Force and Motion
If I have seen farther than others, it is because I have
stood on the shoulders of giants. -- Newton, referring to Galileo
Even as great and skeptical a genius as Galileo was unable to
make much progress on the causes of motion. It was not until a
generation later that Isaac Newton (1642-1727) was able to attack
the problem successfully. In many ways, Newton's personality was
the opposite of Galileo's. Where Galileo agressively publicized his
ideas, Newton had to be coaxed by his friends into publishing a book
on his physical discoveries. Where Galileo's writing had been popular
and dramatic, Newton originated the stilted, impersonal style that most
people think is standard for scientific writing. (Scientific journals today
encourage a less ponderous style, and papers are often written in the
first person.) Galileo's talent for arousing animosity among the rich
and powerful was matched by Newton's skill at making himself a
popular visitor at court. Galileo narrowly escaped being burned at the
stake, while Newton had the good fortune of being on the winning
side of the revolution that replaced King James II with William and
Mary of Orange, leading to a lucrative post running the English royal
mint.
Newton discovered the relationship between force and motion,
and revolutionized our view of the universe by showing that the same
physical laws applied to all matter, whether living or nonliving, on or
off of our planet's surface. His book on force and motion, the
Mathematical Principles of Natural Philosophy, was uncontradicted
by experiment for 200 years, but his other main work, Optics, was on
the wrong track, asserting that light was composed of
particles rather than waves. Newton was also an avid alchemist, a
fact that modern scientists would like to forget.
Force
aristotelian-arrow
a / Aristotle sa
id motion had to be caused by a force. To explain why an arrow kept flying after the bowstring was no longer pushing on it, he said the air rushed around behind the arrow and pushed it forward. We know this is wrong, because an arrow shot in a vacuum chamber does not instantly drop to the floor as it leaves the bow. Galileo and Newton realized that a force would only be needed to change the arrow's motion, not to make its motion continue.
monet
b / “Our eyes receive blue light reflected from this painting because Monet wanted to represent water with the color blue.” This is a valid statement at one level of explanation, but physics works at the physical level of explanation, in which blue light gets to your eyes because it is reflected by blue pigments in the paint.
sax-pos-and-neg-forces-add
c / Forces are applied to a saxophone. In this example, positive signs have been used consistently for forces to the right, and negative signs for forces to the left. (The forces are being applied to different places on the saxophone, but the numerical value of a force carries no information about that.)
We need only explain changes in motion, not motion itself.
So far you've studied the measurement of motion in some
detail, but not the reasons why a certain object would move
in a certain way. This chapter deals with the “why”
questions. Aristotle's ideas about the causes of motion were
completely wrong, just like all his other ideas about
physical science, but it will be instructive to start with
them, because they amount to a road map of modern students'
incorrect preconceptions.
Aristotle thought he needed to explain both why motion
occurs and why motion might change. Newton inherited from
Galileo the important counter-Aristotelian idea that motion
needs no explanation, that it is only changes in
motion that require a physical cause. Aristotle's needlessly
complex system gave three reasons for motion:
Natural motion, such as falling, came from the tendency of
objects to go to their “natural” place, on the ground, and come to rest.
Voluntary motion was the type of motion exhibited by
animals, which moved because they chose to.
Forced motion occurred when an object was acted on by some
other object that made it move.
Motion changes due to an interaction between two objects.
In the Aristotelian theory, natural motion and voluntary
motion are one-sided phenomena: the object causes its own
motion. Forced motion is supposed to be a two-sided
phenomenon, because one object imposes its “commands” on
another. Where Aristotle conceived of some of the phenomena
of motion as one-sided and others as two-sided, Newton
realized that a change in motion was always a two-sided
relationship of a force acting between two physical objects.
The one-sided “natural motion” description of falling
makes a crucial omission. The acceleration of a falling
object is not caused by its own “natural” tendencies but
by an attractive force between it and the planet Earth. Moon
rocks brought back to our planet do not “want” to fly back
up to the moon because the moon is their “natural” place.
They fall to the floor when you drop them, just like our
homegrown rocks. As we'll discuss in more detail later in
this course, gravitational forces are simply an attraction
that occurs between any two physical objects. Minute
gravitational forces can even be measured between human-scale
objects in the laboratory.
The idea of natural motion also explains incorrectly why
things come to rest. A basketball rolling across a beach
slows to a stop because it is interacting with the sand via
a frictional force, not because of its own desire to be at
rest. If it was on a frictionless surface, it would never
slow down. Many of Aristotle's mistakes stemmed from his
failure to recognize friction as a force.
The concept of voluntary motion is equally flawed. You may
have been a little uneasy about it from the start, because
it assumes a clear distinction between living and nonliving
things. Today, however, we are used to having the human body
likened to a complex machine. In the modern world-view, the
border between the living and the inanimate is a fuzzy
no-man's land inhabited by viruses, prions, and silicon
chips. Furthermore, Aristotle's statement that you can take
a step forward “because you choose to” inappropriately
mixes two levels of explanation. At the physical level of
explanation, the reason your body steps forward is because
of a frictional force acting between your foot and the
floor. If the floor was covered with a puddle of oil, no
amount of “choosing to” would enable you to take a
graceful stride forward.
Forces can all be measured on the same numerical scale.
In the Aristotelian-scholastic tradition, the description of
motion as natural, voluntary, or forced was only the
broadest level of classification, like splitting animals
into birds, reptiles, mammals, and amphibians. There might
be thousands of types of motion, each of which would follow
its own rules. Newton's realization that all changes in
motion were caused by two-sided interactions made it seem
that the phenomena might have more in common than had been
apparent. In the Newtonian description, there is only one
cause for a change in motion, which we call force. Forces
may be of different types, but they all produce changes in
motion according to the same rules. Any acceleration that
can be produced by a magnetic force can equally well be
produced by an appropriately controlled stream of water. We
can speak of two forces as being equal if they produce the
same change in motion when applied in the same situation,
which means that they pushed or pulled equally hard
in the same direction.
The idea of a numerical scale of force and the newton unit
were introduced in chapter 0. To recapitulate briefly, a
force is when a pair of objects push or pull on each other,
and one newton is the force required to accelerate a 1-kg
object from rest to a speed of 1 m/s in 1 second.
More than one force on an object
As if we hadn't kicked poor Aristotle around sufficiently,
his theory has another important flaw, which is important to
discuss because it corresponds to an extremely common
student misconception. Aristotle conceived of forced motion
as a relationship in which one object was the boss and the
other “followed orders.” It therefore would only make
sense for an object to experience one force at a time,
because an object couldn't follow orders from two sources at
once. In the Newtonian theory, forces are numbers, not
orders, and if more than one force acts on an object at
once, the result is found by adding up all the forces. It is
unfortunate that the use of the English word “force” has
become standard, because to many people it suggests that you
are “forcing” an object to do something. The force of the
earth's gravity cannot “force” a boat to sink, because
there are other forces acting on the boat. Adding them up
gives a total of zero, so the boat accelerates neither up nor down.
Objects can exert forces on each other at a distance.
Aristotle declared that forces could only act between
objects that were touching, probably because he wished to
avoid the type of occult speculation that attributed
physical phenomena to the influence of a distant and
invisible pantheon of gods. He was wrong, however, as you
can observe when a magnet leaps onto your refrigerator or
when the planet earth exerts gravitational forces on objects
that are in the air. Some types of forces, such as friction,
only operate between objects in contact, and are called
contact forces. Magnetism, on the
other hand, is an example of a noncontact
force. Although the magnetic force gets stronger when the
magnet is closer to your refrigerator, touching is not required.
Weight
In physics, an object's weight, FW, is defined as the
earth's gravitational force on it. The SI unit of weight is
therefore the Newton. People commonly refer to the kilogram
as a unit of weight, but the kilogram is a unit of mass, not
weight. Note that an object's weight is not a fixed property
of that object. Objects weigh more in some places than in
others, depending on the local strength of gravity. It is
their mass that always stays the same. A baseball pitcher
who can throw a 90-mile-per-hour fastball on earth would not
be able to throw any faster on the moon, because the ball's
inertia would still be the same.
Positive and negative signs of force
We'll start by considering only cases of one-dimensional
center-of-mass motion in which all the forces are parallel
to the direction of motion, i.e., either directly forward or
backward. In one dimension, plus and minus signs can be used
to indicate directions of forces, as shown in figure c. We
can then refer generically to addition of forces, rather
than having to speak sometimes of addition and sometimes of
subtraction. We add the forces shown in the figure and get
11 N. In general, we should choose a one-dimensional
coordinate system with its x axis parallel the direction
of motion. Forces that point along the positive x axis are
positive, and forces in the opposite direction are negative.
Forces that are not directly along the x axis cannot be
immediately incorporated into this scheme, but that's OK,
because we're avoiding those cases for now.
4.2 Newton's First Law
eg-boat
d / Example 4.
dq-saxophone-1
Discussion question B.
dq-saxophone-2
Discussion question C.
We are now prepared to make a more powerful restatement of
the principle of inertia.
Newton's first law
If the total force on an object is zero, its center of mass
continues in the same state of motion.
In other words, an object initially at rest is predicted to
remain at rest if the total force on it is zero, and an
object in motion remains in motion with the same velocity in
the same direction. The converse of Newton's first law is
also true: if we observe an object moving with constant
velocity along a straight line, then the total force on it must be zero.
In a future physics course or in another textbook, you may
encounter the term “net force,” which is
simply a synonym for total force.
What happens if the total force on an object is not zero? It
accelerates. Numerical prediction of the resulting
acceleration is the topic of Newton's second law, which
we'll discuss in the following section.
This is the first of Newton's three laws of motion. It is
not important to memorize which of Newton's three laws are
numbers one, two, and three. If a future physics teacher
asks you something like, “Which of Newton's laws are you
thinking of,” a perfectly acceptable answer is “The one
about constant velocity when there's zero total force.” The
concepts are more important than any specific formulation of
them. Newton wrote in Latin, and I am not aware of any
modern textbook that uses a verbatim translation of his
statement of the laws of motion. Clear writing was not in
vogue in Newton's day, and he formulated his three laws in
terms of a concept now called momentum, only later relating
it to the concept of force. Nearly all modern texts,
including this one, start with force and do momentum later.
Example 1: An elevator
◊ An elevator has a weight of 5000 N. Compare the
forces that the cable must exert to raise it at constant
velocity, lower it at constant velocity, and just keep it hanging.
◊ In all three cases the cable must pull up with a
force of exactly 5000 N. Most people think you'd need at
least a little more than 5000 N to make it go up, and a
little less than 5000 N to let it down, but that's
incorrect. Extra force from the cable is only necessary for
speeding the car up when it starts going up or slowing it
down when it finishes going down. Decreased force is needed
to speed the car up when it gets going down and to slow it
down when it finishes going up. But when the elevator is
cruising at constant velocity, Newton's first law says that
you just need to cancel the force of the earth's gravity.
To many students, the statement in the example that the
cable's upward force “cancels” the earth's downward
gravitational force implies that there has been a contest,
and the cable's force has won, vanquishing the earth's
gravitational force and making it disappear. That is
incorrect. Both forces continue to exist, but because they
add up numerically to zero, the elevator has no center-of-mass
acceleration. We know that both forces continue to exist
because they both have side-effects other than their effects
on the car's center-of-mass motion. The force acting between
the cable and the car continues to produce tension in the
cable and keep the cable taut. The earth's gravitational
force continues to keep the passengers (whom we are
considering as part of the elevator-object) stuck to the
floor and to produce internal stresses in the walls of the
car, which must hold up the floor.
Example 2: Terminal velocity for falling objects
◊ An object like a feather that is not dense or
streamlined does not fall with constant acceleration,
because air resistance is nonnegligible. In fact, its
acceleration tapers off to nearly zero within a fraction of
a second, and the feather finishes dropping at constant
speed (known as its terminal velocity). Why does this happen?
◊ Newton's first law tells us that the total force on
the feather must have been reduced to nearly zero after a
short time. There are two forces acting on the feather: a
downward gravitational force from the planet earth, and an
upward frictional force from the air. As the feather speeds
up, the air friction becomes stronger and stronger, and
eventually it cancels out the earth's gravitational force,
so the feather just continues with constant velocity without
speeding up any more.
The situation for a skydiver is exactly analogous. It's
just that the skydiver experiences perhaps a million times
more gravitational force than the feather, and it is not
until she is falling very fast that the force of air
friction becomes as strong as the gravitational force. It
takes her several seconds to reach terminal velocity, which
is on the order of a hundred miles per hour.
More general combinations of forces
It is too constraining to restrict our attention to cases
where all the forces lie along the line of the center of
mass's motion. For one thing, we can't analyze any case of
horizontal motion, since any object on earth will be subject
to a vertical gravitational force! For instance, when you
are driving your car down a straight road, there are both
horizontal forces and vertical forces. However, the vertical
forces have no effect on the center of mass motion, because
the road's upward force simply counteracts the earth's
downward gravitational force and keeps the car from
sinking into the ground.
Later in the book we'll deal with the most general case of
many forces acting on an object at any angles, using the
mathematical technique of vector addition, but the following
slight generalization of Newton's first law allows us to
analyze a great many cases of interest:
Suppose that an object has two sets of forces acting on it,
one set along the line of the object's initial motion and
another set perpendicular to the first set. If both sets of
forces cancel, then the object's center of mass continues in
the same state of motion.
Example 3: A passenger riding the subway
◊ Describe the forces acting on a person standing in
a subway train that is cruising at constant velocity.
◊ No force is necessary to keep the person moving
relative to the ground. He will not be swept to the back of
the train if the floor is slippery. There are two vertical
forces on him, the earth's downward gravitational force and
the floor's upward force, which cancel. There are no
horizontal forces on him at all, so of course the total
horizontal force is zero.
Example 4: Forces on a sailboat
◊ If a sailboat is cruising at constant velocity
with the wind coming from directly behind it, what must be
true about the forces acting on it?
◊ The forces acting on the boat must be canceling each
other out. The boat is not sinking or leaping into the air,
so evidently the vertical forces are canceling out. The
vertical forces are the downward gravitational force exerted
by the planet earth and an upward force from the water.
The air is making a forward force on the sail, and if the
boat is not accelerating horizontally then the water's
backward frictional force must be canceling it out.
Contrary to Aristotle, more force is not needed in order to
maintain a higher speed. Zero total force is always needed
to maintain constant velocity. Consider the following made-up numbers:
boat moving at
a low, constant
velocity
boat moving at
a high, constant
velocity
forward force of the
wind on the sail …
10,000 N
20,000 N
backward force of
the water on the
hull …
10000
N
20000
N
total force on the
boat …
0 N
0 N
The faster boat still has zero total force on it. The
forward force on it is greater, and the backward force
smaller (more negative), but that's irrelevant because
Newton's first law has to do with the total force, not
the individual forces.
This example is quite analogous to the one about terminal
velocity of falling objects, since there is a frictional
force that increases with speed. After casting off from the
dock and raising the sail, the boat will accelerate briefly,
and then reach its terminal velocity, at which the water's
frictional force has become as great as the wind's force on the sail.
Example 5: A car crash
◊ If you drive your car into a brick wall, what is
the mysterious force that slams your face into the steering wheel?
◊ Your surgeon has taken physics, so she is not going
to believe your claim that a mysterious force is to blame.
She knows that your face was just following Newton's first
law. Immediately after your car hit the wall, the only
forces acting on your head were the same canceling-out
forces that had existed previously: the earth's downward
gravitational force and the upward force from your neck.
There were no forward or backward forces on your head, but
the car did experience a backward force from the wall, so
the car slowed down and your face caught up.
Discussion Questions
◊
Newton said that objects continue moving if no forces are
acting on them, but his predecessor Aristotle said that a
force was necessary to keep an object moving. Why does
Aristotle's theory seem more plausible, even though we now
believe it to be wrong? What insight was Aristotle missing
about the reason why things seem to slow down naturally?
Give an example.
◊
In the figure what would have to be true about the
saxophone's initial motion if the forces shown were to
result in continued one-dimensional motion of its center of mass?
◊
This figure requires an ever further generalization
of the preceding discussion. After studying the forces, what
does your physical intuition tell you will happen? Can you
state in words how to generalize the conditions for
one-dimensional motion to include situations like this one?
4.3 Newton's Second Law
double-pan-balance
e / A simple double-pan balance works by comparing the weight forces exerted by the earth on the contents of the two pans. Since the two pans are at almost the same location on the earth's surface, the value of g is essentially the same for each one, and equality of weight therefore also implies equality of mass.
hang-weights
f / Example 7.
x munit t sunit
10 1.84
20 2.86
30 3.80
40 4.67
50 5.53
60 6.38
70 7.23
80 8.10
90 8.96
100 9.83
Discussion question D.
What about cases where the total force on an object is not
zero, so that Newton's first law doesn't apply? The object
will have an acceleration. The way we've defined positive
and negative signs of force and acceleration guarantees that
positive forces produce positive accelerations, and likewise
for negative values. How much acceleration will it have? It
will clearly depend on both the object's mass and on
the amount of force.
Experiments with any particular object show that its
acceleration is directly proportional to the total force
applied to it. This may seem wrong, since we know of many
cases where small amounts of force fail to move an object at
all, and larger forces get it going. This apparent failure
of proportionality actually results from forgetting that
there is a frictional force in addition to the force we
apply to move the object. The object's acceleration is
exactly proportional to the total force on it, not to any
individual force on it. In the absence of friction, even a
very tiny force can slowly change the velocity of a
very massive object.
Experiments also show that the acceleration is inversely
proportional to the object's mass, and combining these two
proportionalities gives the following way of predicting the
acceleration of any object:
Newton's second law
a=Ftotal/m,
where
m is an object's mass
Ftotal is the sum of the forces acting on it, and
a is the acceleration of the object's center of mass.
We are presently restricted to the case where the forces of
interest are parallel to the direction of motion.
Example 6: An accelerating bus
◊ A VW bus with a mass of 2000 kg accelerates from 0
to 25 m/s (freeway speed) in 34 s. Assuming the acceleration
is constant, what is the total force on the bus?
◊ We solve Newton's second law for Ftotal=ma,
and substitute Δ v/Δ t for a, giving
Ftotal
=mΔv/Δt
=(2000kg)(25m/s-0m/s)/(34s)
=1.5kN.
A generalization
As with the first law, the second law can be easily
generalized to include a much larger class of interesting situations:
Suppose an object is being acted on by two sets of forces,
one set lying along the object's initial direction of motion
and another set acting along a perpendicular line. If the
forces perpendicular to the initial direction of motion
cancel out, then the object accelerates along its original
line of motion according to a=Ftotal/m.
The relationship between mass and weight
Mass is different from weight, but they're related. An
apple's mass tells us how hard it is to change its motion.
Its weight measures the strength of the gravitational
attraction between the apple and the planet earth. The
apple's weight is less on the moon, but its mass is the
same. Astronauts assembling the International Space Station
in zero gravity cannot just pitch massive modules back and
forth with their bare hands; the modules are weightless, but not massless.
We have already seen the experimental evidence that when
weight (the force of the earth's gravity) is the only force
acting on an object, its acceleration equals the constant
g, and g depends on where you are on the surface of the
earth, but not on the mass of the object. Applying Newton's
second law then allows us to calculate the magnitude of the
gravitational force on any object in terms of its mass:
FW=mg .
(The equation only gives the magnitude, i.e. the absolute
value, of FW, because we're defining g as a positive
number, so it equals the absolute value of a falling
object's acceleration.)
◊ Solved problem: Decelerating a car — problem 7
Example 7: Weight and mass
◊ Figure f shows masses of one and
two kilograms hung from a spring scale, which measures force in units of
newtons. Explain the readings.
◊ Let's start with the single kilogram. It's not accelerating,
so evidently the total force on it is zero: the spring scale's upward
force on it is canceling out the earth's downward gravitational force.
The spring scale tells us how much force it is being obliged to supply,
but since the two forces are equal in strength, the spring scale's reading
can also be interpreted as measuring the strength of the gravitational
force, i.e., the weight of the one-kilogram mass. The weight of a one-kilogram
mass should be
FW
=mg
=(1.0kg)(9.8m/s2)=9.8N,
and that's indeed the reading on the spring scale.
Similarly for the two-kilogram mass, we have
FW
=mg
=(2.0kg)(9.8m/s2)=19.6N.
Example 8: Calculating terminal velocity
◊ Experiments show that the force of air friction on
a falling object such as a skydiver or a feather can be
approximated fairly well with the equation
Fair=cρ Av2, where c is a constant, ρ is the density of the
air, A is the cross-sectional area of the object as seen
from below, and v is the object's velocity. Predict the
object's terminal velocity, i.e., the final velocity it
reaches after a long time.
◊ As the object accelerates, its greater v causes
the upward force of the air to increase until finally the
gravitational force and the force of air friction cancel
out, after which the object continues at constant velocity.
We choose a coordinate system in which positive is up, so
that the gravitational force is negative and the force of
air friction is positive. We want to find the velocity at which
Fair+FW
=0,i.e.,
cρAv2-mg
=0.
Solving for v gives
vterminal=mgcρA
self-check:
It is important to get into the habit of interpreting
equations. This may be difficult at first,
but eventually you will get used to this kind of reasoning.
(1) Interpret the equation
vterminal=mg/cρA
in the case of ρ=0.
(2) How would the terminal velocity of a 4-cm steel ball
compare to that of a 1-cm ball?
(3) In addition to teasing out the mathematical meaning of
an equation, we also have to be able to place it in its physical
context. How generally important is this equation?
(answer in the back of the PDF version of the book)
Discussion Questions
◊
Show that the Newton can be reexpressed in terms of the
three basic mks units as the combination kg⋅m/s2.
◊
What is wrong with the following statements?
(1) “g is the force of gravity.”
(2) “Mass is a measure of how much space something takes up.”
◊
Criticize the following incorrect statement:
“If an object is at rest and the total force on it is zero,
it stays at rest. There can also be cases where an object is
moving and keeps on moving without having any total force on
it, but that can only happen when there's no friction,
like in outer space.”
◊
Table g gives laser timing data for Ben
Johnson's 100 m dash at the 1987 World Championship in
Rome. (His world record was later revoked because he tested
positive for steroids.) How does the total force on him
change over the duration of the race?
4.4 What Force Is Not
Violin teachers have to endure their beginning students'
screeching. A frown appears on the woodwind teacher's face
as she watches her student take a breath with an expansion
of his ribcage but none in his belly. What makes physics
teachers cringe is their students' verbal statements about
forces. Below I have listed several dicta about what force is not.
Force is not a property of one object.
A great many of students' incorrect descriptions of forces
could be cured by keeping in mind that a force is an
interaction of two objects, not a property of one object.
Incorrect statement: “That magnet has a lot of force.”
× If the magnet is one millimeter away from a steel ball
bearing, they may exert a very strong attraction on each
other, but if they were a meter apart, the force would be
virtually undetectable. The magnet's strength can be rated
using certain electrical units (ampere-meters2), but
not in units of force.
Force is not a measure of an object's motion.
If force is not a property of a single object, then it
cannot be used as a measure of the object's motion.
Incorrect statement: “The freight train rumbled down the
tracks with awesome force.”
× Force is not a measure of motion. If the freight train
collides with a stalled cement truck, then some awesome
forces will occur, but if it hits a fly the force will be small.
Force is not energy.
There are two main approaches to understanding the motion of
objects, one based on force and one on a different concept,
called energy. The SI unit of energy is the Joule, but you
are probably more familiar with the calorie, used for
measuring food's energy, and the kilowatt-hour, the unit the
electric company uses for billing you. Physics students'
previous familiarity with calories and kilowatt-hours is
matched by their universal unfamiliarity with measuring
forces in units of Newtons, but the precise operational
definitions of the energy concepts are more complex than
those of the force concepts, and textbooks, including this
one, almost universally place the force description of
physics before the energy description. During the long
period after the introduction of force and before the
careful definition of energy, students are therefore
vulnerable to situations in which, without realizing it,
they are imputing the properties of energy to phenomena of force.
Incorrect statement: “How can my chair be making an upward
force on my rear end? It has no power!”
× Power is a concept related to energy, e.g., a 100-watt
lightbulb uses up 100 joules per second of energy. When you
sit in a chair, no energy is used up, so forces can exist
between you and the chair without any need for a source of power.
Force is not stored or used up.
Because energy can be stored and used up, people think force
also can be stored or used up.
Incorrect statement: “If you don't fill up your tank with
gas, you'll run out of force.”
× Energy is what you'll run out of, not force.
Forces need not be exerted by living things or machines.
Transforming energy from one form into another usually
requires some kind of living or mechanical mechanism. The
concept is not applicable to forces, which are an interaction
between objects, not a thing to be transferred or transformed.
Incorrect statement: “How can a wooden bench be making an
upward force on my rear end? It doesn't have any springs or
anything inside it.”
× No springs or other internal mechanisms are required. If
the bench didn't make any force on you, you would obey
Newton's second law and fall through it. Evidently it does
make a force on you!
A force is the direct cause of a change in motion.
I can click a remote control to make my garage door change
from being at rest to being in motion. My finger's force on
the button, however, was not the force that acted on the
door. When we speak of a force on an object in physics, we
are talking about a force that acts directly. Similarly,
when you pull a reluctant dog along by its leash, the leash
and the dog are making forces on each other, not your hand
and the dog. The dog is not even touching your hand.
self-check:
Which of the following things can be correctly described in terms of force?
(1) A nuclear submarine is charging ahead at full steam.
(2) A nuclear submarine's propellers spin in the water.
(3) A nuclear submarine needs to refuel its reactor periodically.
(answer in the back of the PDF version of the book)
Discussion Questions
◊
Criticize the following incorrect statement: “If you
shove a book across a table, friction takes away more and
more of its force, until finally it stops.”
◊
You hit a tennis ball against a wall. Explain any and all
incorrect ideas in the following description of the physics
involved: “The ball gets some force from you when you hit
it, and when it hits the wall, it loses part of that force,
so it doesn't bounce back as fast. The muscles in your arm
are the only things that a force can come from.”
4.5 Inertial and Noninertial Frames of Reference
One day, you're driving down the street in your pickup
truck, on your way to deliver a bowling ball. The ball is in
the back of the truck, enjoying its little jaunt and taking
in the fresh air and sunshine. Then you have to slow down
because a stop sign is coming up. As you brake, you glance
in your rearview mirror, and see your trusty companion
accelerating toward you. Did some mysterious force push it
forward? No, it only seems that way because you and the car
are slowing down. The ball is faithfully obeying Newton's
first law, and as it continues at constant velocity it gets
ahead relative to the slowing truck. No forces are acting on
it (other than the same canceling-out vertical forces that
were always acting on it).1 The ball only appeared to violate
Newton's first law because there was something wrong with
your frame of reference, which was based on the truck.
pickup-truck
g / 1. In a frame of reference that moves with the truck, the bowling ball appears to violate Newton's first law by accelerating despite having no horizontal forces on it. 2. In an inertial frame of reference, which the surface of the earth approximately is, the bowling ball obeys Newton's first law. It moves equal distances in equal time intervals, i.e., maintains constant velocity. In this frame of reference, it is the truck that appears to have a change in velocity, which makes sense, since the road is making a horizontal force on it.
How, then, are we to tell in which frames of reference
Newton's laws are valid? It's no good to say that we should
avoid moving frames of reference, because there is no such
thing as absolute rest or absolute motion. All frames can be
considered as being either at rest or in motion. According
to an observer in India, the strip mall that constituted the
frame of reference in panel (b) of the figure was moving
along with the earth's rotation at hundreds of miles per hour.
The reason why Newton's laws fail in the truck's frame of
reference is not because the truck is moving but
because it is accelerating. (Recall that physicists
use the word to refer either to speeding up or slowing
down.) Newton's laws were working just fine in the moving
truck's frame of reference as long as the truck was moving
at constant velocity. It was only when its speed changed
that there was a problem. How, then, are we to tell which
frames are accelerating and which are not? What if you claim
that your truck is not accelerating, and the sidewalk, the
asphalt, and the Burger King are accelerating? The way to
settle such a dispute is to examine the motion of some
object, such as the bowling ball, which we know has zero
total force on it. Any frame of reference in which the ball
appears to obey Newton's first law is then a valid frame of
reference, and to an observer in that frame, Mr. Newton
assures us that all the other objects in the universe will
obey his laws of motion, not just the ball.
Valid frames of reference, in which Newton's laws are
obeyed, are called inertial frames of reference. Frames of
reference that are not inertial are called noninertial
frames. In those frames, objects violate the principle of
inertia and Newton's first law. While the truck was moving
at constant velocity, both it and the sidewalk were valid
inertial frames. The truck became an invalid frame of
reference when it began changing its velocity.
You usually assume the ground under your feet is a perfectly
inertial frame of reference, and we made that assumption
above. It isn't perfectly inertial, however. Its motion
through space is quite complicated, being composed of a part
due to the earth's daily rotation around its own axis, the
monthly wobble of the planet caused by the moon's gravity,
and the rotation of the earth around the sun. Since the
accelerations involved are numerically small, the earth is
approximately a valid inertial frame.
Noninertial frames are avoided whenever possible, and we
will seldom, if ever, have occasion to use them in this
course. Sometimes, however, a noninertial frame can be
convenient. Naval gunners, for instance, get all their data
from radars, human eyeballs, and other detection systems
that are moving along with the earth's surface. Since their
guns have ranges of many miles, the small discrepancies
between their shells' actual accelerations and the
accelerations predicted by Newton's second law can have
effects that accumulate and become significant. In order to
kill the people they want to kill, they have to add small
corrections onto the equation a=Ftotal/m. Doing their
calculations in an inertial frame would allow them to use
the usual form of Newton's second law, but they would have
to convert all their data into a different frame of
reference, which would require cumbersome calculations.
Discussion Question
◊
If an object has a linear x-t graph in a certain inertial
frame, what is the effect on the graph if we change to a
coordinate system with a different origin? What is the
effect if we keep the same origin but reverse the positive
direction of the x axis? How about an inertial frame
moving alongside the object? What if we describe the
object's motion in a noninertial frame?
Summary
Vocabulary
weight — the force of gravity on an object, equal to mg
inertial frame — a frame of reference that is not accelerating,
one in which Newton's first law is true
noninertial frame — an accelerating frame of reference, in
which Newton's first law is violated
Notation
FW — weight
Other Notation
net force — another way of saying “total force”
Summary
Newton's first law of motion states that if all the forces
on an object cancel each other out, then the object
continues in the same state of motion. This is essentially a
more refined version of Galileo's principle of inertia,
which did not refer to a numerical scale of force.
Newton's second law of motion allows the prediction of an
object's acceleration given its mass and the total force on
it, acm=Ftotal/m. This is only the one-dimensional
version of the law; the full-three dimensional treatment
will come in chapter 8, Vectors. Without the vector
techniques, we can still say that the situation remains
unchanged by including an additional set of vectors that
cancel among themselves, even if they are not in the
direction of motion.
Newton's laws of motion are only true in frames of reference
that are not accelerating, known as inertial frames.
Exploring further
Isaac Newton: The Last Sorcerer, Michael White.
An excellent biography of Newton that brings us closer to the real
man.
Homework Problems
hw-blimp
h / Problem 6.
hw-mass-or-weight
i / Problem 10, part c.
1. An object is observed to be moving at constant speed in a
certain direction. Can you conclude that no forces are
acting on it? Explain. [Based on a problem by Serway and Faughn.]
2. A car is normally capable of an acceleration of
3 m/s2. If it is towing a trailer with half as much mass as
the car itself, what acceleration can it achieve? [Based on
a problem from PSSC Physics.]
3.
(a) Let T be the maximum tension that an elevator's
cable can withstand without breaking, i.e., the maximum force
it can exert. If the motor is programmed to give the car an
acceleration a, what is the maximum mass that the car can
have, including passengers, if the cable is not to break?(answer check available at lightandmatter.com)
(b) Interpret the equation you derived in the special cases
of a=0 and of a downward acceleration of magnitude g.
(“Interpret” means
to analyze the behavior of the equation, and connect that to reality, as
in the self-check on page 134.)
4. A helicopter of mass m is taking off vertically. The
only forces acting on it are the earth's gravitational force
and the force, Fair, of the air pushing up on the
propeller blades.
(a) If the helicopter lifts off at t=0,
what is its vertical speed at time t?
(b) Plug numbers
into your equation from part a, using m=2300 kg,
Fair=27000 N, and t=4.0 s.
5. In the 1964 Olympics in Tokyo, the best men's high jump
was 2.18 m. Four years later in Mexico City, the gold
medal in the same event was for a jump of 2.24 m. Because
of Mexico City's altitude (2400 m), the acceleration of
gravity there is lower than that in Tokyo by about
0.01 m/s2. Suppose a high-jumper has a mass of 72 kg.
(a) Compare his mass and weight in the two locations.
(b) Assume that he is able to jump with the same initial
vertical velocity in both locations, and that all other
conditions are the same except for gravity. How much higher
should he be able to jump in Mexico City?(answer check available at lightandmatter.com)
(Actually, the reason for the big change between '64 and '68
was the introduction of the “Fosbury flop.”)
6. A blimp is initially at rest, hovering, when at t=0 the
pilot turns on the motor of the propeller. The motor cannot
instantly get the propeller going, but the propeller speeds
up steadily. The steadily increasing force between the air
and the propeller is given by the equation F=kt, where k is
a constant. If the mass of the blimp is m, find its position
as a function of time. (Assume that during the period of
time you're dealing with, the blimp is not yet moving fast
enough to cause a significant backward force due to air
resistance.)(answer check available at lightandmatter.com)
∫
7. (solution in the pdf version of the book) A car is accelerating forward along a straight road.
If the force of the road on the car's wheels, pushing it
forward, is a constant 3.0 kN, and the car's mass is 1000
kg, then how long will the car take to go from 20 m/s to 50 m/s?
8. Some garden shears are like a pair of scissors: one sharp
blade slices past another. In the “anvil” type, however, a
sharp blade presses against a flat one rather than going
past it. A gardening book says that for people who are not
very physically strong, the anvil type can make it easier to
cut tough branches, because it concentrates the force on one
side. Evaluate this claim based on Newton's laws. [Hint:
Consider the forces acting on the branch, and the motion of the branch.]
9. A uranium atom deep in the earth spits out an alpha
particle. An alpha particle is a fragment of an atom. This
alpha particle has initial speed v, and travels a distance
d before stopping in the earth.
(a) Find the force, F,
that acted on the particle, in terms of v,d, and its mass,
m. Don't plug in any numbers yet. Assume that the force
was constant.(answer check available at lightandmatter.com)
(b) Show that your answer has the right units.
(c) Discuss how your answer to part a depends on all three
variables, and show that it makes sense. That is, for each
variable, discuss what would happen to the result if you
changed it while keeping the other two variables constant.
Would a bigger value give a smaller result, or a bigger
result? Once you've figured out this mathematical
relationship, show that it makes sense physically.
(d) Evaluate your result for m=6.7×10-27 kg,
v=2.0×104 km/s, and d=0.71 mm.(answer check available at lightandmatter.com)
10. You are given a large sealed box, and are not allowed to
open it. Which of the following experiments measure its
mass, and which measure its weight? [Hint: Which experiments would
give different results on the moon?]
(a) Put it on a frozen
lake, throw a rock at it, and see how fast it scoots away after
being hit.
(b) Drop it from a third-floor balcony, and
measure how loud the sound is when it hits the ground.
(c) As shown in the figure, connect it with a spring to the
wall, and watch it vibrate.
(solution in the pdf version of the book)
11. While escaping from the palace of the evil Martian
emperor, Sally Spacehound jumps from a tower of height h
down to the ground. Ordinarily the fall would be fatal, but
she fires her blaster rifle straight down, producing an
upward force of magnitude FB. This force is insufficient to levitate
her, but it does cancel out some of the force of gravity.
During the time t that she is falling, Sally is unfortunately
exposed to fire from the emperor's minions, and can't dodge
their shots. Let m be her mass, and g the strength of
gravity on Mars.
(a) Find the time t in terms of the other
variables.
(b) Check the units of your answer to part a.
(c) For sufficiently large values of FB, your
answer to part a becomes nonsense --- explain what's going on.(answer check available at lightandmatter.com)
12. When I cook rice, some of the dry grains always stick to the
measuring cup. To get them out, I turn the measuring cup upside-down, and hit the
back of the cup with my hand. Explain why this works, and why its success depends on
hitting the cup hard enough.
Footnotes
[1] Let's assume for simplicity that
there is no friction.
