Chapter 01 Introduction to Conceptual Physicsscience.sbcc.edu/~physics/ferril/Resources/SBCC P 101 Chapter 01.pdfNatural science is composed of physical science and biological science.

Ron Ferril SBCC Physics 101 Chapter 01 2017Jun06AA Page 1 of 26

Chapter 01

Introduction to Conceptual Physics

Introduction to Physics as a ScienceThere is so much confusion regarding what science is. I, like many other scientists, tried to develop a good definition of science but have found all definitions to lack perfection. Thus, we must be content with descriptions of “natural science” and its two major branches “physical science” and “biological science.” Natural sciences involve “scientific laws” which are statements believed to reliably describe properties of the universe and how natural processes occur. Some branches of natural science are burdened by controversies regarding the reliability of proposed scientific laws. There have been unofficial movements in universities to “educate” students (which I call “indoctrination” of students) with what professors consider to be the proper views of science, politics, philosophies and religions.

It is not the purpose of this course to indoctrinate you regarding any particular position in these controversies. This course provides tools of investigation and understanding: knowledge and skill. You are required to gain knowledge and skill rather than being forced to adopt a style of reasoning. The opinions of your instructor are not relevant. You can doubt anything I present but you should consider how you can use the material presented in this course.

This course focuses on a branch of physical science called “physics.” Physics has been called an “exactscience” because its results are supposed to be based on mathematical conclusions rather than viewpoints in controversies. Physics is a branch of physical science and is, thus, also a branch of natural science. The subject of physics is so mathematical in nature that the boundary between mathematics and physics is fairly vague. For example, the subject of “kinematics” is considered to be a branch of physics (by physicists and engineers), but is also considered to be a branch of mathematics. Thus, the principles of kinematics are seen both in physics classrooms and in mathematics classrooms. In order to understand what physics is, we first must understand what mathematics really is.

Mathematics is the art and science of the following three processes.

© Copyright 2015, 2016 and 2017 by Ron Ferril. All rights reserved.


1. Forming postulates. Postulates are statements assumed to be true without proof (or without needof proof). Precise definitions of terms are examples of postulates because they are assumed to be true. You assume your definitions are correct since you adopt them. Some statements that areconsidered to be obviously true are adopted as postulates. For example, the “reflexive property of equality” for real numbers states that “every real number is equal to itself.” The reflexive property is obviously true. 5=5, 100=100, pi=pi and 3.87 = 3.87) This property can be assumed true without need of logical proof. It is a postulate.

2. Showing that other statements, called theorems, logically follow from the postulates. Mathematicians demand that the logic be precise or “rigorous” in order to “prove” theorems logically follow from the postulates. Mathematical rigor avoids gaps in the logic and avoids circular reasoning.

3. Applying the postulates and theorems to solve problems. Every algorithm for doing arithmetic problems or other mathematical problems is based on and justified by a combination of postulates and theorems.

Physics is similar to mathematics and is regarded as the most mathematical of the natural sciences. Physics is the science of “physical laws” of the universe. These laws are scientific laws and describe how the universe and all its parts behave. “Basic” or “fundamental” laws are physical postulates. “Derived” laws are physical theorems. In physics, derived laws logically follow from fundamental lawslike theorems logically follow from postulates in mathematics. Thus, the process of physics is similar to the process of mathematics. However, the process of writing fundamental and derived laws often follows a different order than in mathematics. In physics, the derived laws are often discovered by experiments first, and then discovery of fundamental laws follows. That is the reverse of the mathematical process that begins with postulates and logically leads to theorems. For example, Newton's Laws of Motion and Newton's Law of Gravity (with modern clarifications) are fundamental physical laws but they came after Kepler's Laws of Planetary Motion which are derived laws. Kepler's Laws are theorems that logically follow from Newton's Laws which are postulates. However, Newton used Kepler's Laws to found his postulates. I like to call this process, of working backward from theorems to postulates, “reverse rigor” since it is the reverse of the rigorously logical process of mathematics. Once Newton had established his fundamental laws, he showed that Kepler's Laws logically followed as derived laws. Actually, Newton and other physicists showed that most of the scientific laws of Newton's era were derived laws following from his fundamental laws. Thus, we can see that physics involves running the process both ways: postulates to theorems and theorems to postulates. That is, physics runs from derived laws to fundamental laws (by experiments and discoveries) and fundamental laws to derived laws (by rigorous logic as in mathematics).



The main feature of physics that distinguishes it from other natural sciences is its direct involvement of fundamental physical laws. Studies of other branches of natural science tend to begin with either derived laws or consequences of derived laws. All of natural science has physics as a foundation, and physics has mathematics as a foundation. All scientific laws of physical science are either fundamental physical laws or derived laws that logically follow from the fundamental laws of physics. Thus, physicscan be considered the foundation of physical science, and mathematics can be considered the foundation of physics. Physical science is the foundation of the biological and medical sciences. Thus, physics is important for the natural sciences in general because it is a basis for scientific laws.

Physics is often divided into two disciplines: theoretical physics and experimental physics. Experimental physics involves design of and running experiments. Many scientific laws were



discovered by experiments. Theoretical physics involves the logical processes involving fundamental and derived laws—both the rigorous process from postulates to theorems and the reverse process from derived laws to fundamental laws—without need of direct involvement in experiments. Experimental physics produces evidence to drive the processes of theoretical physics and verifies or refutes proposed scientific laws. The design of experiments is often based on the works of theoretical physics. One difference between experimental physicists and theoretical physicists is that experimental physicists have laboratories, shops and equipment with which they design, build and use experimental apparatus, but a theoretical physicist's laboratory is typically a computer, a notebook, and either a chalk board or amarker board. A typical experimental physicist knows how to use technology and often develops technology much like an engineer does. Some students consider theoretical physics to be the more “pure” form of physics because it is more independent of engineering and technology. However, the two branches of physics are so interdependent that they are one science.

Branches of Natural ScienceNatural science is composed of physical science and biological science. The core sciences of physical science are physics and chemistry. These sciences form the foundation for all other physical sciences such as, for example, geology and astronomy. In past times, chemistry and physics were very separate sciences but, with advances in “physical chemistry” (described next), the two have become significantly connected.

Chemistry is the science of substances and can be divided into three fields of study that describe what chemists do. These three basic fields are

• analytic or analytical chemistry,

• synthetic chemistry (or synthesis) and

• physical chemistry.

Analytical chemistry involves identifying substances. An analytical chemist designs methods of testing substances to identify their composition, and runs tests on a sample to identify its chemical structure. Synthetic chemistry involves generating substances. A synthetic chemist designs ways of making desired substances. I often consider the synthetic chemistry to be what I call “molecular engineering” because it involves engineering of processes for building and manipulating molecules. Physical chemistry is the direct application of physics to chemistry. A physical chemist may be regarded as a chemical physicist. Advances in physics in recent centuries advanced the science of chemistry.



The following table shows the division of natural sciences. There are too many branches of biological sciences to explicitly display in the table. For this course, I thought I should emphasize biochemistry and biophysics. Biochemistry applies chemistry to biology. Biophysics applies physics to biology.

Physical Science

Chemistry

Analytic Chemistry

Synthetic Chemistry or Synthesis

Physical Chemistry

Physics

Theoretical Physics

Experimental Physics

Other Physical Sciences Based on Chemistry and Physics

Biological Science

Biochemistry

Biophysics

Biotechnology

and many other biological fields

About this CourseThis course, Physics 101 “Conceptual Physics,” is a special physics course because it presents conceptsof physics with as little mathematics as practical. This separation of mathematics from physics may seem strange since mathematics is the heart of physics. Indeed, you cannot become a physicist without much involvement with mathematics. This was not always the case. Thousands of years ago, physics was studied as a subject very separate from mathematics. Thus, this course revives the ancient tradition by presenting physics with minimal math. However, you will be learning some mathematical subjects early in this course—actually starting with a discussion of units in the next section of this document.



UnitsSome quantities have units. For example, distances have units such as, for example, meters, feet, inches, centimeters, light years and parsecs. Time may have units such as hours, minutes, seconds or years. Some quantities may be expressed without units such as, for example, the number of students in a classroom. We need to know how to do arithmetic with quantities that have units.

First, you can add or subtract quantities that have the same units. In the following example, “kg” is an abbreviation for kilogram.

5 kg + 3 kg = 8 kg

5 kg – 3 kg = 2 kg

Addition and subtraction of quantities with different units is not defined. In the following example, “m” is an abbreviation for meters.

5 kg + 3 m is undefined

However, sometimes conversions exist to express one unit in terms of another unit. For example, one kilogram is 1000 grams.

1 kg = 1000 g

Thus, before I label an expression like

3.000 kg + 200 g

as undefined, I should make use of the conversion and the property (a postulate) that says that multiplying any real number by 1 does not change the number. Since one kilogram (kg) equals 1000 grams (g), the result of dividing 1000 grams by one kilogram should be one.

[1000 g] / [1 kg] = 1

We convert kilograms to grams by multiplying by 1 written as [1000 g] / [1 kg].

3.000 kg + 200 g = 3.000 kg x [1000 g] / [1 kg] + 200 g = 3000 g + 200 g = 3200 g

Notice the kg in the numerator “canceled” the kg in the denominator leaving only grams. (This idea is easier to understand when you see me do this problem on the board in class since you can see me canceling the units that appear in both the numerator and denominator.) Thus, we transformed the original “3.000 kg + 200 g” to an expression where each term had the same units and, thus, we could add them. We could not properly add them without the conversion.



Products of quantities are defined even when the units are different. Likewise quotients of quantities with different units are defined as long as division by zero is not involved. (Division by zero is not defined.) Consider the following examples. The abbreviation of seconds is “s” in these examples.

3 kg x 2 m/s = 6 kg m/s

[8 kg m] / [2 kg s] = 4 m/s

Notice that the kg units canceled each other in the last example because kg appeared in the numerator and the denominator. Sometimes units actually cancel leaving a quantity without units.

[6 kg] / [3 kg] = 2

UncertaintiesNormally measurements of physical quantities involve “uncertainties” which are limits of precision. For example, I might be able to use a ruler to measure a length to the nearest 0.01 meter. Thus, the “uncertainty” in the measurement is 0.01 meter. One method of determining uncertainties is to determine the smallest increment of the quantity your method can measure. For example, instructions for some laboratory instruments state a “limit of precision” which is the uncertainty in readings the instrument yields. When using a ruler, we can judge how precisely we can read the ruler. If you can read the ruler to a precision no better than 1 millimeter (1 mm), then the uncertainty in your length measurement is 1 millimeter.

Another method of determining the uncertainty in a measurement is to repeat the measurement repeatedly. Each repetition is called a “trial” or a “trial run.” For example, forensic scientists can measure the “muzzle velocity” of a rifle by firing a bullet into a device called a “ballistic pendulum.” However, a common practice is to fire the rifle several times and record the readings. Suppose the values obtained are 725 mph, 712 mph, 730 mph and 718 mph. (The abbreviation “mph” indicates “miles per hour” which can also be written as a fraction or quotient miles/hour.) The average of these values is

[720 mph + 722 mph + 730 mph + 728 mph] / 4 = 725 mph

but the values spread from 5 mph below this value to 5 mph above. Thus, you can estimate the uncertainty to be 5 mph.



The uncertainties explained so far can all be called “absolute” uncertainties because they have the sameunits as the measured value, are positive by convention, and express an absolute value of a variation from the measured value. Sometimes scientists like to use “relative” uncertainties which are fractions with no units. For example, if a measured value is 10 meters and the absolute uncertainty in the measurement is 0.1 meter, then the relative uncertainty would be

[0.1 meter] / [10 meters] = 1/100 = 0.01 = 1%

Notice that relative uncertainties can be expressed as fractions, decimals or percents.

Physicists often use measured values, with their uncertainties, in calculations. Determining uncertainties in the resulting calculated values is one subject of a branch of mathematics called “mathematical statistics.” (I say “mathematical” statistics because some colleges and universities have courses in social studies or humanities called “statistics” but with little emphasis on the mathematical logic of statistics.) We won't concern ourselves with most of the details of statistics in this course, but we will state some approximate rules useful for estimating uncertainties in calculated results.

First, the absolute uncertainties sum when quantities are added or subtracted. Thus, if 15 kg with an uncertainty of 2 kg is to be added to 4 kg with an uncertainty of 1 kg, the result is 19 kg with an uncertainty of 3 kg.

15 kg +/- 2 kg

+ 4 kg +/- 1 kg

------------------------

19 kg +/- 3 kg

This summation of uncertainties works for either addition or subtraction. If I subtract 4 kg with an uncertainty of 1 kg from 15 kg with an uncertainty of 2 kg, the result is 11 kg with an uncertainty of 3 kg.

15 kg +/- 2 kg

- 4 kg +/- 1 kg

------------------------

11 kg +/- 3 kg



This summation of uncertainties is an approximate rule suitable for this course. The subject of mathematical statistics gives more precise procedures for handling uncertainties in addition and subtraction.

For multiplication and division, relative uncertainties are summed rather than absolute uncertainties. If I multiply a mass of 12.0 kg with an uncertainty of 0.6 kg by a length of 2.0 meters with an uncertainty of 0.5 meters, the product will be 24 kg m but I need to convert the absolute uncertainties in the mass and length to relative uncertainties to estimate the uncertainty in the product.

[0.6 kg] / [12.0 kg] = 0.05 = 5% = 1/20

[0.5 m] / [2.0 m] = 0.25 = 25% = ¼

0.05 + 0.25 = 0.30 = 30% = 3/10 which is the relative uncertainty in the product 24 kg m

24 kg m x 0.30 = 7.2 kg m which is the absolute uncertainty in the product 24 kg m

Thus, the product 24 kg m has an absolute uncertainty of 7.2 kg m. It is common for people toround uncertainties and write the result as 24 kg m +/- 7 kg m.

Suppose we divide the mass 12.0 kg with an uncertainty of 0.6 kg by the length 2.0 m with an uncertainty of 0.5 m. The quotient will be 6.0 kg/m but we need to calculate the uncertainty by summing relative uncertainties.

[0.6 kg] / [12.0 kg] = 0.05 = 5% = 1/20

[0.5 m] / [2.0 m] = 0.25 = 25% = ¼

0.05 + 0.25 = 0.30 = 30% = 3/10 which is the relative uncertainty in the quotient 6.0 kg/m

6.0 kg/m x 0.30 = 1.8 kg/m which is the absolute uncertainty in the quotient 6.0 kg/m

Thus, the result can be written as 6.0 kg/m +/- 1.8 kg/m. Many people would not round the uncertainty to 2 kg/m since the first digit of 1.8 is 1. For this course, your instructor may leave decisions of rounding to the students. (Check with your instructor in class regarding policies for rounding uncertainties.)



Large and Small NumbersConsider the value 5.213 kg. How precise do you think this quantity was measured? We don't know, but the number of digits given is a clue. Since the value was written with the thousandths digit (thousandths of a kg) included, we suspect that the precision or, in other words, the uncertainty is thousandths of a kg. Scientists have chosen conventions by which they don't write more digits that the precision would allow. Thus writing 5.213000 kg is not proper unless the intention is to suggest that themeasurement was precise to the nearest millionth of a kg. If the uncertainty is thousandths, then the three zeros at the right side are not “significant” digits and should not be included when the value is written.

With such conventions in mind, consider the large value 5,213,000,000,000 kg. Writing such a value suggests that the value was measured with an uncertainty of one or a few kilograms. If the uncertainty is really 10,000,000,000 kg, then those extra zeros are misleading. Thus, we can perform the following procedure for writing a large number in better “scientific notation.”

1. Count all the digits left of the decimal point except for the left-most digit. Thus, for 5,213,000,000,000 kg we count the twelve digits 213,000,000,000 but not the 5 at the far left end. Our count is 12 digits. This count will be used as an exponent in a later step.

2. Write the number with only the first digit left of a decimal point and round off the insignificant digits. This gives us 5.21 since the uncertainty was 10,000,000,000 kg.

3. Write the product of this resulting number with base 10 and exponent 12 since our earlier count was 12. Thus, the final scientific notation for our value is 5.21 x 1012 kg.

For small numbers the procedure is similar. For 0.00000521300 inch with an uncertainty of 0.00000001 inch, we should round to keep only the significant digits. This gives 0.00000521 inch as the correctly written value. We can write this in scientific notation by the following procedure.

1. Round the small number to the correct number of significant digits. This gave us 0.00000521 inch.

2. Count the digits right of the decimal point including the first nonzero digit which is 5 in this case. The count is 6.

3. Write the number with only the first nonzero digit left of the decimal. This gives us 5.21 which is larger compared to our original 0.00000521 inch.



4. Write the product of this resulting number with 10 to the exponent -6 since our count in an earlier step was 6. This gives us 5.21 x 10-6 inch as our final scientific notation for the value.

Notice the large numbers written in scientific notation had 10 to positive exponents and small numbers had 10 to negative exponents.

Absolute and Relative QuantitiesThe terms “absolute” and “relative” have more than one meaning in physics. Relative quantities are differences from a reference. The quantity is taken to be zero at the reference. For example, in class I sometimes hold up an object and ask how high the object (my hand or my carrying case) is, and students give different answers or ask me “how high above what?” which is essentially asking for the reference (the table top or the floor). The object's height was about one foot (above the table top) or four feet (above the floor) depending on the reference. When we specify a distance to some place, we are specifying that distance relative to some reference place. For example, one day I was in Los Angeles County and I inquired about the distance to Disneyland. I wanted to know the distance from my current location but the person apparently gave me the distance from his house. The distance was a relative quantity because it needed to be specified relative to some reference.

Absolute quantities are quantities independent of reference. For example, the number of people in a room can be considered an absolute quantity. The mass of an object and a volume of water are absolute quantities. Some people may deny that there are any absolute quantities because all values can be considered to be relative to zero or, in some cases, another value. That is a valid viewpoint, but physicists often benefit by considering some quantities to be absolute and others to be relative.

Galilean relativity introduces another way quantities can be relative. Galileo realized that speeds and velocities are relative to a reference. (The difference between speed and velocity is explained later.) In class, I often use the example of standing in a train moving forward at 40 meters per second and rollinga ball toward the front of the train at a speed of one meter per second relative to myself. I would say theball was rolling at one meter per second. People standing at a train station could watch the train and mepass. They would say the ball was moving at 41 meters per second (= 40 m/s + 1 m/s). Galileo would say both the people and I were correct. I would specify the speed relative to me and the other people would specify the speed relative to their position or “frame of reference.”



Vectors and ScalarsVectors were introduced to mathematics by physicists. A vector is a quantity that has “magnitude” and direction. For example, a man may walk five kilometers North, and the quantity “five kilometers North” would be a vector because it has magnitude “five kilometers” and direction “North.” (Notice that the magnitude can have units. In this example the units are kilometers.) A scalar has magnitude but no direction. Thus, for example, mass and volume are scalars.

Speed and velocity are not the same quantity because speed is a scalar and velocity is a vector. An example of a speed is “50 miles per hour” (or “50 mph” as it can be abbreviated). An example of a velocity is “ 50 miles per hour East.” The magnitude of this velocity is the speed “50 miles per hour” and the direction is East. For our view of vectors, the magnitude of any vector is a corresponding scalar.

Arrows in diagrams are often used to represent vectors. Consider the following diagram showing two vectors, one with magnitude 5 meters and the other with magnitude 2 meters, but with different directions.

Notice that the lengths of the arrows are proportional to the magnitudes of the vectors the arrows represent. This graphical representation of vectors by arrows allows graphical methods of doing arithmetic with vectors. For example, consider multiplication of a scalar and a vector. The product of a scalar and a vector is defined. Multiplication of a vector by two (a scalar without any units) can be represented by doubling the length of the arrow representing the original vector. Multiplication of the original vector by three can be represented by an arrow with the same direction as the original arrow but three times longer than the original arrow. Multiplication of a vector by negative one (-1) turns the vector around so the arrow points the opposite way.


2 meters5 meters


A very important point about vectors is that they have magnitude and direction but not location. That is,a vector specifies a magnitude and direction but is the same vector no matter where it is moved. Physicists often speak of moving a vector (but keeping its magnitude and direction) as “parallel displacement.” Thus, all the following arrows represent the same vector even though each is at a different location.

Such diagrams and this idea of “parallel displacement” allow us to graphically add vectors. In order for vectors to be added, they must have the same units. In our example of vectors with magnitudes 2 meters and 5 meters, the two vectors both have units of meters. To graphically add the vectors, we place the tail of one vector at the head (arrow end) of the other as in the following diagram. The vector sum is called the resultant vector or the net vector. The resultant is drawn from the tail of the first vector to the head of the last vector.



Sums of Vectors

We can add two or more vectors by merely placing them head-to-tail as in the following figure. We can use geometry to prove that such graphical vector addition is “commutative” meaning we can add the vectors in any order. It can also be shown to be “associative” so we can group vectors together and sumgroups to get subtotals and then add the subtotals to get the complete total vector sum.

Letters or variables are often used to denote quantities. Thus, a distance may be denoted by D. In many books (and in lecture in this class), vector quantities are often written as variables with arrows over the tops of the letters to indicate that they are vectors. In typed notes such as the notes you are reading here,it is more common to use a boldface font to indicate that the quantity is a vector. Thus, a scalar may be written as x but a vector written as x with the bold attribute.


2 meters

5 meters

Resultant

AB

CResultant


There is another method of adding vectors that involves “coordinates.” A common coordinate system isthe “rectangular” or “Cartesian” coordinate system that involves one, two or three axes: an x-axis which by tradition is drawn pointing to the right on a page of paper, a y-axis (if there are at least two axes drawn) which is drawn by tradition upward toward the top of a printed page, and a z-axis (if there are three axes drawn) pointing out of the page toward the reader. One way of specifying a vector’s magnitude and direction is to place the tail of the vector at the origin of this coordinate system and use x, y and z components to specify the vector. Three special vectors, called “unit vectors,” help us specifyany other vectors. These special unit vectors have length one without any units. One, which I will call ex, points in the direction of the positive x-axis. Another, called ey, points in the direction of the positive y-axis, and the third, ez, points in the direction of the positive z-axis. The notation for a vector V is of the form

V = Vx ex + Vy ey + Vz ez

The factors Vx, Vy and Vz are called components of the vector V along the x-axis, y-axis and z-axis, respectively. (Some books use the term “components” in reference to the terms Vx ex, Vy ey and Vz ez.) Consider what each of these components is. I like to use the term “shadow” to explain the significance of these components. The component Vx can be thought of as the “shadow” of the vector on the x-axis as shown in the following diagram.

Here a vector V of magnitude L is shown above the x-axis which points to the right. (The “cos θ” factorwill be explained in a later section.) The shadow Vx is labeled “L cos θ” in the diagram. There is such a shadow of the vector along each axis, and the length of this shadow indicates the component. Components can be positive or negative. If the angle labeled θ in the diagram is greater than 90



degrees, then the shadow will be along the negative x-axis instead of the positive x-axis and the component Vx will be negative.

For example, a vector of magnitude 5 meters pointing along the x-axis could be written as

5 m ex

A vector of magnitude 5 m but pointing along the negative x-axis could be written as

− 5 m ex

and a vector of magnitude 5 meters in the xy-plane, bisecting the angle between the x-axis and the y-axis can be written as

√5 m ex + √5 m ey

Notice that the √5 needs to be there so the Pythagorean theorem will give the magnitude of the vector as

√[(√5 m)2 + (√5 m)2 ] = 5 m

Suppose two vectors A and B with

A = Ax ex + Ay ey + Az ez

B = Bx ex + By ey + Bz ez

are added. The sum of the vectors is (Ax + Bx) ex +(Ay + By) ey +(Az + Bz) ez as can be seen here.


+ B = Bx ex + By ey + Bz ez

---------------------------------------------------------------------------------

A + B = (Ax+Bx) ex + (Ay+By) ey + (Az+Bz) ez

Subtraction works similarly,




− B = Bx ex + By ey + Bz ez

---------------------------------------------------------------------------------

A − B = (Ax−Bx) ex + (Ay−By) ey + (Az−Bz) ez

For example, add and subtract the vectors A = 5 m ex + 2 m ey and B = 2 m ex + 3 m ey.

A = 5 m ex + 2 m ey

+ B = 2 m ex + 3 m ey

---------------------------------------------------------------------------------

A + B = 7 m ex + 5 m ey

A = 5 m ex + 2 m ey

− B = 2 m ex + 3 m ey

---------------------------------------------------------------------------------

A − B = 3 m ex − 1 m ey

Products of VectorsWe already know that a vector can be multiplied by a scalar. Consider the idea of products of vectors. Mathematicians and physicists have considered various concepts of products of vectors. In this course, two types of products of vectors will be useful: “dot products” and “cross products.” Many physics books introduce these products by use of trigonometry. In this course, you are not expected to be experts on trigonometry. However, some simple concepts of two trigonometric functions—the “sine” and the “cosine” functions—should be introduced because they simplify some explanations. Consider avector with magnitude that is a length L and with a direction at angle θ from being horizontal as shown in the following figure. Imagine that the vector can be illuminated by a light and cast a shadow on a floor or wall. These shadows, shown in the following figure, are called “projections” of the vector along horizontal and vertical directions. (Various books differ on how they define projections. Some books treat projections as vectors and some treat projections as scalars.)



The length of the horizontal projection is the product of the length L and the “cosine” of the angle θ which is written as “cos θ” and the vertical projection is the product of length L and the “sine” of the angle which is written as “sin θ” as shown in the diagram. The nature of these two functions is described in courses in trigonometry. In this course, it may be enough for us to know that they can be used to write the horizontal and vertical projections (or “shadows”) of vectors.

There are conventions regarding the sign of an angle. Notice the difference in spelling between “sign” and “sine” even though the two words are pronounced the same. The “sine” is the trigonometric function and the “sign” of a nonzero quantity is positive or negative (+ or -). Angles opening counterclockwise as we view them are taken as having positive measure. Angle opening clockwise are have negative measure. Thus, I can open an angle counterclockwise by +45 degrees or clockwise by -45 degrees. (A later Lecture Summary uses a “right-hand rule” to specify the direction of the opening of the angle by a vector rather than by a sign.) The cosine of an angle θ is the same as the cosine of -θ, but the sine of -θ has sign opposite the sine of θ.

sin(− θ) = − sin θ

cos(− θ) = cos θ

The signs matter to specification of projections and the vector products that are introduced next.



The “dot product” of two vectors A and B is defined to be the scalar AB cos θ where A and B are the magnitudes of vectors A and B, and θ is the angle between the two vectors. We write the dot product asfollows.

A ● B = A B cos θ

Notice the dot ● can be considered good reason for calling the product the “dot” product. (This productcan be considered to be a type of product of magnitude of the first vector with the magnitude of the projection of the second vector along the direction of the first vector.) Notice that the dot product is said to be “commutative” because A ● B = B ● A.

For example, the dot product of a vector along the x-axis with magnitude 3 m/s and a vector with a direction 45 degrees from the x-axis and magnitude 2 m/s is

3 m/s 2 m/s cos(45 degrees) = 6 m2/s2 cos(45 degrees)

I can use a calculator or a table of sines and cosines to find that this dot product is 3√2 m2/s2. Notice that the dot product has magnitude but no direction. The dot product is a scalar.

The dot product can also be written in terms of components.

A ● B = AxBx + AyBy + AzBz

For example, the dot product of the vectors A = 5 m ex + 2 m ey and B = 2 m ex + 3 m ey is

5 m x 2 m + 2 m x 3 m = 16 m2

As in a previous example, we see that the dot product is a scalar (having no direction).

The “cross product” of two vectors A and B is a vector that is perpendicular to both A and B. This cross product has magnitude |AB sin θ|. However, there are two directions perpendicular to both A and B. We want to know which of the two directions is the cross product. This direction is determined by a choice mathematicians made a long time ago regarding how the axes of a Cartesian coordinate system are to be drawn. Mathematicians chose to draw the x-axis horizontal and pointing to the right, the y-axis vertical and pointing upward, and the z-axis pointing directly toward the viewer of the axes. The directions of the three axes can be remembered by laying the fingers of your right hand along the x-axis, then curling your fingers into the direction of the y-axis and your thumb points in the direction of the z-axis. Thus, if you look at an x-axis and y-axis drawn on a page, the z-axis would be pointing out of the page toward you. Because of this choice of drawing convention, the direction of the cross



product is given by the “right-hand rule” similar to the one used for the coordinate axes. This rule has you lay the fingers of your right hand along the direction of the first vector A and then curl your fingerstoward the direction of the second vector B so your thumb points in the direction of the cross product. Your instructor demonstrates this right-hand rule in class. The cross product of two vectors A and B is often written in the following notation.

A X B

The X symbol denotes the cross product. Notice that this product is not commutative. Mathematicians say it is “anti-commutative” because A X B = - B X A. The magnitude of A X B is

|A X B| = A B sin θ

Remember that

• the dot product of two vectors is a scalar and involves the cosine of the angle between the two

vectors, and

• the cross product of two vectors is a third vector that is perpendicular to the original two

vectors. The magnitude of the cross product involves the sine of the angle between the two original vectors.

These strange products are useful in explaining concepts like “work” and angular quantities like “torque” as we can see in later chapters.

KinematicsGalileo, in his book “Two New Sciences,” developed a branch of physics called “kinematics.” Kinematics is so basic to motion that it is taught in physics classes and in mathematics classes. For example, the formula

distance = speed x time

taught in high school mathematics courses is part of kinematics. Kinematics is concerned with three vector quantities of motion: displacement (or position) x, velocity v and acceleration a. The scope of kinematics excludes discussion of causes of motion such as force and resistance to motion such as inertia and mass. In this document, as in many physics books, boldface letters are used to denote vectors and scalars are denoted by letters without the boldface. Thus, you may notice that I write the kinematic vectors as x, v and a. When vector quantities are written by hand (as in writing vectors on



paper or on a board in a classroom), little arrows, pointing to the right, are placed over the letters instead of using boldface type.

In the times of Galileo, people avoided discussion of the directions of the vectors and focused on the magnitudes of the vectors. Thus, in the distance formula from high school, the distance is the magnitude x of displacement x and speed v is the magnitude of velocity v. In modern times, we still use the magnitudes for “one-dimensional problems” in which the motion is all along a single line.

The velocity v is the rate of change of the displacement vector x. Thus, if the displacement changes by an amount Δx in time Δt, then the average velocity vavg over time Δt is is vavg = Δx/Δt. In order to get an estimate of the “instantaneous” velocity v at a time t, the interval of time Δt is taken as small as possible. If the velocity is constant, then the average velocity Δx/Δt is equal to the instantaneous velocity and displacement Δx=vΔt.

The acceleration a is the rate of change of velocity. Thus, if the velocity changes by an amount Δv in a time interval Δt, then in the time interval Δt the average acceleration is aavg = Δv/Δt. In order to get a good estimate of the instantaneous acceleration at a time t, the time interval Δt should be as small as possible. Suppose the acceleration is constant so the average acceleration is the instantaneous acceleration. Then Δv = aΔt. Galileo used mathematical arguments to show that the change in displacement for constant acceleration is

Δx = v0Δt + ½ a (Δt)2 (for constant acceleration)

where v0 is the initial velocity at the start of the time interval Δt. Actually, Galileo showed this for a one-dimensional case where all the motion—displacement, velocity and acceleration—is along a line. His argument was easily extended so the equation applies to three-dimensional cases. If we limit our discussion to one dimension so the motion is along a single line and in a single direction, then we can write this as

Δx = v0Δt + ½ a (Δt)2 (for constant acceleration and one direction)

Kinematics is useful in problems of “free fall” in which a body freely falls. Galileo did experiments to show that all bodies fall downward with the same acceleration g.

g = 9.8 m/s2 downward



The vector g is called the “acceleration due to gravity.” For the one dimension along a vertical line, the change in the height of a falling body can be expressed as

Δh = -v0Δt - ½ g (Δt)2 (free fall, initial velocity downward)

where v0 is the initial vertical speed downward, g is the magnitude of g, and the two minus signs (-) in the equation indicates that the initial velocity and acceleration are both downward but the height is measured upward. Thus, height and acceleration must have opposite signs. The sign of the v0Δt depends on whether the initial velocity is upward or downward. As written, the equation is for initial velocity downward, and the negative sign (-) should be excluded if the initial velocity is upward.

An easy convention for problems of motion in one dimension is to choose one direction to be positive and the other negative. Then the one-dimensional displacement x, velocity v and acceleration a can be treated as one-dimensional vectors with their directions indicated by their signs. Then Galileo's equation for constant acceleration becomes

Δx = v0Δt + ½ a (Δt)2 (for constant acceleration and one dimension)

regardless of which of the two directions the displacement, velocity and acceleration point. Many problems involve only one dimension. In other words, many problems have motion along only a single line.

In our course, we will consider the case of free fall from a fixed height with the body dropped from restwithout the initial velocity. Thus, for free fall from rest without initial velocity and ignoring the signs, Galileo's equation reduces to

Δh = ½ g (Δt)2

For example, if a body, initially at rest free falls for one second, the body will drop (in that one second)

Δh = ½ 9.8 m/s2 (1 s)2 = 4.5 m

The body drops 4.5 meters in one second.

Newton's First Law of MotionBefore and in the days of Isaac Newton, natural scientists were known as “natural philosophers.” Therewas a controversy among natural philosophers regarding how motions of bodies end. If a block of wood was sent sliding along a level floor, its speed diminished until the block stopped. If a ball rolled



on a flat level surface, it was seen to eventually stop rolling. Thus, many natural philosophers, including Aristotle, decided that motion of any body stops of its own accord without any intervention. However, many natural philosophers questioned this idea and suggested that forces were acting on objects to stop their motions. Thus, the force of friction was likely to be the force that stopped the blockof wood, and the viscosity (resistance to flow) of air could produce a force contributing to the stopping of the rolling ball. People suspected that a body would continue moving forever if all the forces acting on it were eliminated.

Newton wanted to write down some basic physical laws of motion, and realized that he could write fundamental laws that were consistent with each other and with observations if he took the viewpoint that only forces start and stop motion. Thus, he sided with the philosophers who believed that forces were the cause of changes in motion. Newton actually advanced beyond the question of whether a bodycontinues moving in the absence of forces and spoke of the vector sum of the forces. Newton's First Law of Motion states, in modern language, that

A body at rest remains at rest, and a body in motion remains in motion with the same speed and direction of travel, as long as the net force on the body is zero.

Notice the “net force” is a vector sum of forces. The law does not require the absence of forces, but does require their vector sum to be zero for the motion to maintain the speed and direction the same. Newton's First Law of Motion is a fundamental physical law—the first fundamental law presented in this course. If this law was wrong, then at least some of Newton's other fundamental physical laws would also be wrong. Thus, Newton had confidence that he should assume this law as his first fundamental law of physics.

In class, I like to use a block of wood, a ball and a micro-sized hovercraft to show how an object moveswhen affected by different amounts of net force. The block of wood slides only a small distance before stopping and ball rolls further since friction was reduced, but the hovercraft travels freely across the room and maintains speed very well since the net force on it was small. Space exploration depends on Newton's First Law of Motion. NASA sent probe spacecraft away from Earth and even out of the Solar System by getting the crafts up to speeds great enough to allow them to escape the gravity of the Earth and Solar System. Once they were up to speed, the probes did not need engines (other than for adjusting their paths) to continue their journeys. Thus, the First Law can be seen to be important, but itsmajor significance is probably that Newton's other fundamental physical laws depend on its validity.



InertiaThe “inertia” of a body is its resistance to changes in motion. Later In this chapter, we will speak of “inertial mass” which is a measure of inertia. In class, I often pull and break a string attached to the bottom of an object with significant mass without the object moving enough to break an upper string from which the object hangs. The string is pulled so fast that the inertia of the object prevents it from getting enough speed to break the upper string. Demonstrations of inertia often involve moving something so quick that an object does not have time to get up to significant speed. (An example seen in movies is a tablecloth being pulled off a table with such speed and skill that the dishes remain on the table.) However, inertia is also demonstrated when a massive object resists being stopped because of itsinertia. Consider how hard it is to stop a rolling truck even on level ground. The reason I mention inertia now instead of waiting until inertial mass is discussed is that historically the concept of inertia was introduced about the time that the subject of “kinematics” was introduced.

Kinematics and Dynamics Kinematics deals with motion apart from its causes. Kinematic properties include the vector quantities displacement x, velocity v and acceleration a. The equations of kinematics don't have forces and mass in them. Dynamics deals with the causes of motion by relating motion to the forces involved. Dynamicsis sometimes called “mechanics.” Equations of mechanics can involve force and mass. Newton's three laws of motion form the foundation of dynamics.

Newton's Second Law of MotionNewton stated his Second Law of Motion in a very general form in terms of the rate at which the linear momentum changes. Linear momentum is a subject of Chapter 6 in our textbook. We won't discuss what this momentum is until a later chapter. Thus, we need a less general form of the Second Law for now. The less general form we use for a net force (a vector sum of forces) F on a body, mass m of the body and acceleration a of the body, is

F = ma

which holds as long as the mass remains constant. In terms of just the magnitudes F of net force F and a of acceleration a, this Second Law can be used to define inertial mass m, for a body initially at rest but suddenly pushed by a net force with magnitude F, as

m = F / a



Inertial mass is a scalar quantity and is a measure of inertia. The term “inertia” is sometimes regarded as a qualitative property while “inertial mass” is the corresponding quantitative quantity and is measurable. (In a later chapter of this textbook, the connection between “inertial” mass and “gravitational” mass will be presented.)

Newton's First Law of Motion and Newton's Second Law of Motion are strongly related. Consider a body with nonzero mass. If the net force (the vector sum of forces) on the body is zero, then the acceleration of the body is zero. Thus, the body will remain at rest if it is already at rest and remain in motion with the same velocity (magnitude and direction) if already in motion. We can compute the net force on a body by measuring its mass (a scalar) and its acceleration (magnitude and direction) and using the Second Law to compute the net force. If a body has mass 2 kg and acceleration 2 m/s2 towardthe East, then the net force has magnitude

2 kg x 2 m/s2 = 4 kg m/s2

and its direction East. The unit kg m/s2 is called a newton. Thus, the net force is 4 newtons East. Notice the net force and the acceleration have the same direction.

Suppose a net force with magnitude 20 newtons (N) accelerates a body with an acceleration of magnitude 10 m/s2. We can calculate the (inertial) mass of the body. The mass is

m = F / a = 20 N / [10 m/s2] = 20 kg m/s2 / [10 m/s2] = 2 kg

Suppose the net force on this body is changed to a force with magnitude 30 newtons (30 N). Then the magnitude of the acceleration would be

a = F / m = 30 N / [2 kg] = 30 kg m/s2 / [2 kg] = 15 m/s2

The directions of the net force and acceleration are always the same.

Example CalculationsNotice that some example calculations were done in previous sections. For example, the section on units showed some sample calculations with units. The section on vectors and scalars showed graphicalsummation of vectors. The reader should notice both the calculations presented in earlier sections and the calculations presented in this section.



Example 01. Suppose a car has a constant acceleration of 2 m/s2 to the right and an initial velocity, at a time we call time zero t0=0, of 4 m/s to the left. We want to know how far from its initial position, at t0=0, in 3 seconds. This is a one-dimensional problem so let us treat velocity and acceleration to the right as positive and the direction to the left as negative. Thus, the constant acceleration is +2 m/s2 and

the initial velocity is -4 m/s.

Δx = v0Δt + ½ a (Δt)2 = (-4 m/s) 3 s + ½ 2 m/s2 (3 s)2 = -12 m + 9 m = 3 m

Thus, at time 3 seconds after the initial time (which was t0=0), the car is 3 m to the right from its initial position (which was its position at t0=0).

Example 02. A ball is thrown upward with an initial velocity of 10 m/s upward. How high will the ball be, relative to its initial position (where it was thrown upward) one second later?

Δh = v0Δt - ½ g (Δt)2 = 10 m/s x 1 s – ½ 9.8 m/s2 (1 s)2 = 10 m – 4.9 m = 5.1 m

Thus, the height will be 5.1 meters at time one second after being thrown. The ball will be at this heightat two different times. To see this, solve the equation for time Δt.

Δh = v0Δt - ½ g (Δt)2

½ g (Δt)2 - v0Δt – Δh = 0

Δh = [v0 +/- √(v02 – 4 x ½ g Δh) ] / [g]

Notice the plus-or-minus symbol “+/-” shows that there can be two values of Δh.

Example 03. Suppose a body with mass m = 2.0 kg moves with a constant acceleration a = 10 m/s2 East. We can calculate the net force Fnet.

Fnet = ma = 2.0 kg x 10 m/s2 East = 20 N East

Thus, the net force on the body is 20 newtons East.

Example 04. Suppose the magnitude Fnet of the net force on a boy is 10 N and the magnitude a of the acceleration of the body is 2 m/s2. We can calculate the (inertial) mass of the body.

m = Fnet / a = 10 N / [2 m/s2] = 5 kg

Thus, the mass is five kilograms.


Chapter 01 Introduction to Conceptual Physicsscience.sbcc.edu/~physics/ferril/Resources/SBCC P 101 Chapter 01.pdfNatural science is composed of physical science and biological science.

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