Physics In Daily Life

&

Simple College Physics

Volume-I

(Classical Mechanics)

by

Murat Uhrayoglu

Smashwords Edition

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Published By:

Murat Uhrayoglu on Smashwords

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Physics In Daily Life

&

Simple College Physics

Volume-I

(Classical Mechanics)

Copyright, 2011 by M. Uhrayoglu

ISBN: 978-1-4661-1578-1

Istanbul, 2011

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Smashwords Edition License Notes

This book is licensed for your personal enjoyment only. This book may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each person you share it with. If you're reading this book and did not purchase it, or it was not purchased for your use only, then you should return to Smashwords and purchase your own copy. Thank you for respecting the author's work.

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Note To the Reader:

"Physics is all around us. It is in the electric light you turn on in the morning; the car you drive to work; your wristwatch, cell phone, CD player, radio, and that big plasma TV set you got for Christmas. It makes the stars shine every night and the sun shine every day, and it makes a baseball soar into the stands for a home run. So also Mechanics is an important field of physics. Developed by Sir Isaac Newton in the 17th century, the laws of mechanics and the law of gravity successfully explained the orbits of the moon around the earth and the planets around the sun. This study teaches simple physics principles to the college-level students and other people interest in daily-life physics.."

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About the Book

"Physics is all around us. It is in the electric light you turn on in the morning; the car you drive to work; your wristwatch, cell phone, CD player, radio, and that big plasma TV set you got for Christmas. It makes the stars shine every night and the sun shine every day, and it makes a baseball soar into the stands for a home run.

Physics is the science of matter, energy, space, and time. It explains ordinary matter as combinations of a dozen fundamental particles (quarks and leptons), interacting through four fundamental forces. It describes the many forms of energy—such as kinetic energy, electrical energy, and mass—and the way energy can change from one form to another. It describes a malleable space-time and the way objects move through space and time.

There are many fields of physics, for example: mechanics, electricity, heat, sound, light, condensed matter, atomic physics, nuclear physics, and elementary particle physics. Physics is the foundation of all the physical sciences—such as chemistry, material science, and geology—and is important for many other fields of human endeavor: biology, medicine, computing, ice hockey, television… the list goes on and on.

So also Mechanics is an important field of physics. Developed by Sir Isaac Newton in the 17th century, the laws of mechanics and the law of gravity successfully explained the orbits of the moon around the earth and the planets around the sun. This study teaches simple physics principles to the college-level students and other people interest in daily-life physics.."

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Table of Contents

Chapter I

What is Physics?

In Our Daily-Life Physics

Chapter II

Fundamental Mathematics Lessons for Learning Physics Units of Measurement

Chapter III

Introduction to the Physics Lessons

Physics Learning Lessons

(Classical Mechanics)-I

Chapter IV

Physics Learning Lessons (Classical Mechanics)-II

Distance and Displacement

Chapter V

Physics Learning Lessons (Classical Mechanics)-III

Newton’s Laws of Motion

Chapter VI

Physics Learning Lessons (Classical Mechanics)-IV

Satellite and Simple Harmonic Motion (SHM)

Chapter I

What Is the Physics?

Physics is all around us. It is in the electric light you turn on in the morning; the car you drive to work; your wristwatch, cell phone, CD player, radio, and that big plasma TV set you got for Christmas. It makes the stars shine every night and the sun shine every day, and it makes a baseball soar into the stands for a home run.

Physics is the science of matter, energy, space, and time. It explains ordinary matter as combinations of a dozen fundamental particles (quarks and leptons), interacting through four fundamental forces. It describes the many forms of energy—such as kinetic energy, electrical energy, and mass—and the way energy can change from one form to another. It describes a malleable space-time and the way objects move through space and time.

There are many fields of physics, for example: mechanics, electricity, heat, sound, light, condensed matter, atomic physics, nuclear physics, and elementary particle physics. Physics is the foundation of all the physical sciences—such as chemistry, material science, and geology—and is important for many other fields of human endeavor: biology, medicine, computing, ice hockey, television… the list goes on and on.

A physicist is not some geek in a long white coat, working on some weird experiment. Physicists look and act like you or me. They work for research laboratories, universities, private companies, and government agencies. They teach, do research, and develop new technologies. They do experiments on mountaintops, in mines, and in earth orbit. They go to movies and play softball. Physicists are good at solving problems—all kinds of problems, from esoteric to mundane. How does a mirror reflect light? What holds an atom together? How fast does a rocket have to go to escape from earth? How can a worldwide team share data in real time? (Solving this last problem led physicists to invent the World Wide Web.)

Mechanics is an important field of physics. Developed by Sir Isaac Newton in the 17th century, the laws of mechanics and the law of gravity successfully explained the orbits of the moon around the earth and the planets around the sun. They are valid over a large range of distances: from much less than the height of an apple tree to much more than the distance from the earth to the moon or the sun. Newton’s laws are used to design cars, clocks, airplanes, earth satellites, bridges, buildings—just about everything, it seems, except electronics.

Electricity is another example of physics, one that you may experience as a spark when you touch a doorknob on a dry winter day. The electrical attraction of protons and electrons is the basis for chemistry. Magnetism is another force of nature, familiar to us from refrigerator magnets and compasses. In the 19th century, James Clerk Maxwell combined electricity and magnetism. He showed that light is an electromagnetic wave that travels through empty space. (Waves had always required a medium, for example, water is the medium for ocean waves.) Other electromagnetic waves besides light also travel through empty space; hence radio signals can reach us from a Mars explorer.

Maxwell’s theory also showed that electromagnetic waves travel with the same speed (the speed of light), even if the person who sees it is moving. This is in conflict with Isaac Newton’s principle of relativity, which said a train’s headlight beam would have one speed as seen by the engineer and a different speed as seen by a person watching the train go by. Newton and Maxwell could not both be right about this matter, and in 1905, Albert Einstein resolved the conflict by allowing space and time to change, depending on motion. His special theory of relativity predicted that an object passing by would look shorter and a passing clock would run slower. These changes are too small to notice unless the object is moving very fast—Newton’s laws work just fine at the speeds of ordinary moving objects. But space really does shrink and time really does expand for particles moving at speeds near the speed of light (300,000 kilometers per second).

Another remarkable consequence of special relativity is the famous equation E=mc², which says that mass is just another form of energy. This equivalence of mass and energy is the source of the energy that comes to earth as sunlight. In the intense heat at the core of the sun, four hydrogen nuclei fuse into one helium nucleus and the mass difference is converted into radiant energy, which emerges as sunlight. E=mc2 is also responsible for the release of energy from fission of uranium in a nuclear reactor, and this energy is used around the world to make large amounts of electric power.

Einstein went on to replace Newton’s theory of gravity with his general theory of relativity, which says that space and time are changed not only by speed, but also by the presence of matter. Imagine space-time as a large sheet of rubber, and set a bowling ball on the sheet; it will be dimpled near the ball. A tennis ball rolled slowly near the bowling ball will curve around it and may settle into an orbit, just as the earth orbits the sun. Today, the general theory of relativity is well-tested and is used to accurately determine the location of your car if you have a GPS (Global Positioning System) device.

Newton’s laws also break down on the tiny distance scales of atoms and molecules, and must be replaced by the theory of quantum mechanics. For example, quantum mechanics describes how electrons can only travel around the nucleus of an atom in orbits with certain specific energies. When an electron jumps from one of these orbits to another, the atom will absorb or emit energy in discrete bundles of electromagnetic radiation. Because the energies of different states of an atom are known with high precision, we can create highly accurate devices such as atomic clocks and lasers.

Quantum mechanics is also necessary to understand how electrons flow through solids. Materials that normally do not conduct electric current can be made to conduct when “doped” with atoms of a particular element. This is how we make transistors, microscopic electrical on-off switches, which are the basis of your cell phone, your iPod, your PC, and all the modern electronics that has transformed our lives and our economy.

There are still profound questions in physics today: what are the mysterious dark matter and energy that make up most of the universe? Are there more than three dimensions of space? The more we learn about physics, the more it will help us every day, and the better we will understand our place in the universe.

Use of Physics In Our Daily Life?

Following of our actions in our daily life are parts of physics study. When we walk or run, our motion is part of laws of mechanics and thermodynamics.

1. We eat food which undergoes chemical reactions producing heat energy which is converted into mechanical energy,

2. Use of refrigerator, pressure cookers, washing machines, television, music system, computers, etc. are all designed on the principles of physics,

3. When we speak, we produce sound properties of which like pitch and intensity are studied in physics,

4. Electricity that we use in household is a gift of physics,

5. Automobiles design is based on physics..

The list is endless. So had there been no study of the science of physics, its development and application in providing all these facilities, we would have remained tribals forever.

Physics Exam Advice

Here are some simple pieces of advice that can help you to squeeze the last mark from your exam paper. Every mark is important because you could be on the boardline between grades.

Revision Tips

Give yourself Time

Don’t start revising a week before the exam give yourself time to be organised. Cramming works for some people but it is often the strategy of last-resort.

Write Notes

Your physics course may seem like a lot of work to revise but repeatedly making notes from the notes will bring your course notes down to a few pages after several revisions. Make use diagrams and spider diagrams to condense the information down to the most salient points. The act of repeatly making notes will also dramatically increase your retention of the information.

Memorising Equations

I beleive it is better to understand how the equations work in an intuitive way but this is not always possible. One can become familiar with the equations by writting them down and using them.

Use the Internet to join a Forum

You will be able to discuss problems and get useful information. Popular forums include Physic Forums. Not only can you find answers to your questions but you can also find links to past papers and mark schemes. Which brings me nicely to my next point.

Do Past Papers

This is the most important point. There are several reasons for this. You can read a book on Judo or playing the piano but you will not master them just by doing this alone. Both Judo and playing the piano are physical tasks. Similarly, if you are taking an exam it is no good just reading the subject from a book. Do the past papers and, if possible, do them under exam conditions, i.e. set yourself the same time-limit as in the real exam and try and do the paper without any books. Use the course book and the mark scheme to check your answers afterwards.

Past papers will help you get used to the format of the exam, such as how many questions you need to answer. You will also get a feel for the level of the questions. Many new questions will be variations on old questions since they have to cover the same material. So for example, a long question on capacitors is likely to involve the charging or discharging or the RC time constant.

Relax (if you have done the above)

If you have planned ahead and organised your revision then you will have all the knowledge to handle the exam. However, students are often worried by the exam process. Try and relax. Try not to worry that failure is going to dramatically alter your career path and destroy any potential of having a happy and successful life. Seriously.

No not really. You can probably take the exam again. With modular courses, failure is an option and it is no biggy.

During the Exam

Don’t Waste Time

Don’t Waste Time on answering a single question. Remember, time is ticking away, always go for the easiest questions first. You can return to the tricky questions later. In this way you make the best use of your time.

Show your working

If you are asked to calculate something you will often get marks even if you make a mistake during a calculation and you obtain the wrong answer as long as the process by which you obtain the answer is correct. Therefore it is vital to show your thought process. Your calculator may store several results but you should go through a calculation one stage at a time. This allows you track any mistakes more easily. If you have done the past papers you will have an idea about what the examiners are expecting.

Look at the number of marks for the question. If the question is worth 2 marks, then it is going to be fairly simple and so do not spend time writing lots of information in the answer.

Some exam papers have a space for you to write the answer and it may already tell you the units to use. This can be useful even if you didn’t know how to calculate the answer. As a simple example if you are given the distance in m and a time in seconds and it asks for you to calculate the velocity, with a unit ms^-1. It is obvious that you can only get units of ms^-1 by dividing m (distance) by s (time).

If the unit is not given, often marks are given for the answer AND the appropriate unit. A number is meaningless on its own and you will not get full marks.

Does your answer make sense?

It takes a while to get a feel for whether the answer is reasonable but if you are asked to calculate the age of the Universe and you get an answer of 2 seconds then something stinks with your calculation. Often students just write an answer without thinking. The process may be correct but the answer can be wrong because they have not converted into the appropriate units.

Another way in which your answer can be wrong is because your equation does not make sense. You have apples on one side and oranges on the other. You can use dimensional analysis to see if the equation makes sense.

Answer as many Questions as Possible

Don’t leave unanswered questions. Many students feel embarrased about not knowing the answer and they leave blank questions. It is the most stupidest thing to do. If you don’t write anything, you will get nothing. If you write something, then sure, it might be complete nonsense for the most part but there might be some nuggest of truth in your frantic scribbling and that means there is an extra mark or two to be had.

Time to Spare?

If you finish before the exam ends, you can draw charicatures of the invigilators on the working out paper or stare at the backs of the other people in the exam. Well you could, and that is fun, but it would be better to use this time to check over the questions.

The Scientific Method

Belief and Faith

Deciding the truth of a statement about the world is at the heart of science. We may beleive certain ideas for many reasons. We may not even question those ideas, if we arrive at them through direct experience, if they come from people we trust or respect. It would be extremely difficult to go through life without excepting some ideas on the basis of trust or experience. When we except an idea, for whatever reason, we are said to believe.

Since there are an infinite number of beliefs, from the ridiculous to the plausable we need a method to distinguish those beliefs which are consistant with reality. The scientific method is our best solution to this problem.

How Science Works

The scientific method has four steps:

1-Observation

Before we make any assumptions about the phenomenon we need to observe and describe the phenomena clearly.

2-Hypothesis

Once we have identified a phenomena that needs explanation, the next step is to think about how the phenomena can happen either by a causal mechanism, or mathematical relation pehaps even an educated guess. In technical language, we form an hypothesis. In forming an hypothesis we may perform tests by gathering data and performing experiments.

If the result of our tests agrees with our hypothesis, the results add weight to our initial hypothesis it does not mean that our hypothesis is true. If the results of the experiments disagree with our hypothesis, the initial hypothesis is WRONG. The initial hypothesis must be modified or rejected. This one statement sums up the power of the scientific method. It doesn’t matter how smart you are or how elegant the theory. If it disagrees with experiment, it is wrong. This one statement sums up the power of the scientific method over other systems of reasoning such as faith or belief.

3-Prediction

Our theory attains more credibility when it is used it to predict the result of another related phenomena or explain observed results the test against reality agrees with prediction.

4-Reproducibility of Results

The final stage of the scientific method is to ensure that the predicted results are reproducible by several independent experimenters with properly performed experiments. The point about experiments is that the results should be repeatable which is why scientists go to great pains to explain what equipment was used, how the experiment was performed along with the results we obtained and a conclusion about the results obtained.

As more evidence is gathered in support of a theory in time it may become regarded as a theory, model or even a law. It may become established to the point that it becomes an accepted scientific fact. Even when an idea is taken as fact, new evidence may require that the range of application over which it agrees with reality to be changed. A good example of this is Newton’s laws of motion. While Newton’s laws work fine when applied to the world of our everyday experience, these laws break down at speeds close to the speed of light. Einstein’s, theory of relativity refined the existing laws of motion to work at speeds close to the speed of light. The new theory also had to agree with Newton’s laws at low speeds. Even with all the weight of evidence from experiments and predictions, a scientific theory is never proven. It can only ever be disproved. And weel Done..

Chapter II

Fundamental Mathematics Lessons for Learning Physics Units of Measurement

Introduction

Physics is about the study of energy and forces. In order to test and measure physical quantities we need to define some standard measures which we everyone can agree on. These standards can never perfectly accurate because they are rooted in the physical world but every endevour is made to make them as precise as possible. The internationally recognised authority for the definition of these standards is the Conference Generale des Poids et Measures (CGPM).

Units

We must ensure that the result we use in our calculations are in the correct units. The consequence of getting it wrong can be very expensive as with the loss of the NASA Mars Climate Orbiter spacecraft in 1999. It spun out of control because part of the software assumed Imperial units and another part assumed metric units.

The units by which we now measure physical quantities is called the S.I. (System International) system established in 1960. Within this system, the most commonly used set of units in physics are M.K.S (Metres, Kilograms, Seconds) system.

Base Units

The basic units are shown in Table- 1:

Table 1. Base Units

Derived Units

Multiplication of physical quantities creates new units.

When you calculate the area, the unit becomes multiplied by itself to become, m². The unit of area is an example of a derived unit.

Other derived units occur so often they are named after illustrious scientists, in honour of their work. Table 2, list derived units and their special names.

Table 2. Derived units and their special names.

Other Derived Units

Table 3. Other derived units

Suplementary Units

These angular units and solid-angle unit are often used but are actually dimensionless.

Table 4. Dimensionless angular units

Dimensions

The dimensions of a unit describe what kind of measurement it is. For example inches, miles and meters are all different units but they all mesure length. Simillarly, the kilogram, the gram and the pound (lb) are all different units of mass but they all share the dimension of mass. Conventially, the dimension of a unit is written between square brackets. The fundamental dimensional quantities are [M], [L], [T] and [A] to represent mass, length, time and charge respectively. All other quantities can be derrived in terms of these dimensions. Dimensions are useful to derive formula and as a check on whether formula has the correct form.

Use of Dimensions to Derive Equations

If we have some idea or can make an educated guess as to how one physical quantity relates to another we can use dimensions to derive the form of the equation. As an example, consider the equation for the period of pendulum bob. We might suppose that the period depends on the mass of the bob, the length of the pendulum and the acceleration due to gravity

We can express this as T=m^xl^yg^z. Where x, y and z are as yet undetermined indices.

To find the values of x, y and z we convert the formula into its dimensions. On the left-hand side the dimension of the period is [T], the dimension of mass is [M]^x, the dimension of the length of the pendulum is [L]^y and the dimension of g is [LT^-2].

[T]=[M]^x[L]^y[LT^-2]^z.

Equating left-hand indices with matching dimensions on the right-hand side.

[M]: 0=x

[L]: 0=y+z

[T]: 1=-2z

From this we can deduce that z=-1/2, while y=1/2 and x=1/2

Substituting these values into the original equation we obtain. T=m⁰l^1/2g^-1/2= k(l/g)^1/2. Where k is a constant of proportionality. Compare this with the equation for the period of a pendulum T=2π(l/g)^1/2. The form of the equation is correct, but it cannot determine the constant of proportionality.

Prefixes and Magnitudes

To make sense of the vast range over which physical quantities are measured, prefixes are used as a short-cut to writting the magnitude using scientific notation

Other prefixes which are commonly used but are not strictly part of the SI system.

Scientific Notation

Measuring physical phenomena in the real world we enevitably encounter numbers of large magnitude. The estimated (insert example here) is 1,000000000 * n zeros while the time taken for a photon of light to cross the radius of an atom is 0.0000000000000000000000000000000000000001.

Clearly, performing calculations with such unweildy numbers is entirely impractical. Therefore, we need ‘short-hand’ method for writing these numbers. For number with upto 18 significant figures (18 zeros to the right). We can use the prefixes. For anything that falls between the gaps, we write the number of zeros as the index of a power of 10.

For example, 1,000,000 is 1 x 10⁶ as there are 6 zeros after the 1. and for 0.1 is 1 x 10^-1. 10⁰ = 1.

Questions

Write the following numbers using scientific notation

• The age of the universe is approximately 14,000,000,000 years

• Using the physics we know today, we can calculate what happened 0.000, 000, 000, 000,000, 000, 000, 000, 000, 000, 000, 000, 000, 001 seconds after the ‘big-bang’.

• There are number of atoms in a metre³ is approximately, 1,000,000,000,000,000,000,000,000,000,000,000,000.

• One mole of atoms contains 624, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000,000, 000, 000, 000, 000, 000.

• An atomic nucleus is approximately 0.000,000,000,000,01 metres.

• Your voice is amplified 1,000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000,000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000,000, 000, 000, 000, 000, 000 times by the valves when making a long distance phone call from california to New-York in 1950s.

• The redshift on a quasi is measured as 10 billion light-years. In meters this is 9,470,000,000,000,000,000,000,000,000 metres.

• Google is one of the most popular search engines, but a Googol is also the name given to the number: 10,000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000.

Scalars and Vectors

Introduction

Some physical quantities can be expressed by a single number, such as mass, temperature, volume, voltage, energy, pressure and charge. These are known as Scalars. Vector quantities need both a magnitude and direction. Examples include velocity, force, momentum, angular velocity and electric field.

Vectors

If we are in a large room at point A and I move ten metres in straight line to point B then this can be represented as a vector. Starting from point A and moving in a new direction and moving 10 metres in a different direction results in a new vector AC.

Since the vector has direction we cannot specify it with one number, we need to define the position of the endpoint from some origin.

Notation

Vectors are usually distinguished from scalar values by writing them in boldface. When writing them on paper, we usually denote them with an arrow or line above the letter. Eg, a or a.

Vector Components

A single vector can be decomposed into two independent vectors in the same the direction as the coordinate axes. These are known as the components of the vector. The mqgnitude of the horizontal component is |a_x| = |a| cos θ and the magnitude of the vertical component is |a_y| = |a| sin θ, where θ is the angle between the vector a and the horizontal axis

Figure. Vector and its components.

Vectors provide a shortcut when thinking about equations in more than one dimension, we can write F = ma to describe the force but in reality we are thinking about three equations,

F_x = ma_x

F_y = ma_y

F_z = ma_z

Where F_x, F_y and F_z are the three components of the vector F in the x, y and z directions respectively and a_x, a_y and a_z are the three components of the vector a in the x, y and z directions respectively. Sometimes we can reduce the problem to one dimension and we can drop the vector notation.

Unit Vectors

The components of a vector are in the same direction as the axes of the coordinate system, if they are of unit length then they are also unit vectors in the direction of the axes.

Addition

Head to Tail Rule

Two vectors can be added together to make a new vector. Graphically, this is done by moving the head of one of the vectors up to the tail of another. A line drawn from the tail of one to the head of the other is the resultant.

Figure 1. Head to tail addition of vectors

Parallogram Rule

Figure 2. Parallelogram addition of vectors

Writing Vectors

While drawings are great for showing the concept of vectors, drawing arrows to scale quickly becomes tedious and so we seek a less tiresome way to represent vectors. Luckly, we can also write vectors algebraicaly in terms of their components. Thus a vector may be written as a coordinates (x,y) which gives the position of the tip of the arrow from the origin. If we wish to use more than 2-dimensions we can simply add another component to the list. Alternatively, we may represent the components of a vector as a multiple of a vector with unit distance.

These multiplying vectors are known as unit vectors. Conventionally, they are denoted by i and j (also k in 3-D). In this way, a vector a may be represented as a=xi + yj. Where x and y.

Two vectors add to produce another vector.

c=a+b

c=(a_xi + a_yj) + (b_xi + b_yj)

c=(a_x+b_x)i + (b_x+b_y) j

Subtraction of vectors is treated in a similar way. So that, c=a-b

c=(a_x-b_x)i + (b_x-b_y) j

Magnitude

Having defined the vector in space how do we find out its length when it is pointing anywhere? At this point we remember, Pythagoras whose theorem, (he didn’t discover it) we need to obtain the distance. The components of a vector are perpendicular to each other and therefore we can determine the length of the vector from these.

Pythagoras’s theorem is c²=a²+b². Where a and b are the perpendicular to each other and c is the hypoteneus.

Figure. Vector and components to determine magnitude.

Relating this to the vector, the magnitude of vector a is then |a|=√(a_x²+a_y²). We use the | lines around the vector to indicate the magnitude.

Direction Cosines

Dot Product

The dot product or scalar product is a special form of multiplying a vector. The dot product gives the projection of the vector onto the x-axis. It is represented by a dot between the two vectors one is multiplying, a.b. The result is not a vector but a scalar. a.b=|a||b| cos θ

Figure. Scalar product of two vectors a and b.

Also, a.b=a_x.b_x+a_y.b_y+a_z.b_z. They both give the same result however, the latter may sometimes be more conveinient if one doesn’t known the angle between the two vectors.

Vector Product

The vector product is the other form of multiplication of vectors the result of a vector product is another vector that is perpenducular to the two vectors being multiplied. To obtain the vector product we must move into three-dimensional space.

Figure. Vector product of two vectors a and b.

We always work in a right-handed coordinate system. (ie the crossproduct of a and b is 1. a x b =c) and it is a good idea to draw the axis to make it clear.

Differentiation

Differentiation or the derivative is the instantaneous rate of change of a function with respect to one of its variables. This is equivalent to finding the slope of the tangent line to the function at a point.

Derivative of Curve

lim _Δx→0 (f(x₀+Δx) – f(x))/Δx = df(x)/dx.

It is possible to haves differentiate the function f(x) more than once. The second-derivative is the derivative of the derivative of a function.

Notation

There are many popular notations for writing the derivative. The usefulness of each notation varies with the context and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.

Leibniz’s Notation

Leibniz’s notation, is d dx f(x).

The second derivative d²dx²f(x)

and Higher orders of differentiation are shown as dⁿdxⁿ f(x)

Lagrange’s Notation

Lagrange’s notation is a common notation and perhaps a more convinient form for writting in HTML. It denotes the derrivative by a superscripted prime mark ‘ .

For example, f^‘ for the first derivative. Higher order derivatives up to the third order, are written by adding prime-marks. Thus f ^” and f ^”’ are written for the second and third order, respectively. Even higher orders may be represented by an arabic number in brackets. f ⁽ⁿ⁾ is the nth derivative.

Newton’s Notation

x ^. for the first derivative.

x ^.. for the second derivative.

x^… for the third derivative.

Euler’s Notation

Euler’s notation is represented by a capital D. For example, D_x²f(x).

General Power Rule

If we have some function of f(x) = xⁿ where n is a integer.

dxⁿdx = n x^n-1

Example

Differentiate, x³, 4x³, 4x²-6x + 6

Answers:

3x², 12x², 8x – 6

Chain Rule

Given f(x) which is a function of another function g(x). Then h(x)=f(g(x)). The derivative is given by the chain rule:

h’(x) = g’(f(x)) f’(x)

Examples

Differentiate the following functions with respect to x

(3x⁵ + 9x³ + 3x² -62)²

Answers

(15x⁴ + 27x² + 6x) (3x⁵ + 9x³ + 3x² – 62)
90x⁹ + 432x⁷ + 126x⁶ + 486x⁵ – 1590x⁴ + 36x³ – 3348x² – 744x

Product Rule

The chain rule allows us to differentiate, a product of terms that depend on the differentiation variable. For example, U(x) and V(x) are function that depend on x. Then the differential is given by:

d dx (UV) = U dVdx + V dUdx

Quotient Rule

Where we have a quotient, the rule for differentiation is

d dx (U/V) = [V dUdx - U dVdx ]/ V²

Other Functions

Differentiation of trigonometric and other functions:

Integration

Introduction

Integration is a major parts of calculus. It is an extension of the concept of summation. In fact the integration symbol derives from an elongated letter S, first used by Leibniz, to stand for Summa meaning sum in Latin.

Integration is widely used throughout mathematics and physics and so is an important concept to grasp. In fact you can’t really do any serious physics without knowing about it.

The Area Under a Curve

One way of understanding integration is to see how it is used to find the area under a curve defined by a function f(x). Imagine trying to find the area under a curve.

Clearly, this is a very poor approximation but we could do a little better by adding together the area of two rectangles of half the widths with different heights corresponding to the value of the function at the points where corner meets the function.

We define an interval [a,b] which is divided into n equal parts. Δx = (b-a)/n.

x₀ = a, x₁ = a+Δx, x₁ = a+ 2Δx, … ,x_n = a+nΔx = b

Having partitioned the interval into n equal parts, the area of the kth strip is Δx f(x_k), where f(x_k) is the value of the function at the x_k. Summing all the areas of all the strips gives the approximate area under the curve. The left-Riemann sum starts with the zeroth strip and ends at the (n-1)th strip. In this case resulting area will be an under-estimate of the true area.

Left-Riemann Sum

A_l = f(x₀)Δx + f(x₁)Δx + … + f(x_n-1)Δx

= [f(x₀) + f(x₁) + ... + f(x_n-1)] Δx

= Σ^n-1_k=0f(x_k) Δx

We could also have represented the area by starting our summation of rectangles at k=1 and ending at k=n. This known as the right-Riemann sum. In this case, it over-estimates the area under the curve.

Right-Riemann Sum

A_r = f(x₁)Δx + f(x₂)Δx + … + f(x_n)Δx

= [f(x₁) + f(x₂) + ... + f(x_n)] Δx

= Σⁿ_k=1 f(x_k) Δx

As the number of strips increase, the value of Δx decreases. In the limit, Δx tends to zero as n tends to ∞. The areas calculated by the left and right Riemann sums become closer to the true value of the area. In the limit, the sum becomes the definite integral of f(x).

Σ^∞_k=0f(x_k)Δx = ∫^b_af(x) dx

The accuracy for larger rectangles widths can be improved by choosing the mid-point of each rectangle, to corespond to the value of the function at that point. An overestimate is cancelled out to some degree by the underestimate.

Definite and Indefinite Integrals

Definite Integrals

The integration symbol sometimes has numbers or other letters alongside the integral sign. These are called the the limits of integration, the top one is known as the uppper limit and the bottom one is the lower limit.

Essentially, these numbers are substituted into the integral after the integration has been performed. For example:

f(x) = ∫⁷₂x² dx

The integral of the term x² is x³/3. The value of the top limit is substituted in for x in the result. Therefore, 7³/3 = (343/3). Next the lower limit, 2, is substituted for x giving, 2³/3 = 8/3.

The final result is given by taking the first limit from the second. So (343/3) – (8/3) = 335/3 = 111.3333^‘

Figure 1. Approximating the area under a curve by rectangular elements.

Indefinite Integral

Where there are no limits on the integral sign, the integral is called indefinite, meaning there is no specific value. Rather, the result is a family of functions. The integration is performed in the same way but we must remember to add an arbitrary constant known as the constant of integration. For example,

∫x² dx = x³/3 + C

Why is this? If we take our answer x³/3 and differentiate with respect to x, we obtain x². We would also obtain the same answer for x³/3 + 5 or x³/3 – 2 or x³/3 + any constant. Therefore, when we integrate, we have to add a constant because differentiation of a constant is zero.The value of the constant has to be determined by additional information about the equation, for example where it intercepts the y-axis.

Common Integrals

∫k f(u) du = k ∫f(u)du

∫[f(u)±g(u)] du = ∫f(u) du ± ∫g(u) du

∫du=u + C

∫uⁿ du = uⁿ⁺¹/(n+1) + C, (n ≠ -1)

∫du/u = ln|u| + C

∫e^u du = e^u + C

∫sin(u) du = – cos(u) + C

∫cos(u) du = sin(u) + C

∫tan(u) du = – ln|cos(u)| + C

∫cot(u) du = ln|sin(u)| + C

∫sec(u) du = ln|sec(u) + tan(u)| + C

∫cosec(u) du = ln|cosec(u) + cot(u)| + C

∫sec²(u) du = tan(u) + C

∫cosec²(u) du = -cot(u) + C

∫sec(u) tan(u) du =sec(u) + C

∫cosec(u) cot(u) du = – cosec(u) + C

∫(a² – u²)^-1/2du = – arcsin(u/a) + C

∫(a² + u²)^-1/2du = (1/a) arctan(u/a) + C

∫(a² – u²)^-1/2du = – arcsin(u/a) + C

d/dx[∫^x_af(t)dt] = f(x)

Table 1. Common integrals.

Integration by Substitution

The substitution of a function, may simplify the integral allowing it to be calculated easily.

Examples

∫²₁5x² cos(x³) dx

Try u = x³, therefore, du = 3x²dx.

x²dx = (1/3) du

We must change the limits of integration, the new values come from u = x³, therefore when x= 1, u = 1 and when x= 2, u = 8. The integral becomes,

∫⁸₁(5/3) cos(u) du

(5/3) sin(u)|⁸₁ = (5/3)[sin(8) - sin(1)]

Integration by Parts

Let U and V be functions of x. From the product rule:

d(UV)/dx = V (dU/dx) + U (dV/dx)

 

Integrating both sides with respect to x and rearranging,

∫ U(dV/dx).dx = UV – ∫ V (dU/dx) dx

Given some product to integrate, we arrange for U and dV to make the integral on the right-hand side, V(dU/dx) more simple than the integral U (dV/dx) we started with.

Example

 

∫x exp(-x) dx

Let u =x, dv = exp(-x), therefore du = dx, v = -exp(-x)

∫x exp(-x) = -x exp(-x) + ∫-exp(-x) dx =

-x exp(-x) – exp(-x) + C

The Fundamental Theorem of Calculus

Integration is the inverse process of differentiation:

∫^b_af(x)dx = F(b)- F(a)

dF(x)/dx = f(x)

Therefore if we recognise that the function to be integrated as a derivative, then we can say the integral is the function that gaves that derivative. For this reason we sometime call the integral the anti-derivative.

Graphs

When performing experimental work, we test the hypothesis and measure observable quantities. Often, we also plot graphs to demonstrate a relationship between the results and our theory.

A graph plots the relationship of one quantity against another on two axes at right-angles to each other. Usually, we have control over one of the quantities and this is known as the independent variable, the other quantity is determined by the outcome of the experiment or some mathematical relationship. We call this the dependent variable because it depends on the independent variable. We usually plot the independent variable on the x-axis and the dependent variable on the y-axis.

Common Graphical Relationships

The Straight Line Graph

The equation for a straight line is y = mx +c (1)

Where y is the dependent variable, m is the gradient or slope of the graph, x is the independent variable and c is the intercept this is the value at which the line crosses the y-axis.

The gradient is determined by the (change in y)/(change in x) or m = (y₂-y₁)/(x₂-x₁) = Δy/Δx.

If the independent variable, x has a relationship that depends on the x² then it is a quadrative dependence.

y = ax² + bx + c(2)

The quadratic dependence has three separate variable and so can produce a variety of parabolas. Figure 2. show some of the behaviours for different values of a, b, and c. The a variable controls how quickly the parabola increases with x. If it is positive the parabola goes up and is trough-shaped as in example a). A negative value of a inverts the parabola as in b). the variable c controls the height of the parabola on the y-axis. c) shows the a positive value of a and a negative value of c which makes the parabola dip below the x-axis y-region between x=±3. d) shows a negative value for a and a positive value for c which make the parabola rise into the positive y region between x=±3

Figure 1. Straight Line Graph

Quadratic Relationship

Figure 2. A Quadratic Dependence Graph

In examples, g) and h) we set c = 0 and a = 1. The value of b controls the sideways displacement of the parabola. If b is 0, then the parabola is symmetric about the y-axis. A negative value of b shifts the parabola to the left while a positive value shifts the parabola to the right. The degree of shifting is half the value of b. Thus for a value of ±2 the shift is 1 unit to the right or left, respectively.

Curves following exponential growth or decay occur in many areas of physics such as the charging and discharging of capacitors, or in the decay of radioactive material.

A general exponential growth is represented by

y =k exp(ax)(3)

A general exponential decay is written,

y = k exp(-ax)(4)

Where y is the dependent variable, k is a multiplying constant, a is a positive constant and x is the independent variable.

Exponential Graph

This graphs is in below and becoming to use simplify most useful gravitation curved space-time.

Straight Line Graphs

A straight-line graph is the most useful type of graph to show the relationship between the dependent and independent variable. It allows one to see a glance, how well the experimental data-points fit the predicted behaviour. It is possible to convert the different types of curve we have discussed to have a linear relationship.

For a quadratic relationship, it is possible to plot a straight-line as long as the curve is symmetric about the y-axis. i.e. b = 0 in equation, (2). In this case, we plot y against x², c will the intercept, (the point on the y-axis when x = 0). a will be the gradient of the curve.

Figure 3. Exponential Graphs

Since the natural log is the inverse function of an exponential function, ln(exp(x)) = x. Therefore, an exponential curve can be changed into a linear relationship by taking natural logs of both sides of (3). A refresher of logarithms is given here.

ln(y) = ax + ln(k) (5)

Similarly, taking logs of (4) gives,

ln(y) = ln(k) – ax (6)

This has the same form as a straight-line with gradient a and intercept ln(k) if we plot ln(y) against x.

The same procedure works when the dependent variable has the form of a power law. For example, if we have a graph has the following form: y = ax^k. It can be turned into the same form as a straight-line graph by taking logs to the base 10 of both sides.

This gives, log₁₀(y) = log₁₀(a) + k log₁₀(x).

This is now also in the form of a straight-line graph by plotting log₁₀(y) against log₁₀(x). The gradient is k with log₁₀(a) for the intercept.

Other Useful Mathematics

Algebra

The Binomial Theorem

(1 + x)ⁿ = 1 + nx + [n(n-1) x²]/(2!) + [n(n-1)(n-2) x³]/(3!) + …

If x << 1, then

(1 + x)ⁿ ≅ 1 + n x

(1 + x)^-n ≅ 1 – n x

These approximations are useful when x² is negliable.

Quadratic Equations

ax² + bx + c = 0 has the solution,

x ={[-b ± (b² - 4ac)]^1/2} / (2a)

Trigonometry

π rad = 180 °

1 rad = 57.3 °

The quadrants in which trigonometrical functions are positive. Is shown below:

Figure 1. Signs of trigonometric functions.

A good way to remember this is the phrase clockwise ACTS. Clockwise gives the direction from the first quadrant is clockwise and each letter from the word ACTS stands for a trigonometric function: All, Cos, Tan and Sin. The direction of the angle increases in an anti-clockwise sense.

If A and B are angles then

tan A = sin A/cos A

sin² A + cos² A = 1

sec²A = 1 + tan² A

cosec² A = 1 + cot² A

sin (A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B -/+; sin A sin B

tan (A ± B) = (tan A ± tan B)/(1 + tan A tan B)

If t= tan (1/2) A, sin A = (2t) / (1 + t²),

cos A = (1 – t²) / (1 + t²)

2 sin A cos B = sin (A + B) + sin (A – B)

2 cos A cos B = cos (A + B) + cos (A – B)

2 sin A sin B = cos (A – B) – cos (A + B)

sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]

sin A – sin B = 2 cos [(A + B)/2] sin [(A - B)/2]

cos A + cos B = 2 cos [(A + B)/2] cos [(A - B)/2]

cos A – cos B = 2 sin [(A + B)/2] sin [(A - B)/2]

Power Series

e^x = exp x = 1 + x + x²/(2!) + … + x^r/(r!) + … for all x

ln (1 + x) = x – x²/ 2 + x³/3 – … + (-1)^r+1x^r/r + … (-1 < x &ltet; 1)

cos x = (e^ix + e^-ix)/2 = 1 – x²/(2!) + x⁴/(4!) – … + (-1)^rx^2r/(2r)! + … for all x

sin x = (e^ix – e^-ix)/(2i) = x – x³/(3!) + x⁵/(5!) – … + (-1)^rx^2r+1/(2r + 1)! + … for all x

cosh x = (e^x + e^-x)/2 = 1 + x²/(2!) + x⁴/(4!) + … + x^2r/(2r)! + … for all x

sinh x = (e^x – e^-x)/2 = x + x³/(3!) + x⁵/(5!) + … + x^2r+1/(2r + 1)! + …

Chapter III

Introduction to the Physics Lessons

Physics Learning Lessons

(Classical Mechanics)-I

Basic Physics Rules

Distance is usually measured in metres (SI units) and is frequently represented by the variable d. Time is usually measured in seconds and is represented by the variable t. Speed is usually measured in metres per second (m/s) and is represented by v. I use Vi and Vf to denote initial speed and final speed, respectively. Remember that variables can be represented by any combination of letters, they are just names for quantities e.g. d = 55m, d = 0.3cm, potato = 43m, time-initial = 4s.

In all cases a movement is measured from a “reference point”, if you measure the distance you walk to school from home you will measure from your house – which is the reference point in this case, it is the point from which you start measuring.

Standard International Units

When you are performing physics calculations you should always keep similar types of measurements in the same units. If you are performing a calculation with two masses which weigh 2kg and 2000g you should convert both of them into one unit type, either convert them into kilograms or convert them into grams. Do not mix units of measurement.

SI (Standard International) units are special conventions for measuring which simplify physics calculations considerably. The metre, second and kilogram are SI units. If you perform your calculation using the above standard units you will be able to refer to your answer’s units by an alias, e.g. 88 kg/m/s² turns into 88 “Newtons”.

Kinematics and motion lack space-saving shorthands for units you should nevertheless use SI units (m, kg, s) when practical. Whenever you do not use SI units make sure that your measurements are all in the units which you are using.

Basic Equations

Physics relies heavily on mathematics. When solving a problem involving physics you will need to convert words and events into mathematical concepts, this conversion results in an equation.

Speed Speed is the rate at which distance changes. How many distance units are passed during each unit of time?
speed = distance / time

v = d / t

Lets do an example:

A car travelled 50km in 120 minutes while keeping a constant speed. What was its speed? Before we start solving for the speed we need to convert our units into useable form. A car’s speed will usually be represented in km/h, we need to convert the time of the trip into hours. We do this by dividing by 60, 120/60 = 2 hours. 

Lets declare our variables

d = 50 km

t = 2h

v = ? km/h

The speed is “how many kilometres are passed in one hour” and we come up with an equation to model that: v = d / t
We input our values into the equation:

v = 50 / 2

v = 25km/h

We always state the answer to the question in a sentence.
“The speed of the car during the trip was 25km/h”.

Acceleration Acceleration is the rate of change in speed, how the speed differs from one moment to the next.
acceleration = speed / time

a = v / t

Also, the total acceleration undergone is widely thought of as the final speed – the initial speed, all divided by the time. This is a useful way to model acceleration because an object might have been already moving before it began accelerating.
a = (V_final – V_initial) / t

Example

A car speeds up from rest up to 60km/h in 6.2 seconds, what is its average acceleration? Convert all measurements into metres and seconds to prevent confusion and/or awkward result units. 

t = 6.2s

V_initial = 0m/s

V_final = 16.6m/s

a = ? m/s²

a = 16.6 / 6.2

= 2.677 m/s²

We need to round the answer to two Significant Digits (what are significant digits?) therefore our word answer will be “The car accelerates at the rate of 2.7 m/s²”

Scalar Quantities are quantities which represent only magnitude, such as kilograms and metres. This way of presenting quantities only tells you how much of something has happened, it does not tell you in which direction it happened. To solve problems which involve a direction, we use:

Vector Quantities. These are simply distance, speed and acceleration quantities which have a specific direction. Here are some typical vector quantities: 7m [east], 56km/h [north], 4m/s [E56°N].

Concepts and Notation

Lets learn about some useful vector concepts:

• Position: This is the location of an object relative to a reference point. An example of using position would be saying “My cat is located 2m [east] of me”.

• Displacement: Closely tied to position. Displacement is the “space between positions”, if you are situated 7m [north] of your house ( position 1 ), you walked for a while and you stopped 17m [north] or your house (position 2) then your displacement is 10m [north]. Displacement is different from distance, this will be discussed later.

• Velocity: Simply displacement over time. How much has your position changed over time. Here are some velocities: 44km/h [south], 14m/s [southeast]. It differs from speed.

• Acceleration: velocity over time. What sets it apart from scalar acceleration is that acceleration will occur even when the magnitude of the velocity is identical – when the direction of the velocity changes.

Notation:

A vector variable will always have a line/little arrow on top of its variable:

I use the above variable to represent velocity. The vector quantity itself will have its direction written in square brackets after the magnitude: 75m [east], 33km/h [west]. When the direction does not directly correspond to one of the four directions (NSEW) we can use angles to describe the direction. The notation that we use is:

[ direction1 angle from direction1 to direction2 ]

Lets analyze a vector quantity, 23m [S55°E]. 23m is the displacement, this is the scalar (directionless) portion of the vector quantity. [S55°E] is the direction portion, which turns this into a vector quantity. The “S” stands for south, 55 is the angle which the direction takes from south to “E”, east. The actual direction is 55 degrees east of the south direction. The above displacement can also be written as 23m [E35°S] ( we go from east 35 degrees to the south ).

Differences Between Vectors and Scalars

You already know the most important difference between vectors and scalars – vectors have a direction and scalars don’t. However there are other important distinctions. Displacement is very different than distance; distance is the actual amount of travel that has happened during motion, displacement is how far you are from your reference point. So that if you start off in your home, go around and return you have passed a certain distance, but your displacement is zero because you have returned to your starting point. No matter how you got to a certain point, displacement will always be the most direct distance to that point (from where you started). Keep in mind that displacement will also have an angle associated with it.

Velocity is not speed either. Velocity is the change in displacement, so that if you went in zigzags and spent an hour getting to a location 1km away from your starting point then your velocity will be 1km/1hour [direction]. The velocity takes into account only the final position and the total time it got to get there.

Using Vectors

You should use vectors any time you use angles and directions of motion, whenever your motion is not forward/backward in the same direction. Vectors are highly useful when they are drawn out, then you can use trigonometric ratios to find out resultant vectors.

What do vectors look like?

When you draw vectors (which are just a way to describe motion in a visual way) they look like a line with a “head” and a “tail”. The line begins in a tail, which shows where you started from and ends in a head (which is an arrow) that shows where the motion ended.

The little d with an arrow on top shows that this vector is a displacement vector, it looks like its in the direction [Right 20° Up] (we are going 20 degrees Up from the Right direction). The length of the line is ideally the magnitude of the vector (the scalar part of it), so when drawing vectors you should make them reasonably proportional (e.g. a 3m vector should be smaller than a 7m one).An important mathematical tool when using vectors is the absolute value notation. An absolute value of a vector is written like so: || and it is a way to represent just the magnitude of the vector, without direction. We will be taking the absolute value in an example question.

Vector addition

This is the actual application of vectors, adding and subtracting vector quantities. When you have two vectors to add you arrange them head to tail and you draw a line from the first tail to the last head:

The vectors, again, represent displacement. The vectors represent the motion of a person: at first the person moved roughly to the east (diplacement 1, d1 ), then the person moved northeast (d2) and stopped. Now, to figure out where the person is postioned relative to his/her starting point we need to add the two displacements. We draw a line (the blue line) from the starting point to the ending point of the journey, this is the resultant displacement and is labeled .

Addition Example

Q: Billy Bob Joe starts going to school from his house. First he walks north 300m to the bus stop, and then the bus takes him 1km west to school.a) How far is Mr. Joe from his house when he is at school?

b) When Mr. Joe returns home what will his displacement be?
c) How will Mr. Joe’s displacement be affected if he took a different route to school? 

A: First lets draw our situation with vectors. Assuming north is up ( when in doubt draw the north arrow ) we draw Mr. Joe’s northernly walk and then his westerly ride – a vector going up and a vector going left.

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