Notes 4 Study

Economics - Tax and Fiscal Policy

husnainrasheed@gmail.com (Husnain Rasheed) — Wed, 29 Jun 2011 06:55:00 -0700

Summary

Taxes are an integral part of your life as an American. Each April you spend countless hours pouring over records and receipts, or paying an accountant to do this, in preparation for income taxes. Similarly, in most states whenever you purchase something, like clothing or a car, you are required to pay sales tax. These are just two of the most common taxes faced by the American people. Others include luxury tax, inheritance tax, and corporate income tax. What is all of this tax money used for? Why does the amount of tax change from place to place and from year to year?

Tax revenues are used to support government spending. Health care, defense, social security, and politicians' salaries are all government expenses. From an economic standpoint, it is reasonable to think of the American government as one large company. The total amount of government spending is dictated by the governmental budget, just as the spending of a company is dictated by the budget.

Through taxes and government spending, the American government has a direct hand in the workings of the economy. By changing either taxes or government spending, the government affects the amount of money available to the public. Changes in taxation and in government spending are called fiscal policy. The government actively uses fiscal policy to steer the American economy. In this SparkNote, you will learn both how and why the government utilizes fiscal policy.

But fiscal policy is not the only means that the government possesses to steer the economy. Through monetary policy, the Fed is able to affect output. The key factor that the Fed uses to affect the economy is the interest rate. Because the growth of the economy is dependent upon the interest rate, by manipulating this variable the Fed can effect an increase or decreases in output to help maintain stable growth and low inflation. The workings of monetary policy will also be revealed in this SparkNote. Together, monetary policy and fiscal policy work together as reigns to steer the mighty horse of the economy in the right direction.

Terms

Introduction and Section 1, Bourgeois and Proletarians (Part 1)

husnainrasheed@gmail.com (Husnain Rasheed) — Wed, 9 Sep 2009 09:12:00 -0700

Introduction and Section 1, Bourgeois and Proletarians (Part 1)

The Manifesto begins by announcing, "A spectre is haunting Europe--the spectre of Communism." All of the European powers have allied themselves against Communism, frequently demonizing its ideas. Therefore, the Communists have assembled in London and written this Manifesto in order to make public their views, aims and tendencies, and to dispel the maliciously implanted misconceptions. The Manifesto begins by addressing the issue of class antagonism. Marx writes, "The history of all hitherto existing society is the history of class struggles." Throughout history we see the oppressor and oppressed in constant opposition to each other. This fight is sometimes hidden and sometimes open. However, each time the fight ends in either a revolutionary reconstruction of society or in the classes' common ruin.

In earlier ages, we saw society arranged into complicated class structures. For example, in medieval times there were feudal lords, vassals, guild-masters, journeymen, apprentices and serfs. Modern bourgeois society sprouted from the ruins of feudal society. This society has class antagonisms as well, but it is also unique: class antagonisms have become simplified, as society increasingly splits into two rival camps--Bourgeoisie and Proletariat.

The Manifesto then shows how the modern bourgeoisie is the product of several revolutions in the mode of production and of exchange. The development of the bourgeoisie began in the earliest towns, and gained momentum with the Age of Exploration. Feudal guilds couldn't provide for increasing markets, and the manufacturing middle class took its place. However, markets kept growing and demand kept increasing, and manufacture couldn't keep up. This led to the Industrial Revolution. Manufacture was replaced by "Modern Industry," and the industrial middle class was replaced by "industrial millionaires," the modern bourgeois. With these developments, the bourgeoisie have become powerful, and have pushed medieval classes into the background. The development of the bourgeoisie as a class was accompanied by a series of political developments. With the development of Modern Industry and the world-market, the bourgeoisie has gained exclusive political sway. The State serves solely the bourgeoisie's interests.

Historically, the bourgeoisie has played a quite revolutionary role. Whenever it has gained power, it has put to an end all "feudal, patriarchal, idyllic relations." It has eliminated the relationships that bound people to their superiors, and now all remaining relations between men are characterized by self-interest alone. Religious fervor, chivalry and sentimentalism have all been sacrificed. Personal worth is now measured by exchange value, and the only freedom is that of Free Trade. Thus, exploitation that used to be veiled by religious and political "illusions" is now direct, brutal and blatant. The bourgeoisie has changed all occupations into wage-laboring professions, even those that were previously honored, such as that of the doctor. Similarly, family relations have lost their veil of sentimentality and have been reduced to pure money relations.

In the past, industrial classes required the conservation of old modes of production in order to survive. The bourgeoisie are unique in that they cannot continue to exist without revolutionizing the instruments of production. This implies revolutionizing the relations of production, and with it, all of the relations in society. Thus, the unique uncertainties and disturbances of the modern age have forced Man to face his real condition in life, and his true relations with others.

Because the bourgeoisie needs a constantly expanding market, it settles and establishes connections all over the globe. Production and consumption have taken on a cosmopolitan character in every country. This is true both for materials and for intellectual production, as national sovereignty and isolationism becomes less and less possible to sustain. The bourgeoisie draws even the most barbaric nations into civilization and compels all nations to adopt its mode of production. It "creates a world after its own image." All become dependent on the bourgeoisie. It has also increased political centralization.

Thus, we see that the means of production and of exchange, which serve as the basis of the bourgeoisie, originated in feudal society. At a certain stage, however, the feudal relations ceased to be compatible with the developing productive forces. Thus the "fetters" of the feudal system had to be "burst asunder," and they were. Free competition replaced the old system, and the bourgeoisie rose to power.

Marx then says that a similar movement underway at the present moment. Modern bourgeois society is in the process of turning on itself. Modern productive forces are revolting against the modern conditions of production. Commercial crises, due, ironically, to over-production, are threatening the existence of bourgeois society. Productive forces are now fettered by bourgeois society, and these crises represent this tension. Yet in attempting to remedy these crises, the bourgeoisie simply cause new and more extensive crises to emerge, and diminish their ability to prevent future ones. Thus, the weapons by which the bourgeoisie overcame feudalism are now being turned on the bourgeoisie themselves.

Commentary

The Communist Manifesto opens with a statement of its purpose, to publicize the views, aims and tendencies of the Communists. As such it is a document intended to be read by the public, and it is meant to be easily grasped by a general audience. It is also meant to be a broad description of what Communism is, both as a theory and as a political movement.

In this first section, Marx already introduces several of the key ideas of his theory. One main idea is that all of history until now is the story of a series of class struggles. Underlying all of history, then, is this fundamental economic theme. The most important concept being discussed here is the concept that each society has a characteristic economic structure. This structure breeds different classes, which are in conflict as they oppress or are oppressed by each other. However, this situation is not permanent. As history "marches" on, eventually the means of production cease to be compatible with the class structure as-is. Instead, the structure begins to impede the development of productive forces. At this point, the existing structure must be destroyed. This explains the emergence of the bourgeoisie out of feudalism. It will also explain the eventual destruction of the bourgeoisie. Marx believes that all of history should be understood in this way--as the process in which classes realign themselves in compliance with changing means of production.

Perhaps the most significant aspect of this theory of history is what it doesnot deem important. In Marx's theory, history is shaped by economic relations alone. Elements such as religion, culture, ideology, and even the individual human being, play a very little role. Rather, history moves according to impersonal forces, and its general direction is inevitable.

Marx believes that this type of history will not go on forever, however. The Manifesto will later argue that the modern class conflict is the final class conflict; the end of this conflict will mark the end of all class relations. This section begins to suggest why this might be, positing some of the ways in which the modern era is unique. First, class antagonisms have been simplified, as two opposing classes, the bourgeoisie and the proletariat, emerge. Secondly, while exploitative relationships were previously hidden behind things like ideology, now the veil has been lifted and everything is seen in terms of self- interest. Thirdly, in order for the bourgeoisie to continue to exist, they must continually revolutionize the instruments of production. This leaves social relations in an unprecedentedly unstable state...

The Communist Manifesto

husnainrasheed@gmail.com (Husnain Rasheed) — Wed, 9 Sep 2009 09:05:00 -0700

The Communist Manifesto:

The Communist Manifesto reflects an attempt to explain the goals of Communism, as well as the theory underlying this movement. It argues that class struggles, or the exploitation of one class by another, are the motivating force behind all historical developments. Class relationships are defined by an era's means of production. However, eventually these relationships cease to be compatible with the developing forces of production. At this point, a revolution occurs and a new class emerges as the ruling one. This process represents the "march of history" as driven by larger economic forces. Modern Industrial society in specific is characterized by class conflict between the bourgeoisie and proletariat. However, the productive forces of capitalism are quickly ceasing to be compatible with this exploitative relationship. Thus, the proletariat will lead a revolution. However, this revolution will be of a different character than all previous ones: previous revolutions simply reallocated property in favor of the new ruling class. However, by the nature of their class, the members of the proletariat have no way of appropriating property. Therefore, when they obtain control they will have to destroy all ownership of private property, and classes themselves will disappear.

The Manifesto argues that this development is inevitable, and that capitalism is inherently unstable. The Communists intend to promote this revolution, and will promote the parties and associations that are moving history towards its natural conclusion. They argue that the elimination of social classes cannot come about through reforms or changes in government. Rather, a revolution will be required.

The Communist Manifesto has four sections. In the first section, it discusses the Communists' theory of history and the relationship between proletarians and bourgeoisie. The second section explains the relationship between the Communists and the proletarians. The third section addresses the flaws in other, previous socialist literature. The final section discusses the relationship between the Communists and other parties.

Introduction to Organic Chemistry

husnainrasheed@gmail.com (Husnain Rasheed) — Wed, 8 Jul 2009 06:26:00 -0700

Introduction

Chemical reactions involve the making and breaking of bonds. It is essential that we know what bonds are before we can understand any chemical reaction. To understand bonds, we will first describe several of their properties. The bond strength tells us how hard it is to break a bond. Bond lengths give us valuable structural information about the positions of the atomic nuclei. Bond dipoles inform us about the electron distribution around the two bonded atoms. From bond dipoles we may derive electronegativity data useful for predicting the bond dipoles of bonds that may have never been made before.
From these properties of bonds we will see that there are two fundamental types of bonds--covalent and ionic. Covalent bonding represents a situation of about equal sharing of the electrons between nuclei in the bond. Covalent bonds are formed between atoms of approximately equal electronegativity. Because each atom has near equal pull for the electrons in the bond, the electrons are not completely transferred from one atom to another. When the difference in electronegativity between the two atoms in a bond is large, the more electronegative atom can strip an electron off of the less electronegative one to form a negatively charged anion and a positively charged cation. The two ions are held together in an ionic bond because the oppositely charged ions attract each other as described by Coulomb's Law.

Ionic compounds, when in the solid state, can be described as ionic lattices whose shapes are dictated by the need to place oppositely charged ions close to each other and similarly charged ions as far apart as possible. Though there is some structural diversity in ionic compounds, covalent compounds present us with a world of structural possibilities. From simple linear molecules like H2 to complex chains of atoms like butane (CH3CH2CH2CH3), covalent molecules can take on many shapes. To help decide which shape a polyatomic molecule might prefer we will use Valence Shell Electron Pair Repulsion theory (VSEPR). VSEPR states that electrons like to stay as far away from one another as possible to provide the lowest energy (i.e. most stable) structure for any bonding arrangement. In this way, VSEPR is a powerful tool for predicting the geometries of covalent molecules.

The development of quantum mechanics in the 1920's and 1930's has revolutionized our understanding of the chemical bond. It has allowed chemists to advance from the simple picture that covalent and ionic bonding affords to a more complex model based on molecular orbital theory. Molecular orbital theory postulates the existence of a set of molecular orbitals, analogous to atomic orbitals, which are produced by the combination of atomic orbitals on the bonded atoms. From these molecular orbitals we can predict the electron distribution in a bond about the atoms. Molecular orbital theory provides a valuable theoretical complement to the traditional conceptions of ionic and covalent bonding with which we will start our analysis of the chemical bond..

Terms

Anion - A negatively charged ion.

Bond - That which holds together atoms in molecules and ions in lattices.

Cation - A positively charged ion.

Coulomb's Law - A mathematical formula whose consequence is that negatively and positively charged particles attract each other and similarly charged species repel each other.

Covalent Bond - A bond that results from a sharing of electrons between nuclei.

Ion - A charged species created by the gain or loss of an electron from an atom or neutral molecule.

Ionic Bond - A bond that results from electrostatic attraction between oppositely charged ions. The cation is positively charged, while the anion is negatively charged.

Lattice - A regularly repeating three-dimensional array of atoms, molecules, or ions.

Molecular Orbital - A combination of atomic orbitals in molecular orbital theory that provides an orbital description of a molecule analogous to the atomic orbital description of atoms.

Molecular Orbital Theory - A description of bonding that combines atomic orbitals from each bondedatom to produce a set of molecular orbitals.

Molecule - A chemical species containing a covalent bond.

Valence Shell Electron Pair Repulsion Theory - A theory used to predict bonding geometries that states that electron pairs will be distributed about the central atom to minimize electron pair repulsions

Gravitation

husnainrasheed@gmail.com (Husnain Rasheed) — Fri, 26 Jun 2009 23:13:00 -0700

Newton and Gravitation

SUMMARY

In 1687 Sir Isaac Newton first published his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) which was a radical treatment of mechanics, establishing the concepts which were to dominate physics for the next two hundred years. Among the book's most important new concepts was Newton's Universal Law of Gravitation. Newton managed to take Kepler's Laws governing the motion of the planets and Galileo's ideas about kinematics and projectile motion and synthesize them into a law which governed both motion on earth and motion in the heavens. This was an achievement of enormous importance for physics; Newton's discoveries meant that the universe was a rational place in which the same principles of nature applied to all objects. The Universal Law of Gravitation has several important features. First, it is an inverse square law, meaning that the strength of the force between two massive objects decreases in proportion to the square of the distance between them as they move farther apart. Second, the direction in which the force acts is always along the line (or vector) connecting the two gravitating objects. Moreover, because there is no "negative mass," gravity is always an attractive force. It is also noteworthy that gravity is a relatively weak force. Modern physicists consider there to be four fundamental forces in nature (the Strong and Weak Nuclear forces, the Electromagnetic force and gravity), of which gravity is the weakest. This means that gravity is only significant when very large masses are being considered.

Terms and Formulae

Terms

Universal Law of Gravitation - Newton's Universal Law of Gravitation states that

where m₁ and m₂ are the masses of any two objects under consideration and r₁ and r₂ are their respective position vectors.

Gravitational Constant - This the G that appears as a constant of proportionality in Newton's Universal Law of Gravitation. It has a value of 6.67×10^-11 Nm²/kg².

Formulae

Equation for the gravitational constant

g =

Newton's Law

Qualitatively Newton's Law of gravitation states that:

Every massive particle attracts every other massive particle with a force directly proportional to the product of their masses and inversely proportional to the square of distance between them

In vector notation, if is the position vector of mass m₁ and is the position vector of mass m₂, then the force on m₁ due to m₂ is given by:

= =

The difference of the two vectors in the numerator gives the direction of the force. The appearance of a cube, instead of a square, in the denominator is in order to cancel this direction-giving factor of | - | in the numerator.

Figure 1.1: Direction of force is the difference of the position vectors.

This force has some remarkable properties. First, we note that it acts at a distance , meaning that irrespective of any intervening matter, every particle in the universe exerts a gravitational force on every other particle. Furthermore, gravity obeys a principle of superposition. This means that to find the gravitational force on any particle it is necessary only to find the vector sum of all the forces from all the particles in the system. For example, the force of the earth on the moon is found by vector summing all the forces between all the particles in the moon and earth. This sounds like an immense task, but actually simplifies calculation.

Gravity as a central force

Newton's Universal Law of Gravitation produces a central force. The force is in the radial direction and depends only on the distance between objects. If one of the masses is at the origin, then () = F(r). That is, the force is a function of the distance between the particles and completely in the direction of . Obviously, the force is also dependent on G and the masses, but these are just constant--the only coordinate on which the force depends is the radial one.

It is easy to show that when a particle is in a central force, angular momentum is conserved, and motion takes place in a plane. First, let us consider the angular momentum:

= (×) = × + × = ×(m) + × = 0

The last equality follows because the cross product of with itself is zero, and since is entirely in the direction of , the cross product of these two vectors is zero also. Since angular momentum does not change over time it is conserved. This is essentially a more general expression of Kepler's Second Law, which we saw (here) also asserted the conservation of angular momentum.

At some time t₀, we have the position vector and velocity vector of the motion that define a plane P with a normal given by = ×. In the previous proof we showed that × does not change in time. This means that = × does not change in time either. Therefore, × = for all t. Since must be orthogonal to , it must always lie in the plane P.

Deducing with Sociological Imagination

husnainrasheed@gmail.com (Husnain Rasheed) — Mon, 20 Apr 2009 09:48:00 -0700

Sociology is the scientific study of human groups and social behavior. Sociologists focus primarily on human interactions, including how social relationships influence people's attitudes and how societies form and change. Sociology, therefore, is a discipline of broad scope: Virtually no topic—gender, race, religion, politics, education, health care, drug abuse, pornography, group behavior, conformity—is taboo for sociological examination and interpretation.

Sociologists typically focus their studies on how people and society influence other people, because external, or social, forces shape most personal experiences. These social forces exist in the form of interpersonal relationships among family and friends, as well as among the people encountered in academic, religious, political, economic, and other types of social institutions. In 1959, sociologist C. Wright Mills defined sociological imagination as the ability to see the impact of social forces on individuals' private and public lives. Sociological imagination, then, plays a central role in the sociological perspective.

As an example, consider a depressed individual. You may reasonably assume that a person becomes depressed when something “bad” has happened in his or her life. But you cannot so easily explain depression in all cases. How do you account for depressed people who have not experienced an unpleasant or negative event?

Sociologists look at events from a holistic, or multidimensional, perspective. Using sociological imagination, they examine both personal and social forces when explaining any phenomenon. Another version of this holistic model is the biopsychosocial perspective, which attributes complex sociological phenomena to interacting biological (internal), psychological (internal), and social (external) forces. In the case of depression, chemical imbalances in the brain (biological), negative attitudes (psychological), and an impoverished home environment (social) can all contribute to the problem. The reductionist perspective, which “reduces” complex sociological phenomena to a single “simple” cause, stands in contrast to the holistic perspective. A reductionist may claim that you can treat all cases of depression with medication because all depression comes from chemical imbalances in the brain.

On a topic related to depression, French sociologist Emile Durkheim studied suicide in the late 19th century. Being interested in the differences in rates of suicide across assorted peoples and countries and groups, Durkheim found that social rather than personal influences primarily caused these rates. To explain these differences in rates of suicide, Durkheim examined social integration, or the degree to which people connect to a social group. Interestingly, he found that when social integration is either deficient or excessive, suicide rates tend to be higher. For example, he found that divorced people are more likely to experience poor social integration, and thus are more likely to commit suicide than are married people. As another example, in the past, Hindu widows traditionally committed ritualistic suicide (called “suttee” meaning “good women”) because the cultural pressure at the time to kill themselves overwhelmed them.

Social forces are powerful, and social groups are more than simply the sum of their parts. Social groups have characteristics that come about only when individuals interact. So the sociological perspective and the social imagination help sociologists to explain these social forces and characteristics, as well as to apply their findings to everyday life.

Introduction to Accounting

husnainrasheed@gmail.com (Husnain Rasheed) — Mon, 20 Apr 2009 09:40:00 -0700

Introduction to Accounting

Accounting is the language of business. It is the system of recording, summarizing, and analyzing an economic entity's financial transactions. Effectively communicating this information is key to the success of every business. Those who rely on financial information include internal users, such as a company's managers and employees, and external users, such as banks, investors, governmental agencies, financial analysts, and labor unions. These users depend upon data supplied by accountants to answer the following types of questions:

Is the company profitable?
Is there enough cash to meet payroll needs?
How much debt does the company have?
How does the company's net income compare to its budget?
What is the balance owed by customers?
Has the company consistently paid cash dividends?
How much income does each division generate?
Should the company invest money to expand?

Accountants must present an organization's financial information in clear, concise reports that help make questions like these easy to answer. The most common accounting reports are called financial statements.

Understanding Financial Statements

The financial statements shown on the next several pages are for a sole proprietorship, which is a business owned by an individual. Corporate financial statements are slightly different. The four basic financial statements are the income statement, statement of owner's equity, balance sheet, and statement of cash flows. The income statement, statement of owner's equity, and statement of cash flows report activity for a specific period of time, usually a month, quarter, or year. The balance sheet reports balances of certain elements at a specific time. All four statements have a three-line heading in the following format.

Income statement. The income statement, which is sometimes called the statement of earnings or statement of operations, is prepared first. It lists revenues and expenses and calculates the company's net income or net loss for a period of time. Net income means total revenues are greater than total expenses. Net loss means total expenses are greater than total revenues. The specific items that appear in financial statements are explained later.

The Greener Landscape Group Income Statement For the Month Ended April 30, 20X2

Revenues
Lawn Cutting Revenue		$845
Expenses
Wages Expense	$280
Depreciation Expense	235
Insurance Expense	100
Interest Expense	79
Advertising Expense	35
Gas Expense	30
Supplies Expense	25
Total Expenses		784
Net Income		$ 61

Statement of owner's equity. The statement of owner's equity is prepared after the income statement. It shows the beginning and ending owner's equity balances and the items affecting owner's equity during the period. These items include investments, the net income or loss from the income statement, and withdrawals. Because the specific revenue and expense categories that determine net income or loss appear on the income statement, the statement of owner's equity shows only the total net income or loss. Balances enclosed by parentheses are subtracted from unenclosed balances.

The Greener Landscape Group Statement of Owner's Equity For the Month Ended April 30, 20X2

J. Green, Capital, April 1		$ 0
Additions
Investments	$15,000
Net Income	61	15,061
Withdrawals		(50)
J. Green, Capital, April 30		$ 15,011

Balance sheet. The balance sheet shows the balance, at a particular time, of each asset, each liability, and owner's equity. It proves that the accounting equation (Assets = Liabilities + Owner's Equity) is in balance. The ending balance on the statement of owner's equity is used to report owner's equity on the balance sheet.

The Greener Landscape Group Balance Sheet April 30, 20X2

ASSETS
Current Assets
Cash		$ 6,355
Accounts Receivable		200
Supplies		25
Prepaid Insurance		1,100
Total Current Assets		7,680
Property, Plant, and Equipment
Equipment	$18,000
Less: Accumulated Depreciation	(235)	17,765
Total Assets		$25,445
LIABILITIES AND OWNER'S EQUITY
Current Liabilities
Accounts Payable		$ 50
Wages Payable		80
Interest Payable		79
Unearned Revenue		225
Total Current Liabilities		434
Long-Term Liabilities
Notes Payable		10,000
Total Liabilities		10,434
Owner's Equity
J. Green, Capital		15,011
Total Liabilities and Owner's Equity		$25,445

Statement of cash flows. The statement of cash flows tracks the movement of cash during a specific accounting period. It assigns all cash exchanges to one of three categories—operating, investing, or financing—to calculate the net change in cash and then reconciles the accounting period's beginning and ending cash balances. As its name implies, the statement of cash flows includes items that affect cash. Although not part of the statement's main body, significant non-cash items must also be disclosed.

According to current accounting standards, operating cash flows may be disclosed using either the direct or the indirect method. The direct method simply lists the net cash flow by type of cash receipt and payment category. For purposes of illustration, the direct method appears below.

The Greener Landscape Group Statement of Cash Flows For the Month Ended April 30, 20X2

Cash Flows from Operating Activities
Cash from Customers	$ 870
Cash to Employees	(200)
Cash to Suppliers	(1,265)
Cash Flow Used by Operating Activities	(595)
Cash Flows from Investing Activities
Purchases of Equipment	(8,000)
Cash Flows from Financing Activities
Investment by Owner	15,000
Withdrawal by Owner	(50)
Cash Flow Provided by Financing Activities	14,950
Net Increase in Cash	6,355
Beginning Cash, April 1	0
Ending Cash, April 30	$6,355

Solving Simple Linear Equations

husnainrasheed@gmail.com (Husnain Rasheed) — Sun, 19 Apr 2009 23:44:00 -0700

Solving Simple Linear Equations

Algebraic equations are translated from complete English sentences. These equations can be solved. In fact, in order to successfully solve a word problem, an equation must be written and solved.

Look at these two definitions in the following sections and compare the examples to ensure you know the distinction between an expression and an equation.

Defining an Algebraic Expression

An algebraic expression is a collection of constants, variables, symbols of operations, and grouping symbols, as shown in Example 1.

Example 1: 4( x − 3) + 6

Defining an Algebraic Equation

An algebraic equation is a statement that two algebraic expressions are equal, as shown in Example 2.

Example 2: 4( x − 3) + 6 = 14 + 2 x

The easiest way to distinguish a math problem as an equation is to notice an equals sign.

In Example 3, you take the algebraic expression given in Example 1 and simplify it to review the process of simplification. An algebraic expression is simplified by using the distributive property and combining like terms.

Example 3: Simplify the following expression: 4( x − 3) + 6

Here is how you simplify this expression:

Remove the parentheses using the distributive property.

4 x + −12 + 6

Combine like terms.

The simplified expression is 4 x + −6.

Note: This problem does not solve for x. This is because the original problem is an expression, not an equation, and, therefore, cannot be solved.

Four Steps for Solving Simple Linear Equations

In order to solve an equation, follow these steps:

Simplify both sides of the equation by using the distributive property and combining like terms, if possible.

Move all terms with variables to one side of the equation using the addition property of equations, and then simplify.

Move the constants to the other side of the equation using the addition property of equations and simplify.

Divide by the coefficient using the multiplication property of equations.

In Example 4, you solve the equation given in Example 2, using the four preceding steps to find the solution to the equation.

Example 4: Solve the following equation: 4( x − 3) + 6 = 14 + 2 x

Use the four steps to solving a linear equation, as follows:

Distribute and combine like terms.

2a.

Move all terms with variables to the left side of the equation.

In this example, add a −2x to each side of the equation.

The addition property of equations states that if the same term is added to both sides of the equation, the equation remains a true statement. The addition property of equations also holds true for subtracting the same term from both sides of the equation.

2b.

Place like terms adjacent to each other and simplify.

Note: Subtracting 6 is changed to adding −6 because the commutative property of addition works only if all operations are addition.

Move the constants to the right side of the equation and simplify.

Note: The opposite operation was used to move the constant.

Divide by the coefficient and simplify.

The solution is x = 10.

Example 5: Solve the following equation: 12 + 2(3 x − 7) = 5 x − 4

Use the four steps to solving a linear equation, as follows:

1a.

Distribute and combine like terms.

1b.

Place like terms adjacent to each other and simplify.

2a.

Move variables to the left side of the equation.

In this example, add −5 x to each side of the equation.

2b.

Place like terms adjacent to each other and simplify.

Note: All subtractions are changed to addition of a negative number.

Move the constants to the right side of the equation and simplify.

Note: The opposite operation was used to move the constant.

4.
Because the coefficient is 1, Step 4 is not necessary.

The solution is x = −2.

Example 5: Solve the following equation: 6 − 3(2 − x) = −5 x + 40

Use the four steps to solving a linear equation, as follows:

Distribute and combine like terms.

Did you remember to distribute the negative three?

2a.

Move variables to the left side of the equation.

In this example, add 5 x to each side of the equation.

2b.

Place like terms adjacent to each other.

2c.

Simplify by combining like terms.

3.
This step is not necessary in this example because all of the constants are on the right side of the equation.

Divide by the coefficient and simplify.

The solution is x = 5.

Remember: The four steps for solving equations must be done in order, but not all steps are necessary in every problem.

Solving Simple Linear Equations

husnainrasheed@gmail.com (Husnain Rasheed) — Sun, 19 Apr 2009 23:44:00 -0700

Solving Simple Linear Equations

Algebraic equations are translated from complete English sentences. These equations can be solved. In fact, in order to successfully solve a word problem, an equation must be written and solved.

Look at these two definitions in the following sections and compare the examples to ensure you know the distinction between an expression and an equation.

Defining an Algebraic Expression

An algebraic expression is a collection of constants, variables, symbols of operations, and grouping symbols, as shown in Example 1.

Example 1: 4( x − 3) + 6

Defining an Algebraic Equation

An algebraic equation is a statement that two algebraic expressions are equal, as shown in Example 2.

Example 2: 4( x − 3) + 6 = 14 + 2 x

The easiest way to distinguish a math problem as an equation is to notice an equals sign.

Example 3: Simplify the following expression: 4( x − 3) + 6

Here is how you simplify this expression:

Remove the parentheses using the distributive property.

4 x + −12 + 6

Combine like terms.

The simplified expression is 4 x + −6.

Note: This problem does not solve for x. This is because the original problem is an expression, not an equation, and, therefore, cannot be solved.

Four Steps for Solving Simple Linear Equations

In order to solve an equation, follow these steps:

Simplify both sides of the equation by using the distributive property and combining like terms, if possible.

Move all terms with variables to one side of the equation using the addition property of equations, and then simplify.

Move the constants to the other side of the equation using the addition property of equations and simplify.

Divide by the coefficient using the multiplication property of equations.

In Example 4, you solve the equation given in Example 2, using the four preceding steps to find the solution to the equation.

Example 4: Solve the following equation: 4( x − 3) + 6 = 14 + 2 x

Use the four steps to solving a linear equation, as follows:

Distribute and combine like terms.

2a.

Move all terms with variables to the left side of the equation.

In this example, add a −2x to each side of the equation.

2b.

Place like terms adjacent to each other and simplify.

Note: Subtracting 6 is changed to adding −6 because the commutative property of addition works only if all operations are addition.

Move the constants to the right side of the equation and simplify.

Note: The opposite operation was used to move the constant.

Divide by the coefficient and simplify.

The solution is x = 10.

Example 5: Solve the following equation: 12 + 2(3 x − 7) = 5 x − 4

Use the four steps to solving a linear equation, as follows:

1a.

Distribute and combine like terms.

1b.

Place like terms adjacent to each other and simplify.

2a.

Move variables to the left side of the equation.

In this example, add −5 x to each side of the equation.

2b.

Place like terms adjacent to each other and simplify.

Note: All subtractions are changed to addition of a negative number.

Move the constants to the right side of the equation and simplify.

Note: The opposite operation was used to move the constant.

4.
Because the coefficient is 1, Step 4 is not necessary.

The solution is x = −2.

Example 5: Solve the following equation: 6 − 3(2 − x) = −5 x + 40

Use the four steps to solving a linear equation, as follows:

Distribute and combine like terms.

Did you remember to distribute the negative three?

2a.

Move variables to the left side of the equation.

In this example, add 5 x to each side of the equation.

2b.

Place like terms adjacent to each other.

2c.

Simplify by combining like terms.

3.
This step is not necessary in this example because all of the constants are on the right side of the equation.

Divide by the coefficient and simplify.

The solution is x = 5.

Remember: The four steps for solving equations must be done in order, but not all steps are necessary in every problem.

Keywords Indicating Equality (Maths)

husnainrasheed@gmail.com (Husnain Rasheed) — Sun, 19 Apr 2009 23:42:00 -0700

Keywords Indicating Equality

One of the hardest parts of solving word problems is creating the correct equation after reading the problem. Many people agree that, in comparison to setting up the correct equation, solving the equation is easy.

The following keywords indicate equality and, when translated, become the equal symbol, =.

IS
IS EQUAL TO
EQUALS
YIELDS
RESULTS IN
THE RESULT IS
IS THE SAME AS

The direct translation strategy works for all the equality keywords, which means you do not need to change the order. The equal sign is placed exactly where the keyword is located.

Example 1: Translate the following sentence into an algebraic equation: Twice the difference between a number and five is equal to negative fourteen

Start by making the appropriate markings on the English sentence as you read it. Did you notice the adjacent keywords, the leading keywords, and the IS? Here's how you make the appropriate markings:

Put an open parenthesis between the adjacent keywords and close the parentheses before IS.

Twice (the difference between a number and five) is equal to negative fourteen.
Underline the two expressions to be subtracted as indicated by the leading keyword, DIFFERENCE BETWEEN.

Twice (the difference between a number and five) is equal to negative fourteen.

Use an arrow to remind you that AND is replaced by a minus sign.

Note: The TO that is part of IS EQUAL TO is not a turnaround word. Remember that the direct translation strategy works for all the equality keywords. The expression on the left side of the equation does not have to be turned around with the expression on the right side.

Translate the equation.

The equation translates to 2( x − 5) = −14.

One of the most confusing keywords indicating equality is IS. In a translation problem, you may see more than one IS, as Example 2 demonstrates.

Example 2: Translate the following sentences into an algebraic equation: When a number is subtracted from twelve, the result is five.What is the number?

The second IS is the only one translated to the = symbol. Any IS placed directly in front of a keyword for an operation does not indicate equality. When a number directly follows IS, however, IS does indicate equality.

Remember: Mark the turnaround word to remind you to translate in the correct order.

The equation translates to 12 − x = 5.

Check more of your translation abilities with Example 3.

Example 3: Translate the following statement into an equation: The product of eight and a number yields twenty-four.

What operation should replace AND in this equation? The leading keyword, PRODUCT OF, indicates the eight is multiplied by a number.

Note: The first expression is translated, then the equal sign is translated, and then the expression on the right of the equal sign. Often, in mathematics, breaking the problem up into smaller pieces can increase your success. Instead of translating the whole equation at one time, translate the first expression, the equal sign, and then the second expression.

Keywords for Mathematical Operations

husnainrasheed@gmail.com (Husnain Rasheed) — Sun, 19 Apr 2009 23:36:00 -0700

Keywords for Mathematical Operations

The first step in solving a word problem is always to read the problem. You need to be able to translate words into mathematical symbols, focusing on keywords that indicate the mathematical procedures required to solve the problem—both the operation and the order of the expression. In much the same way that you can translate Spanish into English, you can translate English words into symbols, the language of mathematics. Many (if not all) keywords that indicate mathematical operations are familiar words.

To begin, you translate English phrases into algebraic expressions. An algebraic expression is a collection of numbers, variables, operations, and grouping symbols. You will translate an unknown number as the variable x or n. The grouping symbols are usually a set of parentheses, but they can also be sets of brackets or braces.

In translating expressions, you want to be well acquainted with basic keywords that translate into mathematical operations: addition keywords, subtraction keywords, multiplication keywords, and division keywords, which are covered in the four following sections.

Addition keywords

Some common examples of addition keywords are as follows:

SUM OF_____ AND _____

TOTAL OF _____ AND _____

_____ PLUS _____

_____ INCREASED BY _____

GAIN

RAISE

INCREASE OF

The first two keywords (SUM and TOTAL) are called leading keywords because they lead the expression. The second two keywords (PLUS and INCREASED BY) are keywords that indicate the exact placement of the plus sign. The last four keywords can be found in word problems and may indicate addition.

When an expression begins with the leading keywords SUM or TOTAL, the leading keyword defines the corresponding AND. The plus sign then physically replaces the AND in the expression.

Example 1: Translate the following: the sum of five and a number

The following steps help you translate this problem:

Underline the words before and after AND when it corresponds to the leading keyword SUM OF.
- the sum of five and a number

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a plus sign.
- The expression translates to 5 + x.

Example 2: Translate the following: the total of a number and negative three

Use the following steps to translate this problem:

The keyword TOTAL OF is a leading keyword that defines AND, so underline the words before and after AND: “a number” and “negative three.”
- the total of a number and negative three

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a plus sign.
- The expression translates to x + −3.

Example 3: Translate the following: the sum of seven and negative four

Translate this example in the following way:

The word SUM OF is a leading keyword that defines AND, so underline the words before and after AND: “seven” and “negative four.”
- the sum of seven and negative four

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a plus sign.
- The expression translates to 7 + −4.

Reminder: The AND keyword translates to mean “plus” because the leading keyword is SUM OF. With other leading keywords (discussed in the following sections), AND can mean other things. Also notice that you do not simplify the expression and get “3” for the answer because you are just translating words into symbols and not performing the math.

Two other keywords on the addition keyword list, PLUS and INCREASED BY, can be correctly translated by the direct translation strategy. In the direct translation strategy, you translate each word into its corresponding algebraic symbol, one at a time, in the same order as written, as shown in Example 4.

Example 4: Translate the following: a number increased by twenty-four

The expression translates to x + 24.

Some additional keywords, such as GAIN, MORE, INCREASE OF, and RAISE, are commonly found in story problems, as in Example 5.

Example 5: Translate the following story problem into a mathematical expression about the weight of the linebacker: The defensive linebacker weighed two hundred twenty-two pounds at the beginning of spring training. He had a gain of seventeen pounds after working out with the team for four weeks.

The expression translates to 222 + 17.

Note: Not all numbers mentioned in a word problem should be included in the mathematical expression. The number “four” is just interesting fact, but it is not information you need in order to write an expression about the linebacker's weight.

You may also be wondering why the answer isn't 239 pounds. That's because the question asks you to translate the story problem into a mathematical expression, not to evaluate the expression.

Example 6: Translate the following word problem into a mathematical expression about the cashier's current hourly wage: A cashier at the corner grocery was earning $6.25 an hour. He received a raise of 25 cents an hour.

The expression translates to 6.25 + 0.25.

Note: The hourly wage is stated in dollars, and the raise is stated in cents. Any time you are adding two numbers that have units, make sure both numbers are measured with the same units; if they aren't, convert one of the numbers to the same units as the other. Having both numbers measured with the same units is called homogeneous units. In this example, you convert his raise, the 25 cents, to $0.25 because his hourly wage is measured in dollars, not cents, so the raise must also be in dollars.

Subtraction keywords

Subtraction keywords also include leading keywords, keywords that can be translated one word at a time, and keywords that are found in story problems. Look at the following list of subtraction keywords:

DIFFERENCE BETWEEN _____ AND _____

_____ MINUS _____

_____ DECREASED BY _____

LOSS

LESS

FEWER

TAKE AWAY

One subtraction keyword (DIFFERENCE BETWEEN) is a two-part expression that begins with a leading keyword that defines the corresponding AND. You can use the same methods of underlining and circling the keywords shown in the preceding section to translate these expressions.

Example 7: Translate the following: the difference between four and six

Here is how you translate Example 7:

Because the keyword DIFFERENCE BETWEEN is a leading keyword that defines the corresponding AND, underline the words before and after AND: “four” and “six.”
- the difference between four and six

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a minus sign.
- The expression translates to 4 – 6.

Note: AND is not always translated to mean addition. Here, the DIFFERENCE BETWEEN is the leading keyword that defines the AND to mean subtraction.

Other subtraction keywords, such as MINUS and DECREASED BY, use the direct translation strategy. Example 8 is a subtraction word problem that is translated one keyword at a time, in the exact order of the expression.

Example 8: Translate the following: twenty-four decreased by a number

The expression translates to 24 – x.

In a story problem, you may find the subtraction keywords LOSS, LESS, FEWER, and TAKE AWAY, as shown in Example 9.

Example 9: Translate the following word problem into a mathematical expression about the current value of materials at the job site: A construction company stored $1,253 worth of materials at the job site. The company suffered a loss of $300 due to storm damage.

The expression translates to 1,253 – 300.

Multiplication keywords

Some common examples of multiplication keywords are as follows:

MULTIPLY _____ BY _____

PRODUCT OF _____ AND _____

_____ TIMES _____

DOUBLE _____

TWICE _____

TRIPLE _____

PERCENT OF _____

FRACTION OF _____

For two of the multiplication keywords, MULTIPLY and PRODUCT OF, a leading keyword defines the corresponding BY or AND, as shown in Example 10.

Example 10: Translate the following: the product of seven and a number

Translate this example in the following way:

Because PRODUCT OF is a leading keyword that corresponds to AND, underline the words before and after AND: “seven” and “a number.”
- the product of seven and a number

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a times sign.
- The expression translates to 7 × x.

Note: Keep in mind that AND does not always indicate addition. The keyword PRODUCT OF defines the AND in this expression to mean multiplication.

A multiplication expression that is translated by the direct translation method is shown in Example 11.

Example 11: Translate the following: a number times fifteen

The expression translates to x × 15.

Some multiplication keywords, such as DOUBLE, TWICE, and TRIPLE, translate into a number and the operation of multiplication, as shown in Examples 12 and 13.

Example 12: Translate the following: twice a number

The expression translates to 2 × x.

Example 13: Translate the following word problem into a mathematical expression: Jennifer had $15 dollars in the bank. Over the next two weeks she doubled her money.

The expression translates to 2 × 15.

One of the keywords that indicates multiplication is OF. In word problems, however, you may see more than one use of the word “of.” The only OF that indicates multiplication is the one that follows the keyword PERCENT, the percent sign, the keyword FRACTION, or a fraction. See Examples 14 and 15.

Example 14: Translate the following: twenty five percent of four hundred dollars

The expression translates to 0.25 × 400.

Note: Remember that a percent is changed to a decimal before multiplying.

Example 15: Translate the following: one-third of twenty-seven

The expression translates to .

Division keywords

Some common examples of division keywords are as follows:

QUOTIENT OF _____ AND _____

DIVIDE _____ BY _____

_____ DIVIDED BY _____

DIVIDED EQUALLY

The keywords PRODUCT OF and QUOTIENT OF are difficult for some people to differentiate. Here is a hint to help you remember which one indicates division and which one indicates multiplication: QUOTIENT is a “harder” word than “PRODUCT,” and division is a “harder” operation than multiplication.

Remember: Leading keywords define the corresponding AND or BY to mean divide, usually designated with the symbol ÷.

Example 16: Translate the following: the quotient of seven and a number

Because the keyword QUOTIENT OF is a leading keyword that defines AND, underline the words before and after AND: “seven” and “a number.”
- the quotient of seven and a number

Circle the leading keyword and indicate the corresponding AND that it defines.

Translate each underlined expression and replace AND with a division sign.
- The expression translates to 7 ÷ n.

Note: Here, the keyword QUOTIENT OF defines AND to mean division.

Example 17: Translate the following: divide negative thirty-six by nine

Because the word DIVIDE is a leading keyword that defines the BY, underline the words before and after BY: “negative thirty-six” and “nine.”
- divide negative thirty-six by nine

Circle the leading keyword and indicate the corresponding BY that it defines.

Translate each underlined expression and replace BY with a division sign.
- The expression translates to .

Note: The first number goes in the numerator when using a fraction bar to indicate division. The number in the numerator (the −36) goes inside the “house” when using the long division symbol.

Some division keywords can be translated one word at a time. Instead, you just follow the sentence and replace with algebraic notations as you go along.

Example 18: Translate the following: a number divided by 16

The expression translates to .

Often, in story problems, the keyword that indicates division is PER. When a story problem asks for the speed of a vehicle in miles per hour, set up the expression to divide the number of miles by the number of hours. You not only directly translate “miles” ÷ “hours,” but also identify the number of miles and number of hours by finding them elsewhere in the problem. See Example 19.

Example 19: Translate the following word problem into a mathematical expression about speed: It takes three hours to travel 150 miles to grandmother's house. How do you find your average speed in miles per hour?

You find “miles” ÷ “hours” in the question. In the first part of the word problem, you find the number of miles, 150 miles, and the number of hours, three hours.

The expression translates to 150 ÷ 3.

(http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Keywords-for-Mathematical-Operations.topicArticleId-8823,articleId-8818.html)

Nouns

husnainrasheed@gmail.com (Husnain Rasheed) — Sun, 19 Apr 2009 21:50:00 -0700

Introduction to Nouns

A noun is a part of speech that names a person, place, or thing. Many different kinds of nouns are used in the English language. Some are specific for people, places, events, and some represent groups or collections. Some nouns aren't even nouns; they're verbs acting like nouns in sentences.

Nouns can be singular, referring to one thing, or plural, referring to more than one thing. Nouns can be possessive as well; possessive nouns indicate ownership or a close relationship. Regardless of the type, nouns should always agree with their verbs in sentences; use singular verbs with singular nouns and plural verbs with plural nouns. You have to know how a noun works in order to write an effective sentence.

Proper Nouns

If a noun names a specific person or place, or a particular event or group, it is called a proper noun and is always capitalized. Some examples are Eleanor Roosevelt, Niagara Falls, Dracula, the Federal Bureau of Investigation, the Great Depression, and Desert Storm. This seems simple enough.

Unfortunately, some writers assign proper-noun status fairly indiscriminately to other words, sprinkling capital letters freely throughout their prose. For example, the Manhattan Project is appropriately capitalized because it is a historic project, the name given to the specific wartime effort to design and build the first nuclear weapons. But project should not be capitalized when referring to a club's project to clean up the campus. Similarly, the Great Depression should be capitalized because it refers to the specific historical period of economic failure that began with the stock market collapse in 1929. When the word depression refers to other economic hard times, however, it is not a proper noun but a common noun and should not be capitalized. Some flexibility in capitalizing nouns is acceptable. A writer may have a valid reason for capitalizing a particular term, for example, and some companies use style guides that dictate capital letters for job titles such as manager. But often the use of a capital outside the basic rule is an effort to give a word an air of importance, and you should avoid it.

Verbs Used as Nouns

One special case is when a verb is used as a noun. Here the verb form is altered and it serves the same function as a noun in the sentence. This type of noun is called a gerund.

The gerund

A noun created from the - ing form of a verb is called a gerund. Like other nouns, gerunds act as subjects and objects in sentences.

Sleeping sometimes serves as an escape from studying.

The gerunds sleeping and studying are - ing forms of the verbs sleep and study. Sleeping is the noun functioning as the subject of this sentence, and studying is an object (in this case, the object of a preposition).

The problem gerund

Gerunds can sometimes be difficult to use properly in a sentence. What problems will you have with gerunds?

When a noun or pronoun precedes a gerund, use the possessive case of the noun or pronoun.
- Jane's sleeping was sometimes an escape from studying.

Even when you think that the word before the gerund looks like an object, use the possessive case.

Newton's Three Laws

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 23:41:00 -0700

Newton's Three Laws:

Newton's Three Laws, named after Sir Isaac Newton, who derived the laws, provide the basis for the study of Dynamics, and describe the fundamental laws of motion. These three laws will serve as a springboard for all other topics concerning dynamics.
Mathematically, the laws can be written as the following:

First Law: If F = 0 then a = 0 and v =constant
Second Law: F = ma
Third Law: F_AB = - F_BA

We will spend the nest three section of this SparkNote examining the ideas behind Newton's Three Laws, and explaining the derivation of their mathematical formulas...

The Concept of Force and Newton's First Law

Definition of a Force

Since force is the fundamental concept of Dynamics, we must give a clear definition of this concept before we proceed with Newton's Laws. A force is defined (very practically) as a push or a pull. Of course, we experience forces all the time in everyday lives. Whenever we lift something, push something or otherwise manipulate other objects, we are exerting a force. A force is a vector quantity, as it has both a magnitude and a direction. Let us show vector quality of a force practically: when exerting a force, for example pushing a crate, we can change the magnitude of our force by pushing harder or softer. We can also change the direction of our force, as we can push it one way or another. Since a force is a vector, all the rules of vector addition and subtraction, seen in Vectors apply. The vector quality of force allows us to manipulate forces in exactly the same way we manipulated velocity and acceleration in Kinematics.

With a formal definition of force, we can now examine its relation to motion through Newton's laws.

Newton's First Law

So how exactly does a force relate to motion? Intuitively, we can say that a force, at least in some way, causes motion. When I kick a ball, it moves. Newton makes this relation more precise in his first law:

An object moves with constant velocity unless acted upon by a net external force.

What does this mean? Let's start by looking at a special case where the constant velocity is zero, i.e. the object is simply at rest. Newton's First Law states that the object will stay at rest unless a force acts upon it. This makes sense: the soccer ball isn't going anywhere unless someone kicks it. This concept is true not only for v = 0, but for any constant velocity. Consider now a ball rolling with a constant velocity. Neglecting friction, the ball will continue to roll with the same velocity until it hits something, or someone kicks it. In physics terminology, it will keep the same velocity until acted upon by a net external force.

What does Newton mean by a net force? Consider a rope being used in a tug of war. There are definitely forces being applied to the rope but, if the two sides pull with the same force, the rope won't move. In this example, the two forces on the rope exactly cancel each other out, and there is no net force on the rope. It is thus possible for forces to act on an object, yet have the net force be zero. When evaluating the motion caused by forces acting on an object, remember to find the vector sum of those forces.

Also included in Newton's First Law, though not explicitly, is the concept of inertia. Inertia is defined as the tendency of an object to remain at a constant velocity. It is a fundamental property of all matter. In a sense, the idea of inertia is unnecessary; it just gives a name to the concept Newton describes in his First Law. However, you're bound to hear the word over and over in physics, so it is important to know to what it refers.

From our concept of inertia, we can develop the idea of an inertial reference frame, meaning a frame in which a body has no observed acceleration. This concept has limited application to classical mechanics, yet is essential for the study of Relativity. Consider a body with no net force acting upon it. For example, imagine yourself in an accelerating automobile. You look out the window, and the ground seems to be accelerating in a direction opposite the motion of the car. Clearly no net forces act upon the ground, yet from the frame of the car the ground is accelerating; in this case the car represents a non-inertial frame, and measurements of inertial fields from non-intertial fields do not conform to the rules of Newton's Laws. If, however, the car is traveling at a constant velocity, the ground will also appear to be moving back with a constant velocity. In this case, the frame of the car is inertial, as no net acceleration is observed. Any inertial reference frame is thus valid one in which to make calculations based on Newton's laws. Before we use these force laws, we must make sure we are making measurements from an inertial frame.

The concept of Mass and Newton's Second Law

Now we have both a definition of force, and a vague idea of how forces relate to motion. What we need is a precise way of relating the two. But even before we do this, we need to define another concept that plays a role in the relation between force and motion, that of mass.

Mass

Mass is defined as the amount of matter in a given body. This definition seems a little vague, and needs some explanation. Mass is a scalar quantity, meaning it has no direction, and is a property of the object itself, not its location. Mass is measured in kilograms (kg). Given a certain object, its mass will be the same on earth, on the moon, or in empty space. In contrast, the weight of the object in these different circumstances will change. We will explore further the relation between mass and weight when we have completed discussing Newton's laws. Yet even without a complete understanding of weight we can use weight to better understand the concept of mass. In our everyday experience, the heavier an object is (the more weight it has), the more mass it has. Thus our experience tells us that a baseball has more mass than a balloon, for example. As long as we do not think of them as the same concept, describing mass in terms of weight allows us to conceptualize mass in practical terms. From this concept of mass, we can more exactly relate force and motion.

Given a certain force, how does an object's motion correspond to its mass? Our intuition tells us that a more massive object moves slower if given the same force as a less massive object. We can throw a baseball with much greater speed than we can throw a massive ball of lead. Our intuition is correct, and is stated in Newton's Second Law.

Newton's Second Law

Newton's Second Law gives us a quantitative relation between force and motion:

secondlaw

F = ma

Stated verbally, Newton's Second Law says that the net force (F) acting upon an object causes acceleration (a), with the magnitude of the acceleration directly proportional to the net force and inversely proportional to the mass (m). Learn it and love it. Like it or not, this equation will be used at almost all times in virtually every physics course you take.

The Second Law relates two vector quantities, force and acceleration. Because both force and acceleration are vector quantities, it is important to understand that the acceleration of an object will always be in the same direction as the sum of forces applied to the object. The magnitude of acceleration depends on the mass of the object, but is always proportional to the force. Newton's Second Law gives an exact relation between the vectors force and motion. Thus we can use this law to predict the motion of an object given forces acting upon it, on a quantitative level.

Free Body Diagrams

The best method for calculating acceleration from force is through a free body diagram. This process, though fairly complicated, is extremely useful. We will go through it step by step:

Step 1: Draw the physical situation in which an object exists. It may lie on an incline, be attached to a string, or simply be resting on the ground. Whatever the situation, draw it complete with any angles or distances that apply.
Step 2: From the center of the body being examined, draw vectors representing each force acting upon the body, giving the magnitude of each one.
Step 3: Sum all horizontal components of forces acting upon the object (this may require resolving a vector into its components (see Vectors).
Step 4: Sum all vertical components of forces acting upon the object (using the same method as step 3).
Step 5: Find the net force acting on the object, using the sum of the vectors found in steps 3 and 4.
Step 6: Divide the net force by the object's mass to find the acceleration vector of the object.
Step 7: From the acceleration vector, compute velocity, position, or any other necessary kinematic quantity.

There we have it! Finally, we can compute an exact relation between force and motion. With Newton's second law, we can take a given physical situation and find the acceleration, and thus the motion, of an object in the situation. In addition, using the method of free-body diagrams, we can evaluate any number of distinct forces. Such an ability is powerful, and will be used over and over in physics courses. We can now move on to Newton's Third Law, which further clarifies the nature of forces..

Newton's Third Law and Units of Force

Newton's Third Law

All forces result from the interaction of two bodies. One body exerts a force on another. Yet we haven't discussed what force, if any, is felt by the body giving the original force. Experience tells us that there is in fact a force. When we push a crate across the floor, our hands and arms certainly feel a force in the opposite direction. In fact, Newton's Third Law tells us that this force is exactly equal in magnitude and opposite in direction of the force we exert on the crate. If body A exerts a force on body B, let us denote this force by F_AB. Newton's Third Law, then, states that:

thirdlaw

F_AB = - F_BA

Stated in words, Newton's third law proclaims: to every action there is an equal and opposite reaction. This law is quite simple and generally more intuitive than the other two. It also gives us a reason for many observed physical facts. If I am in a sailboat, I cannot move the boat simply by pushing on the front. Though I do exert a force on the boat, I also feel a force in the opposite direction. Thus the net force on the system (me and the boat) is zero, and the boat doesn't move. We need some external force, like wind, to move the boat. Though this law seems obvious and unnecessary, we will see its importance when we apply Newton's laws.

Newton's third law also gives us a more complete definition of a force. Instead of merely a push or a pull, we can now understand a force as the mutual interaction between two bodies. Whenever two bodies interact in the physical world, a force results. Whether it be two balls bouncing off each other or the electrical attraction between a proton and an electron, the interaction of two bodies results in two equal and opposite forces, one acting on each body involved in the interaction.

Amazingly enough, Newton's Three Laws provide all the necessary information to describe the motion involved in any given situation. We will soon study the applications of Newton's Laws, but we first need to take care of the units of force.

Units of Force

The unit of force is defined, quite appropriately, as a Newton. What is a Newton in terms of fundamental units? Given that acceleration (a) = m/s² and mass = 1 kg, we can find out from Newton's Second Law: F = ma implies that a Newton, N = kg (m/s²) = (kg ƒ m)/s². Therefore, one Newton causes a one kilogram body to accelerate at a rate of one meter per second per second. Our definition of units becomes important when we get into practical applications of Newton's Laws.

Summary of Newton's Laws

We can now give an equation summary of Newton's Three Laws: First Law: If F = 0 then a = 0 and v =constant
Second Law: F = ma
Third Law: F_AB = - F_BA

(http://www.sparknotes.com/physics/dynamics/newtonsthreelaws/section3.rhtml)

A.S.P.I.R.E. - a study system

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 23:35:00 -0700

A: Approach/attitude/arrange

Approach your studies with a positive attitude
Arrange your schedule to eliminate distractions

S: Select/survey/study!

Select a reasonable chunk of material to study
Survey the headings, graphics, pre- and post questions
to get an overview
Study marking any information you don’t understand

P: Put aside/piece together:

Put aside your books and notes
Piece together what you've studied, either alone, with a study pal or group, and summarize what you understand.

I: Inspect/Investigate/inquire/:

Inspect what you did not understand.
Investigate alternative sources of information you can refer to:
other text books, websites, experts, tutors, etc.
Inquire from support professionals (academic support, librarians, tutors, teachers, experts,) for assistance

R: Reconsider/reflect/relay

Reconsider the content:
If I could speak to the author, what questions would I ask or what criticism would I offer?
Reflect on the material:
How can I apply this material to what I am interested in?
Relay understanding:
How would I make this information interesting and understandable to other students?

E: Evaluate/examine/explore:

Evaluate your grades on tests and tasks
look for a pattern
Examine your process
toward improving it
Explore options
with a teacher, support professional, tutor, etc.

A.S.P.I.R.E. - a study system

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 23:35:00 -0700

A: Approach/attitude/arrange

Approach your studies with a positive attitude
Arrange your schedule to eliminate distractions

S: Select/survey/study!

Select a reasonable chunk of material to study
Survey the headings, graphics, pre- and post questions
to get an overview
Study marking any information you don’t understand

P: Put aside/piece together:

Put aside your books and notes
Piece together what you've studied, either alone, with a study pal or group, and summarize what you understand.

I: Inspect/Investigate/inquire/:

Inspect what you did not understand.
Investigate alternative sources of information you can refer to:
other text books, websites, experts, tutors, etc.
Inquire from support professionals (academic support, librarians, tutors, teachers, experts,) for assistance

R: Reconsider/reflect/relay

Reconsider the content:
If I could speak to the author, what questions would I ask or what criticism would I offer?
Reflect on the material:
How can I apply this material to what I am interested in?
Relay understanding:
How would I make this information interesting and understandable to other students?

E: Evaluate/examine/explore:

Evaluate your grades on tests and tasks
look for a pattern
Examine your process
toward improving it
Explore options
with a teacher, support professional, tutor, etc.

The Decimal System

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 09:05:00 -0700

The Decimal System

The system of numbers that you use is called the decimal system and is based on powers of ten ( base ten system). Each place in the place value grid is ten times the value of the place to the right of it. Every number to the right of the decimal point is a decimal fraction (a fraction with a denominator of 10, 100, 1,000, and so on).

On the place value grid (see the following chart), notice that¹/₁₀ can be written as ten to a negative exponent, 10⁻¹. Similarly,¹/₁₀₀ can be written as ten to a negative exponent, 10⁻².

Value of money (Theory)

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 08:59:00 -0700

Quantity theory of money

Value of money

What gives money value? We know that intrinsically, a dollar bill is just worthless paper and ink. However, the purchasing power of a dollar bill is much greater than that of another piece of paper of similar size. From where does this power originate?

Like most things in economics, there is a market for money. The supply of money in the money market comes from the Fed. The Fed has the power to adjust the money supply by increasing or decreasing the number of bills in circulation. Nobody else can make this policy decision. The demand for money in the money market comes from consumers.

The determinants of money demand are infinite. In general, consumers need money to purchase goods and services. If there is an ATM nearby or if credit cards are plentiful, consumers may demand less money at a given time than they would if cash were difficult to obtain. The most important variable in determining money demand is the average price level within the economy. If the average price level is high and goods and services tend to cost a significant amount of money, consumers will demand more money. If, on the other hand, the average price level is low and goods and services tend to cost little money, consumers will demand less money.

Figure 2.1: Sample money market

The value of money is ultimately determined by the intersection of the money supply, as controlled by the Fed, and money demand, as created by consumers. Figure 1 depicts the money market in a sample economy. The money supply curve is vertical because the Fed sets the amount of money available without consideration for the value of money. The money demand curve slopes downward because as the value of money decreases, consumers are forced to carry more money to make purchases because goods and services cost more money. Similarly, when the value of money is high, consumers demand little money because goods and services can be purchased for low prices. The intersection of the money supply curve and the money demand curve shows both the equilibrium value of money as well as the equilibrium price level.

Figure 2.2: Sample shift in the money market

The value of money, as revealed by the money market, is variable. A change in money demand or a change in the money supply will yield a change in the value of money and in the price level. Notice that the change in the value of money and the change in the price level are of the same magnitude but in opposite directions. An increase in the money supply is depicted in Figure 2. Notice that the new intersection of the money supply curve and the money demand curve is at a lower value of money but a higher price level. This happens because more money is in circulation, so each bill becomes worth less. It takes more bills to purchase goods and services, and thus the price level increases accordingly.

The quantity theory of money is based directly on the changes brought about by an increase in the money supply. The quantity theory of money states that the value of money is based on the amount of money in the economy. Thus, according to the quantity theory of money, when the Fed increases the money supply, the value of money falls and the price level increases. In the SparkNote on inflation we learned that inflation is defined as an increase in the price level. Based on this definition, the quantity theory of money also states that growth in the money supply is the primary cause of inflation.

Velocity

While the relationship between money supply, money demand, the price level, and the value of money presented above is accurate, it is a bit simplistic. In the real world economy, these factors are not connected as neatly as the quantity theory of money and the basic money market diagram present. Rather, a number of variables mediate the effects of changes in the money supply and money demand on the value of money and the price level.

The most important variable that mediates the effects of changes in the money supply is the velocity of money. Imagine that you purchase a hamburger. The waiter then takes the money that you spent and uses it to pay for his dry cleaning. The dry cleaner then takes that money and pays to have his car washed. This process continues until the bill is eventually taken out of circulation. In many cases, bills are not removed from circulation until many decades of service. In the end, a single bill will have facilitated many times its face value in purchases.

Velocity of money is defined simply as the rate at which money changes hands. If velocity is high, money is changing hands quickly, and a relatively small money supply can fund a relatively large amount of purchases. On the other hand, if velocity is low, then money is changing hands slowly, and it takes a much larger money supply to fund the same number of purchases.

As you might expect, the velocity of money is not constant. Instead, velocity changes as consumers' preferences change. It also changes as the value of money and the price level change. If the value of money is low, then the price level is high, and a larger number of bills must be used to fund purchases. Given a constant money supply, the velocity of money must increase to fund all of these purchases. Similarly, when the money supply shifts due to Fed policy, velocity can change. This change makes the value of money and the price level remain constant.

The relationship between velocity, the money supply, the price level, and output is represented by the equation M * V = P * Y where M is the money supply, V is the velocity, P is the price level, and Y is the quantity of output. P * Y, the price level multiplied by the quantity of output, gives the nominal GDP. This equation can thus be rearranged as V = (nominal GDP) / M. Conceptually, this equation means that for a given level of nominal GDP, a smaller money supply will result in money needing to change hands more quickly to facilitate the total purchases, which causes increased velocity.

The equation for the velocity of money, while useful in its original form, can be converted to a percentage change formula for easier calculations. In this case, the equation becomes (percent change in the money supply) + (percent change in velocity) = (percent change in the price level) + (percent change in output). The percentage change formula aids calculations that involve this equation by ensuring that all variables are in common units.

The velocity equation can be used to find the effects that changes in velocity, price level, or money supply have on each other. When making these calculations, remember that in the short run, output (Y), is fixed, as time is required for the quantity of output to change.

Let's try an example. What is the effect of a 3% increase in the money supply on the price level, given that output and velocity remain relatively constant? The equation used to solve this problem is (percent change in the money supply) + (percent change in velocity) = (percent change in the price level) + (percent change in output). Substituting in the values from the problem we get 3% + 0% = x% + 0%. In this case, a 3% increase in the money supple results in a 3% increase in the price level. Remember that a 3% increase in the price level means that inflation was 3%.

In the long run, the equation for velocity becomes even more useful. In fact, the equation shows that increases in the money supply by the Fed tend to cause increases in the price level and therefore inflation, even though the effects of the Fed's policy is slightly dampened by changes in velocity. This results a number of factors. First, in the long run, velocity, V, is relatively constant because people's spending habits are not quick to change. Similarly, the quantity of output, Y, is not affected by the actions of the Fed since it is based on the amount of production, not the value of the stuff produced. This means that the percent change in the money supply equals the percent change in the price level since the percent change in velocity and percent change in output are both equal to zero. Thus, we see how an increase in the money supply by the Fed causes inflation.

Let's try another example. What is the effect of a 5% increase in the money supply on inflation? Again, we being by using the equation (percent change in the money supply) + (percent change in velocity) = (percent change in the price level) + (percent change in output). Remember that in the long run, output not affected by the Fed's actions and velocity remains relatively constant. Thus, the equation becomes 5% + 0% = x% + 0%. In this case, a 5% increase in the money supply results in a 5% increase in inflation.

The velocity of money equation represents the heart of the quantity theory of money. By understanding how velocity mitigates the actions of the Fed in the long run and in the short run, we can gain a thorough understanding of the value of money and inflation..

(Spark notes)

Money

husnainrasheed@gmail.com (Husnain Rasheed) — Sat, 18 Apr 2009 08:55:00 -0700

Terms

Bartering - The trading of one good for another. This requires the double Coincidence of wants, a condition met when two individuals each have different goods that they other wants.

Commodity Money - Money that has an intrinsic value, that is, value beyond any value given to it because it is money. An example of this would be a gold coin that has value because it is a precious metal.

Compound Interest - Interest that is paid on a sum of money where the interest paid is added to the principal for the future calculation of interest. Click here to see the Formula.

Consumption - The purchase and use of goods and services by consumers.

Currency - The form of money used in a country.

Defaulting on the Loan - When a borrower fails to repay a loan leaving the lender without the money loaned.

Demand for Money - The amount of currency that consumers use for the purchase of goods and services. This varies depending mainly upon the price level.

Equilibrium - The state in a market when supply equals demand.

Fiat Money - Money that has no intrinsic value, that is, its only value comes from the fact that a governing body backs and regulates the currency.

Fischer Effect - The point for point relationship between changes in the money supply and changes in the inflation rate.

Inflation - The increase of the price level over time.

Interest - Money paid by a borrower to a lender for the use of a sum of money.

Interest Rates - The percent of the amount borrowed paid each year to the lender by the borrower in return for the use of the money.

Liquidity - The ease with which something of value can be exchanged for the currency of an economy.

Medium of Exchange - An item used commonly to trade for goods and services.

Money Supply - The quantity of money in an economy. In the US this is controlled through policy by the Fed.

Nominal GDP - The total value of all goods and services produced in a country valued at current prices.

Nominal Interest - The percent of the amount borrowed paid each year to the lender by the borrower in return for the use of the money not taking inflation into account.

Nominal Value - The value of something in current dollars without taking into account the effects of inflation.

Output - The amount of goods and services produced within an economy.

Price Level - The overall level of prices of goods and services in an economy. This is used in the calculation of inflation rates.

Purchasing Power - The real value of a dollar. This describes the quantity of goods and services that can be purchased for a dollar, taking into account the effects of inflation.

Quantity Theory of Money - The theory that says that the value of money is based on the amount of money in circulation, that is, the money supply.

Real Interest - The percent of the amount borrowed paid each year to the lender by the borrower in return for the use of the money adjusted for inflation.

Real Value - The value of something in taking into account the effects of inflation.

Store of Value - A good that holds a value in such a way that its price is fairly insensitive inflation.

Unit of Account - Something that is used universally in the description of money matters such as prices. The unit of account most commonly used in the US is the dollar.

Value of Money - The purchasing power of the dollar. The amount of goods and services that can be purchased for a fixed amount of money.

Velocity - The speed with which a dollar bill changes hands. The higher the velocity of money, the quicker that a given piece of currency will be traded for goods and services.

Wage - The amount of money paid to workers by employers valued in current dollars.

Velocity of Money

M * V = P * Y where M is the money supply, V is the velocity, P is the price level, and Y is the quantity of output. P * Y, the price level multiplied by the quantity of output, gives the nominal GDP. This equation can be rearranged as V = (nominal GDP) / M. It can also be converted into a percentage change formula as (percent change in the money supply) + (percent change in velocity) = (percent change in the price level) + (percent change in output).

Compound Interest

First, calculate the value of the loan, by adding one to the interest rate, raising it to the number of years for the loan, and multiplying it by the loan amount. Then, to calculate the amount of interest, simply subtract the original loan amount from the total due.

Real Interest Rate

The real interest rate is equal to the nominal interest rate minus the inflation rate.

Difference Between Input and Output Devices of a Computer

husnainrasheed@gmail.com (Husnain Rasheed) — Tue, 31 Mar 2009 22:44:00 -0700

Overview:
This study note differentiate between input and output devices of a computer.

Input Devices:Input is the first stage of computing, referring to any means that moves data (information) from the outside world into the processor or from one component of the computer to another.

Keyboard
The primary input device for a computer, allowing users to type information just as they once did on a typewriter.

Mouse
Used with graphical interface environments to point to and select objects on the system's monitor. Can be purchased in a variety of sizes, shapes, and configurations.

Scanner
Converts printed or photographic information to digital information that can be used by the computer. Works similar to the scanning process of a photocopy machine.

Microphone
Works like the microphone on a tape recorder. Allows input of voice or music to be converted to digital information and saved to a file.

CD-ROM/DVD drive
Compact disc–read only memory: stores large amounts of data on a CD that can be read by a computer.

Terms and Formulae

husnainrasheed@gmail.com (Husnain Rasheed) — Tue, 31 Mar 2009 09:15:00 -0700

Terms

Base year - The year from which constant prices or quantities are taken in calculations of such indices as real GDP and CPI.

Bureau of Labor Statistics - The government organization responsible for regularly gathering data about the economic status of the population.

Consumer price index (CPI) - A cost of living index that measures the total cost of goods and services purchased by a typical consumer within a country.

Fixed basket - A set group of goods and services whose quantities do not change over time. This is used, for instance, in the calculation of the CPI.

Gross domestic product (GDP) - The sum of the market values of all final goods and services produced within a particular country during a period of time.

Gross domestic product deflator (GDP deflator) - The ratio of nominal GDP to real GDP for a given year minus 1. The GDP deflator shows how much of the change in the GDP from a base year is reliant on changes in the price level.

Gross domestic product per capita (GDP per capita) - GDP divided by the number of people in the population. This measure describes what portion of the GDP an average individual gets.

Gross national product (GNP) - An alternative measure of economic activity to GDP. GNP is the sum of the market values of all goods and services produced by the citizens of a country regardless of their physical location.

Nominal gross domestic product (nominal GDP) - The sum value of goods and services produced in a country and valued at current prices.

Real gross domestic product (real GDP) - The sum value of goods and services produced in a country and valued at constant prices, calibrated from some base year. Real GDP frees year-to-year comparisons of output from the effects of changes in the price level.

Formulae

Gross Domestic Product GDP = [(quantity of A X price of A) + (quantity of B X price of B) + ... + (quantity of N X price of N)] for every good and service produced within the country

GDP = (national income) = Y = (C + I + G + NX)

GDP Growth Rate GDP growth rate = [(GDP for year N) / (GDP for year N-1)] - 1

GDP Deflator GDP deflator = [(nominal GDP) / (real GDP)] - 1

GDP Per Capita GDP per capita = (GDP) / (population)

Measuring the Economy

husnainrasheed@gmail.com (Husnain Rasheed) — Tue, 31 Mar 2009 09:08:00 -0700

Macroeconomists use a variety of different observational means in their effort to study and explain how the economy as a whole functions and changes over time. One such method relies on personal experience. It is relatively simple to notice that your company is producing more than it has in the past or that a paycheck does not go as far as it used to. Yet while personal observations do provide information about the economy, that information can often be localized rather than universal, and may not accurately reflect the state of the economy as a whole.

In order to move beyond the limitations inherent in personal experiences, macroeconomists begin by systematically measuring the basic elements of the economy in order to derive standard and comprehensive statistics. This data provides information about the entire economy rather than simply about a single household or firm. Two of the most fundamental elements macroeconomists study are the total output of an economy (GDP) and the cost of living within an economy (CPI). Gross domestic product, or GDP, is an indicator of economic performance that measures the market value of goods and services produced within a country. This measurement is of great importance to consumers since it also equals the total income within an economy. The consumer price index, or CPI, is a cost of living indicator; it measures the total cost of goods and services purchased by a typical consumer within a country. This index allows economists and consumers to see just how much purchasing power a dollar yields, and to compare that power between different years and eras. Together, GDP and CPI show how much income exists within an economy and how much this income can purchase.

The concepts of GDP and CPI open the door to a scientific understanding of the functioning of the economy on a large, or macro, level. These are the most basic tools of measurement used by macroeconomists, policy makers, and consumers to understand and describe the economy. In fact, GDP and CPI are published and discussed regularly in the media. Through understanding the concepts of GDP and CPI, the world of macroeconomics begins to unfold...