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Chapter 1: Units, Scalars, Vectors Version 2
The Big Idea
Since physics depends fundamentally on measurements that are interpreted through math, the first distinction we have to make is between different types of measurements and their properties. First, all measurements must have units. Units identify what a specific number refers to. For instance, the number 42 can be used to represent 42 miles, 42 pounds, or 42 elephants! Numbers are mathematical objects, but units give them physical meaning. Keeping track of units can help you avoid mistakes when you work out problems.
Units Concepts
Every answer to a physics problem must include units. Even if a problem explicitly asks for a speed in meters per second (m/s), the answer is 5 m/s, not 5.
When you’re not sure how to approach a problem, you can often get insight by considering how to obtain the units of the desired result by combining the units of the given variables. For instance, if you are given a distance (in meters) and a time (in hours), the only way to obtain units of speed (meters/hour) is to divide the distance by the time. This is a simple example of a method called dimensional analysis, which can be used to find equations that govern various physical situations without any knowledge of the phenomena themselves. To use dimensional analysis, assume that the answer to a problem consists of a product of all the variables given raised to various powers. Many times, there will be only one such combination that gives the desired result.
This textbook uses SI units (La Système International d’Unités), the most modern form of the metric system.
When converting speeds from metric to American units, remember the following rule of thumb: a speed measured in mi/hr is about double the value measured in m/s (i.e., 10 m/s is equal to about 20 MPH). Remember that the speed itself hasn’t changed, just our representation of the speed in a certain set of units.
If a unit is named after a person, it is capitalized. So you write “10 Newtons,” or “10 N,” but “10 meters,” or “10 m.”
Scalars
The simplest kind of measurement is a single number, or scalar. Scalars are all one needs to describe temperature, density, length, and many other phenomena in physics. The mathematics used in the manipulation of scalars -- addition, subtraction, multiplication, and division -- come naturally to humans, and, to a large extent, to other animals. Many mammals have an innate ability to divide a pile of food into relatively equal pieces, to distinguish between objects of different size, and to perform other tasks that seemingly require intelligence. It would seem crazy to suggest that the animals are performing mathematical operations based on formal logic, but that is not the point. Much more likely is the idea that formal mathematics is an extension of our natural abilities. In fact, the way math has been taught throughout history and across the world -- think of your own elementary and middle school classes -- seems to reflect this underlying property of human nature.
Vectors
The first new concept introduced here is that of a vector: a scalar magnitude with a direction. In a sense, we are almost as good at natural vector manipulation as we are at adding numbers. Consider, for instance, throwing a ball to a friend standing some distance away. To perform an accurate throw, one has to figure out both where to throw and how hard. We can represent this concept graphically with an arrow: it has an obvious direction, and its length can represent the distance the ball will travel in a given time. Such a vector (an arrow between the original and final location of an object) is called a displacement:
Vector Addition and Subtraction
Like scalars, vectors have a branch of mathematics dedicated to them; and again, the basics can be considered an extension of our natural abilties, while the more advanced parts are quite foreign to our intuition. The first concept is that of vector addition. Think about throwing a pass, but this time to a moving target. If we use our original arrow, the target will have moved by the time the ball reaches its endpoint. To be accurate, we need to consider the displacement of the target and add it to the original arrow. The picture can be presented this way:
It should be apparent that if we throw the ball according to the dashed arrow, we will hit the target. This third vector is the sum of the first two displacements, and of course, also a displacement vector. This is how vectors are added graphically: if the end of the first vector is drawn at the beginning of the second, the arrow linking the beginning of the first with the end of the second will be their sum. Alternatively, the two vectors can be moved to become the legs of a parallelogram. Their sum is then the diagonal:
To subtract vectors, you can simply flip the vector you are subtracting by 180 degrees and add them. This is essentially the vector version of saying that subtracting a positive number is the same as adding a negative one:
Vector Components
From the above examples, it should be clear that two vectors add to make another vector. Sometimes, the opposite operation is useful: we often want to represent a vector as the sum of two other vectors. This is called breaking a vector into its components. When vectors point along the same line, they essentially add as scalars. If we break vectors into components along the same lines, we can add them by adding their components. The lines we pick to break our vectors into components along are often called a basis. Any basis will work in the way described above, but we usually break vectors into perpendicular components, since it will frequently allow us to use the Pythagorean theorem in time-saving ways. Specifically, we usually use the and axes as our basis, and therefore break vectors into what we call their and components:
A final reason for breaking vectors into perpendicular components is that they are in a sense independent: adding vectors along a component perpendicular to an original component one will never change the original component, just like changing the -coordinate of a point can never change its -coordinate.
Frequently Used Measurements, Greek Letters, and Prefixes
Measurements
Types of Measurements Type of measurement Commonly used symbols Fundamental units
length or position meters (m)
time seconds (s)
velocity or speed meters per second (m/s)
mass kilograms (kg)
force Newtons (N)
energy Joules (J)
power Watts (W)
electric charge Coulombs (C)
temperature Kelvin (K)
electric current Amperes (A)
electric field Newtons per Coulomb (N/C)
magnetic field Tesla (T)
Prefixes
SI prefix In Words Factor
nano (n) billionth
micro (µ) millionth
milli (m) thousandth
centi (c) hundreth
deci (d) tenth
deca (da) ten
hecto (h) hundred
kilo (k) thousand
mega (M) million
giga (G) billion
Greek Letters
Frequently used Greek letters. “mu” “tau” “Phi”* “omega” “rho”
“theta” “pi ” “Omega”* “lambda” “Sigma”*
“alpha” “beta” “gamma” “Delta”* “epsilon”
Applications and Examples
Here
are some situations where the ideas covered in the chapter are useful.
Question: The lengths of the sides of a cube are doubling each second. At what rate is the volume increasing?
Solution:The cube side length, , is doubling every second. Therefore after 1 second it becomes . The volume of the first cube of side is . The volume of the second cube of side is . The ratio of the second volume to the first volume is . Thus the volume is increasing by a factor of 8 every second.
Fermi Questions
The late great physicist Enrico Fermi used to solve problems by making educated guesses. Say you want to guesstimate the number of cans of soda drunk in San Francisco in one year. You’ll come pretty close if you guess that there are about 800,000 people in S.F. and that one person drinks on average about 100 cans per year. So, around 80,000,000 cans are consumed every year. Sure, this answer is not exactly right, but it is likely not off by more than a factor of 10 (i.e., an “order of magnitude”). That is, even though we guessed, we’re going to be in the ballpark of the right answer. This is often the first step in working out a physics problem.
Dimensional Analysis
Question: find (up to a proportionality constant) the period of a pendulum hanging on a string; that is, find how long it takes such a pendulum to swing through one cycle, knowing that it depends only the acceleration due to gravity and its length, , which is measured in meters.
Solution: since the period is in units of time, the answer needs to have units of time (seconds). The only way to obtain seconds from the given quantities is to take the square root of the reciprocal of (which will have units of seconds over square root of meters) and multiply it by the square root of , which has units of square root of meters --- this will get rid of the meters altogether. In other words, the period will be proportional to the square root of divided by the square root of :
Units and Problem Solving Problem Set
Estimate or measure your height. Convert your height from feet and inches to meters.
Convert your height from feet and inches to centimeters
Estimate or measure the amount of time that passes between breaths when you are sitting at rest. Convert the time from seconds into hours
Convert the time from seconds into milliseconds
Convert the French speed limit of into .
Estimate or measure your weight. Convert your weight in pounds into a mass in
Convert your mass from into
Convert your weight into Newtons
Find the unit for pressure.
An English lord says he weighs stone. Convert his weight into pounds (you may have to do some research online)
Convert his weight in stones into a mass in kilograms
If the speed of your car increases by every 2 seconds, how many is the speed increasing every second? State your answer with the units .
A tortoise travels meters west, then another centimeters west. How many meters total has she walked?
A tortoise, Bernard, starting at point A travels west and then millimeters east. How far west of point is Bernard after completing these two motions?
. What is ?
A square has sides of length . What is the area of the square in ?
A square with area is stretched so that each side is now twice as long. What is the area of the square now? Include a sketch.
A rectangular solid has a square face with sides in length, and a length of . What is the volume of the solid in ? Sketch the object, including the dimensions in your sketch.
As you know, a cube with each side in length has a volume of . Each side of the cube is now doubled in length. What is the ratio of the new volume to the old volume? Why is this ratio not simply ? Include a sketch with dimensions.
What is the ratio of the mass of the Earth to the mass of a single proton? (See equation sheet.)
A spacecraft can travel . How many km can this spacecraft travel in 1 hour ?
A dump truck unloads kilograms of garbage in . How many are being unloaded?
The lengths of the sides of a cube are doubling each second. At what rate is the volume increasing?
Estimate the number of visitors to Golden Gate Park in San Francisco in one year. Do your best to get an answer that is correct within a factor of .
Estimate the number of water drops that fall on San Francisco during a typical rainstorm.
What does the formula tell you about the units of the quantity (whatever it is)?
Add the following vectors using the parallelogram method.
Answers to Selected Problems
a. A person of height . . is tall
b. The same person is
a. b.
c. if the person weighs . this is equivalent to
Pascals (Pa), which equals
f.
b.
c.
each side goes up by , so it will change by
; is for each second starting with seconds for
About million
About trillion
Chapter 2: Energy Conservation Version 2
The Big Idea
Energy, a scalar quantity measured in Joules, is a measure of the amount of, or potential for, dynamical activity in something. The total amount of energy in the universe is constant. This kind of symmetry is called a conservation law. Conservation of energy is actually just one of five conservation laws that have been identified by scientists.
The concept of energy conservation is one of the most important in physics, since according to our best model of the world, everything in the universe --- even mass --- is a form of energy. Any group of things (we’ll use the word system for this concept in the book) has a certain amount of energy. Energy can be added to a system: when chemical bonds in a burning log break, they release heat. A system can also lose energy: when a spacecraft “burns up” its energy of motion during re-entry, it releases energy to the surrounding atmospher in the form of heat. A closed system is one for which the energy is constant, or conserved. In this chapter, we will often consider closed systems; although the total amount of energy stays the same, it can transform from one kind to another. We will consider transfers of energy between systems – known as work – in more detail in Chapter 8. Needless to say, the universe as a whole is a closed system in this sense.
Types of Energy
Understanding how various processes change energy from one form to another is equivalent to understanding physics. In this class, we will present an overview of various forms of energy but will mainly focus on three: kinetic, gravitational potential, and electrical potential. We will focus on the first two in this chapter; electrical potential energy is covered in later chapters.
Kinetic Energy
The first is Kinetic Energy, or the energy of motion. Any moving object --- from the earth to an individual gas molecule --- has some kinetic energy, which can be calculated by using the following formula: The refers to the object's mass, while the is its speed.
Gravitational Potential Energy
The second type of energy is due to gravity and is therefore called gravitational potential energy. Things with mass have noticeable gravitational potential energy when they are near another object of significant mass, such as the earth, the sun, a black hole, etc. This energy is different from kinetic energy in that it represents potential for motion, rather than motion itself. If I lift a rock away from the surface of the earth to some height and then let it drop, it will gain velocity as it travels downwards. According to the last paragraph, this means it also gains kinetic energy. Assuming no energy is lost to air resistance, there will be a one to one correspondence between gravitational potential energy lost and kinetic energy gained. Near the surface of a planet, the gravitational potential energy gained by an object of mass when raised a height from its original position perpendicular to the surface of the planet is just The constant will vary from planet to planet, star to star. On earth, the acceleration due to gravity is , often rounded to . This is the formula you will likely
use the majority of the time. However, there is a way to express the gravitational potential energy of any two objects in the universe --- any number of objects, in fact. For the two object case, if we call their masses and and the distance between their centers of mass , the formula is: In fact, equation [2] is a special case of equation [3]. That is, it is a version of equation [3] that holds under specific circumstances. See the appendix to this chapter for a derivation.
Key Concepts
Any object in motion has kinetic energy. Kinetic energy increases as the square of the velocity, so faster objects have much more kinetic energy than slower ones.
The energy associated with gravity is called gravitational potential energy. Near the surface of the earth, an object's gravitational potential energy increases linearly with its height.
Molecules store chemical potential energy in the bonds between electrons; when these bonds are broken the released energy can be transferred into kinetic and/or potential energy. 1KCal (1 food Calorie) is equal to 4180 Joules of stored chemical potential energy.
Energy can be transformed from one kind into another and exchanged between systems; if there appears to be less total energy in a system at the end of a process then at the beginning, the “lost” energy has been transferred to another system, often by heat or sound waves.
CK-12 People's Physics Book Version 2 Page 1
