Download In Chapters 2 and 3 of this course the emphasis is

Chapter 4: Forces and Newton's Laws of
Motion (Part 1)
Tuesday, September 17, 2013
10:00 PM
In Chapters 2 and 3 of this course the emphasis is on kinematics, the
mathematical description of motion. In Chapter 4, the heart of the course, we
shift the emphasis to explaining why changes in motions occur. This branch of
mechanics is called dynamics. Together, what we have learned and what we will learn,
kinematics and dynamics, form the foundation of Newtonian mechanics.
"Everything happens for a reason," as they say. In Newtonian mechanics the reason
always has to do with a force. Clarifying these vague statements is the goal of this
chapter. One perspective on these vague statements is causality, one of the
fundamental principles of science: Each effect has a cause.
But what is it about motion that is caused by a force? For Aristotle, one of the greatest
thinkers of the Ancient Greek era, the natural state of an object is to be at rest. Any
deviation from rest needed explaining in terms of some cause. This seems sensible; in
our experience, objects don't suddenly fly around for no reason. Things need to be
pushed or pulled in order to get started moving, and if you push something and then
stop pushing, then our experience is that it eventually stops. So it's natural that
Aristotle, and his followers for nearly two millennia, came to accept this as the truth
about motion.
And it is true, as far as it goes; but there are deeper truths, and it took Kepler, Galileo,
Newton, and others to tease out richer understandings. One of Newton's great
advances was to connect forces to acceleration, whereas previously scientists tried to
connect forces to velocity. It's natural to say that the harder you push something the
faster it goes, and if you stop pushing it slows down and stops. Newton built on the
work of Galileo, who used his imagination to consider motions in an ideal world
without friction. In the absence of friction, Galileo reckoned that an object moving at a
constant speed in a straight line would continue moving in the same straight line at the
same constant speed. Thus, Galileo argued, motion in a straight line at a constant
speed is just as "natural" as a state of rest. He shifted the central question from, "What
to humans do to create motion?" to the much more fertile question, "What are all the
influences on a moving object?" This led to the modern concept of force.
This fundamental shift in understanding helped to change our conception of "force"
from something exerted only by humans (and other animate creatures), to a more
general kind of phenomenon. It was gradually realized that inanimate objects could
also exert forces on other objects.
This was a great advance, but perhaps Galileo's greatest advance was to promote the
idea that understanding the natural world required mathematical and experimental
means. At the time, it was accepted that the highest form of natural philosophy
involved careful reading of Aristotle and commentaries and discussions of his works.
Galileo declared that the world was written in the language of mathematics, and to
understand it we need to perform experiments and make measurements to collect
numerical data above all else, not rely on the opinions of authorities, no matter how
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numerical data above all else, not rely on the opinions of authorities, no matter how
great. Therefore many people consider Galileo the founder of modern science. (The full
story is more complicated than is possible to summarize in these brief notes, but
Galileo does stand out as a decisive figure.)
We'll begin by examining some critical scenarios to determine your prior knowledge;
then we'll discuss the examples, and bring in the key concepts of the chapter.
Almost everyone entering a first-year university physics course has many
misconceptions about Newton's laws; this is normal and natural, because Newton's
laws are extremely non-intuitive. Remember that it took the world's greatest scientific
minds about 2000 years to correct the misconceptions held by virtually every
intelligent person, from the great Aristotle onwards. These great thinkers had these
misconceptions because they were natural; it took the unusual thinking of the great
Newton (and his great predecessors, including Kepler, Galileo, and others) to correct
these misconceptions. We should not feel bad if it takes us a few weeks or months to
correct our own misconceptions about motion.
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Q1: Two small spherical metal balls of the same size and shape are dropped from the
same height, about 10 m from the ground. One of the balls is twice as heavy as the
other. Which ball hits the ground first? How much sooner? A lot sooner, or only a little
sooner? Why?
Sample student responses:
• The balls hit the ground at the same time, because the velocities of the balls are
independent of the balls' masses.
• Without air resistance the acceleration due to gravity is the same for each ball, so their
velocities are the same.
• the heavier object should hit the ground first
• under ideal circumstances, they should both hit the ground at the same time
• if you neglect air resistance, they have the same acceleration (due to gravity),
independent of their masses
Analysis using Newton's laws of motion: The gravitational force exerted by the Earth
on an object of mass m is mg, at least near the Earth's surface. If air resistance is
neglected, then this is the only force acting on each of the balls. Thus, by Newton's
second law of motion, the acceleration of a ball with mass m is
a = F/m
a = mg/m
a=g
Notice that the mass of the ball cancels, and so the acceleration of a freely falling
object is the same, no matter what its mass is. (This is what we mean by "free fall:" no
other forces act except for gravity.) This is the prediction of Newton's second law, and
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other forces act except for gravity.) This is the prediction of Newton's second law, and
it has been verified countless times by very precise experiments.
If the balls both start from rest at the same height, and both have the same constant
acceleration, then they also have the same velocities, and the same position functions.
Thus, the two balls have exactly the same motions, and so hit the ground at the same
time.
Further discussion: Aristotle believed that heavier objects fall faster, because each
object has an inherent desire to return to the Earth, and the heavier the object the
greater the desire. Galileo argued against this as follows: If we take two identical
objects and place them so that they are touching, would they then fall twice as fast,
just by virtue of touching? Attach them with a thread if you like, so that they form one
single object; do you really believe that the single object would fall faster just because
of the thread? And do you really believe that they would suddenly slow down if the
thread were cut? Regardless of our beliefs, the final word belongs to experiment: Do
the experiment and measure for yourself what happens.
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Q2: The same two balls from Q1 are rolled off a horizontal table top with the same
initial speed. Which one lands farther from the edge of the table? How much farther? A
lot farther or only a little farther? Why?
Sample student responses:
• Because they have the same initial velocity, when they leave the table gravity acts on
each ball equally, so the two balls hit the floor the same distance away from the table.
• The x- and y-components of motion are independent; elaborates on first comment.
• The effect of friction is the same for both balls, both on the table and in the air.
• Won't inertia affect the heavier ball more than the lighter ball?
• The heavier ball goes further because its momentum is affected by its mass; doesn't
the heavier ball have greater momentum, and therefore goes further?
• If you slide two objects on a surface, why does the lighter one goes further?
• neglecting air resistance, the distances will be the same; gravity acts on the balls
independent of their masses
• maybe we need to account for friction on the table
• the lighter one travels farther, because there is no acceleration in the x-direction; they
will hit the ground at the same time, but the lighter one travels farther in the xdirection
Analysis using Newton's laws of motion: As we discussed in the previous question, the
two balls have exactly the same acceleration once they leave the table and are falling
freely. (Remember that this is only true provided that there is no air resistance; if there
is air resistance, the situation is much more complicated.) If they leave the table with
the same velocities, at the same time, then their subsequent motions will be identical.
Therefore they travel the same distance (and the same time) before they hit the
ground.
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• Won't inertia affect the heavier ball more than the lighter ball?
Yes. The heavier ball has the greater inertia, but the heavier ball also has the greater
force acting on it. The two factors exactly cancel out. (Remember that for a projectile,
you can treat the two components of the motion independently.) Note that mass is a
measure of inertia, so larger mass means larger inertia. In other words, the larger the
mass, the greater the resistance to acceleration.
a = F/m
a = mg/m
a=g
• The heavier ball goes further because its momentum is affected by its mass; doesn't
the heavier ball have greater momentum, and therefore goes further?
The heavier ball definitely has greater momentum than the lighter ball, as they have
the same velocity. This means that the heavier ball has a greater impact when it hits
the ground, so it can leave a larger dent, cause more damage, etc. However, how far
each ball goes depends on their velocities, not on their momenta; they have the same
velocity at each instant, so they have the same displacements.
• In Q1 and Q2, note that we must carefully distinguish between causes (i.e., forces) and
effects (i.e., accelerations). In Q1 and Q2, the causes are different, because the forces
acting on each ball have different magnitudes, but nevertheless the effects are the
same because the masses are in the same proportion as the forces. Remember:
a = F/m
a = mg/m
a=g
For advanced readers only: Note that there is a logical distinction between two
different concepts, that we use the same label m to represent. Inertia, also called
inertial mass, is the concept that appears in Newton's second law of motion. Inertial
mass represents the property of an object that resists acceleration. Gravitational mass
appears in the formula for the weight of an object, mg. Gravitational mass represents
the property of an object that is responsible for a gravitational force between two
objects.
We use the same symbol m to represent these two distinct concepts because all
experiments done so far support the hypothesis that these two distinct concepts are
numerically equal.
Newton was well aware of the logical distinction between inertial and gravitational
mass, and did experiments to test their equivalence. Many other experiments have
been done over the years to improve the accuracy; here are links to recent tests:
http://physicsworld.com/cws/article/news/2004/nov/16/equivalence-principle-passes-atomic-test
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http://einstein.stanford.edu/STEP/information/data/testq2.html
Einstein made the equivalence principle (the concept that inertial mass and
gravitational mass are equivalent) the foundation stone of his theory of gravity, which
supersedes Newton's theory of gravity and includes the latter as a special case. To
learn more about early experiments supporting the equivalence between inertial and
gravitational mass, try this link:
http://en.wikipedia.org/wiki/E%C3%B6tv%C3%B6s_experiment
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Q3: A heavily-loaded transport truck collides head-on with a very small car. The mass of
the truck is much greater than the mass of the car. Which vehicle exerts the greater
force on the other vehicle during the collision? A lot greater or only a little greater?
Sample student responses:
• The truck exerts a greater force on the car than the car exerts on the truck.
• the small car exerts more force than the truck
• the forces will be dependent on the speeds and the masses
• by Newton's second law (F = ma), the truck will exert a greater force because it has a
greater mass
• they will exert the same force but the smaller car will have the greater acceleration,
because it has the smaller mass
• maybe they exert the same force on each other, because the forces should be equal
and opposite
Analysis using Newton's laws of motion: By Newton's third law of motion, the
magnitude of the force exerted by the small car on the heavy truck is equal to the
magnitude of the force exerted by the heavy truck on the small truck.
It doesn't seem this way, because the effects on each participant in the collision are
different, and this is what is hard to wrap our minds around.
Consider an insect that hits the windshield of a fast-moving car. The force that the
windshield exerts on the insect may be enough to destroy it. The force that the insect
exerts on the windshield has the same magnitude, but has negligible effect on the
windshield. The collision of the heavy truck and the small car is similar.
Just because the force of A on B has the same magnitude as the force of B on A, it's not
necessarily true that the resulting accelerations of A and B will be the same. The
insect's mass is so small that it's acceleration upon collision with the windshield is
enormous. The windshield (and the car its attached to) is so massive that it's
acceleration after collision with the insect is not noticeable. Similarly, the small car
colliding with the heavy truck has a much greater acceleration than the heavy truck.
Don't confuse the equal forces on the small car and heavy truck with the unequal
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Don't confuse the equal forces on the small car and heavy truck with the unequal
effects: the damage done to each, and the acceleration of each.
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Q4: A ball is swung in a horizontal circle at a constant speed. The string suddenly
breaks. As seen from above, which path does the ball follow after the string breaks?
Car moving around a circular curve, hits a patch of ice:
Analysis using Newton's laws of motion: Once the string breaks there are no
longer any forces in the horizontal plane in which the ball was moving. Thus, the
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longer any forces in the horizontal plane in which the ball was moving. Thus, the
ball follows its tendency to continue moving in a straight line at a constant speed,
according to Newton's first law of motion. (The ball "flies off on a tangent.") That
is, the horizontal component of the ball's velocity will remain constant once the
string breaks. (To make the discussion simple, we are ignoring gravity, which acts in
the vertical direction.)
When you are using an electric mixer to mix the batter for a cake, if you bring the
mixer out of the bowl, the batter will splatter from the mixer in all directions. But
observe this carefully and you will see that the batter does not splatter radially, but
more in a pinwheel pattern:
The splatter from the mixer can also be explained using Newton's first law.
Just like the ball on a string whirling in a circle, or the car driving around the
curve, the batter is held to the mixer's blades by attractive forces (it's
"sticky"). However, if the sticky forces are not strong enough, then the batter
releases from the blades, and "flies off on a tangent."
Industrial centrifuges work similarly; the lettuce spinner you have in your
kitchen works in the same way.
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Q5: A hockey puck is sliding with constant speed in a straight line from A to B on a
frictionless horizontal ice rink. When the puck reaches B, it receives a brief hit from
a hockey stick in the direction of the arrow. Here's a view from above:
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Which path does the puck follow after being struck by the hockey stick? Explain.
Sample student responses:
• If the applied force from the stick is greater than the previous force, then the puck will
go in the direction of the greater force. (No. To apply Newton's second law, you
substitute the vector sum of all forces acting on the object for F in the formula. But
once the puck is moving towards the right, the first force is gone; the first force is not
carried along with the puck, and no longer acts on the puck. When the stick hits the
puck later, it is the net force acting on the puck, and it only acts for a short time.)
• If the two forces are perpendicular, then wouldn't the result be somewhere in
between? (Remember, the two forces do not act at the same time. To analyze the
subsequent motion of the puck after the second force acts, you only need to use the
second force.)
• Wouldn't you treat this using vectors? (Yes.)
• The change in direction and force would be the "net" of the initial force and the
applied force from the stick. (No. When the second force acts, it is the net force acting.
The first force no longer acts.)
• Wouldn't Newton's third law be applicable here? There is a countering of forces; when
the stick hits the puck, the puck also acts on the stick, and when we look at the
direction that the puck is going in, we're looking at acceleration not force. (No.
Consider the force that the puck exerts on the stick and the force that the stick exerts
on the puck. These two forces act on different objects, and so we don't add them up.
Only add forces acting on one object.)
• Isn't it possible to make the puck go North if you hit the puck very hard? (Yes, but you
can't hit the puck in the direction shown. You must hit the puck in a different direction,
as shown in the figures below:
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Analysis using Newton's laws of motion: Initially the puck has an x-component of
velocity, but zero y-component of velocity. The force exerted by the stick on the puck is
entirely in the y-direction; there is no x-component of force. By Newton's second law of
motion, the puck will experience a brief acceleration in the direction of the applied force
(that is, the y-direction) while it is in contact with the stick. There is no acceleration in the
x-direction. Thus, the x-component of the puck's velocity remains constant, and the ycomponent of the puck's velocity increases while the puck is in contact with the stick.
Once the puck leaves contact with the stick, the net force acting on it is zero (gravity and
the normal force exerted by the ice on the puck balance to zero). Thus, after the puck
leaves contact with the stick, its velocity remains constant, and so it continues to move in
a straight line at a constant speed.
Thus, "b" is the correct path.
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Q6: When the puck is moving on the frictionless path you have chosen in the previous
question, the speed of the puck
1. is constant.
2. gradually increases.
3. gradually decreases.
4. increases for a while and then decreases.
5. is constant for a while and then decreases.
Explain.
Analysis using Newton's laws of motion: Some students believe that the speed of the
puck will decrease because the force carrying the puck along in its motion dissipates.
There is no such force; once the puck leaves the stick, the net force is zero. Thus, the
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There is no such force; once the puck leaves the stick, the net force is zero. Thus, the
speed of the puck is constant after it leaves the stick, in accord with Newton's first law of
motion.
Remember that objects do NOT carry forces along with them. Forces act on objects, but
objects don't have forces. An object can have mass, have a colour, have a shape, have a
position, and so on, but an object cannot have a force. As usual, one of the difficulties in
learning physics is to understand the precise way language is used in physics, which is
sometimes different from the every-day usage of language. In every-day language, we
might say an argument is forceful, or carries a lot of force, or that a person is a force of
nature, but this is not the way we use the word "force" in physics.
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Q7. A ball is thrown horizontally from the top of a cliff. Which path does the ball take?
Explain.
Analysis using Newton's laws of motion: Path number 2 is correct. The horizontal component
of velocity is constant, but the speed in the vertical direction steadily increases. Thus, the path
of the object is not a straight line, and so this eliminates path 1. Paths 3, 4, and 5 can be
eliminated because they suffer from the "Coyote and Roadrunner" problem; that is, the object
goes straight out from the cliff for a while. This could only happen if there were no net force
acting on the object; we know this is incorrect, because gravity acts on the object, so it begins
to accelerate immediately. Thus path 2 is correct.
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Q8: An engine accidently falls off an airplane as it is in flight. Which is the path of the engine as
it falls to Earth (from the perspective of an outside observer)? Explain.
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Sample student responses: 4 is the most popular choice.
Analysis using Newton's laws of motion: This is virtually the same situation as
in the previous problem. When the engine leaves the airplane, its initial
velocity is the same as the airplane's velocity. Thus, the engine behaves in the
same way as the ball thrown from the top of a cliff in the previous problem, for
the same reasons.
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Q9: You are below-deck on a ship, or in an airplane. The ship or airplane moves
at a constant speed in a straight line. You reach overhead and drop a ball. The
ball lands … behind you? in front of you? at your feet? Depends on the speed
of the ship/airplane? Depends on something else? Explain.
Analysis using Newton's laws of motion: You will certainly have experienced
being in a car moving at an approximately constant velocity on a highway, or
being on an airplane moving at an approximately constant velocity. (And by
"constant velocity" we mean "constant speed moving in a straight line".) If you
toss an object inside the car or inside the airplane, it behaves as if you were at
rest in your living room at home.
In other words, being in a car or airplane moving at a constant velocity, objects
inside the car behave just as they do "at rest" in your living room. Another way
to say this is that the car and the airplane are examples of inertial reference
frames.
An inertial reference frame is one in which Newton's first law is valid. Once
you find one such reference frame, any other reference frame moving with
constant velocity relative to it is also an inertial reference frame.
Thus, the ball drops straight down, hitting the ground at your feet.
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Q10: A large truck breaks down on the road and receives a push back to the
station by a small car. Compare the force that the car exerts on the truck with
the force that the truck exerts on the car when the car is accelerating up to its
cruising speed.
Analysis using Newton's laws of motion: By Newton's third law of motion, the
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Analysis using Newton's laws of motion: By Newton's third law of motion, the
magnitude of the force exerted by the car on the truck is the same as the
magnitude of the force exerted by the truck on the car.
Q 10A: But don't these two forces cancel, so that the net force is zero? How
then could the vehicles accelerate?
No, the forces do not cancel, because they act on different object. Remember
that all forces occur in pairs, acting on different objects. If A exerts a force on
B, then B exerts an equal and oppositely-directed force on A. One force acts on
A, the other force acts on B, so the two forces do not cancel.
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Q11: A large truck breaks down on the road and receives a push back to the
station by a small car. Compare the force that the car exerts on the truck with
the force that the truck exerts on the car when the car is moving at a constant
cruising speed.
Analysis using Newton's laws of motion: By Newton's third law of motion, the
magnitude of the force exerted by the car on the truck is the same as the
magnitude of the force exerted by the truck on the car.
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Q12: An elevator is being lifted up at a constant speed by a cable. Compare the
force exerted by the cable on the elevator to the gravitational force exerted by
the Earth on the elevator. (All other forces are negligible.) Explain.
Analysis using Newton's laws of motion: The only forces acting on the
elevator are gravity and the force exerted by the cable on the elevator.
Because the elevator is moving at a constant speed in a straight line, the net
force acting on the elevator is zero, according to Newton's first law. Thus, the
gravitational force on the elevator and the force exerted by the cable on the
elevator add up to zero; thus, the two forces have the same magnitude and
opposite directions.
Note the difference between this question and Q10A. In Q10A, the two forces
being considered act on different objects, so we are not allowed to add them
together. In Q12, the two forces being considered act on the same object, so
we are allowed to add them to determine the total force acting on the object.
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Q13: Despite a very strong wind, a tennis player hits a tennis ball with her
racquet so that the ball passes over the net and lands in her opponent's court.
After the ball leaves the racquet, and before it hits the ground, which forces
act on the ball? Gravity? The force from being struck by the racquet? A force
exerted by the air? Other forces? All of the above? None of the above? Explain.
Analysis using Newton's laws of motion: Once the ball leaves the racquet, and
before it hits the ground, the only forces acting on the ball are the
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before it hits the ground, the only forces acting on the ball are the
gravitational force exerted by the Earth on the ball, and the force from the
wind. There is no force that "follows" the ball in its motion; once it leaves the
racquet, there is no more force exerted by the racquet on the ball.
Remember, forces are not carried by objects; forces are exerted on objects.
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OK, now that we have looked at a number of specific examples, let's dig into
the key concepts of Newtonian dynamics.
Force
What is a force? One definition of a force is a physical influence of one object acting on another
object.
An object can "have" energy, an object can "have" momentum, an object can "have" mass or
velocity, but an object CANNOT "have" force in the same way. Remember we are speaking
about the technical, physics definitions of these words, not the every-day meanings of these
words.
An object can exert a force on another object, but an object cannot have force.
A force always acts between two objects; one object exerts a force on another object, and the
second object "feels" the force exerted by the first object on it.
We often think of forces as pushes or pulls. A force requires an agent; that is, something must
be doing the pushing or pulling.
Forces can be modelled as vectors; that is, a force has magnitude and direction. This is
confirmed by numerous careful experiments.
The SI unit of force is the newton (N).
A short catalogue of forces (Section 4.3 in the textbook):
• weight; i.e. the force that Earth's gravity exerts on an object
The magnitude of the gravitational force that the Earth exerts on an object of mass m can be
written as W = mg or, equivalently, as F = mg; this is an excellent approximation provided that
the object is not too far from the Earth's surface.
• spring force
If you pull on a spring, the spring pulls back on you. If you push on a spring, it pushes back on
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If you pull on a spring, the spring pulls back on you. If you push on a spring, it pushes back on
you. The magnitude of the force exerted by the spring is proportional to the amount by which it
has been stretched or compressed, and in the opposite direction of the stretch or compression:
F = kx
This relationship (known as Hooke's law) is an excellent approximation provided that the
magnitude x of the stretch or compression is not too large. The constant of proportionality k is
called the stiffness of the spring, and is larger for stiffer springs.
Physics majors: It turns out that many situations can be modelled by forces that satisfy the
same relationship as Hooke's law, even though no springs are involved. For example, one can
model inter-atomic forces using a similar relation, with good results; that is, just pretend that
the electrical forces acting between atoms are just springs, use the same mathematics as for
springs, and sure enough your predictions about how the atoms behave is pretty good.
This happens over and over again in physics; the same mathematical models come up over and
over again in different situations. The model of small-amplitude vibrations experienced by a
small mass attached to the end of a spring, called simple harmonic motion, is a classic paradigm
in physics. We'll explore this model in Chapter 14, later in the course. The same mathematics is
also used to describe everything from electrical oscillations in tuned circuits, reception and
transmission by antennas, and many other kinds of oscillations. There are even quantum
versions of simple harmonic motion that are applied to microscopic systems, such as the
vibrations of atoms and molecules. Even the fundamental behaviour of electromagnetic fields
in quantum field theory uses the quantal version of simple harmonic motion as a basic model.
The moral: Learn the simple harmonic motion paradigm well, because it is used all over physics.
• tension force
When a string is pulled from each end, or is attached to a fixed object and is pulled from the
other end, we say the string is under tension. If you hang a chandelier from the ceiling, the
string or chain that holds the chandelier up exerts an upward force on the chandelier. Such
forces are called tension forces.
• normal force
When you sit on a chair the Earth exerts a gravitational force on you downwards. You would fall
through the chair unless the chair exerted an upward force on you to balance the gravitational
force. The force exerted on you by the chair is called a normal force, because it acts
perpendicular to the surface of the chair; "normal" is another word for "perpendicular".
• friction
Each surface, no matter how smooth, as microscopic protrusions. When two surfaces rub
against each other, the protrusions from each surface bump into each other; this manifests
macroscopically as a resistive force that we call friction.
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macroscopically as a resistive force that we call friction.
• drag
Drag is another word for "air resistance", except that drag is more general, because it can apply
to other fluids too. For example, when you feel resistance while walking through water in a
swimming pool, this is a type of drag.
• engine thrust
• electric forces
• magnetic forces
Classification of force types
• contact forces and non-contact forces
Forces such as normal forces, tension forces, spring forces, friction and drag, and similar forces,
are called contact forces because they arise because two objects are touching. However,
electrical forces, magnetic forces, and gravitational forces are more mysterious, because they
operate even though the object exerting the force and the object on which the force is exerted
are not in contact.
• four fundamental types of forces:
gravitational force, strong nuclear force, weak nuclear force, electromagnetic force
The two "nuclear" forces are "short-range" forces that act inside atoms; we won't discuss them
any further in this course. We call them "short-range" forces because they diminish in
magnitude so rapidly with distance that for all practical purposes we can pretend that they
don't exist outside of atoms.
The other two types of forces, gravitational and electromagnetic, are "long-range" forces. In
other words, although their magnitudes diminish with distance, the decrease in their strength is
not so rapid, so that their effects can still be felt over very long distances.
Virtually all of the forces that we deal with in this course that are not gravitational are
ultimately electromagnetic. For example, the forces that hold atoms together in molecules (and
that hold molecules together) are electromagnetic. Thus, you can pretend that your chair is
somewhat like the top of a trampoline, where the atoms are connected by minute springs.
When you sit in the chair, it's as if the surface of the chair flexes a little, and exerts an elastic
force upward on you. In reality, there are no springs; what holds the atoms together is
electromagnetic forces. Thus, normal forces are ultimately electromagnetic in origin. Similarly,
friction, tension, and spring forces are all electromagnetic in origin.
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friction, tension, and spring forces are all electromagnetic in origin.
• the theme of unification in physics
In modern physics, a theme that has resulted in a lot of progress in our understanding of the
universe is that of unification. For example, in the 1860s, Maxwell unified (to some extent)
electricity and magnetism, which were before then thought to be two separate interactions.
Thanks to his work, in which he (among other things) predicted that light is an electromagnetic
wave, a sort of dance involving oscillations in both electric and magnetic fields, electricity and
magnetism were recognized as being intimately connected. This intimate union was recognized
as being even closer thanks to the work of Einstein (with his theory of special relativity in 1905),
where the difference between electric and magnetic fields was seen to be simply one of
perspective.
Einstein then embarked on the task of unifying electromagnetic fields with gravitational fields;
that is, he hoped to show that each type of field was simply an aspect of some deeper kind of
field, which could appear to be electromagnetic from some perspectives and could appear to
be gravitational from other perspectives.
Alas, Einstein was never able to unify electromagnetic fields with gravitational fields. To make
matters more complicated, the two nuclear force fields were discovered late in his life.
Interestingly, the two nuclear force fields were unified with electromagnetic fields thanks to the
work of many physicists (crowned by the work of Weinberg, Glashow, and Salam in the 1970s).
Gravity remains the oddball.
A related problem is to develop quantum versions of all physics theories about fundamental
force fields. We have excellent quantum theories for electromagnetic fields, and for the strong
and weak nuclear force fields, but as yet there is no quantum field theory of gravitational fields.
(String theory is an attempt, but it is not yet a theory, and so remains only a tantalizing search
in the dark.)
Newton's first law of motion
Modern statement:
If the total force (net force) acting on an object is zero, then (a) if the object started out at rest,
it will remain at rest, and (b) if the object was moving in a straight line at a constant speed (i.e.,
if its velocity was constant) then the object will remain moving in a straight line at a constant
speed.
In Newton's own words:
Every body perseveres in its state of being at rest or of moving uniformly straight forward
except insofar as it is compelled to change its state by forces impressed.
(see <http://plato.stanford.edu/entries/newton-principia/#NewLawMot>)
Newton's first law of motion is also called the law of inertia. Inertia is the tendency of an object
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Newton's first law of motion is also called the law of inertia. Inertia is the tendency of an object
to remain at rest or moving in a straight line at a constant speed. That is, if an object is not
acted upon by external forces, then it will remain in a state of inertia, which means it will
continue in a straight line at constant speed (if that was its initial state), or it will remain at rest
(if that was its initial state).
Physics majors: Note that there is more to Newton's first law of physics than we have described
so far. Newton's first law is not just a special case of Newton's second law; that is, if we
substitute 0 for F in Newton's second law, we obtain a = 0, which is equivalent to Newton's first
law: If F = 0, then a = 0.
Newton's first law is more than this. Newton's first law asserts the existence of a reference
frame in which it is valid. That is, there exists a reference frame in which if the net force on an
object is 0, then its acceleration is also 0. Such a reference frame is called an inertial reference
frame. Furthermore, any reference frame that moves with constant velocity relative to an
inertial reference frame is also an inertial reference frame. Thus, as soon as we find one inertial
reference frame, there are in infinite number of others.
In practice, we typically work with reference frames that are only approximately inertial. For
example, the reference frame that is at rest with respect to you while you are sitting at a desk
in your home is approximately inertial, but not exactly so, because the Earth rotates; rotation is
a kind of acceleration, and fouls things up. Imagine being on a carousel; strange things happen,
even though there are no forces acting. The carousel is not an inertial reference frame.
You'll learn how to deal with non-inertial reference frames in second-year mechanics. There are
important consequences to being in a rotating reference frame; for example, the typical
cyclone action of winds in the northern hemisphere is due to the rotation of the Earth. Do a
search on the term "Coriolis forces" if you would like to learn more about this, but remember
that Coriolis forces are not real forces, but only apparent forces, and arise because we observe
phenomena from the perspective of a non-inertial reference frame. Similarly, when you feel
yourself thrown forward in a braking car, or thrown towards the passenger door in a car that is
turning, you are feeling fictitious forces because your perspective is a non-inertial reference
frame. The forces are not real, we just think they are because of our perspective.
Examples of Newton's first law:
a book resting on a desk; gravity acts downward and the normal force from the desk acts
upward; the net force on the book is zero, so the book remains in a state of rest
shovelling snow or gravel: you accelerate the gravel, but then stop the shovel abruptly; the
gravel flies off the shovel thanks to its inertia
stamping your feet to get the snow off your boots: similarly to shovelling gravel, you accelerate
your feet (and the snow on your boots), but then your boots abruptly stop when they meet the
ground; the hope is that the snow will continue to move by its inertia, and leave your boots
(sometimes this doesn't work because the force holding the snow on your boots is strong
enough to accelerate the snow and keep it on your boots)
shaking water off your hands before you put them under the dryer in a public washroom: same
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shaking water off your hands before you put them under the dryer in a public washroom: same
as the previous two examples
throwing a ball: same as the previous examples … just like shovelling gravel
further examples: how you feel in the passenger seat when a car brakes suddenly; how you feel
in the passenger seat when a car turns around a curve at high speed; explain both of these
using Newton's first law of motion as we did in class.
• contrast "getting into motion" vs. "remaining in motion"; getting into motion from rest requires a
force; remaining in motion (as long as you're moving in a straight line at a constant speed) does
not require a force; thus, for an object moving in a straight line at a constant speed, the net force
acting on it is zero
Newton's second law of motion
What can forces do to an object? They can cause an object to:
• speed up
• slow down
• change its direction of motion
All three of these possible types of effects can be categorized as accelerations. Thus, acceleration
means more than just speeding up or slowing down.
Newton's second law of motion has several aspects. The first is that the resulting acceleration is
in the same direction as the net force. The second is that the magnitude of the acceleration is
related to the magnitude of the force.
Newton's second law of motion has been accurately verified by numerous experiments.
Newton's second law of motion is (in modern vector form)
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Newton's second law of motion is (in modern vector form)
F = ma
which is equivalent to
a=F/m
Newton's second law in Newton's own words:
A change in motion is proportional to the motive force impressed and takes place along the
straight line in which that force is impressed.
Pasted from <http://plato.stanford.edu/entries/newton-principia/#NewLawMot>
We'll see many examples of applying Newton's second law of motion in Chapter 5.
Newton's third law of motion
Forces always come in pairs that are equal in magnitude and opposite in direction. To be more
specific, Newton's third law states that:
If object A exerts a force FAB on object B, then object B automatically exerts a force FBA on object
A; the magnitude of the two forces are the same, and their directions are opposite:
FBA = FAB
Another common way that some people say Newton's third law: For every action there is an
equal and opposite reaction. If you use this version, remember that the action force and the
reaction force act on different objects!
Examples:
push a skater in the back on an ice rink: the person you push will feel a force from you, and yet
you will also automatically feel a force pushing you in the opposite direction
paddling a canoe: you push the water backwards with your paddle, and automatically the paddle
experiences a force pushing it forward; you and the canoe, which are attached to the paddle,
also experience this forward force, which is why the canoe moves forward when you push the
water backwards with your paddle
Also note that only external forces can move the centre of mass of an object; external forces
change the motion of an object; internal forces are not effective at changing the state of motion
of the centre of mass of an object. For example, if you try to move a sailboat by turning on an
electric fan that is sitting in the boat and directing the fan towards the sail, then this will be
utterly ineffective. Draw a diagram and sketch in the forces to see why this is so!
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utterly ineffective. Draw a diagram and sketch in the forces to see why this is so!
Contrast this situation with an "everglades boat"; in these boats, the electric fan pushes air
backwards; this is exactly like paddling a canoe, except air is pushed backwards instead of water;
the result is the same, the reaction force pushes the boat forward. Airplanes and boats that have
propellers work in the same way; the propellers push air or water backwards, and the reaction
force pushes the plane or boat forwards.
In the early days of rocketry, there were many people who couldn't understand that rockets
could work in space, because there is no air to push against. They were thinking in terms of
propellers or paddles, which push against air or water, and then the reaction force pushes the
plane or boat in the opposite direction. Nevertheless, even though there is no air to push against,
rockets in space work in the same way, according to Newton's third law of motion: Hot
combustion gases are pushed out the back of the rocket at high speed by the combustion
chamber; the reaction force of the gases on the combustion chamber pushes the combustion
chamber (and the rest of the attached rocket) forward.
Examples of force analysis:
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Free-body diagrams
• a tool for analyzing problems involving forces and Newton's laws of motion
• we'll look at them in more depth in Chapter 5
Some further random thoughts:
"Motion is caused by forces." … pretty vague
"Acceleration is caused by forces." … this is more precise, and it is also correct
"Motion in a straight line at a constant speed happens for no reasons." … to contrast with "everything
happens for a reason"; well, this statement is also a bit vague, but it's OK if you interpret it correctly; it
gets back to what is "natural"; according to Newton's first law, an object at rest or moving at a
constant speed in a straight line will keep doing what it's doing unless a force acts … in other words,
there is no need to explain motion in a straight line at a constant speed; it occurs if there are no forces
acting
Chapter Summary
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Chapter Summary
Three key points of this chapter:
1. What is a force?
• be careful to distinguish between force, energy, power, momentum, speed, etc.; remember that these
are all technical terms that have very precise meanings in physics, even though some of them may be
used interchangeably or in similar ways in everyday speech
• some people find it hard to understand that inanimate objects can exert forces; they think that only
living things can exert pushes or pulls. For example, when you sit in a chair, gravity exerts a downward
force on you, and the chair exerts a balancing upward force on you. But some people can't understand
that the chair, an inanimate object, can exert a force on you; they figure the chair "just gets in the
way."
• some people don't recognize that certain forces are really forces; they just think of them as influences.
For example, they might think of friction as "what makes a moving object stop" without
acknowledging that friction is a force. They might think of gravity as "what makes an object fall"
without realizing that gravity is a force.
• some people have a difficult time realizing that forces are not inherent in an object (i.e., objects can't
have forces), but rather that a force is exerted by one object by another object. For example, when a
ball is thrown up, some people believe that after the ball leaves the hand it is still acted upon by a
"force of the throw", as if this "force of the throw" is somehow carried along with the ball; as if the ball
has the force of the throw within it. This is not so; the hand exerts a force on the ball while they are in
contact; once the ball leaves the hand, the hand no longer exerts a force on the ball.
• many people find it easier to understand the perspective of a person that exerts forces on other
objects, but difficult to understand the perspective of an object that has several forces exerted on it,
including forces from inanimate objects. The latter perspective is critical for drawing free-body
diagrams, the first step in an analysis using Newton's laws of motion.
2. What is the connection between force and motion?
Based on our long experience with everyday objects, it's natural for us to think about motion in
Aristotelian terms. Many people have internalized a number of erroneous beliefs, and it is difficult
indeed to replace these erroneous beliefs with the deeper insights of modern physics. Examples of
some of these erroneous beliefs are:
• If there is no net force acting on an object, then the object is either at rest or will immediately come to
rest. (Why is this false? Explain and give examples.)
• If an object is at rest, then the net force acting on it is not necessarily zero. (Explain why this is false.)
• An object in motion requires a force to keep it in motion; that is, force causes motion. (Why is this
false? Explain and give examples.)
• Force produces motion; the greater the force, the greater the velocity. (Why is this false? Explain.)
3. How are forces that act between different objects related?
Newton's third law of motion is difficult to understand. For a collision between two objects of very
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Newton's third law of motion is difficult to understand. For a collision between two objects of very
different mass, it's hard for some people to accept Newton's third law, because they think the differing
effects produced by the collision (different amounts of damage, different resulting accelerations)
implies that the forces acting on each object are different. This is challenging to wrap your head
around, and well worth considerable reflection.
Some people have difficulties with action-reaction force pairs, making a number of common errors
such as placing the two forces on the wrong object, placing the two forces on the same object, or not
realizing that long-range force such as gravity also participate in action-reaction force pairs.
Some final comments on the mechanical world view according to Newton:
• determinism vs. free will
"We ought then to regard the present state of the universe as the effect of its anterior state and as the
cause of the one which is to follow. Given for one instant an intelligence that could comprehend all the
forces by which nature is animated and the respective situation of the beings who compose it an
intelligence sufficiently vast to submit these data to analysis it would embrace in the same formula
the movements of the greatest bodies of the universe, and those of the lightest atom; for it, nothing
would be uncertain and the future, as the past, would be present to its eyes." Laplace, 1814
This deterministic mechanical world view was the dominant paradigm throughout the 1800s.
However, it was already shown to be impossible in principle by Einstein's special theory of relativity
(1905), and was subsequently strongly contradicted (for a quite different reason) by quantum
mechanics (1920s); and let's not forget chaos theory, developed starting in the late 20th century,
where small changes in initial conditions lead to widely divergent final conditions; this means that
measuring the initial positions and velocities of all the particles in a system (a la Laplace) is insufficient
for determining all later positions, even approximately; that is, inevitable small measurement
uncertainties in initial positions and velocities make the final positions and velocities unpredictable.
Interestingly, Newton was a dualist. He believed that the universe contained two types of substances,
matter (which is subject to the laws of physics) and mental substance, such as the soul (which is
subject to free will).
Currently, the fundamental laws of microscopic physics are non-deterministic, and the question of free
will is still open, although it seems that we have free will. It's also not clear whether science can shed
light on the issue, but one never knows what will be discovered by one of us inventive humans down
the road.
Moving beyond Newton's laws of motion
Although Newton's laws of motion have been very accurately verified in an uncountable number of
experiments, both experimental and theoretical developments in the 20th century have helped us to
realize the limitations of Newtonian mechanics. Newtonian mechanics is a tremendously good theory
for may macroscopic applications, such as constructing buildings, bridges, machines, and for predicting
the motions of planets, spacecraft, baseballs, cars, and so on.
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However, Newtonian mechanics fails utterly in the subatomic realm, and also in the realm of
extremely fast motions (once speeds that are a significant fraction of the speed of light are achieved).
New theories of mechanics have been created in the 20th century, and continue to be developed
today: Quantum mechanics helps us explain the innards of atoms (and also the inter-atomic forces
that form the basis of chemistry), and relativistic mechanics (special relativity) helps us explain
phenomena that occur at very high speeds. (Perhaps surprisingly, special relativity is important in
making GPS accurate, due to the need for making very high-precision time measurements.)
• the mechanistic program according to Newtonian mechanics (for calculus lovers/physics majors only):
An advanced way to think about Newton's second law of motion is as a differential equation; the
acceleration is the second derivative of position, so a = F/m relates the second derivative of the
position function to the force function and the mass of the object. Depending on the nature of the
force function (is it constant? does it vary with position? does it vary with time? does it vary with
velocity?), there are different techniques for solving the resulting differential equation. You'll learn
about this in second-year mechanics.
A differential equation is not like an algebraic equation; in an algebraic equation, the solution is a
number or some numbers. In a differential equation, the solution is a function; in this case, what we
are after is the position function of the moving object.
Thus, Newton's second law does not tell you what is; it tells you how things change. To be more
specific, Newton's second law does not specify the position function of a moving object, but only
specifies (once the forces are given) the second derivative of the position function. To determine the
position function, which is often the quantity of interest, one must solve the differential equation, in a
process akin to integration (anti-differentiation). The constants of integration that result from the
solution process are essential, because they allow us to model all possible motions; which motion
actually occurs depends on the initial conditions of the motion, which then specify the constants of
integration.
The same paradigm occurs over and over again in physics. Many of the most important laws of physics
can be expressed as differential equations (some ordinary differential equations, some partial
differential equations), and many of them are second-order differential equations (i.e., involving the
second derivative). As such, they have determinism built into them, because there are mathematical
theorems that state that for such equations, once initial conditions (and perhaps also/or boundary
conditions) are specified, then the equations have unique solutions. Thus, this fits in with Laplace's
dream that once all initial conditions are known, the future is determined.
It is worth thinking about this state of affairs, and as you learn more about both the mathematics and
physics involved, observe how your perspective on these issues of determinism evolves. For example,
how can a deterministic law of physics be useful when we know that in principle the world is not
deterministic?
Such conundrums are worth pondering, and helps one to engage with the mysteries of the universe,
and also with the depths of our best theories of physics, one of our avenues for probing the mysteries
of the universe.
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