Download Chapter 1 Falling Chapter Check In You have two balls of the same

Chapter 1 Falling
Chapter Check In
You have two balls of the same size and smoothness. You’re standing on the
surface of the earth, but you can ignore air resistance. The heavier ball is made of
lead, the lighter one of wood. You hold one ball in each hand at the same height
above the ground. You release them at the same time. What will happen?
A. The heavier one will hit the ground first.
B. They both will hit the ground at the same time.
C. The lighter one will hit the ground first.
If you answered B, congratulations! You have in mind the most fundamental
concept in this book: Falling. If you can understand the physics of falling, you can
understand gravity, spacetime, black holes and the universe. Understanding is the
central learning goal of this book. If you learn the main concepts that we present,
you will see the big picture of the cosmos.
Let’s look at the nature of falling in more detail, so that you can see how will it is a
core concept. Falling fills the universe from the earth, out to the moon, to the solar
system, to binary stars, to galaxies, and even to the cosmos itself.
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Section 1.1
Consider these examples. An egg falling off the edge of the counter. Your child
dropping french fries. An apple tumbling from a tree. What do these have in
common? They’re all examples of objects falling near the surface of the earth.
Take a simple example. You have a tennis ball and you hold it about shoulder
height. Drop the ball, and it falls to the ground. Now take a baseball. If you heft the
baseball and tennis ball, you will sense that they baseball is heavier than the tennis
ball. Drop them both at the same time from shoulder height. What do you predict
happens? Right, they both hit the ground at the same time.
You have a sheet of paper and a tennis ball. You drop them at the same time from
shoulder height. Which hits the ground first?
A. The sheet of paper.
B. They both it the ground at the same time.
C. The tennis ball.
Right, the answer is C, because friction with the air slows down the fall of the
paper. Now try this question.
You have a sheet of paper and tennis ball. You squeeze the sheet of paper into a
ball. You drop the ball of paper and the tennis ball at the same time from shoulder
height. Which hits the ground first?
A. The sheet of paper.
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B. They both hit the ground at the same time.
C. The tennis ball.
Right, the answer is B, because we have reduced the effect of air friction.
What is the cause of the falling? Gravity! But what is gravity? You are probably
assuming that gravity is a force. Why do you believe forces are acting on the balls?
Well, because they start with no speed toward the ground, and then speed up until
they hit. You have just invoked Newton’s first law of motion! To deepen your
understanding of falling, we need to review, investigate, and apply all of Newton’s
laws of motion.
Section 1.2
You have already applied Newton’s first law to examples of falling objects.
Initially, you saw them at rest. Then, you observed them moving, falling down. So,
you now know the first part of Newton’s first law: Bodies will remain at rest
unless acted upon by an outside force.
Now imagine a ball rolling along a straight line at a constant speed. If the speed of
the ball changes, you’d probably conclude that a force is acting on the ball. If the
direction of the ball motion changed, you’d also conclude that a force act on the
ball. Then you’d have the second part of Newton’s first law: A mass will travel at
constant speed along a straight line unless acted upon by a force.
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In both cases, it is the net force that counts. Forces can act on objects in different
directions. In a tug-of-war, if both sides exert equal forces, the net result is no
motion because the net force is zero.
Let’s put this all together. Newton’s first law really tells you when you can declare
that a force is acting or has acted on an object. To get an object to move requires a
force overcomes an object’s resistance to motion, that is its inertia. So we can call
Newton’s first law the inertial law:
ÿ Newton’s First Law (Inertial Law):
A body at rest or in motion at a constant speed along a straight line remains in
that state of rest or motion unless acted on by an outside force.
The key here: change in motion. When you drop an object, it accelerates. How so?
Its speed increases, just like a car does when you step on its accelerator! Now
imagine you are driving a car around a circle at a constant speed. The direction of
your speed changes at every point on the circle. Then the car is also accelerating
because it is no longer moving along a straight line (Figure 1.1).
Figure 1.1 Acceleration of a car. (a) Along a straight line, (b) On a circular path.
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So motion involves two distinct characteristics. One, a direction. Two, a change of
position over time. Both together define an object’s velocity. Then we can recast
our thinking about motion as the change in an object’s velocity: acceleration. Note
that acceleration can mean a change in direction or a change in speed or both.
Newton’s second law pulls these ideas together with forces and masses. You have
probably heard of Newton’s second law stated as:
• Force equals mass times acceleration.
I think it is conceptually clearer to state Newton’s second law as
• Acceleration equals force divided by mass.
This form emphasizes to look at accelerations as the clue to applied forces. Note
that Newton’s second law states that the acceleration of the body from an applied
net force is in the same direction as the force and proportional to it, but is inversely
proportional to the body’s mass. So I’ll call the Newton’s second law the
acceleration law:
ÿ Newton’s Second Law (Acceleration law):
Acceleration = force/mass
a = F/m
So for the same mass, as the force increases its acceleration it also increases. How?
In direct proportion to the force. So if you double the force, you get twice the
acceleration. If you triple the force, you get three times the acceleration (Figure
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1.2). What about ten times the force on the same mass? Right, ten times the
acceleration! As a proportionality:
aµF
10
9
8
Acceleration
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6
5
4
3
2
1
0
0
2
4
6
8
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Net Force
Figure 1.2 Plot for increasing the net force applied to a 1 kg mass and the resulting
acceleration of the mass; a direct proportion. With a ruler, draw a straight line
through the points.
In contrast, for the same force, if you increase the mass, the acceleration decreases.
And if you decrease the mass, the acceleration increases. So you have been in
inverse relationship between acceleration and mass. If one goes up the other goes
down; if one goes down, the other goes up (Figure 1.3). As a proportionality:
a µ1/m
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12
10
Acceleration
8
6
4
2
0
0
2
4
6
8
10
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Mass
Figure 1.3 As mass increases, the acceleration for the same net force (10 N)
decreases; an inverse proportion. Draw a smooth, freehand line through the points.
OK, now you’re ready for Newton’s third law. What do you measure when you
step on a scale? Your weight! What is weight? The force of the earth’s gravity
pulling you down. Now let’s turn this around: what is your force pulling on the
earth? Hint: you are not accelerating. So the net force between you and the earth
must be zero. That means the force you exert on the earth must be equal in
size—but opposite in direction—of the force that the earth exerts on you. And that
is your weight! So if you measure a weight of about 150 lbs (XX kg), that is the
force of you on the earth (upward!) as well as the force of the earth on you
(downward)
Newton’s most creative leap was to recognize that forces occur in matched pairs.
This is Newton’s third law, which was really his novel addition to our
understanding of forced motion. I’ll call it the reaction law:
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ÿ Newton’s Third Law (Reaction Law):
Forces act in simultaneous pairs, equal in size, opposite in direction
To sum up. You can basically think about motion in two distinct ways. Natural
motion—motion without forces—no accelerations! Forced motion—accelerated
motion!
Section 1.3
Now to get down to the nitty gritty. We have looked at gravity as some kind of
mysterious force. Can we understand it better?
Newton’s conceptual solution about gravity marked a stunning breakthrough,
sparked by the fall of an apple. I believe that Newton’s law of gravitation should
really be considered his “fourth” law. Go back to measuring weight on a scale. If
you add more mass—say twice as much—the weight reads twice as much. Now
what if you kept the mass on a scale the same, but increased the mass of the earth.
The weight measured would also increase, in direct proportion to the amount of
mass added to the earth. So we can surmise that the force of gravitation is directly
proportional to the to masses involved. If you increase either one the force goes up
by the amount of increase. If you decrease either one, the force goes down by the
amount of decrease.
• Gravitational force is directly proportional to the product of the masses.
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How about the distance between the masses? Well, your intuition probably tells
you that as you move the masses apart, the gravitational force between them
decreases. That’s correct. But exactly how? That’s what Newton was able to figure
out. It is an inverse proportion of a very special kind, the inverse square of the
distance.
• Gravitational force is inversely proportional to the square of the distance
between the masses.
Figure 1.4 shows an inverse square relationship. Note that it looks a lot like an
inverse relationship but the y-axis values fall off much faster to zero.
Figure 1.4
1
0.9
Gravitational Force
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
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Distance
Figure 1.4 Inverse-square law for gravitation. Draw a smooth, freehand curve
through the points.
Now let’s to pull these two ideas together in Newton’s law of gravitation:
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ÿ Newton’s Fourth Law (Gravitation Law)
Gravitational force = G Mm/R2
That’s it! You can see the product of the masses in the numerator and the square of
the distance between them in the denominator. What about this big G? That’s
Newton’s gravitational constant, a number whose value is assumed not to vary
with time or location in the universe. The value for G relates the size of the force to
the sizes of the other quantities. It is a conversion of exchange, just like finding out
how many U.S. dollars will buy so many Mexican pesos.
Newton’s view of gravitation contains some odd features. First, gravitation is a
very, very, very weak force. Imagine two 1 kg masses 1 m apart. Then the
gravitational force between them is a mere 6.67 x 10-11 newtons. (One newton, a
standard unit of force, is the amount of force need to accelerate a mass of 1 kg at a
rate of 1 m per second per second.) That is 0.0000000000667 Newton!
Second, gravitation has a long reach, because it obeys an inverse square law.
Figure 1.4 shows that the force decreases quickly with small distances but then
kind of levels out at far distance. It then slowly approaches zero. Mathematically,
the force will reach zero only at an infinite distance.
Third, if you have a small mass right at the surface of a large mass, both masses
attract as if their masses were concentrated in the center. This simplification was a
great insight of Newton’s. But, it raises the question of the size of the gravitational
force if the two masses are placed closer and closer together. It looks like it can go
to infinity, which is an unphysical result!
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Fourth, all masses in the cosmos must be interacting gravitationally at the same
time. Even though, as we know today, that these distances are enormous!
The fifth, gravitation is only an attractive force and only an attractive one. And, it
does not depend on the types of material that is involved. It can be apples, people,
tennis balls, planets, stars, galaxies, or the whole cosmos. Newton worried about
this point. He reasoned that, if the universe were finite in size, eventually mutual
gravitational forces would pull all the mass in the cosmos together in one lump. So
Newton concluded that the universe must be infinite in size.
Try this application. Astronomers discover a new moon orbiting the earth! It has
the same mass as our current moon, but it’s three times as far away. How does the
gravitational force of the earth on this new moon compare to the gravitational force
of the earth on our moon?
A. It is the same.
B. It is three times as much.
C. It is one-third as much.
D. It is one-ninth as much.
How you know?
These issues show that Newton’s law of gravitation, despite its simplicity and
power, contains problems. Despite them, Newton’s gravitation has been tried and
tested again and again with great success (otherwise we could not fly space probes
to other planets in the solar system or send astronauts to the moon).
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Section 1.4
Now to pull together these concepts about falling. Focus on the following
experiment, the one given at the start of this chapter. You have two balls, same
size, different masses, held at arm’s length, and you drop them simultaneously.
They hit the ground at the same time.
What can you conclude about the velocities of the two objects?
• The speeds were zero as you held them up.
o They gained speed at the same rate to reach the ground at the same time.
• What can you conclude about the accelerations of the two objects?
o The speeds increase, so there was an acceleration of each object.
• How do the sizes of the accelerations compare?
o They must be the same; otherwise the balls would not hit the ground at the
same time.
• How you know the masses differed?
o They have different weights when placed on a scale.
• What does this measurement tell you about the forces on the balls?
o It indicates that the forces are different.
What can you infer overall? The accelerations are the same, so the forces differ.
How much? As you can see from Newton’s second law, the force on the heavier
ball must be just enough greater so that it falls with the same acceleration as the
lighter ball. Recall that gravitational forces are directly proportional to the masses.
But accelerations are inversely proportional to masses. When we put these two
together that means that the mass just cancels out. What a neat result! The
acceleration of different masses near the surface of the earth is the same.
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We can measure the amount: it comes out to about 10 meters per second (m/s) per
second. That means for every second of fall, a falling object gains 10 meters per
second of speed. At the end of second 1, the object’s speed is 10 m/s; at the end of
second 2, 20 m/s; at the end of 3, 30 m/s, until it hits the ground.
Math Wizard
The distilled language of mathematics cuts right to the answer. Recall that Newton
second law is
• a = F/m
where a is the acceleration, F is the net force, and m is the mass of our little test
particle. Now Newton’s law of gravitation for two objects, m and M is
• F = GMm/R2
where R is the distance between the centers of the two masses. Here M is the mass
of the earth and m our little test particle, sitting on the surface of the earth.
Combine these two equations
• acceleration = F/m = GMm/ mR2 = GM/R2
The little mass cancels out! So, finally,
• a = g = GM/R2
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The mass of our test particle is gone. That means that masses dropped near the
surface of the earth have the same acceleration. If you calculate the acceleration
using the mass of the earth, the value of Newton’s constant of gravitation, and the
radius of the earth, you get 9.8 meters per second per second; that’s the value for g.
It is the special symbol for gravity’s acceleration near the earth’s surface. I have
been rounding it off to 10 meters per second per second.
q
Student Astra: I think that Professor Mike has gone too far here. He forgot to
tell you about air friction! Otherwise we would get pounded when it rains.
q
Professor Mike: Astra is right! But most of our future examples will contain
situations where there’s no air and so no friction because of the air.
Now to check your understanding of these concepts. Imagine that the earth had the
same mass but a much smaller radius. What would happen to the gravity, g, at the
surface?
A. It would increase.
B. It would stay the same.
C. It would decrease.
The correct answer is A. Here’s the reasoning: Newton’s law of gravitation is an
inverse square law. If the earth were smaller, then a test mass at the surface is
much closer to the earth’s center. So the force of gravity on it must increase. Note
that we have implicitly used the notion that, for gravitational purposes, the mass of
the earth acts as if it were all concentrated at the center.
What would happen if the earth’s radius remains the same, but its mass increased?
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Chapter Check Out
¸ Masses falling near the surface of the earth all have the same
acceleration.
¸ Newton’s first law tells us that there must be force acting on these
objects.
¸ Newton’s second law states that for a given force and mass, they will
have the same acceleration.
¸ The gravitational force of the earth on these masses causes this
acceleration. We informally call this the earth’s gravity.
¸ Newton’s law of gravitation is directly proportional to the product of the
masses involved, and is inversely proportional to the distance between
their centers.
¸ Combine Newton’s law of gravitation with Newton’s second law, and
you can explain the observation that all objects close to the surface fall at
the same acceleration.
Note that you have developed your understanding by the application of powerful
concepts to specific situations.
Concept Applications
1. You are standing on the surface of the moon you drop a feather and a hammer at
the same time from shoulder height. What happens to these two objects?
A. The hammer hits the ground first.
B. They both hit the ground at the same time.
C. The feather hits the ground first.
D. I don’t know; I’ve never performed this experiment.
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Explain your choice.
2. If both a feather and a hammer on the moon hit the ground at the same time
when dropped together from the same height, what can you conclude?
A. Both objects fall with the same speed.
B. Both objects fall with the same acceleration.
C. The force on both objects is the same.
Explain your choice.
3. Imagine you are in an elevator that can go any height above the earth’s surface.
You have a basketball that you drop from shoulder height. Position this elevator so
that it is one earth’s radius above the earth’s surface. How does the acceleration of
the basketball dropped at this distance compare to that when the elevator was on
the surface of the earth?
A. Greater.
B. The same.
C. Less
Explain your choice.
Concept Connections
Draw a concept map using ten of the following concepts from this chapter:
1. Speed
2. Acceleration
3. Force
4. Mass
5. Weight
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6. Natural motion
7. Forced motion
8. Newton’s law of gravitation
9. Newton’s first law
10. Newton’s second law
11. Newton’s third law
12. Newton’s “fourth” law
Here’s a sample using some of these:
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ABOUT
Concept Maps and Connected Understanding in Physics
Human knowledge functions at different levels. A simple scheme depicts
that people know what, how, and why. You know what when you show
awareness of an object, event, or concept. You know how when you can
apply what you know to solving problems, for example, or making
arguments. Finally, you know why when you grasp in what way concepts in
a certain area are interrelated. If you are so good in an area of knowledge
that you perform well without consciously thinking much about the what and
the why, you are an expert in that field. You see automatically the complex
interrelationships among important concepts. This is called structural
knowledge or connected understanding. It sets experts apart from
novices.
An expert's knowledge of a field contains rich sets of interconnected
concepts. In contrast, a novice may quickly learn to know what in a new
area but has a very difficult time in gaining the knowledge for knowing why
and how. To use the old metaphor, the expert sees the forest, the novice
the trees. The expert views an ecosystem, all parts interconnected,
sustained by a small number of unifying natural laws. The novice sees a
tree, and another tree, and another tree and is overwhelmed. How can you
memorize all the facts about each tree for so many trees? The novice lacks
the structural knowledge of the expert and does not see the unity and the
beauty of the forest.
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The same expert/novice gulf appears in the study of physics. The expert
physicist and teacher see the physical cosmos: its unity, beauty, and
natural order. The novice student spots one law, then another, and—and is
promptly swamped and confused by the vastness of the physical world.
What marks the key difference between the expert and novice physicist?
Connected understanding! This book will explicitly convey connected
understanding in the form of concept maps. They are two-dimensional
diagrams that reveals the relationships between major ideas in a content
area, such as physics. One of the main features of concepts maps is that
the concepts are linked by labeled lines that identify the relationships
among concepts. Many concepts have multiple links. It is the richness of
interrelationships that characterizes an expert's connected in a field—a
connected understanding.
How to read a concept map? Start with the central concept and follow the
direction of the links. Read the relations like a sentence: Concept A links
(usually a verb) to concept B. It should make sense to you! Each connected
pair of concepts is a basic piece of the map.
I will present these “instructor-generated” maps in this book to preview and
review key topics and concepts. Do not try to memorize these maps! Use
them as a guide to help you make your connections. You may design and
draw your own concept maps to assist your own understanding. Individual
concept maps are idiosyncratic; each person’s map will differ. So your
concept map and mine will not be the same about a any topic.
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The research concept maps has verified their effectiveness in helping
learners understand the “big picture” more like an expert. The work so far
has resulted in some essential inferences about concept maps:
• All knowledge contains structure; abstract ideas rely on structure
• Recall and comprehension require connected understanding
• Learners build connected understanding, which reorganizes their
cognitive structure into a map more like that of an expert's
• Memory structures, as represented by concept maps, reflect the world
• Problem solving requires structural knowledge
• Deep understanding ("expert") of an area requires connected
understanding
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