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Contents

Section 4.1 - Force

Section 4.2 - Newton's first law

Section 4.3 - Newton's second law

Section 4.4 - What force is not

Section 4.5 - Inertial and noninertial frames of reference

Section 4.6 - Summary

Section 4.1 - Force

Section 4.2 - Newton's first law

Section 4.3 - Newton's second law

Section 4.4 - What force is not

Section 4.5 - Inertial and noninertial frames of reference

Section 4.6 - Summary

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.

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.

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.

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.

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.

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.

In physics, an object's weight, \(F_W\), 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.

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.

◊

In chapter 0, I defined 1 N as the force that would accelerate a 1-kg mass from rest to 1 m/s in 1 s. Anticipating the following section, you might guess that 2 N could be defined as the force that would accelerate the same mass to twice the speed, or twice the mass to the same speed. Is there an easier way to define 2 N based on the definition of 1 N?

We are now prepared to make a more powerful restatement of
the principle of inertia.^{1}

In other words, an object initially at rest is predicted to remain at rest if the total force acting 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.

\(\triangleright\) 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.

\(\triangleright\) 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.

\(\triangleright\) 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?

\(\triangleright\) 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.

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.

\(\triangleright\) Describe the forces acting on a person standing in a subway train that is cruising at constant velocity.

\(\triangleright\) 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.

\(\triangleright\) 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 … | − 10,000 N | − 20,000 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.

\(\triangleright\) If you drive your car into a brick wall, what is the mysterious force that slams your face into the steering wheel?

\(\triangleright\) 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.

◊

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?

x textupm | t textups |

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 |

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 (e.g., the one described in example 11 on p. 138) 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:

\[\begin{align*}
&a = F_{total}/m , \\
\text{where}
&\text{$m$ is an object's mass}\\
&\text{$F_{total}$ is the sum of the forces acting on it, and}\\
&\text{$a$ is the acceleration of the object's center of mass.}
\end{align*}\]

We are presently restricted to the case where the forces of interest are parallel to the direction of motion.

We have already encountered
the SI unit of force, which is the newton (N). It is designed so that the
units in Newton's second law all work out if we use SI units: \(\text{m}/\text{s}^2\)
for acceleration and kg (*not* grams!) for mass.

\(\triangleright\) Let's choose our coordinate system such that positive is up. Then the downward force of gravity is considered negative. Using Newton's second law,

\[\begin{align*}
a &= \frac{F_{total}}{m} \\
&= \frac{F_t-F_g}{m} \\
&= \frac{(5.9\times10^6\ \text{N})-(5.0\times 10^6\ \text{N})}{5.1\times 10^5\ \text{kg}} \\
&= 1.6\ \text{m}/\text{s}^2 ,
\end{align*}\]

where as noted above, units of N/kg (newtons per kilogram) are the same as \(\text{m}/\text{s}^2\).

\(\triangleright\) 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?

\(\triangleright\) We solve Newton's second law for \(F_{total}=ma\), and substitute \(\Delta v/\Delta t\) for \(a\), giving

\[\begin{align*}
F_{total} &= m\Delta v/\Delta t \\
&= (2000\ \text{kg})(25\ \text{m}/\text{s} - 0\ \text{m}/\text{s})/(34\ \text{s}) \\
&= 1.5\ \text{kN} .
\end{align*}\]

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 parallel to 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 \(a=F_\parallel/m\), where \(F_\parallel\) is the sum of the forces parallel to the line.

Suppose a coin is sliding to the right across a table, f, and let's choose a positive \(x\) axis that points to the right. The coin's velocity is positive, and we expect based on experience that it will slow down, i.e., its acceleration should be negative.

Although the coin's motion is purely horizontal, it feels both vertical and horizontal forces. The Earth exerts a downward gravitational force \(F_2\) on it, and the table makes an upward force \(F_3\) that prevents the coin from sinking into the wood. In fact, without these vertical forces the horizontal frictional force wouldn't exist: surfaces don't exert friction against one another unless they are being pressed together.

Although \(F_2\) and \(F_3\) contribute to the physics, they do so only indirectly. The only thing that directly relates to the acceleration along the horizontal direction is the horizontal force: \(a=F_1/m\).

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 couldn't just pitch massive modules back and forth with their bare hands; the modules were 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:

\[\begin{equation*}
|F_W|=mg .
\end{equation*}\]

(The equation only gives the magnitude, i.e. the absolute value, of \(F_W\), 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

\(\triangleright\) 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

\[\begin{align*}
F_{W} &= mg \\
&= (1.0\ \text{kg})(9.8\ \text{m}/\text{s}^2)=9.8\ \text{N} ,
\end{align*}\]

and that's indeed the reading on the spring scale.

Similarly for the two-kilogram mass, we have

\[\begin{align*}
F_{W} &= mg \\
&= (2.0\ \text{kg})(9.8\ \text{m}/\text{s}^2)=19.6\ \text{N} .
\end{align*}\]

\(\triangleright\) 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 \(|F_{air}|=c\rho Av^2\), where \(c\) is a constant, \(\rho\) 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.

\(\triangleright\) 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

\[\begin{align*}
F_{air}+ F_W &= 0 , i.e., \\
c\rho Av ^2- mg &= 0 .
\end{align*}\]

Solving for \(v\) gives

\[\begin{equation*}
v_{terminal} = \sqrt{\frac{mg}{c\rho A}}
\end{equation*}\]

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 \(v_{terminal}=\sqrt{mg/c\rho A}\) in the case of \(\rho \)=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?

◊

Show that the Newton can be reexpressed in terms of the three basic mks units as the combination \(\text{kg}\!\cdot\!\text{m}/\text{s}^2\).

◊

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 i 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?

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 six dicta about what force is not.

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 \((\text{ampere}-\text{meters}^2)\), but not in units of force.

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.

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.

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.

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!

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.

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)
◊

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.”

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).^{3} 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.

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 \(a=F_{total}/m\). 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.

◊

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?

*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

\(F_W\) — weight

*net force* — another way of saying “total force”

{}

Newton's first law of motion states that if all the forces acting 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, \(a_{cm}=F_{total}/m\). 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.

**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**. At low speeds, every car's acceleration is limited by traction, not by the
engine's power.
Suppose that at low speeds, a certain car is normally capable of an acceleration of \(3\ \text{m}/\text{s}^2\).
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 \(a\) (\(a>0\) is upward), 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 137.)

**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, \(F_{air}\), 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) Check that the units of your answer to part a make sense.

(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) Plug numbers
into your equation from part a, using \(m=2300\) kg,
\(F_{air}=27000\ \text{N}\), and \(t=4.0\ \text{s}\).
(answer check available at lightandmatter.com)

**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
\(0.01\ \text{m}/\text{s}^2\). 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, \(F\), from the dirt
that stopped 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 \(m=6.7\times10^{-27}\) kg,
\(v=2.0\times10^4\) 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 \(F_B\). 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 \(F_B\), 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
“roof” with my hand so that the grains come off of the “ceiling.”
(a) Explain why static friction is irrelevant here.
(b) Explain why gravity is negligible.
(c) Explain why hitting the cup works, and why its success depends on
hitting the cup hard enough.

**13**. At the turn of the 20th century, Samuel Langley engaged in a bitter rivalry with
the Wright brothers to develop human flight. Langley's design used a catapult for launching. For safety,
the catapult was built on the roof of a houseboat, so that any crash would be into the water.
This design required reaching cruising speed within a fixed, short distance, so large
accelerations were required, and the forces frequently damaged the craft, causing dangerous
and embarrassing accidents. Langley achieved several uncrewed, unguided flights,
but never succeeded with a human pilot. If the force of the catapult is fixed by the
structural strength of the plane, and the distance for acceleration by the size of the
houseboat, by what factor is the launch velocity reduced when the plane's
340 kg is augmented by the 60 kg mass of a small man?(answer check available at lightandmatter.com)

**14**. The tires used in Formula 1 race cars can generate traction (i.e., force from the road) that
is as much as 1.9 times greater than with the tires typically used
in a passenger car. Suppose that we're trying to see how fast a car
can cover a fixed distance starting from rest, and traction is the
limiting factor. By what factor is this time reduced when switching
from ordinary tires to Formula 1 tires?(answer check available at lightandmatter.com)

\begin{handson}{}{Force and motion}{\onecolumn}

Equipment:

1-meter pieces of butcher paper

wood blocks with hooks

string

masses to put on top of the blocks to increase friction

spring scales (preferably calibrated in Newtons)

Suppose a person pushes a crate, sliding it across the floor at a certain speed, and then repeats the same thing but at a higher speed. This is essentially the situation you will act out in this exercise. What do you think is different about her force on the crate in the two situations? Discuss this with your group and write down your hypothesis:

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

1. First you will measure the amount of friction between the wood block and the butcher paper when the wood and paper surfaces are slipping over each other. The idea is to attach a spring scale to the block and then slide the butcher paper under the block while using the scale to keep the block from moving with it. Depending on the amount of force your spring scale was designed to measure, you may need to put an extra mass on top of the block in order to increase the amount of friction. It is a good idea to use long piece of string to attach the block to the spring scale, since otherwise one tends to pull at an angle instead of directly horizontally.

First measure the amount of friction force when sliding the butcher paper as slowly as possible:\_\_\_\_\_\_\_\_\_

Now measure the amount of friction force at a significantly higher speed, say 1 meter per second. (If you try to go too fast, the motion is jerky, and it is impossible to get an accurate reading.) \_\_\_\_\_\_\_\_\_

Discuss your results. Why are we justified in assuming that the string's force on the block (i.e., the scale reading) is the same amount as the paper's frictional force on the block?

2. Now try the same thing but with the block moving and the paper standing still. Try two different speeds.

Do your results agree with your original hypothesis? If not, discuss what's going on. How does the block “know” how fast to go? \end{handson}

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

[1] Page 82 lists places in this
book where we describe experimental tests of the principle of inertia and Newton's first law.