant velocity of l000 km/h when the thrusting force
ofits engines is a constant 100,000 N. What is the acceleration of the jet? What
is the force of air resistance on the jet? When Acceleration ls g—Free Fall i ll ‘5 = g» ..2_F :: m m 9
FIGURE 4.12
Interactive Figure
The ratio ofweight (F) to
mass (m) is the same for all
objects in the same locality;
hence, their accelerations
are the same in the absence
ofair resistance. Only a single Force acts on something in
free fall—the force of gravity. Although Galileo founded both the concept of inertia and the concept of accel
eration, and although he was the first to measure the acceleration of falling
objects, he could not explain why objects of various masses fall with equal
accelerations. Newton‘s second law provides the explanation. We know that a falling object accelerates toward Earth because of the
gravitational force of attraction between the object and Earth. When the force
of gravity is the only force—that is, when friction (such as air resistance) is
negligible—we say that the object is in a state of free fall. The greater the mass of an object, the greater is the gravitational force of
attraction between it and the Earth. The double brick in Figure 4.12, for exam
ple, has twice the gravitational attraction of the single brick. Why, then, as Aris—
totle supposed, doesn’t the double brick fall twice as fast? The answer is that
the acceleration of an object depends not only on the force—in this case, the Ii'llEL“ Free Fall Acceleration
Explained CHECK YOUR ANSWERS 1. Acceleration is deﬁned as the time rate of change of velocity and is produced by a
Force. How much force/mass (the cause) determines the rate change in v/time
(the effect). So, whereas we deﬁned acceleration in Chapter 3, in this chapter we
deﬁne the terms that produce acceleration. 2. The acceleration is zero because the velocity is constant. Since the acceleration is
zero, it follows From Newton’s second law that the net Force is zero, which
means that the force of air drag must just equal the thrusting Force oF100,000 N
and must act in the opposite direction. So the air drag on the jet is 100,000 N.
(Note that we don’t need to know the velocity of the jet to answer this question.
We need only to know that it is constant, our clue that acceleration—and there
Fore net force—is zero.) ,_,_—_—.—._.——————— 66 Part One Mechanics =‘lT é Ulﬂ D FIGURE 4.13 The ratio ofweight (F) to
mass (m) is the same For the
large rock and the small
Feather; similarly, the ratio of
circumference (C) to diame
ter (D) is the same For the
large and the small circle. weight_but also on the object’s resistance to motion, its inertia. Whereas a
force produces an acceleration, inertia is a resistance to acceleration. So twice
the force exerted on twice the inertia produces the same acceleration as half the
force exerted on half the inertia. Both accelerate equally. The acceleration due
to gravity is symbolized by g. We use the symbol g, rather a, to denote that
acceleration is due to gravity alone. The ratio of weight to mass for freely falling objects equals a constant!
g. This is similar to the constant ratio of circumference to diameter for cir
cles, which equals the constant 17. The ratio of weight to mass is the same
both for heavy objects and for light objects, just as the ratio of circumfer
ence to diameter is the same borh for large circles and for small circles
(Figure 4.13). We now understand that the acceleration of free fall is independent of an
object’s mass. A boulder 100 times more massive than a pebble falls with the
same acceleration as the pebble because, although the force on the boulder (its
weight) is 100 times greater than the force on the pebble, its resistance to a
change in motion (its mass) is 100 times that of the pebble. The greater force offsets the equally greater mass. CHECK YOURSELF In a vacuum, a coin and a Feather fall at the same rate, side by side. Would it be
correct to say that equal forces of gravity act on both the coin and the feather when in a vacuum? When Acceleration Is Less Than g—Nonfree Fall ,_: When Galileo tried to
explain why all objects
fall with equal accel
erations, wouldn’t he
have loved to know
the rule a : F/m? Iill§£°s Falling and Air Resistance Objects falling in a vacuum are one thing, but what of the practical cases of
objects falling in air? Although a feather and a coin will fall equally fast in a
vacuum, they fall quite differently in air. How do Newton’s laws apply to objects
falling in air? The answer is that Newton’s laws apply for all objects, whether
freely falling or falling in the presence of resistive forces. The accelerations, how—
ever, are quite different for the two cases. The important thing to keep in mind
is the idea of net force. In a vacuum or in cases in which air resistance can be
neglected. the net force is the weight because it is the only force. In the presence ______________————————— CHECK YOUR ANSWER No, no, no, a thousand times no! These objects accelerate equally not because the
Forces of gravity on them are equal, but because the ratios of their weights to their masses are equal. Although air resistance is net present in a vacuum, gravity is.
(You’d know this if you stuck your hand into a vacuum chamber and the truck shown
in Figure 4.10 rolled over it!) If you answered yes to this question, let this be a warn ing to be more careful when you think physics!
_____,_______—————— FIGURE 4.14 When weight mg is greater
than air resistance R, the
falling sack accelerates. At
higher speeds, R increases.
When R = mg, acceleration
reaches zero, and the sack
reaches its terminal velocity. Air Resistance
Air Resistance T Weight FIGURE 4.15
Interactive Figure k The heavier parachutist must
fall Faster than the lighter parachutist for air resistance
to cancel his greater weight. Chapter 4 Newton’s Second Law of Motion 67 of air resistance, however, the net force is less than the weight—it is the weight
minus air drag, the force arising from air resistance.4 The force of air drag experienced by a falling object depends on two things.
First, it depends on the frontal area of the falling object—that is, on the amount
of air the object must plow through as it falls. Second, it depends on the speed
of the falling object; the greater the speed, the greater the number of air mol
ecules an object encounters per second and the greater the force of molecular
impact. Air drag depends on the size and the speed of a falling object. In some cases, air drag greatly affects falling; in other cases, it doesn’t. Air drag
is important for a falling feather. Because a feather has so much area compared
with its small weight, it doesn’t have to fall very fast before the upwardacting air
resistance cancels the downwardacting weight. The net force on the feather is then
zero and acceleration terminates. When acceleration terminates, we say that the
object has reached its terminal speed. If we are concerned with direction, down for
falling objects, we say the object has reached its terminal velocity. The same idea
applies to all objects falling in air. Consider skydiving. As a falling skydiver gains
speed, air drag may ﬁnally build up until it equals the weight of the skydiver. If and
when this happens, the net force becomes zero and the skydiver no longer acceler—
ates; she has reached her terminal velocity. For a feather, terminal velocity is a few
centimeters per second, whereas, for a skydivet, it is about 200 kilometers per hour.
A skydiver may vary this speed by varying position. Head or feet ﬁrst is a way of
encountering less air and thus less air drag and attaining maximum terminal veloc
ity. A smaller terminal velocity is attained by spreading oneself out like a ﬂying
squirrel. Minimum terminal velocity is attained when the parachute is opened. Consider a man and woman parachuting together from the same altitude
(Figure 4.15). Suppose that the man is twice as heavy as the woman and that
their samensized parachutes are initially opened. Having parachutes of the same
size means that, at equal speeds, the air resistance is the same on both of them.
Who reaches the ground ﬁrst—the heavy man or the lighter woman? The answer
is that the person who falls fastest gets to the ground ﬁrst—that is, the person
with the greatest terminal speed. At ﬁrst we might think that, because the para
chutes are the same, the terminal speeds for each would be the same and, there
fore, that both would reach the ground at the same time. This doesn’t happen,
however, because air drag depends on speed. Greater speed means greater force
of air impact. The woman will reach her terminal speed when the air drag against
her parachute equals her weight. When this occurs, the air drag against the para
chute of the man will not yet equal his weight. He must fall faster than she does
for the air drag to match his greater weight” Terminal velocity is greater for the
heavier person, with the reSult that the heavier person reaches the ground ﬁrst. 4In mathematical flotation,
Fm mg  R u' i
7” H1 where mg 15 the weight and R is the air resistance. Note that when R = mg, a = 0, then, with no acceleration,
the ohiect falls at constant velocity. \Xlith elementary algebra. we can go another step and get we r R R g=—= 7 .8
m m m W’s: see that the acceleration a will always be less than 3 if air resistance R impedes falling. Only when R = 0
does .1 : 3‘ 3Terminal speed for the twiceasheavy man will be about 41 percent greater than the woman‘s terminal speed,
. . . . . . . 7 . . 7
because the retarding force of air resistance is proportional to speed squared. (21,",“7emu“; = 1.41“ = 2.) 68 Part One Mechanics JUL c When the Force 0F
gravity and air drag act on a Falling object,
it is not in Free Fall. FIGURE 4.16 A stroboscopic study oFa golF
ball (leFt) and a StyroFoam
ball (right) Falling in air. The
air resistance is negligible For
the heavier golFball, and its
acceleration is nearly equal to
g. Air resistance is not negligie
ble For the lighter StyroFoam
ball, which reaches its termi
nal velocity sooner. CHECK YOURSELF A skydiver jumps From a highuFlying helicopter. As she Falls Faster and Faster through
the air, does her acceleration increase, decrease, or remain the same? one a regular hollow ball and the other ﬁlled
the ironﬁlled ball is consider—
ably heavier than the regular ball. If you hold them above your head and drop
them simultaneously, you’ll see that they strike the ground at about the same time.
But if you drop them From a greater heightﬁsay, from the top of a building— Consider a pair of tennis balls,
with iron pellets. Although they are the same size, you’ll note the heavier ball strikes the ground First. Why? In the ﬁrst case, the balls
do net gain much speed in their short fall. The air drag they encounter is small
compared with their weights, even {or the regular ball. The tiny difference in their
arrival time is not noticed. But, when they are dropped from a greater height, the
greater speeds of fall are met with greater air resistance. At any given speed, each
ball encounters the same air resistance because each has the same size. This same
air resistance may be a lot compared with the weight of the lighter ball, but only
a little compared with the weight of the heavier ball (like the parachutists in Figure
4.15). For example, 1 N of air drag acting on a 2N object will reduce its accel
eration by half, but 1 N of air drag on a ZOON object will only slightly diminish
its acceleration. So, even with equal air resistances, the accelerations of each are
different. There is a moral to be learned here. Whenever you consider the acceler—
ation of something, use the equation of Newton’s second law to guide your think
ing: The acceleration is equal to the ratio of net force to the mass. For the falling
tennis balls, the net force on the hollow ball is appreciably reduced as air drag
builds up, while the net Force on the iron—filled ball is comparably only slightly
reduced. Acceleration decreases as net force decreases, which, in turn, decreases as
air drag increases. If and when the air drag builds up to equal the weight of the
falling object, then the net force becomes zero and acceleration terminates. _________________—————— CHECK YOUR ANSWER Acceleration decreases because the net Force on her decreases. Net Force is equal to
her weight minus her air resistance, and, because air resistance increases with increas ing speed, net Force and hence acceleration decrease. By Newton’s second law, Fner _ M? R a :
m m
where mg is her weight and R is the air resistance she encounters. As R increases, a 0; then, with no decreases. Note that, iF she Falls Fast enough so that R = mg, a —
acceleration, she Falls at a constant speed. ...
View
Full
Document
 Spring '08
 MODI

Click to edit the document details