Newton’s Laws - Frat Stock · Full file at NEWTON’S LAWS 25 TEACHING SUGGESTIONS AND LECTURE HINTS The idea of weight being a force—a pull—is often new to students and must

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CHAPTER 2

Newton’s Laws

CHAPTER OUTLINE

New and Old Join Forces

2.1 Force

LEARNING CHECK 2.2 Newton’s First Law of Motion

PHYSICS POTPOURRI Friction—A Sticky Subject

LEARNING CHECK 2.3 Mass

LEARNING CHECK 2.4 Newton’s Second Law of Motion

LEARNING CHECK 2.5 The International System of Units (SI)

2.6 Examples: Different Forces, Different Motions

Simple Harmonic Motion

Falling Body with Air Resistance

LEARNING CHECK 2.7 Newton’s Third Law of Motion

LEARNING CHECK 2.8 The Law of Universal Gravitation

Orbits

Gravitational Field

LEARNING CHECK

PHYSICS POTPOURRI Hooke–d!

2.9 Tides

LEARNING CHECK Physics Family Album

SUMMARY

IMPORTANT EQUATIONS

MAPPING IT OUT!

QUESTIONS

PROBLEMS

CHALLENGES

SUGGESTED READINGS


24 CHAPTER 2

CHAPTER OVERVIEW

The concept of force is presented and it is argued that a force is needed to cause any change

in motion—in speed or direction. Mass is introduced again and shown to also be involved in

changing motions of objects. Newton’s second law brings the above concepts together.

Applications of Newton’s second law are given in example calculations. A short section presents

the SI system of units. Motion examples from Chapter 1 are now analyzed further in terms of

forces. Projectile motion, simple harmonic motion, and motion including air resistance are

discussed. The concept of action-reaction forces and Newton’s third law are presented. Newton’s

law of universal gravitation, the Cavendish experiment, orbits, and the idea of a gravitational

field are explained. The chapter concludes with a discussion of the tides.

Two new Physics Potpourris discuss friction and the life and work of Robert Hooke.

LEARNING OBJECTIVES

A student who has mastered this material should be able to:

1. Develop a better understanding of the specific meaning of force in mechanics.

2. Understand that gravity causes weight.

3. Explain the mechanisms underlying friction, the laws of friction, and the difference

between static and kinetic friction.

4. State Newton’s first law and explain the meaning of net force.

5. Compare mass, inertia and weight and explain the differences.

6. State Newton’s second law in words as well as mathematically, and be able to perform

calculations using it.

7. Be comfortable with the SI system of units and give some examples.

8. Give examples of free fall.

9. Explain projectile motion, especially the fact that the vertical and horizontal components

are independent.

10. Explain simple harmonic motion.

11. Add air resistance to a free fall situation and explain the changes, and explain terminal

speed.

12. State Newton’s third law, and give several examples of action-reaction force pairs.

13. State Newton’s law of universal gravitation, in words and mathematical notation.

14. Describe the Cavendish experiment.

15. Explain how gravity is involved in orbits, and relate Newton’s “cannon” idea for a satellite

launch.

16. Make an ellipse with tacks and a loop of string.

17. Picture a gravitational field, and explain the benefit of this concept over “action at a

distance.”

18. Explain how tides are created.

19. Give some details of Newton’s life and work.


NEWTON’S LAWS 25

TEACHING SUGGESTIONS AND LECTURE HINTS

The idea of weight being a force—a pull—is often new to students and must be stressed.

You may not wish to dwell on friction to the extent presented in Section 2.1, although kinetic

friction is mentioned by name in later chapters. See the interesting article “Soft Matter in a Tight

Spot” by Steve Granick in the July 1999 Physics Today about current work on friction and

lubrication—he even shows evidence that kinetic friction behaves chaotically.

Question 5 about pressing and pushing on a book resting on a table is a “must do” in class.

Have everyone try this, pressing lightly at first, then harder and harder. Then try it again with a

sheet of paper between your hand and the book. (When I tried this variation on my computer

table using the fourth edition of this book, I couldn’t get the book to slip no matter how hard I

pressed!)

Use a weak spring to show how pulling distorts it, and that the amount of stretch can be used

to measure the size of a force. Show a spring-type force scale for comparison.

The key idea to bring across in connection with Newton’s first law is that a force is needed to

cause any change in motion—in speed or direction. The concept of a net force and an external

force may have to be explained. Describe several situations in which a centripetal force keeps

something moving along a circular path. Figure 2.12 and Explore-It-Yourself 2.3 with the sock

should be pointed out and used to dispense with the common misconception that the object

would move radially rather than tangentially. If the students don’t get around to trying it right

away, tie a rubber stopper to a weak string (strong thread) and swing it in a circle overhead. Hold

a sharp knife in front of you and ask the class to predict where the stopper will go after you raise

the knife so it will cut the string.

Do the “yank the tablecloth without spilling the wine” demonstration, or some variant. This

is as much a demonstration of friction as it is of the third law.

Attach a weak spring to a loaded dynamics cart and show how it speeds up or slows down

when the stretched spring shows that a force is acting.

Swing a bucket of water in a circle about a horizontal axis fast enough to keep the water from

falling out at the high point. You could calculate how fast the bucket had to be going.

Use a VCR to show scenes of the rotating space station in the movie 2001: A Space

Odyssey. Ask the class if the space station in Star Trek: Deep Space 9 rotated.

Artificial gravity in rotating space stations can be a starting point for discussion of O’Neill’s

The High Frontier (See the Suggested Readings at the end of the chapter). Or see his cover

article in the September 1974 issue of Physics Today. Research with human subjects suggests

that the rotation period would have to be two minutes or larger. With the recent push for human

exploration of Mars, ways to manage the effects of long duration space flight are of renewed

interest.

The distinction between mass and weight should be stressed again, and the fact that weight is

caused by something outside of an object. The concept of weightlessness in an orbiting

spacecraft is very tricky to explain to students at this level (see Common Misconceptions below).


26 CHAPTER 2

Lift, then shove a dynamics cart while describing how weight is involved in the former and

mass in the latter.

Hang 1 kg and 2 kg masses from a demonstration force scale to show their weights. Pass

them around the class, also.

Talk about the Apollo astronauts’ experience on the moon and how they developed their own

way of “walking”. A film loop or videotape would be very useful.

I usually work out several examples and/or problems on the board. Incidentally, the

momentum form of the second law is presented in Section 3.2.

DHP, page M-18, item Md-2 shows one setup using an air track and photogate timers to

demonstrate the second law.

DHP, page M-16, item Mc-2 shows a classic demonstration of inertia, where you hang a

heavy ball from a thread and hang another thread from the ball, and you can break the top thread

with a gradually increasing downward pull, or break the bottom thread with a quick jerk.

Discuss challenges 1 and 2 at the end of the chapter (the force of a bat hitting a baseball and

banked highway curves).

LAB EXERCISE. Attach a rubber stopper to one end of a 1.5-meter piece of strong fishing line.

Pass the other end through a 20-cm length of glass tubing and attach it to a weight hanger. The

weight of the hanger and masses supplies the centripetal force on the stopper as it is swung in a

circle overhead. A small alligator clip can be used as a marker on the string below the tube so

that different radii can be used. The period of rotation is measured, and from this, the speed and

acceleration of the stopper. Then m times a is compared with the centripetal force. A 10-g

stopper, a radius of 1 meter, and a total of 200 g supplying the centripetal force yields reasonable

accuracy.

I use the term SI a great deal and try to get the students to automatically use SI units.

The fact that the states of no motion and of uniform motion are equivalent as far as forces

are concerned needs to be pointed out again. Projectile motion, simple harmonic motion, and

falling body with air resistance can be treated lightly if desired. The sinusoidal graphs for simple

harmonic motion do appear in later chapters.

DHP, pages M-7 to M-11, gives several demonstrations related to projectile motion. Item

Mb-14 (a spring gun that launches one ball horizontally and drops one straight down at the same

time) is a nice way to show that horizontal motion does not change the effect of the force of

gravity.

Use a mass on a spring, a simple pendulum, a metronome, a glider on an air track attached to

two springs, or other systems to show simple harmonic motion.

Drop a ping-pong ball and a golf ball simultaneously from 2 meters or more and listen to the

former hit the floor later.

Figure 2.26 can lead to a discussion of physics in space. NASA has available many films of

physics demonstrations in Skylab and Space Shuttle missions. Page 334 in Sport Science has a

table that gives the terminal velocities of several different balls used in sports. It is interesting


NEWTON’S LAWS 27

that a baseball can leave a pitcher’s hand fast enough to equal its terminal velocity when falling,

so it is decelerating at 1 g. It must be decelerating at 1 g because the force of air resistance on the

ball must be the same as when it is falling at its terminal speed (how could the ball tell the

difference between being thrown at its terminal speed vs. falling at that speed?). Here, though,

the force of gravity is not present to balance the air resistance force. When falling at terminal

speed, the air resistance force is equal to the force of gravity, so with air resistance only (at the

same speed), the acceleration is the same magnitude as when you have gravity only.)

LAB EXERCISE. A calibration curve for an inertial balance can be plotted after the periods of

known masses are measured. One can plot period versus time, or compute the frequencies and

plot them.

Even though chaos is mentioned only briefly on page 66, fun can be had with computer

programs dealing with fractals and chaotic dynamical systems. Added fun can be had with actual

devices that exhibit chaos—see for example Appendix C, “Chaotic Toys” in Chaotic and Fractal

Dynamics, an Introduction for Applied Scientists and Engineers by Francis C. Moon (1992,

Wiley) for several interesting ideas.

The fact that a wall or other passive object can exert a force is a new and important concept

for students.

In addition to the demonstrations with dynamics carts shown in Figure 2.33, you might look

at the many action-reaction and thrust demonstrations in DHP, pages M-17 to M-25.

Discuss challenge 4 (the tiny gravitational force between two people).

The concepts of an inverse square force and a field are important and will be seen again in

Chapter 7. The text relates Newton’s reasoning about gravity being an inverse square law early

in section 2.8 (the moon is 60 times farther away from the center of the earth than something on

the earth’s surface is from the center of the earth, and the acceleration of the moon is 1/3600th—

1/60th squared—of the acceleration of, say, a falling apple) but does not go into the more

difficult problem of proving that it does not matter that the mass of the earth is spread throughout

the interior of a huge planet. It is quite remarkable that at the earth’s surface, with the ground

right under your feet and nearby soil attracting you much more strongly (in a relative sense) than

matter at places inside and on the far side of the earth—each bit pulling in a different direction—

that the overall effect is exactly the same as if all the matter in the earth were concentrated at its

center. (It might be interesting to have students chase down people who can do calculus and are

willing to show them the details of the proof.) The Physics Family Album makes brief mention

of Newton’s use of the calculus to prove this, but more clarification is probably needed.

Use a string and a volunteer’s fingers instead of tacks to draw elliptical orbits on the

chalkboard as in Explore-It-Yourself 2.9. Show that the ellipse is difficult to distinguish from a

circle unless the two foci are rather far apart.

Discuss challenges 5, 6, 7, and 8. Challenge 7 on the stationing of a geosynchronous

communication satellite is very interesting and practical. Use the internet to download some

“live” (well, 30 minute or so “old”) weather satellite images of your region.


28 CHAPTER 2

You may have students who have never experienced tides, so a simple description of the

phenomenon might be in order. Have some students find (or make) friends who live near the

coast and call to ask them about the tides.

COMMON MISCONCEPTIONS

What? —When there isn’t force, objects can still make with constant velocity?

How can the net force of a car be zero if it is moving?

Newton’s first law can lead to confusion if students place it in an unintended context. They

might be misled by the need to exert a force on an object to bring it up to speed from rest. But

Newton’s first law does not refer to an object’s history, only what is going on right now. If no

forces are acting the object simply coasts at constant velocity. They also may be unwittingly

thinking of a concept closer to that of energy or momentum when they mistakenly use the term

force.

Strongly tied to this confusion is what E. (“Joe”) Redish calls Newton’s Zeroth Law of

Motion: “At a time t, an object only responds only to forces that are exerted on it itself at time

t.” (Am. J. Phys., July 1999, p. 571). Students need to see that if a force acts on an object over a

time interval, the object will accelerate during that time interval, but if the force stops, so will the

acceleration. So a car must have a net force acting on it while it is speeding up, but not after it

has reached and maintains cruising speed. Similarly an object going in a circle must have a net

force on it all the time, since it always has a centripetal acceleration (it the force were to quit,

then the object would necessarily instantly quit moving in a circle and head off in a straight line).

If a skier is going downhill at a constant speed there’s no force exerted because there’s no acceleration, right?

The only flaw in the above question is that it should say “no net force exerted”—there are

clearly forces exerted on the skier by the snow surface (partly frictional, partly supportive),

gravity, and air resistance, but they add vectorially to zero (I’m assuming the hill has a constant

slope—otherwise constant speed wouldn’t imply constant velocity and therefore zero

acceleration).

What is the difference between a vector and a force? Will a force produce a vector?

A force is one particular example of a vector quantity. I think this questioner is suffering

from attempting to learn the general concept of vector before seeing some examples.

How can a ball have inertia if it is still? Can there be inertia with an object that is standing still?

Inertia, just like mass, is a very elusive concept. I see very little distinction between inertia

and mass until relativity is discussed. Mass is an invariant, yet objects get harder and harder to

accelerate as their speeds approach the speed of light, so I would like to say their inertia

increases at relativistic speeds even though their mass stays constant.


NEWTON’S LAWS 29

Thinking of inertia as a “resistance” to changes in motion simply does not satisfactorily

address questions regarding what it is. Perhaps spending some time simply struggling with the

concept will convey the deep issue it really is. Encourage research into this subject.

I don’t really understand inertia. How can someone hold a brick on their head and have someone hit it with a sledgehammer and have it not hurt.

Perhaps this demonstration should be preceded (or followed, if you prefer suspense) by

hanging the brick from a rope and hitting it on the side with the hammer. If the brick is massive

enough, the hammer will more or less bounce off, and it will be clear that a person’s head on the

far side would be in little danger. You need something a lot bigger than a chimney brick if you

want to hit it with a sledgehammer! Experiment carefully.

When I begin discussing Newton’s third law I always start by pushing hard on the

demonstration table (it’s bolted down). The students can easily be convinced that the table

pushes back just as hard on me. Then I use a movable chair or desk and push gently on it (not

hard enough to move it). I ask the students about the chair’s force on me, and they say it is the

same strength as my push. Then I push hard enough to actually move the chair—how do the

forces compare now? Most of them confidently report that now I am pushing harder than the

chair is pushing back. They are very surprised when I tell them they are wrong—that the forces

are still the same size.

I understand Newton’s 3rd law the best, I think. It’s neat how you can be pushing a huge wall and the reason it doesn’t move is basically because it’s pushing on you, too.

This is wrong. The reason the wall doesn’t move is because its attachments are able to exert

forces on the wall equal and opposite to your push. If you push hard enough (with some sort of

help, obviously) the wall will move.

If the moon pulls on the earth as hard as the earth pulls on the moon why isn’t the earth’s orbit around the sun affected by the moon.

It is! The earth and moon actually both orbit about the center of mass of the earth-moon

system once a month. This point is actually beneath the earth’s surface, so the earth’s true motion

looks more like a wobble than an orbit, but this motion is fundamentally involved in causing the

tides.

I don’t understand the part where the author talks about the third law with the ball falling towards earth. How can the ball exert a force on the earth when there is air between the ball and the earth?

How can the earth exert an equal force as a ball?

The force the ball exerts on the earth is a gravitational force, no more (or less) mysterious

than the gravitational force the earth exerts on the ball. It does seem strange that a tiny ball can

pull on the earth just as strongly as the enormous earth pulls on the ball. Is it at all comforting to

note that the effect of the earth’s gravity on the ball (an acceleration of 9.8 m/s2) is much more


30 CHAPTER 2

dramatic than the effect of the ball’s gravity on the earth (an acceleration, but one that is

immeasurably small because of the earth’s huge mass)?

G is so small! Does it really make that much of a difference in calculations?

I once had my algebra students compare the cost (per ounce) of different kinds of pop in

order to discover which brand was the best deal. One of them reacted to the tiny values (1.28

cents/oz for cheap cola vs. 2.56 cents/oz for brand name colas in convenience stores, for

example), saying something like “who cares about pennies?”, totally missing the point that the

differences really add up when you purchase in quantity. The same goes for the gravitational

constant, G. It needs to emphasized that its small value does imply a kind of “weakness” about

gravity (you need a lot of mass to see any effect at all), but on the other hand one could say that

we just happened to choose inconvenient force, mass and length units when defining newtons,

kilograms and the meter, all of which conspire to require a tiny G value to make Newton’s law of

gravitation work out right. New units could be invented that would make a G value of 1 possible

(you could leap ahead for a moment and talk about the definition of the ampere). But make the

point that the value of G, even though its numerical size depends on the above factors, is a deep

universal fact of nature—something worth pondering.

It says that the only force on a spacecraft in orbit is gravity and that the objects inside are weightless. I don’t understand. If there’s gravity, why is everything weightless?

We are stuck with two meanings of the term “weight”—weight as the force exerted by

gravity on an object (the one we try to convey in physics) vs. weight as the force supporting an

object as it sits on a bathroom scale, for instance. If you are simply standing motionless on a

scale, the downward force of gravity is exactly balanced by the equal strength upward force of

the scale—there seems little point in fussing a lot about the distinction between the two. We may

as well think of the scale force as representing the weight. But this is a very specific situation. In

orbit I suppose you could glue your feet to a bathroom scale and push yourself toward a wall of

your ship, feet first—the scale would register a force as you made contact with the wall, and the

strength and duration of the force would vary depending on whether you bent your knees to

soften the blow or had them “locked” as well as whether or not you decide to “kick off” from the

wall in some way. Here the scale force is indeed what causes your acceleration during your

impact with the wall—it has nothing to go with gravity at all. After you break contact with the

wall the scale will read zero—you are “weightless” in the second sense above. Yet all this while

gravity has been acting on you—you are definitely not “weightless” in the first sense. The point

is that in orbit, gravity gives rise to a centripetal acceleration, whereas on the earth’s surface it

gives rise to a contact force that counters the gravitational force. It is hard not to think of this

contact force as “weight”, and that’s what leads to all the confusion.

I don’t understand why if gravity is the only force acting upon a spacecraft, you are weightless. Wouldn’t gravity pull you down?

Gravity does pull you down—it is causing the centripetal acceleration that both you and your

craft experience. “Down” must be thought of as “toward the center of the earth,” however, not

“pulled out of orbit” or something.


NEWTON’S LAWS 31

In section 2.9 it talks about shooting a cannonball into orbit. Why do things stay in orbit? Why don’t they go like this?

They can! To achieve a perfectly circular

orbit the launch velocity has to be a very precise

value—if it’s slower than that the cannonball will

fall to the ground (as shown in Figure 2.38). Any

faster and the trajectory will curve outward as

shown here. But even then there are two

possibilities: the orbit can curve back around

following an ellipse or it can (if the cannonball is

launched fast enough) continue outward forever

following a hyperbolic path.

Is the gravitational force on a person in space zero because of weightlessness?

The premise is in error—the gravitational

force is not zero on a person in space (though it

might be vanishingly small way out in deep space,

far from any other masses). Also, weightlessness

is not a cause of anything, just a characterization of a particular situation.

When an object reaches terminal speed does the object’s speed and air resistance equal out.

No. The forces of gravity and of air resistance upon the object are equal in strength. It

doesn’t make sense to compare a speed with air resistance (a force).

Terminal velocity is basically your highest speed.

Not quite. If you fall from rest your terminal velocity will be the highest speed you attain, but

it is possible to throw something downward fast enough so that the air resistance force is larger

than the force of gravity at the start, slowing the fall to terminal velocity.

I’m a little confused on how the tides work. When the water is closer to the moon and is “heaping up”, does that mean the tide is out?

“Out” as in “the tide is out” means that the water has flowed oceanward—“out” away from

the beach, meaning low tide. “Out” as in “pulled out into a bulge” means upward, away from the

earth’s surface. At the location of a bulge the tide would high, or “in.”

Does the space shuttle shut off its engines in orbit?

Yes. But often movies show spaceships with engines blazing continuously. I enjoy finding

physics flubs in science fiction shows that demonstrate misunderstandings such as this one (some

of my students thought this rather weird).


32 CHAPTER 2

What exactly moves a rocket forward in space? What does the thrust push against?

The thrust pushes against the rocket! Of course, the question may imply a mistaken belief

that the thrust needs a way to get “leverage” or a sort of “foothold” on some external medium,

which is not the case. This is a good place to discuss the idea of a system again. When the rocket

and its exhaust are considered together, the system as a whole doesn’t get anywhere—it just

spreads out—the rocket going forward and the exhaust particles going backward (and the forces

are all internal). Thinking of the rocket by itself is actually more complicated.

CONSIDER THIS—

Where does gravity come from and why is it there?

How is the earth pulling on the moon?

I don’t think questions of this type should be ducked. Instead, point out that this sort of

curiosity and wonder is exactly what drives physicists. (Good luck finding the answers.)

How come gravity pulls harder on objects with greater mass?

This is actually a very deep question. It is fascinating that gravity pulls harder in exact

proportion to how much more massive the object is—that’s why everything free falls (near the

earth’s surface) with an acceleration of 9.8 m/s2.

If gravity affects a person with a larger mass more, and the force of gravity works on them harder, why can’t smaller people consistently jump higher and farther than big people? Are there maybe too many variables in people for this to hold true?

Investigate this directly with a bunch of large and small student volunteers.

Since we’re on the subject of forces, what would happen if all the people in China jumped at the same time and landed at the same exact moment?

This invites experimentation—how high can people jump?; calculation—what force do they

exert on the earth during a jump?; and a bit of research—how many people are there in China?

Collect the answers, then do F = ma for the earth. Getting the force exerted during a jump really

brings together a lot of material: Use the free fall equations to calculate the jumper’s initial

velocity from their jump height. Then time the jump—more specifically, time from when they

first start pushing up with their legs to when their feet leave the floor (videotaping the jump and

counting video frames works well). Use this time plus the initial velocity to get the average

acceleration. From that get the average net force on the jumper from Newton’s second law. This

net force is how much the force of their legs exceeds their weight, so it must be added to their

weight to get the force generated by their legs. Newton’s third law then says the same strength

force acts on the earth. Multiplying by the number of people in China will give the force on the

earth, and F = ma can again be applied using the earth’s mass this time. The result will be tiny,


NEWTON’S LAWS 33

of course—a good demonstration of the enormity of the earth, but hopefully also a demonstration

that physics can provide explicit quantitative answers to many questions.

Several students pointed out to me that even landing with a parachute is not exactly a soft

landing (the text mentions a landing speed of 10 mph, another I checked said 25 mph). Urge your

students to make some phone calls to skydiving outfits to get a fuller story. To get a real feel for

how hard such landings are, compute the height of a table you might jump from in order to hit

the floor with these speeds (it is a little scary). Chapter 3 shows how to do this calculation using

energy conservation.

Consider two satellites in orbit around earth. Both are in the same orbit, but on opposite sides of the earth. Both satellites have a limited amount of maneuvering fuel available. What kinds of maneuvers would be done to get these satellites close enough to each other to “link up”? When the maneuvering is done they will be in the same orbit as the beginning. The operation must be done as quickly as possible, with a minimum of fuel use. What would be a likely approach?

I have a feeling the requirements “as quickly as possible” and “minimum of fuel use” are

incompatible (and too vague). This is a very difficult problem. Nevertheless, it could be used to

motivate detailed study of real space missions. Request a press kit for a space shuttle mission—I

found it a trivial matter to get the phone number for Johnson Space Center in Houston from

Directory Assistance and got a lot of free information with just a phone call (OK, two phone

calls).

If you were in the space shuttle and some outside force pushed the shuttle (you, floating in midair inside still), would you stay in the same place as the shuttle moved until some part in the interior hit you, or would you float in the same spot in relation to the interior?

The wall would move over and hit you. I often wonder if some television reporters

mistakenly think the other answer makes sense—I have seen them act as if they expect

astronauts doing untethered EVA’s in the shuttle’s cargo bay to be suddenly whisked away if

they drift a bit too far out. Impossible. Now, if one of their little maneuvering thrusters got stuck

on—that would be another matter.

How does a probe that is in orbit fall out of orbit and then tumble to earth?

Air resistance in low earth orbit is slight, but not zero. Over time the orbit slowly decays

lower into gradually thicker atmosphere where the air resistance is greater, speeding up the rate

of decay until the reentry. Communication satellites in much higher geosynchronous orbits do

not suffer this fate. By the way, the fact that geosynchronous orbits have to be above the equator

of a planet is often missed (especially on sci fi shows). Have the students try to see why there is

no other possibility. Are weather satellites geosynchronous? If they are and therefore orbit

above the equator, can we not get weather photos of the polar regions of the earth, then?


34 CHAPTER 2

Since there is always air resistance why do physicists ignore it when they study projectile motion? Wouldn’t whatever they find be wrong because they didn’t take air resistance into account?

The book keeps wanting to “ignore air resistance”, or “ignore gravity” or “ignore friction.” Well, those things do exist and do play a part in daily life. Why calculate things or talk about things that don’t really happen on earth? I don’t care to talk about imaginary probabilities.

This point was addressed back in Chapter 1. To get a handle on the more difficult real life

problems we have to start simple. Also, more fundamental behavior may be hidden by real life

complexity. Stripping that away may lead to better understanding. The first question is correct,

though—ignore air resistance in a real life situation and you will go wrong.

INFOTRAC® PROJECTS

You may wish to explore more details of spacecraft mission planning. Look up the New

Horizons mission and the recent robotic missions to Mars as well as plans to send humans there.

Find out more about the history of friction and lubrication. Research friction, wear and

lubricants in modern engineering and biomechanics. Look into how car braking systems work.

Try searches for anti-lock and ABS brakes and tribology.

Investigate the conflict between Newton and others like Hooke and Leibniz.


NEWTON’S LAWS 35

ANSWERS TO MAPPING IT OUT!

1. Universal gravitation concepts:

Every object exerts a gravitational force on every other object via the gravitational field it

creates.

Gravitational forces between masses are

proportional to each object’s mass,

inversely proportional to the square of the distance between them,

given by the formula F GmM

r2 .

G was measured by Cavendish using a torsion balance.

G = 6.67 10-11 Nm2/kg2.

Earth’s mass (6 1024 kg) can be computed from G and g.

Every planet has a different “g.”

Orbits of planets and comets (and everything else) around the sun are ellipses.

Achieving earth orbit is like throwing a projectile horizontally at just the right speed

(about 7,900 m/s).

Here’s an attempt at prioritizing—

Law of universal gravitation; mass creates a gravitational field; the field obeys an inverse

square law; gravity is one of the four fundamental forces.

The gravity force between a pair of objects (weight is an example) is proportional to each

mass in the pair; gravity can be measured with a Cavendish torsion balance; it reveals G =

6.67•10-11 Nm2/kg2. G and g are related; every planet has its own “g.”

Orbits: satellite orbits imagined in Newton’s ‘cannon’ thought-experiment; orbital velocities

can be calculated; orbits of planets, comets, etc. are ellipses with sun at one focus.

Tides: on earth, primarily due to moon’s gravity and the fact it is different on the near and

far sides of the earth; neap and spring tides are due to sun’s added influence.


36 CHAPTER 2

2.

Net force

which may be

yields

Simple

Harmonic

Motion

Constant

and

Opposite to

velocity

yields

Straight

line motion

with

Perpendicular

to velocity

yields

Parallel to

velocity

yields

Decreasing

acceleration

and

Terminal

speed

Decreasing

speed

Proportional to

and opposite to

displacement

Circular

motion

Proportional to

and opposite to

velocity

yields

Straight

line motion

Increasing

speed

with

ANSWERS TO QUESTIONS

1. Force is a push or a pull that usually causes distortion and/or acceleration. Examples:

weight (force of gravity) pulling down on everything, force of friction between you and your

seat, force of friction holding nails and screws in desks, cabinets, walls, etc., and so on.

2. Weight is the force of gravity acting on a body. An object is truly weightless only if there is

no other body around to exert a gravitational force on it.

3. Force of air resistance acting toward the rear of the car and a force of static friction between

the car’s roof and the book acting forward.

4. Static friction is the type of friction that acts when there is no relative motion between two

bodies. Kinetic friction acts when there is relative motion between two bodies in contact.

One example of both acting at the same time is a moving car: static friction acts between the


NEWTON’S LAWS 37

road and the tires, and kinetic friction acts between the air and the outside of the car. (See

pages 49-50.)

5. There are friction forces at both contact surfaces—between the hand and the book on top,

and between the book and the table underneath. If the friction force between the hand and

the book is larger than that between the table and the book, the book will be dragged along

by the hand. In this case static friction acts between the book and the hand, and kinetic

friction acts between the table and the book.

In the case where the book stays put and the hand slips, it’s very tempting to think that the

friction between the book and the hand is less than that between the book and the table. But

that is not true—the two friction forces must be equal! The book is motionless, so it has

zero acceleration and therefore zero net force. The two friction forces (acting in opposite

directions) must exactly cancel each other. Here static friction acts between the book and the

table, and kinetic friction acts between the hand and the book.

Do several trials with your hand and a real book. Press gently at first so your hand slips, then

gradually increase the pressure. In successive trials where the book does not move, the static

friction between the book and table simply grows in strength to match the increasing kinetic

friction force applied by your hand. When you reach the pressure where the book begins to

slide, your hand is exerting a friction force larger than the maximum possible static friction

force between the book and table at that pressure.

6. An external force is one caused by something outside of the body under consideration. An

internal force can’t accelerate an object because, by Newton’s third law, any internal force

acting in some direction on one part of a body would produce an equal but opposite force

acting on another part. The two forces would cancel each other.

7. The combined forces of A and the teammate will the same as a single force 1.41 (the square

root of 2) times as strong on B toward the southeast. What B’s subsequent motion will be

depends on whether he was already moving or not (and how fast and in which direction),

how massive he is, whether other forces in addition to the two mentioned also act on B, and

how long the forces continue to act. Only in the simple case of B being at rest when the

other players hit him can we say he will go toward the southeast. Newton’s second law can

be applied to solve for the motion in all the other cases, with more work involved.

8. A steady centripetal force causes an object to move along a circular path. If the force

disappears, the object moves in a straight line with constant speed, its direction being that in

which it was traveling at the instant the force disappeared. (See page 54.)

9. Constant speed and heading together imply constant velocity and therefore zero

acceleration. Newton’s second law then says the net force on the train car is zero.

10. Mass is an intrinsic property of a body that determines the acceleration when a net force

acts. Weight is a force acting on a body caused by something outside the body, like the

earth. Weight depends on where an object is—mass does not. (See pages 48-49.)

11. To accelerate the ball during the act of throwing, the astronaut’s hand will have to apply the

same force to the ball in orbit as on the ground, since Newton’s Second Law says this force

depends only on the mass of the ball and the acceleration, both of which will be the same in

orbit as on the ground for an identical horizontal throw. The same goes for the force during


38 CHAPTER 2

the catching process, so throwing and catching will “feel” the same in orbit as on the

ground.

What’s missing in orbit are the friction and support forces of the ground on the astronauts,

so repeated throws or catches in orbit will make each astronaut move away from the other at

an increasing speed. They will start spinning head over heels slowly as well, if they catch

and throw at roughly head level in the usual way.

12. The track is shaped in such a way that it is accelerating the roller coaster downward. Riders

would feel "negative g's". One shape that would cause this is a sharp hill: the track slants

upward, then quickly slants downward.

13. If the propeller exerts the same forward force on the craft in both cases, it won’t matter

whether it pushes from behind or pulls from the front—the motion of the plane or boat will

be the same.

14. The rocket’s mass decreases as its fuel is consumed. The same net force acting on a smaller

mass results in a larger acceleration.

15. The SI is a system of units within the metric system that is internally consistent. The units in

the answer of any legitimate calculation are SI provided the units of the input quantities are

also SI.

16. The watch and the arrow hit the ground at the same time. The acceleration of the arrow is

the same as that of the watch because horizontal motion does not affect vertical motion.

17. For a mass oscillating vertically, as in Figure 2.24, the force varies continuously from the

maximum (directed downward) at the high point of the motion, to zero at the equilibrium

point, to the maximum (directed upward) at the low point. The size of the force is

proportional to the distance from the equilibrium position. By Newton’s second law, the

acceleration undergoes the same variation.

18. As the speed of a falling body increases due to the force of gravity, the size of the force of

air resistance (acting upward) increases as well. Therefore the net force decreases. This

continues as the speed increases until the force of air resistance is the same size as the force

of gravity and the net force is zero. Then the acceleration is zero and the speed is constant,

equal to the terminal speed.

19. The ping-pong ball decelerates initially because the force of air resistance is greater than the

weight. After it slows down to its terminal speed (20 mph) it falls at that speed until it hits

the ground.

20. When sitting, the two forces on you are the force of gravity (weight) pulling downward and

the force of the chair pushing upward. The equal and opposite force to that of the earth’s

gravity on you is the pull (also due to gravity) that your body exerts on the earth. For the

chair, it is your downward push (caused by contact) that is the equal and opposite force.

21. The road exerts a force upward (supporting the car so gravity doesn't accelerate it downward)

and a force forward (the equal and opposite force to the force of the tires pushing backward).

22. Your legs exert a downward force on the ground. The ground exerts an equal and opposite

force on you and that force accelerates you upward.


NEWTON’S LAWS 39

23. They move the same in the two cases. By the third law of motion the forces are always equal

and opposite, no matter which skater initiates a force.

24. At a distance R above the earth’s surface, your distance from the earth’s center is 2R. There

the gravitational force would be one-fourth your weight on earth. (This is shown in Figure

2.36.)

25. Cavendish balanced two masses on a rod suspended by a thin wire. Two larger masses were

placed next to them so that the gravitational forces acting on the two smaller masses caused

the wire to twist. Cavendish used the amount of twist to measure the size of the gravitational

force acting on each mass. (See Figure 2.37.)

26. The weight of everything on earth would increase a billion times so buildings would

collapse, people would be pulled to the ground and be killed as internal organs would

suddenly weigh tons, satellites would be pulled into much lower orbits, the air pressure

would increase dramatically and crush many things. Overall it would be a catastrophe.

27. Events that involve jumping or throwing things for distance would be greatly enhanced and

records would fall. Swimming, bicycling (on level terrain), and similar races would not be

affected very much. Running races may be adversely affected because the runner would go

high in the air with each step and perhaps would not be able to keep up a fast pace.

28. Tides are caused by the gravitational pull of the moon on the earth and its oceans. Since the

water on the side of the earth closest to the moon experiences a stronger force and the water

on the opposite of the earth experiences a weaker force, tidal bulges appear. As the earth

rotates, parts of its surface are alternately in a tidal bulge (resulting in high tide) and between

the bulges (low tide).

29. What matters is the difference in the gravitational forces on the near and far sides of the

earth—the size of the earth is a much more substantial fraction of the earth-moon distance

than it is of the earth-sun distance, so it makes a bigger difference (via the 1/r2 dependence)

for the moon’s gravity than the sun’s gravity. The moon’s gravity is about 7% weaker on the

far side of the earth compared to the near side, while the sun’s gravity only varies about

0.02% over this range. But this is only part of the answer, since this effect is largely

compensated for by the sun being much more massive than the moon (7% of the moon’s

gravity is roughly only double 0.02% of the sun’s gravity). A full explanation of the tides

must include the motion of the earth about the earth-moon center of mass. See the special

topic on pp. 273-75 of University Physics by Harris Benson for a good discussion.

30. a) Newton’s second law

b) the law of universal gravitation

c) Newton’s second law (F = ma W = mg)

d) Newton’s first law

e) Newton’s third law

f) Newton’s third law

31. Calculus.


40 CHAPTER 2

ANSWERS TO EVEN NUMBERED PROBLEMS

2. m 30.6 kg

4. a) W 11,100 N

b) W 2,500 lb

6. F 1,500 N

8. m 15 kg

10. a) m 81.6 kg

b) F 963 N

c) F 800 N

12. a) F 2,450 N 551 lb

b) F 2,450 N

14. a) a 6,000 m/s2 612 g , opposite in direction to the baseball’s original velocity.

b) F 870 N 196 lb

16. F 39,200 N

18. a) F 4,900 N

b) W 110 lb

F 1,100 lb

20. a) F 1,500 N

b) a 12.5 m/s2

22. v 20 m/s

24. 5000 s

ANSWERS TO CHALLENGES

1. The force on a baseball as it collides with a bat is not transmitted from the person swinging

the bat. This is a collision between two objects moving in opposite directions. As soon as

their surfaces are in contact, the forces arise because of Newton’s first law—they can’t

occupy the same space, so they have to change their speeds, which requires forces. As the

ball and bat deform each other, their stiffness causes equal and opposite forces to act.

2. On a banked curve the force of the road acting on the car is not vertical so it has a

component towards the inside of the curve, supplying an addition to the centripetal force.

This force adds to the frictional force, allowing a larger total centripetal force and, therefore,

a larger speed.

3. Two equal and opposite forces cancel each other if they are acting on the same object. Here,

one 400 N force acts on the carriage and the other acts on the horse. The actual motion of

the horse and carriage depends on the net force exerted on each. The frictional forces acting

between the road and the horse and between the road and the carriage have to be taken into

account. In this case the latter force is less than 400 N so there is a net force on the carriage

and it accelerates.


NEWTON’S LAWS 41

4. F (6.67 10–11 70 kg 70 kg) (1 m)2

F 3.3 10–7 N

5. Because of the earth’s rotation, objects on its surface (except at the poles) are moving to the

east. The closer they are to the equator, the higher their speed. So objects launched towards

the east from near the equator require a bit less fuel to reach orbital speed since they have a

running start. For example, rockets sitting on the launch pad in Florida are already moving

400 m/s towards the east.

6. Two factors are involved here, both caused by the earth’s rotation. The main factor is that

part of the gravitational force acting on objects is needed as a centripetal force since objects

are moving in circles around the earth’s axis. The closer an object is to the earth’s equator,

the higher its speed and the greater the centripetal force required. The remaining

gravitational force is what causes objects to accelerate while falling. The second factor is

that the rotation makes the earth bulge at the equator so that objects successively closer to

the equator are farther away from the earth’s center. In particular, objects at the earth’s

equator are about 13 miles farther from its center than objects at the poles. By the universal

law of gravitation, the gravitational force is therefore less strong near the equator so that

falling bodies have lower acceleration there.

7. a) F (6.67 10–11 200 6 1024) (4.23 107)2

F 44.7 N

b) F mv2 r v2 Fr m

v2 (44.7 N 4.23 107) (200 kg)

v2 9.46 106 m2/s2

v 3,075 m/s

c) d v t

t d v

t circumference v

t (2 r) v

t (2 3.14 4.23 107 m) (3,075 m/s)

t (2.66 108 m) (3,075 m/s)

t 86,500 s

1 day 24 60 60 s 86,400 s

(The discrepancy is due to roundoff error.)

8. g (G M) R2 M (g R2) G

M (9.8 m/s2 (6.4 106 kg)2) (6.67 10-11 N-m2/kg2)

M (4.0 1014 m-kg2/s2) (6.67 10-11 N-m2/kg2)

M 6.0 1024kg

9. Yes, there is a contradiction, but as a practical matter it can be ignored. When we consider

real life free fall problems we usually think of motion near the surface of the earth over


42 CHAPTER 2

vertical distances on the order of meters. The r in the Law of Universal Gravitation is the

distance from the falling body to the center of the earth, which is thousands of miles. The

range of variation of r during the free fall is so small compared to r itself that we can treat r

as constant in our approximate calculations and introduce very little error.

The exact calculation can be done using more sophisticated mathematics or a computer

program.

Documents

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