Take the fear out of Physics I If the thought of studying physics makes you sweat, you can finally have something to rest easy about! U Can: Physics I For Dummies takes the intimidation out of this tough subject, offering approachable lessons, examples, and practice opportunities--as well as access to additional practice problems online. With this one-stop resource, you'll find friendly and accessible instruction on everything you'll encounter in your Physics I course and will gain the practice and confidence you need to score high at exam time. Inside this comprehensive study resource, how-to lessons are thoughtfully blended with practical examples and problems to help you put your knowledge to practice and gauge your comprehension of the physics topics presented. Lessons and practice problems are fully integrated and track to a typical Physics I course, giving you one mega-resource that combines the 'how-to' you need with the 'do it' practice you want to keep the physics anxiety at bay. * Get up to speed on the basic concepts of physics * Grasp physics formulas in a clear and concise manner * Explore the newest discoveries in the field * Access additional practice problems online If you're looking for an all-inclusive product to help with your Physics I coursework, U Can: Physics I For Dummies has it all--and then some!
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U Can: Physics I For Dummies®
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Table of Contents
About This Book
Beyond the Book
Where to Go from Here
Part I: Getting Started with Physics
Chapter 1: Using Physics to Understand Your World
What Physics Is All About
Observing Objects in Motion
When Push Comes to Shove: Forces
Feeling Hot but Not Bothered: Thermodynamics
Chapter 2: Reviewing Physics Measurement and Math Fundamentals
Measuring the World around You and Making Predictions
Eliminating Some Zeros: Using Scientific Notation
From Meters to Inches and Back Again: Converting Between Units
Checking the Accuracy and Precision of Measurements
Arming Yourself with Basic Algebra
Tackling a Little Trig
Interpreting Equations as Real-World Ideas
Chapter 3: Exploring the Need for Speed
Going the Distance with Displacement
Speed Specifics: What Is Speed, Anyway?
Speeding Up (Or Down): Acceleration
Relating Acceleration, Time, and Displacement
Linking Velocity, Acceleration, and Displacement
Chapter 4: Following Directions: Motion in Two Dimensions
Putting Vectors on the Grid
A Little Trig: Breaking Up Vectors into Components
Featuring Displacement, Velocity, and Acceleration in 2-D
Accelerating Downward: Motion under the Influence of Gravity
Part II: May the Forces Be with You
Chapter 5: When Push Comes to Shove: Force
Newton’s First Law: Resisting with Inertia
Newton’s Second Law: Relating Force, Mass, and Acceleration
Newton’s Third Law: Looking at Equal and Opposite Forces
Chapter 6: Getting Down with Gravity, Inclined Planes, and Friction
Acceleration Due to Gravity: One of Life’s Little Constants
Finding a New Angle on Gravity with Inclined Planes
Getting Sticky with Friction
On the Move: Understanding Static and Kinetic Friction
Chapter 7: Circling around Rotational Motion and Orbits
Centripetal Acceleration: Changing Direction to Move in a Circle
Getting Angular with Displacement, Velocity, and Acceleration
Getting into Angular Velocity
Whipping Around with Angular Acceleration
Connecting Angular Velocity and Angular Acceleration to Angles
Connecting Angular Acceleration and Angle to Angular Velocity
Seeking the Center: Centripetal Force
Letting Gravity Supply Centripetal Force
Chapter 8: Go with the Flow: Looking at Pressure in Fluids
Mass Density: Getting Some Inside Information
Buoyancy: Float Your Boat with Archimedes’s Principle
Fluid Dynamics: Going with Fluids in Motion
Getting Up to Speed on Flow and Pressure
Part III: Working Energetically
Chapter 9: Getting Some Work Out of Physics
Looking for Work
Making a Move: Kinetic Energy
Energy in the Bank: Potential Energy
Choose Your Path: Conservative versus Nonconservative Forces
Keeping the Energy Up: The Conservation of Mechanical Energy
Powering Up: The Rate of Doing Work
Chapter 10: Putting Objects in Motion: Momentum and Impulse
Looking at the Impact of Impulse
The Impulse-Momentum Theorem: Relating Impulse and Momentum
When Objects Go Bonk: Conserving Momentum
When Worlds (Or Cars) Collide: Elastic and Inelastic Collisions
Chapter 11: Winding Up with Angular Motion
Going from Linear to Rotational Motion
Understanding Tangential Motion
Applying Vectors to Rotation
Doing the Twist: Torque
Spinning at Constant Velocity: Rotational Equilibrium
Chapter 12: Round and Round with Rotational Dynamics
Rolling Up Newton’s Second Law into Angular Motion
Moments of Inertia: Looking into Mass Distribution
Wrapping Your Head around Rotational Work and Kinetic Energy
Can’t Stop This: Angular Momentum
Chapter 13: Springs ’n’ Things: Simple Harmonic Motion
Bouncing Back with Hooke’s Law
Getting Around to Simple Harmonic Motion
Factoring Energy into Simple Harmonic Motion
Swinging with Pendulums
Part IV: Laying Down the Laws of Thermodynamics
Chapter 14: Turning Up the Heat with Thermodynamics
The Heat Is On: Thermal Expansion
Heat: Going with the Flow (of Thermal Energy)
Chapter 15: Here, Take My Coat: How Heat Is Transferred
Convection: Letting the Heat Flow
Too Hot to Handle: Getting in Touch with Conduction
Radiation: Riding the (Electromagnetic) Wave
Chapter 16: In the Best of All Possible Worlds: The Ideal Gas Law
Digging into Molecules and Moles with Avogadro’s Number
Relating Pressure, Volume, and Temperature with the Ideal Gas Law
Tracking Ideal Gas Molecules with the Kinetic Energy Formula
Chapter 17: Heat and Work: The Laws of Thermodynamics
Thermal Equilibrium: Getting Temperature with the Zeroth Law
Conserving Energy: The First Law of Thermodynamics
Flowing from Hot to Cold: The Second Law of Thermodynamics
Going Cold: The Third (And Absolute Last) Law of Thermodynamics
Part V: The Part of Tens
Chapter 18: Ten Common Mistakes People Make When Solving Problems
Expressing the Answer in the Wrong Units
Swapping Radians and Degrees
Getting Sines and Cosines Mixed Up
Not Treating Vectors as Vectors
Mixing Up the Signs of the Components of Vectors
Getting the Direction of Forces Wrong
Neglecting Latent Heat
Using the Wrong Temperature in the Ideal Gas Law
Getting the Signs Wrong in the First Law of Thermodynamics
Chapter 19: Ten Wild Physics Theories
Heisenberg Says You Can’t Be Certain
Black Holes Don’t Let Light Out
Gravity Curves Space
You Can Measure a Smallest Distance
Matter and Antimatter Destroy Each Other
Supernovas Are the Most Powerful Explosions
The Universe Starts with the Big Bang and Ends with the Gnab Gib
Most Matter Is Invisible
Microwave Ovens Are Hot Physics
Is the Universe Made to Measure?
About the Authors
Connect with Dummies
End User License Agreement
Table of Contents
Physics is what it’s all about. What what’s all about? Everything. Physics is present in every action around you. And because physics is everywhere, it gets into some tricky places, which means it can be hard to follow. Studying physics can be even worse when you’re reading some dense textbook that’s hard to follow.
For most people who come into contact with physics, textbooks that land with 1,200-page whumps on desks are their only exposure to this amazingly rich and rewarding field. And what follows are weary struggles as the readers try to scale the awesome bulwarks of the massive tomes. Has no brave soul ever wanted to write a book on physics from the reader’s point of view? Yes, and here we come with such a book.
This book is different. Instead of writing it from the physicist’s or professor’s point of view, we wrote it from the reader’s point of view. After thousands of one-on-one tutoring sessions, we know where the usual book presentation of this stuff starts to confuse people, and we’ve taken great care to jettison the top-down kinds of explanations. You don’t survive one-on-one tutoring sessions for long unless you get to know what really makes sense to people — what they want to see from their points of view. In other words, we designed this book to be crammed full of the good stuff — and only the good stuff. You also discover unique ways of looking at problems that professors and teachers use to make figuring out the problems simple.
This book is crammed with physics examples and physics problems. It’s designed to show you how to tackle the kinds of problems you may encounter in physics classes.
In this book, you can find solutions to problems similar to the ones you’re asked to solve elsewhere. And when you see how it’s done, solving similar problems should be a breeze.
From time to time, we include sidebars to provide a little more insight into what’s going on with a particular topic. They give you a little more of the story, such as how some famous physicist did what he did or an unexpected real-life application of the point under discussion. You can skip these sidebars, if you like, without missing any essential physics.
Some books have a dozen conventions that you need to know before you can start. Not this one. All you need to know is that variables and new terms appear in italics, like this, and that vectors — items that have both a magnitude and a direction — appear in bold. Web addresses appear in monofont.
In writing this book, we made some assumptions about you:
You have no or very little prior knowledge of physics.
You have some math prowess. In particular, you know algebra and a little trig. You don’t need to be an algebra pro, but you should know how to move items from one side of an equation to another and how to solve for values.
You want physics concepts explained clearly and concisely, and you want examples that let you see those concepts in action.
You want to work through some physics problems on your own. You’ll find plenty of practice problems in this book along with step-by-step solutions in case you get stuck.
In addition to the material in the print or e-book you’re reading right now, this product also comes with some access-anywhere goodies on the web. Check out these features:
Cheat Sheet (
When you need a quick refresher, refer to the Cheat Sheet for some handy equations and the values of important constants.
Dummies.com articles (
Each part in this book is supplemented by a relevant online article that provides additional tips and techniques related to the subject of that part. Read helpful articles that reveal how to draw a free-body diagram, how to use energy diagrams, the Carnot cycle, and ten physics heroes.
Online practice and study aids:
The online practice that comes free with this book offers 500 questions and answers that allow you to gain more practice with physics concepts. The beauty of the online questions is that you can customize your online practice to focus on the topics that give you the most trouble. So if you need help with projectile motion or angular momentum, just select those question types online and start practicing. Or if you’re short on time but want to get a mixed bag of a limited number of questions, you can specify the number of questions you want to practice. Whether you practice a few hundred questions in one sitting or a couple dozen, and whether you focus on a few types of questions or practice every type, the online program keeps track of the questions you get right and wrong so you can monitor your progress and spend time studying exactly what you need.
To gain access to the online practice, all you have to do is register. Just follow these simple steps:
Find your access code.
If you purchased a hard copy of this book, turn to the inside of the front cover to find your access code.
If you purchased this book as an e-book, you can get your access code by registering your e-book at
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Click on the product you want to access, and then click Login.
On the Register tab, enter your access code, and click Go.
Follow the instructions to create an account and set up your personal login.
Now you’re ready to go! You can come back to the online program as often as you want — simply log on with the username and password you created during your initial login. No need to enter the access code a second time.
Tip: If you have trouble with your access code or can’t find it, contact Wiley Product Technical Support at 877-762-2974 or go to http://wiley.custhelp.com.
You can leaf through this book; you don’t have to read it from beginning to end. Like other For Dummies books, this one was designed to let you skip around as you like. This is your book, and physics is your oyster. You can jump into Chapter 1, which is where all the action starts; you can head to Chapter 2 for a discussion of the necessary algebra and trig you should know; or you can jump in anywhere you like if you know exactly what topic you want to study.
Visitwww.dummies.comfor free access to great Dummies content online.
In this part …
See what physics is all about.
Brush up on basic algebra, trig, and physics measurements and units.
Master the motion of displacement, velocity, and acceleration.
Point yourself in the right direction with vectors.
In This Chapter
Recognizing the physics in your world
Handling the force and energy around you
Getting hot under the collar with thermodynamics
Physics is the study of the world and universe around you. Luckily, the behavior of the matter and energy — the stuff of this universe — is not completely unruly. Instead, it strictly obeys laws, which physicists are gradually revealing through the careful application of the scientific method, which relies on experimental evidence and sound rigorous reasoning. In this way, physicists have been uncovering more and more of the beauty that lies at the heart of the workings of the universe, from the infinitely small to the mind-bogglingly large.
Physics is an all-encompassing science. You can study various aspects of the natural world (in fact, the word physics is derived from the Greek word physika, which means “natural things”), and accordingly, you can study different fields in physics: the physics of objects in motion, of energy, of forces, of gases, of heat and temperature, and so on. You enjoy the study of all these topics and many more in this book. In this chapter, we give an overview of physics — what it is, what it deals with, and why mathematical calculations are important to it — to get you started.
Many people are a little on edge when they think about physics. For them, the subject seems like some highbrow topic that pulls numbers and rules out of thin air. But the truth is that physics exists to help you make sense of the world. Physics is a human adventure, undertaken on behalf of everyone, into the way the world works.
Remember: At its root, physics is all about becoming aware of your world and using mental and mathematical models to explain it. The gist of physics is this: You start by making an observation, you create a model to simulate that situation, and then you add some math to fill it out — and voilà! You have the power to predict what will happen in the real world. All this math exists to help you see what happens and why.
In this section, we explain how real-world observations fit in with the math. The later sections take you on a brief tour of the key topics that comprise basic physics.
You can observe plenty going on around you in your complex world. Leaves are waving, the sun is shining, light bulbs are glowing, cars are moving, computer printers are printing, people are walking and riding bikes, streams are flowing, and so on. When you stop to examine these actions, your natural curiosity gives rise to endless questions such as these:
Why do I slip when I try to climb that snow bank?
How distant are other stars, and how long would it take to get there?
How does an airplane wing work?
How can a thermos flask keep hot things warm
keep cold things cool?
Why does an enormous cruise ship float when a paper clip sinks?
Why does water roll around when it boils?
Any law of physics comes from very close observation of the world, and any theory that a physicist comes up with has to stand up to experimental measurements. Physics goes beyond qualitative statements about physical things — “If I push the child on the swing harder, then she swings higher,” for example. With the laws of physics, you can predict precisely how high the child will swing.
Physics is simply about modeling the world (although an alternative viewpoint claims that physics actually uncovers the truth about the workings of the world; it doesn’t just model it). You can use these mental models to describe how the world works: how blocks slide down ramps, how stars form and shine, how black holes trap light so it can’t escape, what happens when cars collide, and so on.
When these models are first created, they sometimes have little to do with numbers; they just cover the gist of the situation. For example, a star is made up of this layer and then that layer, and as a result, this reaction takes place, followed by that one. And — pow! — you have a star. As time goes on, those models become more numeric, which is where physics students sometimes start having problems. Physics class would be a cinch if you could simply say, “That cart is going to roll down that hill, and as it gets toward the bottom, it’s going to roll faster and faster.” But the story is more involved than that — not only can you say that the cart is going to go faster, but in exerting your mastery over the physical world, you can also say how much faster it’ll go.
There’s a delicate interplay between theory, formulated with math, and experimental measurements. Often experimental measurements not only verify theories but also suggest ideas for new theories, which in turn suggest new experiments. Both feed off each other and lead to further discovery.
Many people approaching this subject may think of math as something tedious and overly abstract. However, in the context of physics, math comes to life. A quadratic equation may seem a little dry, but when you’re using it to work out the correct angle to fire a rocket at for the perfect trajectory, you may find it more palatable! Chapter 2 explains all the math you need to know to perform basic physics calculations.
So what are you going to get out of physics? If you want to pursue a career in physics or in an allied field such as engineering, the answer is clear: You’ll need this knowledge on an everyday basis. But even if you’re not planning to embark on a physics-related career, you can get a lot out of studying the subject. You can apply much of what you discover in an introductory physics course to real life:
In a sense, all other sciences are based upon physics. For example, the structure and electrical properties of atoms determine chemical reactions; therefore, all of chemistry is governed by the laws of physics. In fact, you could argue that everything ultimately boils down to the laws of physics!
Physics does deal with some pretty cool phenomena. Many videos of physical phenomena have gone viral on YouTube; take a look for yourself. Do a search for “non-Newtonian fluid,” and you can watch the creeping, oozing dance of a cornstarch-water mixture on a speaker cone.
More important than the applications of physics are the problem-solving skills it arms you with for approaching any kind of problem. Physics problems train you to stand back, consider your options for attacking the issue, select your method, and then solve the problem in the easiest way possible.
Some of the most fundamental questions you may have about the world deal with objects in motion. Will that boulder rolling toward you slow down? How fast do you have to move to get out of its way? (Hang on just a moment while we get out our calculator… .) Motion was one of the earliest explorations of physics.
When you take a look around, you see that the motion of objects changes all the time. You see a motorcycle coming to a halt at a stop sign. You see a leaf falling and then stopping when it hits the ground, only to be picked up again by the wind. You see a pool ball hitting other balls in just the wrong way so that they all move without going where they should. Part I of this book handles objects in motion — from balls to railroad cars and most objects in between. In this section, we introduce motion in a straight line and rotational motion.
Speeds are big with physicists — how fast is an object going? Thirty-five miles per hour not enough? How about 3,500? No problem when you’re dealing with physics. Besides speed, the direction an object is going is important if you want to describe its motion. If the home team is carrying a football down the field, you want to make sure they’re going in the right direction.
When you put speed and direction together, you get a vector — the velocity vector. Vectors are a very useful kind of quantity. Anything that has both size and direction is best described with a vector. Vectors are often represented as arrows, where the length of the arrow tells you the magnitude (size), and the direction of the arrow tells you the direction. For a velocity vector, the length corresponds to the speed of the object, and the arrow points in the direction the object is moving. (To find out how to use vectors, head to Chapter 4.)
Everything has a velocity, so velocity is great for describing the world around you. Even if an object is at rest with respect to the ground, it’s still on the Earth, which itself has a velocity. (And if everything has a velocity, it’s no wonder physicists keep getting grant money — somebody has to measure all that motion.)
If you’ve ever ridden in a car, you know that velocity isn’t the end of the story. Cars don’t start off at 60 miles per hour; they have to accelerate until they get to that speed. Like velocity, acceleration has not only a magnitude but also a direction, so acceleration is a vector in physics as well. We cover speed, velocity, and acceleration in Chapter 3.
Plenty of things go round and round in the everyday world — CDs, DVDs, tires, pitchers’ arms, clothes in a dryer, roller coasters doing the loop, or just little kids spinning from joy in their first snowstorm. That being the case, physicists want to get in on the action with measurements. Just as you can have a car moving and accelerating in a straight line, its tires can rotate and accelerate in a circle.
Going from the linear world to the rotational world turns out to be easy, because there’s a handy physics analog (which is a fancy word for “equivalent”) for everything linear in the rotational world. For example, distance traveled becomes angle turned. Speed in meters per second becomes angular speed in angle turned per second. Even linear acceleration becomes rotational acceleration.
So when you know linear motion, rotational motion just falls in your lap. You use the same equations for both linear and angular motion — just different symbols with slightly different meanings (angle replaces distance, for example). You’ll be looping the loop in no time. Chapter 7 has the details.
Forces are a particular favorite in physics. You need forces to get motionless things moving — literally. Consider a stone on the ground. Many physicists (except, perhaps, geophysicists) would regard it suspiciously. It’s just sitting there. What fun is that? What can you measure about that? After physicists had measured its size and mass, they’d lose interest.
But kick the stone — that is, apply a force — and watch the physicists come running over. Now something is happening — the stone started at rest, but now it’s moving. You can find all kinds of numbers associated with this motion. For instance, you can connect the force you apply to something to its mass and get its acceleration. And physicists love numbers, because numbers help describe what’s happening in the physical world.
Physicists are experts in applying forces to objects and predicting the results. Got a refrigerator to push up a ramp and want to know if it’ll go? Ask a physicist. Have a rocket to launch? Same thing.
Have you ever watched something bouncing up and down on a spring? That kind of motion puzzled physicists for a long time, but then they got down to work. They discovered that when you stretch a spring, the force isn’t constant. The spring pulls back, and the more you pull the spring, the stronger it pulls back.
So how does the force compare to the distance you pull a spring? The force is directly proportional to the amount you stretch the spring: Double the amount you stretch the spring, and you double the amount of force with which the spring pulls back.
Physicists were overjoyed — this was the kind of math they understood. Force proportional to distance? Great — you can put that relationship into an equation, and you can use that equation to describe the motion of the object tied to the spring. Physicists got results telling them just how objects tied to springs would move — another triumph of physics.
This particular triumph is called simple harmonic motion. It’s simple because force is directly proportional to distance, and so the result is simple. It’s harmonic because it repeats over and over again as the object on the spring bounces up and down. Physicists were able to derive simple equations that could tell you exactly where the object would be at any given time.
But that’s not all. Simple harmonic motion applies to many objects in the real world, not just things on springs. For example, pendulums also move in simple harmonic motion. Say you have a stone that’s swinging back and forth on a string. As long as the arc it swings through isn’t too high, the stone on a string is a pendulum; therefore, it follows simple harmonic motion. If you know how long the string is and how big of an angle the swing covers, you can predict where the stone will be at any time. We discuss simple harmonic motion in Chapter 13.
You don’t have to look far to find your next piece of physics. (You never do.) As you exit your house in the morning, for example, you may hear a crash up the street. Two cars have collided at a high speed, and locked together, they’re sliding your way. Thanks to physics (and more specifically, Part III of this book), you can make the necessary measurements and predictions to know exactly how far you have to move to get out of the way.
Having mastered the ideas of energy and momentum helps at such a time. You use these ideas to describe the motion of objects with mass. The energy of motion is called kinetic energy, and when you accelerate a car from 0 to 60 miles per hour in 10 seconds, the car ends up with plenty of kinetic energy.
Where does the kinetic energy come from? It comes from work, which is what happens when a force moves an object through a distance. The energy can also come from potential energy, the energy stored in the object, which comes from the work done by a particular kind of force, such as gravity or electrical forces. Using gasoline, for example, an engine does work on the car to get it up to speed. But you need a force to accelerate something, and the way the engine does work on the car, surprisingly, is to use the force of friction with the road. Without friction, the wheels would simply spin, but because of a frictional force, the tires impart a force on the road. For every force between two objects, there is a reactive force of equal size but in the opposite direction. So the road also exerts a force on the car, which causes it to accelerate.
Or say that you’re moving a piano up the stairs of your new place. After you move up the stairs, your piano has potential energy, simply because you put in a lot of work against gravity to get the piano up those six floors. Unfortunately, your roommate hates pianos and drops yours out the window. What happens next? The potential energy of the piano due to its height in a gravitational field is converted into kinetic energy, the energy of motion. You decide to calculate the final speed of the piano as it hits the street. (Next, you calculate the bill for the piano, hand it to your roommate, and go back downstairs to get your drum set.)
Ever notice that when you’re 5,000 feet down in the ocean, the pressure is different from at the surface? Never been 5,000 feet beneath the ocean waves? Then you may have noticed the difference in pressure when you dive into a swimming pool. The deeper you go, the higher the pressure is because of the weight of the water above you exerting a force downward. Pressure is just force per area.
Got a swimming pool? Any physicists worth their salt can tell you the approximate pressure at the bottom if you tell them how deep the pool is. When working with fluids, you have all kinds of other quantities to measure, such as the velocity of fluids through small holes, a fluid’s density, and so on. Once again, physics responds with grace under pressure. You can read about forces in fluids in Chapter 8.
Heat and cold are parts of your everyday life. Ever take a look at the beads of condensation on a cold glass of water in a warm room? Water vapor in the air is being cooled when it touches the glass, and it condenses into liquid water. The condensing water vapor passes thermal energy to the glass, which passes thermal energy to the cold drink, which ends up getting warmer as a result.
Thermodynamics can tell you how much heat you’re radiating away on a cold day, how many bags of ice you need to cool a lava pit, and anything else that deals with heat energy. You can also take the study of thermodynamics beyond planet Earth. Why is space cold? In a normal environment, you radiate heat to everything around you, and everything around you radiates heat back to you. But in space, your heat just radiates away, so you can freeze.
Radiating heat is just one of the three ways heat can be transferred. You can discover plenty more about heat, whether created by a heat source like the sun or by friction, through the topics in Part IV.
In This Chapter
Mastering measurements (and keeping them straight as you solve equations)
Accounting for significant digits and possible error
Brushing up on basic algebra and trig concepts
Physics uses observations and measurements to make mental and mathematical models that explain how the world (and everything in it) works. This process is unfamiliar to most people, which is where this chapter comes in.
This chapter covers some basic skills you need for the coming chapters. We cover measurements and scientific notation, give you a refresher on basic algebra and trigonometry, and show you which digits in a number to pay attention to — and which ones to ignore. Continue on to build a physics foundation, solid and unshakable, that you can rely on throughout this book.
Physics excels at measuring and predicting the physical world — after all, that’s why physics exists. Measuring is the starting point — part of observing the world so you can then model and predict it. You have several different measuring sticks at your disposal: some for length, some for mass or weight, some for time, and so on. Mastering those measurements is part of mastering physics.
To keep like measurements together, physicists and mathematicians have grouped them into measurement systems. The most common measurement system you see in introductory physics is the meter-kilogram-second (MKS) system, referred to as SI (short for Système International d’Unités, the International System of Units), but you may also come across the foot-pound-second (FPS) system. Table 2-1 lists the primary units of measurement in the MKS system, along with their abbreviations.
Table 2-1 MKS Units of Measurement
Magnetic flux density
These are the measuring sticks that will become familiar to you as you solve problems and triumph over the math in this book. Also for reference, Table 2-2 shows the primary units of measurement (and their abbreviations) in the CGS system. (Don’t bother memorizing the ones you’re not familiar with now; you can come back to them later as needed.)
Table 2-2 CGS Units of Measurement
Magnetic flux density
Warning: Because different measurement systems use different standard lengths, you can get several different numbers for one part of a problem, depending on the measurement you use. For example, if you’re measuring the depth of the water in a swimming pool, you can use the MKS measurement system, which gives you an answer in meters, or the less common FPS system, in which case you determine the depth of the water in feet. The point? When working with equations, stick with the same measurement system all the way through the problem. If you don’t, your answer will be a meaningless hodgepodge, because you’re switching measuring sticks for multiple items as you try to arrive at a single answer. Mixing up the measurements causes problems — imagine baking a cake where the recipe calls for 2 cups of flour, but you use 2 liters instead.
Q. You’re told to measure the length of a racecar track using the MKS system. What unit(s) will your measurement be in?
A. The correct answer is meters. The unit of length in the MKS system is the meter.
Q. You’re told to measure the mass of a marble using the CGS system. What unit(s) will your measurement be in?
A. The correct answer is grams. The unit of mass in the CGS system is the gram.
1. You’re asked to measure the time it takes the moon to circle Earth using the MKS system. What will your measurement’s units be?
2. You need to measure the force a tire exerts on the road as it’s moving using the MKS system. What are the units of your answer?
3. You’re asked to measure the amount of energy released by a firecracker when it explodes using the CGS system. What are the units of your answer?
1.seconds. The unit of time in the MKS system is the second.
2.newtons. The unit of force in the MKS system is the newton.
3.ergs. The unit of energy in the CGS system is the erg.
Physicists have a way of getting their minds into the darndest places, and those places often involve really big or really small numbers. Physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest, it uses scientific notation.
Remember: In scientific notation, you write a number as a decimal (with only one digit before the decimal point) multiplied by a power of ten. The power of ten (10 with an exponent) expresses the number of zeroes. To get the right power of ten for a vary large number, count all the places in front of the decimal point, from right to left, up to the place just to the right of the first digit (you don’t include the first digit because you leave it in front of the decimal point in the result).
For example, say you’re dealing with the average distance between the sun and Pluto, which is about 5,890,000,000,000 meters. You have a lot of meters on your hands, accompanied by a lot of zeroes. You can write the distance between the sun and Pluto as follows:
The exponent is 12 because you count 12 places between the end of 5,890,000,000,000 (where a decimal would appear in the whole number) and the decimal’s new place after the 5.
Scientific notation also works for very small numbers, such as the one that follows, where the power of ten is negative. You count the number of places, moving left to right, from the decimal point to just after the first nonzero digit (again leaving the result with just one digit in front of the decimal):
Remember: If the number you’re working with is larger than ten, you have a positive exponent in scientific notation; if it’s smaller than one, you have a negative exponent. As you can see, handling super large or super small numbers with scientific notation is easier than writing them all out, which is why calculators come with this kind of functionality already built in.
Scientists have come up with a handy notation that helps take care of variables that have very large or very small values in their standard units. Say you’re measuring the thickness of a human hair and find it to be 0.00002 meters thick. You could use scientific notation to write this as meters ( meters), or you could use the unit prefix , which stands for micro:. When you put in front of any unit, it represents 10–6 times that unit.
A more familiar unit prefix is k, as in kilo, which represents 103 times the unit. For example the kilometer, km, is 103 meters, which equals 1,000 meters. The following table shows other common unit prefixes that you may see.
Q. How does the number 1,000 look in scientific notation?
A. The correct answer is 1.0 × 103. You have to move the decimal point three times to the left to get 1.0.
Q. What is 0.000037 in scientific notation?
A. The correct answer is 3.7 × 10–5. You have to move the decimal point five times to the right to get 3.7.
1. What is 0.0043 in scientific notation?
2. What is 430,000 in scientific notation?
3. What is 0.00000056 in scientific notation?
4. What is 6,700 in scientific notation?
1.4.3 × 10–3. You have to move the decimal point three places to the right.
2.4.3 × 105. You have to move the decimal point five places to the left.
3.5.6 × 10–7. You have to move the decimal point seven places to the right.
4.6.7 × 103. You have to move the decimal point three places to the left.
1.235 cm. Convert 2.35 meters into centimeters:
2.79 in. Convert 2.0 meters into inches:
3.91.4 cm. One yard is 3 feet, so convert that to inches:
Use a second conversion factor to convert that into centimeters:
4.4.0 × 10–9 in. Convert 1 angstrom to centimeters:
Use a second conversion factor to convert that into inches:
5.8,760 hr. Convert 1 year into days:
Use a second conversion factor to convert that into hours:
6.0.18 km/hr. One foot is 12 inches, so convert that into centimeters:
Use a second conversion factor to convert centimeters into meters and then into kilometers:
Convert 1 hour into minutes:
Use your results to convert 10 feet per minute into kilometers per hour:
Accuracy and precision are important when making (and analyzing) measurements in physics. You can’t imply that your measurement is more precise than you know it to be by adding too many significant digits, and you have to account for the possibility of error in your measurement system by adding a when necessary. This section delves deeper into the topics of significant digits, precision, and accuracy.
This section is all about how to properly account for the known precision of the measurements and carry that through the calculations, how to represent numbers in a way that is consistent with their known precision, and what to do with calculations that involve measurements with different levels of precision.
In a measurement, significant digits (or significant figures) are those that were actually measured. Say you measure a distance with your ruler, which has millimeter markings. You can get a measurement of 10.42 centimeters, which has four significant digits (you estimate the distance between markings to get the last digit). But if you have a very precise micrometer gauge, then you can measure the distance to within one-hundredth of that, so you may measure the same thing to be 10.4213 centimeters, which has six significant digits.
By convention, zeroes that simply fill out values down to (or up to) the decimal point aren’t considered significant. When you see a number given as 3,600, you know the 3 and 6 are included because they’re significant. However, knowing which, if any, of the zeros are significant can be tricky.
Tip: The best way to write a number so you leave no doubt about how many significant digits there are is to use scientific notation. For example, if you read of a measurement of 1,000 meters, you don’t know if there are one, two, three, or four significant figures. But if it were written as meters, you would know that there are two significant figures. If the measurement were written as meters, then you would know that there are four significant figures.
When you do calculations, you often need to round your answer to the correct number of significant digits. If you include any more digits, you claim a precision that you don’t really have and haven’t measured.
For example, if someone tells you that a rocket traveled 10.0 meters in 7.0 seconds, the person is telling you that the distance is known to three significant digits and the seconds are known to two significant digits (the number of digits in each of the measurements). If you want to find the rocket’s speed, you can whip out a calculator and divide 10.0 meters by 7.0 seconds to come up with 1.428571429 meters per second, which looks like a very precise measurement indeed. But the result is too precise — if you know your measurements to only two or three significant digits, you can’t say you know the answer to ten significant digits. Claiming as such would be like taking a meter stick, reading down to the nearest millimeter, and then writing down an answer to the nearest ten-millionth of a millimeter. You need to round your answer.
Remember: The rules for determining the correct number of significant digits after doing calculations are as follows:
When you multiply or divide numbers:
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