1.1: A New Generation
The 1920s was a time of dramatic change in the United States. Many young people, especially those living in big cities, embraced a new morality that was much more permissive than that of previous generations. They listened to jazz music, especially in the nightclubs of Harlem. Although prohibition outlawed alcohol, criminal bootlegging and importing businesses thrived. The decade was not a pleasure cruise for everyone, however; in the wake of the Great War, many were left awaiting the promise of a new generation.
A New Morality
Many Americans were disillusioned in the post-World War I era, and their reactions took many forms. Rebellious American youth, in particular, adjusted to the changes by embracing a new morality that was far more permissive than the social mores of their parents. Many young women of the era shed their mother’s morality and adopted the dress and mannerisms of a flapper, the Jazz Age female stereotype, seeking the endless party. Flappers wore shorter skirts, shorter hair, and more makeup, and they drank and smoked with the boys (Figure 1.1.1). Flappers’ dresses emphasized straight lines from the shoulders to the knees, minimizing breasts and curves while highlighting legs and ankles. The male equivalent of a flapper was a “ sheik ,” although that term has not remained as strong in the American vernacular. At the time, however, many of these fads became a type of conformity, especially among college-aged youths, with the signature bob haircut of the flapper becoming almost universal—in both the United States and overseas.
As men and women pushed social and cultural boundaries in the Jazz Age, sexual mores changed and social customs grew more permissive. “Petting parties” or “necking parties” became the rage on college campuses. Psychologist Sigmund Freud and British “sexologist” Havelock Ellis emphasized that sex was a natural and pleasurable part of the human experience. Margaret Sanger, the founder of Planned Parenthood, launched an information campaign on birth control to give women a choice in the realm in which suffrage had changed little—the family. The popularization of contraception and the private space that the automobile offered to teenagers and unwed couples also contributed to changes in sexual behavior.
Flappers and sheiks also took their cues from the high-flying romances they saw on movie screens and confessions in movie magazines of immorality on movie sets. Movie posters promised: “Brilliant men, beautiful jazz babies, champagne baths, midnight revels, petting parties in the purple dawn, all ending in one terrific smashing climax that makes you gasp.”
EXPLORE
Could you go “on a toot” with flappers and sheiks? Improve your chances with this collection of Jazz Age slang .
New dances and new music—especially jazz—also characterized the Jazz Age. Born out of the African American community, jazz was a uniquely American music. The innovative sound emerged from a number of different communities and from a number of different musical traditions such as blues and ragtime. By the 1920s, jazz had spread from African American clubs in New Orleans and Chicago to reach greater popularity in New York and abroad. One New York jazz establishment, the Cotton Club, became particularly famous and attracted large audiences of hip, young, and white flappers and sheiks to see black entertainers play jazz (Figure 1.1.2).
The "New Woman"
The Jazz Age and the proliferation of the flapper lifestyle of the 1920s should not be seen merely as the product of postwar disillusionment and newfound prosperity. Rather, the search for new styles of dress and new forms of entertainment like jazz was part of a larger women’s rights movement. The early 1920s, especially with the ratification of the Nineteenth Amendment guaranteeing full voting rights to women, was a period that witnessed the expansion of women’s political power. The public flaunting of social and sexual norms by flappers represented an attempt to match gains in political equality with gains in the social sphere. Women were increasingly leaving the Victorian era norms of the previous generation behind, as they broadened the concept of women’s liberation to include new forms of social expression such as dance, fashion, women’s clubs, and forays into college and the professions.
Nor did the struggle for women’s rights through the promotion and passage of legislation cease in the 1920s. In 1921, Congress passed the Promotion of the Welfare and Hygiene of Maternity and Infancy Act, also known as the Sheppard-Towner Act, which earmarked $1.25 million for well-baby clinics and educational programs, as well as nursing. This funding dramatically reduced the rate of infant mortality. Two years later, in 1923, Alice Paul drafted and promoted an Equal Rights Amendment (ERA) that promised to end all sex discrimination by guaranteeing that “Men and women shall have equal rights throughout the United States and every place subject to its jurisdiction.”
Yet, ironically, at precisely the time when the Progressive movement was achieving its long-sought-after goals, the movement itself was losing steam and the Progressive Era was coming to a close. As the heat of Progressive politics grew less intense, voter participation from both sexes declined over the course of the 1920s. After the passage of the Nineteenth Amendment, many women believed that they had accomplished their goals and dropped out of the movement. As a result, the proposed ERA stalled (the ERA eventually passed Congress almost fifty years later in 1972, but then failed to win ratification by a sufficient number of states), and, by the end of the 1920s, Congress even allowed funding for the Sheppard-Towner Act to lapse.
The growing lethargy toward women’s rights was happening at a time when an increasing number of women were working for wages in the U.S. economy—not only in domestic service, but in retail, healthcare and education, offices, and manufacturing. Beginning in the 1920s, women’s participation in the labor force increased steadily. However, most were paid less than men for the same type of work based on the rationale that they did not have to support a family. While the employment of single and unmarried women had largely won social acceptance, married women often suffered the stigma that they were working for pin money—frivolous additional discretionary income.
The Harlem Renaissance and the New Negro
It wasn’t only women who found new forms of expression in the 1920s. African Americans were also expanding their horizons and embracing the concept of the “new Negro.” The decade witnessed the continued Great Migration of African Americans to the North, with over half a million fleeing the strict Jim Crow laws of the South. Life in the northern states, as many African Americans discovered, was hardly free of discrimination and segregation. Even without Jim Crow, businesses, property owners, employers, and private citizens typically practiced de facto segregation, which could be quite stifling and oppressive. Nonetheless, many southern blacks continued to move north into segregated neighborhoods that were already bursting at the seams, because the North, at the very least, offered two tickets toward black progress: schools and the vote. The black population of New York City doubled during the decade. As a result, Harlem, a neighborhood at the northern end of Manhattan, became a center for Afro-centric art, music, poetry, and politics. Political expression in the Harlem of the 1920s ran the gamut, as some leaders advocated a return to Africa, while others fought for inclusion and integration.
Revived by the wartime migration and fired up by the white violence of the postwar riots, urban blacks developed a strong cultural expression in the 1920s that came to be known as the Harlem Renaissance. In this rediscovery of black culture, African American artists and writers formulated an independent black culture and encouraged racial pride, rejecting any emulation of white American culture. Claude McKay’s poem “If We Must Die” called on African Americans to start fighting back in the wake of the Red Summer riots of 1919 (Figure 1.1.3). Langston Hughes, often nicknamed the “poet laureate” of the movement, invoked sacrifice and the just cause of civil rights in “The Colored Soldier,” while another author of the movement, Zora Neale Hurston, celebrated the life and dialect of rural blacks in a fictional, all-black town in Florida. Hurston’s Their Eyes Were Watching God was only published posthumously in 1937.
The new Negro found political expression in a political ideology that celebrated African Americans distinct national identity. This Negro nationalism, as some referred to it, proposed that African Americans had a distinct and separate national heritage that should inspire pride and a sense of community. An early proponent of such nationalism was W. E. B. Du Bois. One of the founders of the NAACP, a brilliant writer and scholar, and the first African American to earn a Ph.D. from Harvard, Du Bois openly rejected assumptions of white supremacy. His conception of Negro nationalism encouraged Africans to work together in support of their own interests, promoted the elevation of black literature and cultural expression, and, most famously, embraced the African continent as the true homeland of all ethnic Africans—a concept known as Pan-Africanism.
Taking Negro nationalism to a new level was Marcus Garvey. Like many black Americans, the Jamaican immigrant had become utterly disillusioned with the prospect of overcoming white racism in the United States in the wake of the postwar riots and promoted a “Back to Africa” movement. To return African Americans to a presumably more welcoming home in Africa, Garvey founded the Black Star Steamship Line . He also started the United Negro Improvement Association (UNIA), which attracted thousands of primarily lower-income working people. UNIA members wore colorful uniforms and promoted the doctrine of a “negritude” that reversed the color hierarchy of white supremacy, prizing blackness and identifying light skin as a mark of inferiority. Intellectual leaders like Du Bois, whose lighter skin put him low on Garvey’s social order, considered the UNIA leader a charlatan. Garvey was eventually imprisoned for mail fraud and then deported, but his legacy set the stage for Malcolm X and the Black Power movement of the 1960s.
Prohibition
At precisely the same time that African Americans and women were experimenting with new forms of social expression, the country as a whole was undergoing a process of austere and dramatic social reform in the form of alcohol prohibition. After decades of organizing to reduce or end the consumption of alcohol in the United States, temperance groups and the Anti-Saloon League finally succeeded in pushing through the Eighteenth Amendment in 1919, which banned the manufacture, sale, and transportation of intoxicating liquors (Figure 1.1.4). The law proved difficult to enforce, as illegal alcohol soon poured in from Canada and the Caribbean, and rural Americans resorted to home-brewed “moonshine.” The result was an eroding of respect for law and order, as many people continued to drink illegal liquor. Rather than bringing about an age of sobriety, as Progressive reformers had hoped, it gave rise to a new subculture that included illegal importers, interstate smuggling (or bootlegging), clandestine saloons referred to as “speakeasies,” hipflasks, cocktail parties, and the organized crime of trafficking liquor.
Prohibition also revealed deep political divisions in the nation. The Democratic Party found itself deeply divided between urban, northern “wets” who hated the idea of abstinence, and rural, southern “dries” who favored the amendment. This divided the party and opened the door for the Republican Party to gain ascendancy in the 1920s. All politicians, including Woodrow Wilson, Herbert Hoover, Robert La Follette, and Franklin D. Roosevelt, equivocated in their support for the law. Publicly, they catered to the Anti-Saloon League; however, they failed to provide funding for enforcement.
Prohibition sparked a rise in organized crime. “Scarface” Al Capone (Figure 1.1.5) ran an extensive bootlegging and criminal operation known as the Chicago Outfit or Chicago mafia. By 1927, Capone’s organization included a number of illegal activities including bootlegging, prostitution, gambling, loan sharking, and even murder. His operation was earning him more than $100 million annually, and many local police were on his payroll. Although he did not have a monopoly on crime, his organizational structure was better than many other criminals of his era. His liquor trafficking business and his Chicago soup kitchens during the Great Depression led some Americans to liken Capone to a modern-day Robin Hood. Still, Capone was eventually imprisoned for eleven years for tax evasion, including a stint in California’s notorious Alcatraz prison.
The Lost Generation
As the country struggled with the effects and side-effects of prohibition, many young intellectuals endeavored to come to grips with a lingering sense of disillusionment. World War I, fundamentalism, and the Red Scare—a pervasive American fear of Communist infiltrators prompted by the success of the Bolshevik Revolution—all left their mark on these intellectuals. Known as the Lost Generation, writers like F. Scott Fitzgerald, Ernest Hemingway, Sinclair Lewis, Edith Wharton, and John Dos Passos expressed their hopelessness and despair by skewering the middle class in their work. They felt alienated from society, so they tried to escape (some literally) to criticize it. Many lived an expatriate life in Paris for the decade, although others went to Rome or Berlin.
The Lost Generation writer that best exemplifies the mood of the 1920s was F. Scott Fitzgerald, now considered one of the most influential writers of the twentieth century. His debut novel, This Side of Paradise , describes a generation of youth “grown up to find all gods dead, all wars fought, all faith in man shaken.” The Great Gatsby , published in 1925, exposed the doom that always follows the fun, fast-lived life. Fitzgerald depicted the modern millionaire Jay Gatsby living a profligate life: unscrupulous, coarse, and in love with another man’s wife. Both Fitzgerald and his wife Zelda lived this life as well, squandering the money he made from his writing.
MY STORY: F. SCOTT FITZGERALD ON THE 1920S
In the 1920s, Fitzgerald was one of the most celebrated authors of his day, publishing This Side of Paradise , The Beautiful and Damned , and The Great Gatsby in quick succession. However, his profligate lifestyle with his wife Zelda sapped their funds, and Fitzgerald had to struggle to maintain their lavish lifestyle. Below is an excerpt from “The Crack-Up,” a personal essay by Fitzgerald originally published in Esquire in which he describes his “good life” during the 1920s.
It seemed a romantic business to be a successful literary man—you were not ever going to be as famous as a movie star but what note you had was probably longer-lived; you were never going to have the power of a man of strong political or religious convictions but you were certainly more independent. Of course within the practice of your trade you were forever unsatisfied—but I, for one, would not have chosen any other.
As the Twenties passed, with my own twenties marching a little ahead of them, my two juvenile regrets—at not being big enough (or good enough) to play football in college, and at not getting overseas during the war—resolved themselves into childish waking dreams of imaginary heroism that were good enough to go to sleep on in restless nights. The big problems of life seemed to solve themselves, and if the business of fixing them was difficult, it made one too tired to think of more general problems.
—F. Scott Fitzgerald, “The Crack-Up,” 1936
How does Fitzgerald describe his life in the 1920s? How did his interpretation reflect the reality of the decade?
Equally idiosyncratic and disillusioned was writer Ernest Hemingway (Figure 1.1.6). He lived a peripatetic and adventurous lifestyle in Europe, Cuba, and Africa, working as an ambulance driver in Italy during World War I and traveling to Spain in the 1930s to cover the civil war there. His experiences of war and tragedy stuck with him, emerging in colorful scenes in his novels The Sun Also Rises (1926), A Farewell to Arms (1929), and For Whom the Bell Tolls (1940). In 1952, his novella, The Old Man and the Sea , won the Pulitzer Prize. Two years later, he won the Nobel Prize in Literature for this book and his overall influence on contemporary style.
EXPLORE
Listen to an audio of Hemingway’s Nobel Prize acceptance speech .
Not all Lost Generation writers were like Fitzgerald or Hemingway. The writing of Sinclair Lewis, rather than expressing a defined disillusionment, was more influenced by the Progressivism of the previous generation. In Babbitt (1922), he examined the “sheep following the herd” mentality that conformity promoted. He satirized American middle-class life as pleasure-seeking and mindless. Similarly, writer Edith Wharton celebrated life in old New York, a vanished society, in The Age of Innocence , in 1920. Wharton came from a very wealthy, socialite family in New York, where she was educated by tutors and never attended college. She lived for many years in Europe; during the Great War, she worked in Paris helping women establish businesses.
Section Summary
Different groups reacted to the upheavals of the 1920s in different ways. Some people, especially young urbanites, embraced the new amusements and social venues of the decade. Women found new opportunities for professional and political advancement, as well as new models of sexual liberation; however, the women’s rights movement began to wane with the passage of the Nineteenth Amendment. For black artists of the Harlem Renaissance, the decade was marked less by leisure and consumption than by creativity and purpose. African American leaders like Marcus Garvey and W. E. B. Du Bois responded to the retrenched racism of the time with different campaigns for civil rights and black empowerment. Others, like the writers of the Lost Generation, reveled in exposing the hypocrisies and shallowness of mainstream middle-class culture. Meanwhile, the passage of prohibition served to increase the illegal production of alcohol and led to a rise in organized crime.
REVIEW QUESTIONS
-
The popularization of which psychologist’s ideas encouraged the new morality of the 1920s?
- Sigmund Freud
- Alice Paul
- W. E. B. Du Bois
- Margaret Sanger
-
Which amendment did Alice Paul promote to end gender discrimination?
- Prohibition Amendment
- Equal Rights Amendment
- Sheppard-Towner Amendment
- Free Exercise Amendment
-
Which novel of the era satirized the conformity of the American middle class?
- This Side of Paradise
- The Sun Also Rises
- A Farewell to Arms
- Babbitt
- Why did the prohibition amendment fail after its adoption in 1919?
- What was the Harlem Renaissance, and who were some of the most famous participants?
- Answer
-
- a
- b
- d
- The prohibition amendment failed due to its infeasibility. It lacked both public support and funds for its enforcement. It also lessened Americans’ respect for law and order, and sparked a rise in unlawful activities, such as illegal alcohol production and organized crime.
- The Harlem Renaissance was a rediscovery and celebration of black culture and race pride. Within this context, black literature and art flourished. Writers such as Claude McKay, Langston Hughes, and Zora Neale Hurston created fiction and poetry that spoke directly to the experiences of black Americans. Meanwhile, black scholars and political leaders, such as W. E. B. Du Bois and Marcus Garvey, created new social and political ideologies and defined a distinct national identity for African Americans.
- bootlegging
- a nineteenth-century term for the illegal transport of alcoholic beverages that became popular during prohibition
- expatriate
- someone who lives outside of their home country
- flapper
- a young, modern woman who embraced the new morality and fashions of the Jazz Age
- Lost Generation
- a group of writers who came of age during World War I and expressed their disillusionment with the era
- Negro nationalism
- the notion that African Americans had a distinct and separate national heritage that should inspire pride and a sense of community
- new morality
- the more permissive mores adopted my many young people in the 1920s


4.2: Electric Field from Electric Potential
By the end of this section, you will be able to:
- Explain how to calculate the electric field in a system from the given potential
- Calculate the electric field in a given direction from a given potential
- Calculate the electric field throughout space from a given potential
Recall that we were able, in certain systems, to calculate the potential by integrating over the electric field. As you may already suspect, this means that we may calculate the electric field by taking derivatives of the potential, although going from a scalar to a vector quantity introduces some interesting wrinkles. We frequently need \(\vec{E}\) to calculate the force in a system; since it is often simpler to calculate the potential directly, there are systems in which it is useful to calculate \(V\) and then derive \(\vec{E}\) from it.
In general, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of \(\vec{E}\) and also in the direction of lower potential \(V\). Furthermore, the magnitude of \(\vec{E}\) equals the rate of decrease of \(V\) with distance. The faster \(V\) decreases over distance, the greater the electric field. This gives us the following result.
In equation form, the relationship between voltage and the \(s\)-component of a uniform electric field is
\[E_s = - \dfrac{\Delta V}{\Delta s}\]
where \(\Delta s\) is the distance over which the change in potential \(\Delta V\) takes place. The minus sign tells us that \(E_s\) points in the direction of decreasing potential. The electric field is said to be the gradient (as in grade or slope) of the electric potential.
For continually changing potentials, \(\Delta V\) and \(\Delta s\) become infinitesimals, and we need differential calculus to determine the electric field. As shown in Figure \(\PageIndex{1}\), if we treat the distance \(\Delta s\) as very small so that the electric field is essentially constant over it, we find that
\[E_s = - \dfrac{dV}{ds}.\]
Therefore, the electric field components in the Cartesian directions are given by
\[E_x = - \dfrac{\partial V}{\partial x}, \, E_y = - \dfrac{\partial V}{\partial y}, \, E_z = - \dfrac{\partial V}{\partial z}.\]
This allows us to define the “grad” or “del” vector operator, which allows us to compute the gradient in one step. In Cartesian coordinates, it takes the form
\[\vec{\nabla} = \hat{i} \dfrac{\partial}{\partial x} + \hat{j} \dfrac{\partial}{\partial y} + \hat{k} \dfrac{\partial}{\partial z}.\]
With this notation, we can calculate the electric field from the potential with
\[\vec{E} = - \vec{\nabla}V, \label{eq20}\]
a process we call calculating the gradient of the potential .
If we have a system with either cylindrical or spherical symmetry, we only need to use the del operator in the appropriate coordinates:
\[\begin{align} \vec{\nabla}_{cyl} &= \underbrace{\hat{r} \dfrac{\partial}{\partial r} + \hat{\varphi}\dfrac{1}{r} \dfrac{\partial}{\partial \varphi} + \hat{z} \dfrac{\partial}{\partial z}}_{\text{Cylindrical}} \label{cylindricalnabla} \\[4pt] \vec{\nabla}_{sph} &= \underbrace{ \hat{r} \dfrac{\partial}{\partial r} + \hat{\theta}\dfrac{1}{r} \dfrac{\partial}{\partial \theta} + \hat{\varphi} \dfrac{1}{r \, sin \, \theta}\dfrac{\partial}{\partial \varphi}}_{\text{Spherical}} \label{spherenabla} \end{align}\]
Calculate the electric field of a point charge from the potential.
Strategy
The potential is known to be \(V = k\dfrac{q}{r}\), which has a spherical symmetry. Therefore, we use the spherical del operator (Equation \ref{spherenabla}) into Equation \ref{eq20}:
\[\vec{E} = - \vec{\nabla}_{sph} V \nonumber.\]
Solution
Performing this calculation gives us
\[\begin{align*} \vec{E} &= - \left( \hat{r}\dfrac{\partial}{\partial r} + \hat{\theta}\dfrac{1}{r} \dfrac{\partial}{\partial \theta} + \hat{\varphi}\dfrac{1}{1 \, \sin \, \theta} \dfrac{\partial}{\partial \varphi}\right) k\dfrac{q}{r} \\[4pt] &= - k\left( \hat{r}\dfrac{\partial}{\partial r}\dfrac{1}{r} + \hat{\theta}\dfrac{1}{r} \dfrac{\partial}{\partial \theta}\dfrac{1}{r} + \hat{\varphi}\dfrac{1}{1 \, \sin \, \theta} \dfrac{\partial}{\partial \varphi}\dfrac{1}{r}\right).\end{align*}\]
This equation simplifies to
\[\vec{E} = - kq\left(\hat{r}\dfrac{-1}{r^2} + \hat{\theta}0 + \hat{\varphi}0 \right) = k\dfrac{q}{r^2}\hat{r} \nonumber\]
as expected.
Significance
We not only obtained the equation for the electric field of a point particle that we’ve seen before, we also have a demonstration that \(\vec{E}\) points in the direction of decreasing potential, as shown in Figure \(\PageIndex{2}\).
Use the potential found previously to calculate the electric field along the axis of a ring of charge (Figure \(\PageIndex{3}\)).
Strategy
In this case, we are only interested in one dimension, the z -axis. Therefore, we use
\[E_z = - \dfrac{\partial V}{\partial z} \nonumber\]
with the potential
\[V = k \dfrac{q_{tot}}{\sqrt{z^2 + R^2}} \nonumber\]
found previously.
Solution
Taking the derivative of the potential yields
\[ \begin{align*} E_z &= - \dfrac{\partial}{\partial z} \dfrac{kq_{tot}}{\sqrt{z^2 + R^2}} \\[4pt] &= k \dfrac{q_{tot}z}{(z^2 + R^2)^{3/2}}. \end{align*}\]
Significance
Again, this matches the equation for the electric field found previously. It also demonstrates a system in which using the full del operator is not necessary.
Which coordinate system would you use to calculate the electric field of a dipole?
- Answer
-
Any, but cylindrical is closest to the symmetry of a dipole.


2.5: Sequences
We have already met
- the sequence of natural numbers (1, 2, 3, 4, 5, ...),
- the sequence of squares (1, 4, 9, 16, 25, ...),
- the sequence of cubes (1, 8, 27, 64, 125, ...),
- the sequence of prime numbers (2, 3, 5, 7, 11, 13, 17, ...),
- the sequence of powers of 2 (1, 2, 4, 8, 16, 32, ...), and the sequence of powers of 4 (1, 4, 16, 64, 256, ...).
We have also considered
- the sequence of units digits of the powers of 4 (1, 4, 6, 4, 6, 4, 6, ...),
- the sequence of leading digits of the powers of 4 (1, 4, 1, 6, 2, 1, 4, ...).
2.5.1 Triangular numbers
Problem 54
(a) Evaluate the first twelve terms of the sequence of triangular numbers:
(b) Find and prove a formula for the n th triangular number
(c) Which triangular numbers are also (i) powers of 2? (ii) prime? (iii) squares? (iv) cubes?
2.5.2 Fibonacci numbers
The Hindu-Arabic numeral system emerged in the Middle East in the 10 th and 11 th centuries. Fibonacci, also known as Leonardo of Pisa, is generally credited with introducing this system to Europe around 1200 – especially through his book Liber Abaci (1202). One of the problems in that book introduced the sequence that now bears his name.
The sequence of Fibonacci numbers begins with the terms F 0 = 0, F 1 = 1, and continues via the Fibonacci recurrence relation:
The sequence was introduced through a curious problem about breeding rabbits; but to this day it continues to feature in many unexpected corners of mathematics and its applications.
Problem 55
(a) (i) Generate the first twelve terms of the Fibonacci sequence:
(ii) Use this to generate the first eleven terms of the sequence of “differences” between successive Fibonacci numbers. Then generate the first ten terms of the sequence of “differences between successive differences”.
(iii) Find an expression for the m th term of the k th sequence of differences.
(b) (i) Generate the first twelve terms of the sequence of powers of 2:
(ii) Use this to generate the first eleven terms of the sequence of “differences” between successive powers of 2. Then generate the first ten terms of the sequence of “differences between successive differences”.
(iii) Find an expression for the m th term of the k th sequence of differences.
The sequence of differences between successive terms in the sequence of triangular numbers is just the sequence of natural numbers (starting with 2):
and the sequence of “second differences” is then constant :
The sequences of powers of 2 and the Fibonacci numbers behave very differently from this, in that taking differences reproduces something very like the initial sequence. In particular, taking differences can never lead to a constant sequence.
Logically the next four problems should wait until Chapter 6 , where we address the delicate matter of “proof by mathematical induction”. However, that would deprive us of the chance to sample the kind of surprises that lie just beneath the surface of the Fibonacci sequence, and to experience the process of fumbling our way towards a structural understanding of the apparent patterns that emerge. Of course, each time we think we have managed to guess what seems to be true, we face the challenge of proof. Those who have not yet mastered “proof by induction” are encouraged to get what they can from the solutions, and to view this as an informal introduction to ideas that will be squarely addressed in Chapter 6 .
Problem 56
(a) (i) Generate the sequence of partial sums of the sequence of powers of 2:
(ii) Prove that each partial sum is 1 less than the next power of 2.
(b) (i) Generate the sequence of partial sums of the Fibonacci sequence:
(ii) Prove that each partial sum is 1 less than the next but one Fibonacci number.
Problem 56 (b) starts out with the observation that
which is a consequence of the first two instances of the fundamental recurrence relation
and derives a surprising value for the n th partial sum:
Fibonacci numbers make their mathematical presence felt in a quiet way – partly through the almost spooky range of unexpected internal relations which they satisfy, as illustrated in Problem 56 (b) and in the next few problems.
Problem 57
(a) Note that
(i) Evaluate the succession of terms:
(ii) Guess a simpler expression for the product . Prove your guess is correct.
(b) Let .
(i) Show that the parallelogram OABC spanned by the origin O , and the points and their sum has area .
(ii) Find the area of the first parallelogram in the sequence of “Fibonacci parallelograms”, spanned by the origin O , and the points , .
(iii) Show that the n th parallelogram OACB in this sequence, spanned by the origin O , and the points and , and the parallelogram OBDC spanned by the origin O , and the points and overlap in the triangle OBC , which is exactly half of each parallelogram.
Conclude that every such parallelogram has area 1. Relate this to the conclusion of (a)(ii).
The basic recurrence relation for Fibonacci numbers specifies the next term as the sum of two successive terms. We now consider what this implies about the sum of the squares of two successive terms.
Problem 58
(a) Evaluate the first few terms of the sequence
(b) Guess a simpler expression for the sum . Prove your guess is correct.
Problem 59
(a) Note that
(i) Evaluate the succession of terms:
(ii) Guess a simpler expression for the product . Prove your guess is correct.
(b) (i) Evaluate the succession of terms:
(ii) Guess a simpler expression for the product . Prove your guess is correct.


13.2: Thermal Expansion of Solids and Liquids
Learning Objectives
By the end of this section, you will be able to:
- Define and describe thermal expansion.
- Calculate the linear expansion of an object given its initial length, change in temperature, and coefficient of linear expansion.
- Calculate the volume expansion of an object given its initial volume, change in temperature, and coefficient of volume expansion.
- Calculate thermal stress on an object given its original volume, temperature change, volume change, and bulk modulus.
The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion , the change in size or volume of a given mass with temperature. Hot air rises because its volume increases, which causes the hot air’s density to be smaller than the density of surrounding air, causing a buoyant (upward) force on the hot air. The same happens in all liquids and gases, driving natural heat transfer upwards in homes, oceans, and weather systems. Solids also undergo thermal expansion. Railroad tracks and bridges, for example, have expansion joints to allow them to freely expand and contract with temperature changes.
What are the basic properties of thermal expansion? First, thermal expansion is clearly related to temperature change. The greater the temperature change, the more a bimetallic strip will bend. Second, it depends on the material. In a thermometer, for example, the expansion of alcohol is much greater than the expansion of the glass containing it.
What is the underlying cause of thermal expansion? As is discussed in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature , an increase in temperature implies an increase in the kinetic energy of the individual atoms. In a solid, unlike in a gas, the atoms or molecules are closely packed together, but their kinetic energy (in the form of small, rapid vibrations) pushes neighboring atoms or molecules apart from each other. This neighbor-to-neighbor pushing results in a slightly greater distance, on average, between neighbors, and adds up to a larger size for the whole body. For most substances under ordinary conditions, there is no preferred direction, and an increase in temperature will increase the solid’s size by a certain fraction in each dimension.
Linear Thermal Expansion—Thermal Expansion in One Dimension
The change in length \(\Delta L\) is proportional to length \(L\). The dependence of thermal expansion on temperature, substance, and length is summarized in the equation
\[\Delta L = \alpha L \Delta T, \label{linear}\]
where \(\Delta L\) is the change in length \(L\), \(\Delta T\) is the change in temperature, and \(\alpha\) is the coefficient of linear expansion , which varies slightly with temperature.
Table \(\PageIndex{1}\) lists representative values of the coefficient of linear expansion, which may have units of \(1/^oC\) or 1/K. Because the size of a kelvin and a degree Celsius are the same, both \(\alpha\) and \(\Delta T\) can be expressed in units of kelvins or degrees Celsius. The equation \(\Delta L = \alpha L \Delta T\) is accurate for small changes in temperature and can be used for large changes in temperature if an average value of \(\alpha\) is used.
| Material | Coefficient of linear expansion (\(\alpha (1/^oC)\)) | Coefficient of volume expansion (\(\beta (1/^oC)\)) |
|---|---|---|
| Solids | ||
| Aluminum | \(25 \times 10^{-6}\) | \(75 \times 10^{-6}\) |
| Brass | \(19 \times 10^{-6}\) | \(56 \times 10^{-6}\) |
| Copper | \(17 \times 10^{-6}\) | \(51 \times 10^{-6}\) |
| Gold | \(14 \times 10^{-6}\) | \(42 \times 10^{-6}\) |
| Iron or Steel | \(12 \times 10^{-6}\) | \(35 \times 10^{-6}\) |
| Invar (Nickel-Iron alloy) | \(0.9 \times 10^{-6}\) | \(2.7 \times 10^{-6}\) |
| Lead | \(29 \times 10^{-6}\) | \(87 \times 10^{-6}\) |
| Silver | \(18 \times 10^{-6}\) | \(54 \times 10^{-6}\) |
| Glass (ordinary) | \(9 \times 10^{-6}\) | \(27 \times 10^{-6}\) |
| Glass (Pyrex) | \(3 \times 10^{-6}\) | \(9 \times 10^{-6}\) |
| Quartz | \(0.4 \times 10^{-6}\) | \(1 \times 10^{-6}\) |
| Concrete, Brick | \(\approx 12 \times 10^{-6}\) | \(\approx 36 \times 10^{-6}\) |
| Marble (average) | \(7 \times 10^{-6}\) | |
| Liquids | ||
| Ether | N/A | \(1650 \times 10^{-6}\) |
| Ethyl alcohol | N/A | \(1100 \times 10^{-6}\) |
| Petrol | N/A | \(950 \times 10^{-6}\) |
| Glycerin | N/A | \(500 \times 10^{-6}\) |
| Mercury | N/A | \(180 \times 10^{-6}\) |
| Water | N/A | \(210 \times 10^{-6}\) |
| Gases | ||
| Air and most other gases at atm | N/A | \(3400 \times 10^{-6}\) |
Example \(\PageIndex{1}\): Calculating Linear Thermal Expansion of The Golden Gate Bridge
The main span of San Francisco’s Golden Gate Bridge is 1275 m long at its coldest. The bridge is exposed to temperatures ranging from \(-15^oC\) to \(40^oC\). What is its change in length between these temperatures? Assume that the bridge is made entirely of steel.
Strategy
Use the equation for linear thermal expansion (Equation \ref{linear}) to calculate the change in length, \(\Delta L\). Use the coefficient of linear expansion, \(\alpha\), for steel from Table \(\PageIndex{1}\), and note that the change in temperature, \(\Delta T\), is \(55^oC\).
Solution
Plug all of the known values into the equation to solve for \(\Delta L\).
\[ \begin{align*} \Delta L &= \alpha L \Delta T \\[5pt] &= \left(\dfrac{12 \times 10^{-6}}{^oC}\right) (1275 \, m)(55^oC) \\[5pt] &= 0.84 \, m \end{align*} \]
Discussion
Although not large compared with the length of the bridge, this change in length is observable. It is generally spread over many expansion joints so that the expansion at each joint is small.
Thermal Expansion in Two and Three Dimensions
Objects expand in all dimensions, as illustrated in Figure \(\PageIndex{2}\). That is, their areas and volumes, as well as their lengths, increase with temperature. Holes also get larger with temperature. If you cut a hole in a metal plate, the remaining material will expand exactly as it would if the plug was still in place. The plug would get bigger, and so the hole must get bigger too. (Think of the ring of neighboring atoms or molecules on the wall of the hole as pushing each other farther apart as temperature increases. Obviously, the ring of neighbors must get slightly larger, so the hole gets slightly larger).
Thermal Expansion in Two Dimensions
For small temperature changes, the change in area \(\Delta A\) is given by
\[\Delta A = 2 \alpha A \Delta T,\]
where \(\Delta A\) is the change in area \(A\), \(\Delta T\) is the change in temperature, and \(\alpha\) is the coefficient of linear expansion, which varies slightly with temperature.
Thermal Expansion in Three Dimensions
The change in volume \(\Delta V\) is very nearly \(\Delta V = 3\alpha V \Delta T\). This equation is usually written as
\[\Delta V = \beta V \Delta T, \label{volume}\]
where \(\beta\) is the coefficient of volume expansion and \(\beta \approx 3\alpha\). Note that the values of \(\beta\) in Table \(\PageIndex{1}\) are almost exactly equal to \(3\alpha\).
In general, objects will expand with increasing temperature. Water is the most important exception to this rule. Water expands with increasing temperature (its density decreases ) when it is at temperatures greater than \(4^oC(40^oF)\). However, it expands with decreasing temperature when it is between \(+4^oC\) and \(0^oC(40^oF\) to \(32^oF)\). Water is densest at \(+4^oC\) (Figure \(\PageIndex{1}\)). Perhaps the most striking effect of this phenomenon is the freezing of water in a pond. When water near the surface cools down to \(4^oC\) it is denser than the remaining water and thus will sink to the bottom. This “turnover” results in a layer of warmer water near the surface, which is then cooled. Eventually the pond has a uniform temperature of \(4^oC\). If the temperature in the surface layer drops below \(4^oC\), the water is less dense than the water below, and thus stays near the top. As a result, the pond surface can completely freeze over. The ice on top of liquid water provides an insulating layer from winter’s harsh exterior air temperatures. Fish and other aquatic life can survive in \(4^oC\) water beneath ice, due to this unusual characteristic of water. It also produces circulation of water in the pond that is necessary for a healthy ecosystem of the body of water.
Real-World Connections - Filling the Tank
Differences in the thermal expansion of materials can lead to interesting effects at the gas station. One example is the dripping of gasoline from a freshly filled tank on a hot day. Gasoline starts out at the temperature of the ground under the gas station, which is cooler than the air temperature above. The gasoline cools the steel tank when it is filled. Both gasoline and steel tank expand as they warm to air temperature, but gasoline expands much more than steel, and so it may overflow.
This difference in expansion can also cause problems when interpreting the gasoline gauge. The actual amount (mass) of gasoline left in the tank when the gauge hits “empty” is a lot less in the summer than in the winter. The gasoline has the same volume as it does in the winter when the “add fuel” light goes on, but because the gasoline has expanded, there is less mass. If you are used to getting another 40 miles on “empty” in the winter, beware—you will probably run out much more quickly in the summer.
Example \(\PageIndex{2}\): Calculating Thermal Expansion: Gas vs. Gas Tank
Suppose your 60.0-L (15.9-gal) steel gasoline tank is full of gas, so both the tank and the gasoline have a temperature of \(15^oC\). How much gasoline has spilled by the time they warm to \(35.0^oC\)?
Strategy
The tank and gasoline increase in volume, but the gasoline increases more, so the amount spilled is the difference in their volume changes. (The gasoline tank can be treated as solid steel.) We can use the equation for volume expansion to calculate the change in volume of the gasoline and of the tank.
Solution
1. Use the equation for volume expansion (Equation \ref{volume}) to calculate the increase in volume of the steel tank:
\[\Delta V_s = \beta_s V_s \Delta T. \nonumber\]
2. The increase in volume of the gasoline is given by this equation:
\[\Delta V_{gas} = \beta_{gas} V_{gas} \Delta T. \nonumber\]
3. Find the difference in volume to determine the amount spilled as
\[V_{spill} = \Delta V_{gas} - \Delta V_S. \nonumber\]
Alternatively, we can combine these three equations into a single equation. (Note that the original volumes are equal.)
\[ \begin{align*} V_{spill} &= (\beta_{gas} - \beta_s)V \Delta T \\[5pt] &= [(950 - 35) \times 10^{-6} /^oC] (60.0 \, L)(20.0^oC) \\[5pt] &= 1.10 \, L. \end{align*} \]
Discussion
This amount is significant, particularly for a 60.0-L tank. The effect is so striking because the gasoline and steel expand quickly. The rate of change in thermal properties is discussed in Heat and Heat Transfer Methods .
If you try to cap the tank tightly to prevent overflow, you will find that it leaks anyway, either around the cap or by bursting the tank. Tightly constricting the expanding gas is equivalent to compressing it, and both liquids and solids resist being compressed with extremely large forces. To avoid rupturing rigid containers, these containers have air gaps, which allow them to expand and contract without stressing them.
Thermal Stress
Thermal stress is created by thermal expansion or contraction (see Elasticity: Stress and Strain for a discussion of stress and strain). Thermal stress can be destructive, such as when expanding gasoline ruptures a tank. It can also be useful, for example, when two parts are joined together by heating one in manufacturing, then slipping it over the other and allowing the combination to cool. Thermal stress can explain many phenomena, such as the weathering of rocks and pavement by the expansion of ice when it freezes.
Example \(\PageIndex{3}\): Calculating Thermal Stress - Gas Pressure
What pressure would be created in the gasoline tank considered in Example \(\PageIndex{2}\), if the gasoline increases in temperature from \(15.0^oC\) to \(35.0^oC\) without being allowed to expand? Assume that the bulk modulus \(B\) for gasoline is \(1.00 \times 10^9 \, N/m^2\). (For more on bulk modulus, see Elasticity: Stress and Strain .)
Strategy
To solve this problem, we must use the following equation, which relates a change in volume \(\Delta V\) to pressure:
\[\Delta V = \dfrac{1}{B}\dfrac{F}{A}V_0, \nonumber \]
where \(F/A\) is pressure, \(V_0\) is the original volume, and \(B\) is the bulk modulus of the material involved. We will use the amount spilled in Example \(\PageIndex{2}\) as the change in volume, \(\Delta V\).
Solution
1. Rearrange the equation for calculating pressure: \[P = \dfrac{F}{A} = \dfrac{\Delta V}{V_0}B. \nonumber\]
2. Insert the known values. The bulk modulus for gasoline is \(B = 1.00 \times 10^9 \, N/m^2\). In the previous example, the change in volume \(\Delta V = 1.10 \, L\) is the amount that would spill. Here, \(V_0 = 60.0 \, L\) is the original volume of the gasoline. Substituting these values into the equation, we obtain
\[P = \dfrac{1.10 \, L}{60.0 \, L}(1.00 \times 10^9 Pa) = 1.83 \times 10^7 \, Pa. \nonumber\]
Discussion
This pressure is about \(2500 \, lb/in^2\), much more than a gasoline tank can handle.
Forces and pressures created by thermal stress are typically as great as that in the example above. Railroad tracks and roadways can buckle on hot days if they lack sufficient expansion joints. (Figure \(\PageIndex{1}\)). Power lines sag more in the summer than in the winter, and will snap in cold weather if there is insufficient slack. Cracks open and close in plaster walls as a house warms and cools. Glass cooking pans will crack if cooled rapidly or unevenly, because of differential contraction and the stresses it creates. (Pyrex is less susceptible because of its small coefficient of thermal expansion.) Nuclear reactor pressure vessels are threatened by overly rapid cooling, and although none have failed, several have been cooled faster than considered desirable. Biological cells are ruptured when foods are frozen, detracting from their taste. Repeated thawing and freezing accentuate the damage. Even the oceans can be affected. A significant portion of the rise in sea level that is resulting from global warming is due to the thermal expansion of sea water.
Metal is regularly used in the human body for hip and knee implants. Most implants need to be replaced over time because, among other things, metal does not bond with bone. Researchers are trying to find better metal coatings that would allow metal-to-bone bonding. One challenge is to find a coating that has an expansion coefficient similar to that of metal. If the expansion coefficients are too different, the thermal stresses during the manufacturing process lead to cracks at the coating-metal interface.
Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion, and are closer to those of teeth.
Exercise \(\PageIndex{1}\)
Two blocks, A and B, are made of the same material. Block A has dimensions \(l \times w \times h = L \times 2L \times L\) and Block B has dimensions \(2L \times 2L \times 2L\). If the temperature changes, what is
- the change in the volume of the two blocks,
- the change in the cross-sectional area \(l \times w\) and
- the change in the height \(h\) of the two blocks?
- Answer a
-
The change in volume is proportional to the original volume. Block A has a volume of \(L \times 2L \times L = 2L^3\). Block B has a volume of \(2L \times 2L \times 2L = 8L^3\),
which is 4 times that of Block A. Thus the change in volume of Block B should be 4 times the change in volume of Block A.
- Answer b
-
The change in area is proportional to the area. The cross-sectional area of Block A is \(L \times 2L = 2L^2\), while that of Block B is \(2L \times 2L = 4L^2\).
Because cross-sectional area of Block B is twice that of Block A, the change in the cross-sectional area of Block B is twice that of Block A.
- Answer c
-
The change in height is proportional to the original height. Because the original height of Block B is twice that of A, the change in the height of Block B is twice that of Block A.
Summary
- Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
- Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
- Linear thermal expansion is \[\Delta L = \alpha L \Delta T, \nonumber\] where \(\Delta L \) is the change in length \(L\), \(\Delta T\) is the change in temperature, and \(\alpha\) is the coefficient of linear expansion, which varies slightly with temperature.
- The change in area due to thermal expansion is \[\Delta A = 2\alpha A \delta T, \nonumber\] where \(\Delta A\) is the change in area.
- The change in volume due to thermal expansion is \[\Delta V = \beta V \delta T, \nonumber\] where \(\beta\) is the coefficient of volume expansion and \(\beta \approx 3\alpha\). Thermal stress is created when thermal expansion is constrained.
Glossary
- thermal expansion
- the change in size or volume of an object with change in temperature
- coefficient of linear expansion
- the change in length, per unit length, per 1 degree Celsius, change in temperature; a constant used in the calculation of linear expansion; the coefficient of linear expansion depends on the material and to some degree on the temperature of the material
- coefficient of volume expansion
- the change in volume, per unit volume, per 1 degree Celsius,change in temperature
- thermal stress
- stress caused by thermal expansion or contraction


2.1: Introduction
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Fieldwork is one of the most important practices of anthropology. While all of the subfields of anthropology conduct fieldwork in some form to gather information, each subfield may use different methods of conducting research. The concept of working in “the field” was traditionally based on the practice of traveling to distant regions to study other cultures within their native environmental contexts. In recent decades, “the field” has broadened to include diverse settings such as one’s hometown (as in urban anthropology), the Internet (visual or virtual anthropology), or collections in university archives and museums (ethnohistory or museum anthropology).


16.6: An Explanation of Substituent Effects
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Objectives
After completing this section, you should be able to
- draw the resonance contributors for the carbocation intermediate formed during the reaction of a given monosubstituted benzene derivative with any of the electrophiles discussed in this chapter.
- classify each of the substituents listed in Objective 2 of Section 16.4 as being either meta or ortho/para directing.
- classify each of the substituents listed in Objective 2 of Section 16.4 as being ortho/para directing activators, ortho/para directing deactivators, or meta directing deactivators.
- predict the product or products formed from the reaction of a given monosubstituted benzene derivative with each of the electrophiles discussed in this chapter.
- explain, by drawing the resonance contributors for the intermediate carbocation, why the electrophilic substitution of an alkyl benzene results in a mixture of mainly ortho- and para- substituted products.
- explain why the electrophilic substitution of phenols, amines and their derivatives proceeds more rapidly than the electrophilic substitution of benzene itself.
- explain, by drawing the resonance contributors for the intermediate carbocation, why meta substitution predominates in electrophilic aromatic substitution reactions carried out on benzene derivatives containing one of the substituents R 3 N + , NO 2 , CO 2 H, CN, CO 2 R, COR or CHO.
- explain why electrophilic aromatic substitution of benzene derivatives containing one of the substituents listed in Objective 7, above, proceeds more slowly than the electrophilic substitution of benzene itself.
- explain, by drawing the resonance contributors for the intermediate carbocation, why the electrophilic aromatic substitution of halobenzenes produces a mixture of mainly ortho- and para-substituted products.
- explain why the electrophilic aromatic substitution of halobenzenes proceeds more slowly than does the electrophilic substitution of benzene itself.
- use the principles developed in this chapter to predict in which of the three categories listed in Objective 3, above, a previously unencountered substituent should be placed.
Key Terms
Make certain that you can define, and use in context, the key terms below.
- steric effect
- steric hindrance
Study Notes
As you saw in Section 16.4, a substituent on a benzene ring can be an activator or a deactivator. At the same time, a substituent can also be a meta director or an ortho/para director. Of the four possible combinations, only three are known—there are no meta directing activators.
If you look at the data for the nitration of toluene, you will see that the yield of o ‑nitrotoluene is 63% and that of p ‑nitrotoluene is 34%. Statistically, we should expect to obtain twice as much ortho product as para product, because the former is produced by attack at either of two carbon atoms whereas the latter is produced by attack at only one carbon atom (see Figure below).
In this instance, the observed ortho/para ratio is almost 2:1, as we might expect. However, if we study the ortho/para ratio found in the nitration of a number of other arenes, we see that this is not always the case. Note that the data for the nitration of toluene given in the table below differ from those presented elsewhere. The variation may result from a difference in temperature, reaction conditions or reagent, and emphasizes the point that it is the trends which are important, not the numbers themselves.
| Substrate | % ortho | % para | ortho/para ratio |
|---|---|---|---|
| toluene | 58 | 37 | 1.57:1 |
| ethylbenzene | 45 | 49 | 0.92:1 |
| isopropylbenzene | 30 | 62 | 0.48:1 |
| tert-butylbenzene | 16 | 73 | 0.22:1 |
[ Source: These data were taken from the audiocassette Some Organic Reaction Pathways , by Peter Sykes. London: Educational Techniques Subject Group, The Chemical Society, 1975.]
Table \(\PageIndex{1}\) : Nitration of arenes
The table above shows us that as the size of the alkyl substituent already present in the ring increases, attack at the ortho position becomes more difficult, and the percentage of ortho isomers in the mixture of products decreases. This is an example of a steric effect —an effect caused by the size of the substituent—and we would say that as the size of the alkyl group increases, attack at the ortho position becomes less favorable as a result of steric hindrance . Note that the size of the electrophile can also be a factor in determining the ortho/para ratio: the larger the electrophile, the less able it is to attack at the ortho position, particularly if the substituent already present in the ring is itself quite bulky.
When drawing the resonance contributors to the carbocation formed during an electrophilic aromatic substitution, bear in mind that those of the type
are particularly important, because in such structures each atom possesses a complete octet of electrons.
Note that, as do the hydroxyl and amino groups, the halogens have an inductive electron-withdrawing effect and a resonance electron-releasing effect on a benzene ring. The difference in behavior during electrophilic substitutions arises because, with the hydroxyl and amino groups, the resonance effect is much greater than the inductive effect, whereas with the halogens, there is a much finer balance. In the case of the latter, the inductive effect reduces the overall reactivity, but the resonance effect means that this reduction is felt less at the ortho and para positions than at the meta position.
Substituted rings are divided into two groups based on the type of the substituent that the ring carries:
- Activated rings : the substituents on the ring are groups that donate electrons.
- Deactivated rings : the substituents on the ring are groups that withdraw electrons.
Introduction
Examples of activating groups in the relative order from the most activating group to the least activating:
-NH 2 , -NR 2 > -OH, -OR> -NHCOR> -CH 3 and other alkyl groups
with R as alkyl groups (C n H 2n +1 )
Examples of deactivating groups in the relative order from the most deactivating to the least deactivating:
-NO 2 , -CF 3 > -COR, -CN, -CO 2 R, -SO 3 H > Halogens
with R as alkyl groups (C n H 2n +1 )
The order of reactivity among Halogens from the more reactive (least deactivating substituent) to the least reactive (most deactivating substituent) halogen is:
F> Cl > Br > I
The order of reactivity of the benzene rings toward the electrophilic substitution when it is substituted with a halogen groups, follows the order of electronegativity. The ring that is substituted with the most electronegative halogen is the most reactive ring (less deactivating substituent) and the ring that is substituted with the least electronegative halogen is the least reactive ring (more deactivating substituent), when we compare rings with halogen substituents. Also the size of the halogen effects the reactivity of the benzene ring that the halogen is attached to. As the size of the halogen increase, the reactivity of the ring decreases.
The direction of the reaction
The activating group directs the reaction to the ortho or para position, which means the electrophile substitutes for the hydrogen that is on carbon 2 or carbon 4. The deactivating group directs the reaction to the meta position, which means the electrophile substitutes for the hydrogen that is on carbon 3 with the exception of the halogens which are deactivating groups but direct the ortho or para substitution.
Substituents determine the reaction direction by resonance or inductive effect
Resonance effect is the conjugation between the ring and the substituent, which means the delocalizing of the \(\pi\) electrons between the ring and the substituent. Inductive effect is the withdraw of the sigma ( the single bond ) electrons away from the ring toward the substituent, due to the higher electronegativity of the substituent compared to the carbon of the ring.
Activating groups (ortho or para directors)
When substituents such as -OH have an unshared pair of electrons, the resonance effect is stronger than the inductive effect which make these substituents stronger activators, since this resonance effect direct the electron toward the ring. In cases where the subtituents is esters or amides, they are less activating because they form resonance structure that pull the electron density away from the ring.
By looking at the mechanism above, we can see how electron donating groups direct electrophilic substitution to the ortho and para positions. Since the extra electron density is localized on the ortho and para carbons, these carbons are more likely to react with the electrophile.
Inductive effects of alkyl groups activate the direction of the ortho or para substitution, which is when s electrons gets pushed toward the ring.
Deactivating group (meta directors)
The deactivating groups deactivate the ring by the inductive effect in the presence of an electronegative atom that withdraws electron density away from the ring.
The mechanism above shows that when electron density is withdrawn from the ring, that leaves the carbons at the ortho, para positions with a parital positive charge which is unfavorable for the electrophile, so the electrophile attacks the carbon at the meta positions.
Halogens are an exception of the deactivating group that directs to the ortho or para substitution. The halogens deactivate the ring by inductive effect not by the resonance even though they have an unpaired pair of electrons. The unpaired pair of electrons gets donated to the ring, but the inductive effect pulls away the s electrons from the ring by the electronegativity of the halogens.
Substituents determine the reactivity of rings
The reaction of a substituted ring with an activating group is faster than the same reaction wtih benzene. On the other hand, a substituted ring with a deactivated group reacts slower than benzene.
Activating groups speed up reaction with electrophiles due to increased electron density on the ring. This stabilizes the intermediate carbocation, which decreases the activation energy for the reaction. On the other hand, deactivating groups withdraw electron density away from the carbocation formed in the intermediate step, increasing the activation energy, which slows down the reaction.
The CH 3 Group is an ortho, para director
Alkyl groups are inductively donating, therefore are activators. This resulsts in o/p attack to form a tertiary arenium carbocation which speeds up the reaction.
The O-CH 3 Group is an ortho, para director
The methoxy group is an example of groups that are ortho, para directors by having and oxygen or nitrogen adjacent to the aromatic ring. This same activation is present with alcohols, amines, esters and amides (with the oxygen or nitrogen attached to the ring, not the carbonyl).
Groups with an oxygen or nitrogen attached to the aromatic ring are ortho and para directors since the O or N can push electrons into the ring, making the ortho and para positions more reactive and stabilizing the arenium ion that forms. This causes the ortho and para products to form faster than meta. Generally, the para product is preferred because of steric effects.
Acyl groups are meta directors
Ketones are an example of groups that deactivate an aromatic ring through resonance. Similar deactivation also occurs with ammonium ions, nitro groups, aldehydes, nitriles, sulfonic acids, and groups with a carbonyl attached to the ring (amides, esters, carboxylic acids, and anhydrides).
Acyl groups are resonance deactivators. Ortho and para attack produces a resonance structure which places the arenium cation next to an additional cation. This destabilizes the arenium cation and slows down ortho and para reaction. By default the meta product forms faster because it lacks this destabilizing resonance structure.
Halogens
Halogens are an interesting hybrid case. They are ortho, para directors, but deactivators. Overall, they remove electron density from the ring, making it less reactive. However, due to their resonance donation to the ring, if it does react, it reacts primarily at ortho and para positions.
References
- Schore, N.E. and P.C. Vollhardt. 2007. Organic Chemistry, structure and function, 5th ed. New York,NY: W.H. Freeman and Company.
- Fryhle, C.B. and G. Solomons. 2008. Organic Chemistry, 9th ed.Danvers,MA: Wiley.
Exercise \(\PageIndex{1}\)
Predict the pattern of the electrophilic substitution on these rings:
- Answer
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The first substitution is going to be ortho and/or para substitution since we have a halogen subtituent. The second substitution is going to be ortho and/or para substitution also since we have an alkyl substituent.
Exercise \(\PageIndex{2}\)
Which nitration product is going to form faster: nitration of aniline or nitration of nitrobenzene?
- Answer
-
The nitration of aniline is going to be faster than the nitration of nitrobenzene, since the aniline is a ring with NH 2 substituent and nitrobenzene is a ring with NO 2 substituent. As described above NH 2 is an activating group which speeds up the reaction and NO 2 is a deactivating group that slows down the reaction.
Exercise \(\PageIndex{3}\)
Predict the product of the following sulfonation reaction:
- Answer
-
Exercise \(\PageIndex{4}\)
Classify these two groups as activating or deactivating groups:
A. alcohol
B. ester
- Answer
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A. alcohol is an activating group.
B. Esters can be either. If the oxygen atom is next to the ring, esters are activating. However if the carbonyl is next to the ring, the ester is a deactivating group.
Exercise \(\PageIndex{5}\)
Does a chloride substituent activate or deactivate an aromatic ring?
- Answer
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Chloride deactivate an aromatic ring due to the inductive effect.
Exercise \(\PageIndex{6}\)
(Trichloromethyl)benzene has a strong concentration of electrons at the methyl substituent. Comparing this toluene, which is more reactive toward electrophilic substitution?
- Answer
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The trichloromethyl group is an electron donor into the benzene ring, therefore making it more stable and therefore more reactive compared to electrophilic substitution.
Exercise \(\PageIndex{7}\)
The following compound is less reactive towards electrophilic substitution than aniline? Explain.
- Answer
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As seen in resonance the electron density is also localized off of the ring, thereby deactivating it compared to aniline.
Exercise \(\PageIndex{8}\)
Consider the intermediates of the following molecule during an electrophilic substitution. Draw resonance structures for ortho, meta, and para attacks.
- Answer
-


2.11: The More Writing Process, the Better
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- Cheryl E. Ball & Drew M. Loewe ed.
- West Virginia University via Digital Publishing Institute and West Virginia University Libraries
Author: Jimmy Butts @jimmy_butts, theyellowrobot.com.
“If you spend too much time thinking about a thing, you’ll never get it done.”—Bruce Lee
Everyone has a writing process. And writing, like anything, takes time. But for quite a while—much too long—writers have increasingly embraced a specific and drawn out model for the writing process, which generally refers to five sure-fire steps: prewriting, drafting, editing, revising, and eventually—if one can muster the endurance to get through each of those earlier stages—publishing. As we have increasingly valued this writing process, we have moved further away from valuing writing itself as a wonderful, finished thing that humans can produce, that is, actually get done.
In other words, we have so fetishized the creative process that we’ve forgotten what we can actually create : words strung out into beautiful constellations. The obsession with the writing process rather than seeing writing as a completed product is an increasingly problematic psychological perspective as we increase the speed of production in the 21st century. Beleaguered revision has become the norm. When any view becomes that dominant, it is important to consider revising it. I am worried that we are taking too much time to write. And time is our most important non-renewable resource.
Meanwhile, well-meaning teachers all over still have colorful little posters up in their rooms with those classic five steps borrowed from some fairly dated views on what writing involves. We often do the various steps of the writing process while we write, of course—just not necessarily in the prescriptive order outlined by this slightly archaic structure. Writing is more complex than a five-step program. These steps come from a time when people typed on charming things called typewriters. Hannah Sullivan blames the introduction of the typewriter along with Modernism’s aim to present an ideal text for our current obsession with punctilious editing in her book The Work of Revision . There, Sullivan warns that “revision can go too far.” The technological shift in what writing involves plays a big part in how we view writing—as process or product. For many, the writing process involves checking social media accounts for something like three days. In an age of mass distraction, it is perhaps increasingly important to see our writing finished, or it will increasingly become our collective pipedream—a thing that we desire to do but never realize. (Indeed, in much online writing, posts end up being fluid iterations, but readable published iterations nonetheless.)
Perhaps we should drop writing as a verb and see it more and more as a noun—the thing that writing is. Writing is no longer merely a thing we do hunched over for hours, but a thing we make, and increasingly quickly. This view, then, attempts to pull the rug out from under process stances from the mid-20th century to the present—embracing instead what might be called a hyperproduct perspective on writing for the fastest century yet.
The process theory of composition arose with a handful of well-meaning thinkers in the 1960s and 70s. There is a bit of history here about how we got ourselves into an idolatry of a slow process theory of writing. Before the middle of the 20th century, the general theories about writing in schools encouraged finishing products—things like writing themes for English classes. This view is now generally known as current-traditional rhetoric. It was the kind of schooling that we might think of when we think of rulers rapping student’s knuckles to finish their work. An alternate model that valued expressivism of creative and unruly thought and the process of writing—and not the product—grew fairly common as a backlash.
Donald Murray and Peter Elbow in particular—those good, friendly, inspiring kinds of writing teachers—advocated seeing “writing as a process, not a product.” Many embraced a kind of slower process in the teaching of writing that resisted the kind of production-line expectation of written work that can sometimes arise along with a lot of anxiety. All of this was good. It was a valuing of the human as a writer, but it began—I suspect—to devalue the written work. As such, an era embracing endless drafting, tinkering, and reworking ad infinitum began.
The Council of Writing Program Administrators, an organization that thinks about how writing works especially on college campuses, claims that one of the primary outcomes in learning to become a writer is to understand writing as a process. The council states this goal in this way: “Writers use multiple strategies, or composing processes , to conceptualize, develop, and finalize projects.” To their credit, the objective is stated in a loose way to accept multiple approaches to writing and flexibility and recursivity in the processes that we develop as writers.
We love tinkering. We love delaying that finished, polished draft. But we must complete work. Otherwise, I think we relegate that work—the work of writing—to something less valuable. We blush at the fact that we scramble to complete drafts up to the moment before they’re due. Here, now, we might embrace that kind of breakneck pace—the work finished in the wee hours of the morning. Meanwhile, process has been critiqued since at least the early 1990s through a set of frameworks called post-process theory. Still, process haunts us. Most post-process critiques of the process formula suggest that a process view is invalid by merely embracing more complex, unknowable processes.
Of course, the idea of following a formula to write a perfect draft is a false construction. We write for specific situations, each unique. A certain set of cognitive steps are involved in writing anything—from academic papers to tweets; however, the set of steps used to compose one thing isn’t necessarily a learnable and reproducible set of steps. We cannot follow a writing process, because writing is messier than that. Instead, then, revision and process become excuses on the part of writers who have taken a bit too long to finish writing projects, never ending or completing compositional tasks. Holding to dying process-based views, endlessly reworking drafts never to completion, does a disservice to writers who are seeking to make good, new things fast.
We don’t often meticulously revise text messages. The written code of iOS 11 revises iOS 10. Each iteration becomes a potential improvement, but always as a valuable—and evaluable—product. In economic perspectives, revision is worthless unless it creates iterations of value. How we view process essentially depends on how much time we spend creating a thing. Is there a process to composing a tweet? Sure. But is there much editing, revision, recursive work? Not anymore.
It is finally time to explore an alternative to our obsession over the writing process in favor of a return to a healthy appreciation of finished drafts—of writing itself. Our culture really does value finished work. The exploratory work done in prewriting is harder to appreciate. And while a concern with overvaluing product is understandable, we should reclaim our writing—those satisfying finished pieces that do what we intend. If we don’t value writing, then we can, of course, relegate it to something we endlessly defer by thinking about it. Making writing achievable and real is the goal.
The outcome of a written product is quite important. While how writing gets done (process) may not be able to be demystified, we might be better able to see writing as a thing we make (product). The process of writing is insignificant, unimportant, and immaterial when the product of writing is really good. The thing is what matters. And more than that, regularly finishing written work makes one a writer. Writers must find their own ways of completing all kinds of writing in their normal, everyday lives. But completed writing is really what we’re after—not worthless drafts. Those drafts can be bad, and shaky, and loathsome. (They sometimes are!) But they must be finished.
This perspective asks us to write and write a lot, but also to finish and publish wherever we can happily find a space for our work. I think that this view also invites us to stop dilly-dallying, and saying that we are writers when we are sometimes not acting like it and to accept our successes along with our failures. For if we are to write things for the world, then some of it may not be perfect, but it will be completed and made public.
Of course, a hyperproduct view of writing is similar to a view sometimes espoused by hundreds of pithy productivity websites— the kind one might read when being unproductive. However, a strong product-based view of writing values the writing as well as what the writer can do. Famous portrait painter Chuck Close has this wonderful line: “Inspiration is for amateurs, I get to work.” Mark Twain and Maya Angelou, among others, notoriously hated the editing process. There are probably more quotations valuing revision from famous writers than not, coming from a natural tendency, which is why this line of thought is so heretical and necessary. Like any creative practice, it is easy to make writing a ponderous sort of artistic practice. And it should involve a bit of pondering and playful exploration, but it cannot stay there. The tension between process and product involves a shift in how we think about time. There is a well-known anecdote about Oscar Wilde confessing that he spent all morning taking out a comma, and all afternoon putting it back. This decadent view of the writing
process simply isn’t where we are as a culture. It’s like spending all day manicuring your nails. Arrant pedantry!
And it is vanity. Time is a luxury. Revising too much can be unethical—a waste. There are diapers to be changed. More than that: People are dying. You are dying. And you need to write as though your next piece could be your last.
For better and worse, we value getting things done. Doing stuff. Making stuff. With words, in our particular case. When we make written work, we fall in line with other productive professions: farmers, smiths, tailors, cobblers, artisans. Writing was analogous to building for the Romans. In this view, writing is construction, but thinking about writing is never arriving at what we might create. Our trade is sentences—complete ones generally. Perhaps we fear finishing drafts because we’re afraid they will fail. We writers might nail Nike’s maxim to our walls: Just do it .
This approach is not the soft handholding of Donald Murray. We should clean the sludgy snailtrail of slow, process-oriented writing that is reminiscent of the Mac Beach Ball of Doom, endlessly spinning. We owe it to the work of writing. We owe it to ourselves. In this view, we say write something next week. And the next! Write all the time. Don’t waste another second. Love deadlines. Eliminate distractions. Get it done. Now, go. Do .
Further Reading
If you’re interested in thinking more about process and getting writing done, you might begin by reading one of the articles that started it all, “Teaching Writing as Process Not as Product” by Donald Murray. Later reconsiderations of process are collected in the edited collection Beyond Postprocess , put together by Sidney Dobrin, J. A. Rice, and Michael Vastola. A few more popular and accessible articles on process include “Why Writers Are the Worst Procrastinators” by Megan McArdle and “The Trick to Being a Prolific Scholar” by Tanya Golash-Boza. Another popular article, “The Dilemmas of Maker Culture” by John Tierney, questions our obsession with getting things done, while Getting Things Done by David Allen has been a bestseller for a long time and influenced our productivity movement. Daniel Kahneman’s book Thinking , Fast and Slow also explores our work habits along these thematic lines. And finally, perhaps the most in-depth study of our historical trajectory toward process and productivity is Hannah Sullivan’s book, The Work of Revision .
Keywords
getting things done, post-process, process, recursivity, revision
Author Bio
Jimmy Butts teaches writing and the teaching of writing to impressive, young humans. He likes to get them doing wonderfully weird things with words. He immensely enjoys finishing a draft. You can find him on Twitter @jimmy_butts or at theyellowrobot.com.


9.11: Images and Clip Art
Learning Objectives
- Add images and clip art
To add an image, insert a new slide. For our purposes, it is helpful to start with a Title and Content slide, but remember you can always change the format of your slide later.
Method 1
- Click inside the content box so that it becomes the active portion of the slide.
- If you hover your mouse over any of the icons shown in the green box below, you will see a description of what type of object you can insert.
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Select Pictures, the bottom left icon.
- By default, your computer’s Pictures folder will pop up. If your images are stored elsewhere, navigate to the appropriate location to find your image.
- Click on the picture or image you wish to insert.
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Click Insert
Your image will be automatically sized by PowerPoint. You can then type in a title, add a text box, or even resize your image if you wish.
Method 2
Another way to add an image to a slide is to use the Pictures option in the Insert tab. Pictures embedded this way are automatically sized as well.
Clicking the Pictures button uses images stored on your computer. Clicking the Online Pictures button allows you to use an embedded search engine to use images directly from the internet. Note that the term pictures is used to represent really any graphic: it can be a photograph, an icon, or a graphic file. Older versions of Microsoft Office included something called clip art. Clip art was basically a graphic icon, such as the image shown below. This type of image can still be used and searched for using the Online Pictures option.
Contributors and Attributions
- Adding Images. Authored by : Shelli Carter. Provided by : Lumen Learning. License : CC BY: Attribution


8.8: Applications
View Nondimensionalization on YouTube
8.8.1. RLC circuit
Consider a resister \(R\) , an inductor \(L\) , and a capacitor \(C\) connected in series as shown in Fig. 8.1. An AC generator provides a time-varying electromotive force (emf), \(\mathcal{E}(t)\) , to the circuit. Here, we determine the differential equation satisfied by the charge on the capacitor.
The constitutive equations for the voltage drops across a capacitor, a resister, and an inductor are given by
\[V_{C}=q / C, \quad V_{R}=i R, \quad V_{L}=\frac{d i}{d t} L, \nonumber \]
where \(C\) is the capacitance, \(R\) is the resistance, and \(L\) is the inductance. The charge \(q\) and the current \(i\) are related by
\[i=\frac{d q}{d t} . \nonumber \]
Kirchhoff’s voltage law states that the \(\operatorname{emf} \mathcal{E}\) applied to any closed loop is equal to the sum of the voltage drops in that loop. Applying Kirchhoff’s voltage law to the \(R L C\) ciruit results in
\[V_{L}+V_{R}+V_{C}=\mathcal{E}(t) \label{8.32}\]
or using Equation \ref{8.30} and Equation \ref{8.31},
\[L \frac{d^{2} q}{d t^{2}}+R \frac{d q}{d t}+\frac{1}{C} q=\mathcal{E}(t) \nonumber \]
The equation for the \(R L C\) circuit is a second-order linear inhomogeneous differential equation with constant coefficients.
The \(A C\) voltage can be written as \(\mathcal{E}(t)=\mathcal{E}_{0} \cos \omega t\) , and the governing equation for \(q=q(t)\) can be written as
\[\frac{d^{2} q}{d t^{2}}+\frac{R}{L} \frac{d q}{d t}+\frac{1}{L C} q=\frac{\mathcal{E}_{0}}{L} \cos \omega t \label{8.33}\]
Nondimensionalization of this equation will be shown to reduce the number of free parameters.
To construct dimensionless variables, we first define the natural frequency of oscillation of a system to be the frequency of oscillation in the absence of any driving or damping forces. The iconic example is the simple harmonic oscillator, with equation given by
\[\ddot{x}+\omega_{0}^{2} x=0, \nonumber \]
and general solution given by \(x(t)=A \cos \omega_{0} t+B \sin \omega_{0} t\) . Here, the natural frequency of oscillation is \(\omega_{0}\) , and the period of oscillation is \(T=2 \pi / \omega_{0}\) . For the \(R L C\) circuit, the natural frequency of oscillation is found from the coefficient of the \(q\) term, and is given by
\[\omega_{0}=\frac{1}{\sqrt{L C}} \nonumber \]
Making use of \(\omega_{0}\) , with units of one over time, we can define a dimensionless time \(\tau\) and a dimensionless charge \(Q\) by
\[\tau=\omega_{0} t, \quad Q=\frac{\omega_{0}^{2} L}{\mathcal{E}_{0}} q \label{8.34}\]
The resulting dimensionless equation for the \(R L C\) circuit is then given by
\[\frac{d^{2} Q}{d \tau^{2}}+\alpha \frac{d Q}{d \tau}+Q=\cos \beta \tau \nonumber \]
where \(\alpha\) and \(\beta\) are dimensionless parameters given by
\[\alpha=\frac{R}{L \omega_{0}}, \quad \beta=\frac{\omega}{\omega_{0}} . \nonumber \]
Notice that the original five parameters in Equation \ref{8.33}, namely \(R, L, C, \mathcal{E}_{0}\) and \(\omega\) , have been reduced to the two dimensionless parameters \(\alpha\) and \(\beta\) . We will return later to solve Equation \ref{8.34} after visiting two more applications that will be shown to be governed by the same dimensionless equation.
8.8.2. Mass on a spring
View Mass on a Spring on YouTube
Consider a mass connected by a spring to a fixed wall, with top view shown in Fig. 8.2. The spring force is modeled by Hooke’s law, \(F_{s}=-k x\) , and sliding friction is modeled as \(F_{f}=-c d x / d t\) . An external force is applied and is assumed to be sinusoidal, with \(F_{e}=F_{0} \cos \omega t\) . Newton’s equation, \(F=m a\) , results in
\[m \frac{d^{2} x}{d t^{2}}=-k x-c \frac{d x}{d t}+F_{0} \cos \omega t \nonumber \]
Rearranging terms, we obtain
\[\frac{d^{2} x}{d t^{2}}+\frac{c}{m} \frac{d x}{d t}+\frac{k}{m} x=\frac{F_{0}}{m} \cos \omega t \nonumber \]
Here, the natural frequency of oscillation is given by
\[\omega_{0}=\sqrt{\frac{k}{m}} \nonumber \]
and we define a dimensionless time \(\tau\) and a dimensionless position \(X\) by
\[\tau=\omega_{0} t, \quad X=\frac{m \omega_{0}^{2}}{F_{0}} x . \nonumber \]
The resulting dimensionless equation is given by
\[\frac{d^{2} X}{d \tau^{2}}+\alpha \frac{d X}{d \tau}+X=\cos \beta \tau \label{8.35}\]
where here, \(\alpha\) and \(\beta\) are dimensionless parameters given by
\[\alpha=\frac{c}{m \omega_{0}}, \quad \beta=\frac{\omega}{\omega_{0}} . \nonumber \]
Notice that even though the physical problem is different, and the dimensionless variables are defined differently, the resulting dimensionless equation Equation \ref{8.35} for the mass-spring system is the same as that for the \(R L C\) circuit Equation \ref{8.34}.
8.8.3. Pendulum
Here, we consider a mass that is attached to a massless rigid rod and is constrained to move along an arc of a circle centered at the pivot point (see Fig. 8.3). Suppose \(l\) is the fixed length of the connecting rod, and \(\theta\) is the angle it makes with the vertical.
We can apply Newton’s equation, \(F=m a\) , to the mass with origin at the bottom and axis along the arc with positive direction to the right. The position \(s\) of the mass along the arc is given by \(s=l \theta\) . The relevant gravitrational force on the pendulum is the component along the arc, and from Fig. \(8.3\) is observed to be
\[F_{g}=-m g \sin \theta \nonumber \]
We model friction to be proportional to the velocity of the pendulum along the arc, that is
\[F_{f}=-c \dot{s}=-c l \dot{\theta} \nonumber \]
With a sinusoidal external force, \(F_{e}=F_{0} \cos \omega t\) , Newton’s equation \(m \ddot{s}=F_{g}+F_{f}+\) \(F_{e}\) results in
\[m l \ddot{\theta}=-m g \sin \theta-c l \dot{\theta}+F_{0} \cos \omega t . \nonumber \]
Rewriting, we have
\[\ddot{\theta}+\frac{c}{m} \dot{\theta}+\frac{g}{l} \sin \theta=\frac{F_{0}}{m l} \cos \omega t . \nonumber \]
At small amplitudes of oscillation, we can approximate \(\sin \theta \approx \theta\) , and the natural frequency of oscillation \(\omega_{0}\) of the mass is given by
\[\omega_{0}=\sqrt{\frac{\bar{g}}{l}} \nonumber \]
Nondimensionalizing time as \(\tau=\omega_{0} t\) , the dimensionless pendulum equation becomes
\[\frac{d^{2} \theta}{d \tau^{2}}+\alpha \frac{d \theta}{d \tau}+\sin \theta=\gamma \cos \beta \tau \nonumber \]
where \(\alpha, \beta\) , and \(\gamma\) are dimensionless parameters given by
\[\alpha=\frac{c}{m \omega_{0}}, \quad \beta=\frac{\omega}{\omega_{0}}, \quad \gamma=\frac{F_{0}}{m l \omega_{0}^{2}} \nonumber \]
The nonlinearity of the pendulum equation, with the term \(\sin \theta\) , results in the additional dimensionless parameter \(\gamma\) . For small amplitude of oscillation, however, we can scale \(\theta\) by \(\theta=\gamma \Theta\) , and the small amplitude dimensionless equation becomes
\[\frac{d^{2} \Theta}{d \tau^{2}}+\alpha \frac{d \Theta}{d \tau}+\Theta=\cos \beta \tau \label{8.36}\]
the same equation as Equation \ref{8.34} and Equation \ref{8.35}.


2.6: A Look at High-Quality Practices
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One of the hallmarks of high-quality practices is the practice of continuously monitoring and evaluating a program’s overall effectiveness for the purpose of accountability. Therefore, in addition to complying with state licensing regulations, high-quality programs may utilize noted assessment tools and resources (e.g. ECERS, CLASS, QRIS, NAEYC’s Developmentally Appropriate Practice) to evaluate their center and staff. To guide high-quality practices, early childhood educators often look to the National Association for the Education of Young Children (NAEYC) to provide ethical pedagogy. NAEYC is a recognized professional organization committed to promoting quality early learning experiences for children, birth through age 8. Grounded by research and theory principles, NAEYC promotes best practices for children, families, teachers, administrators, stakeholders and policy makers. To ensure quality practices are being implemented, it is suggested that teachers follow NAEYC’s Developmentally Appropriate Practice (DAP) Framework: