24.3: 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 24.3.1). 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.”
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 24.3.2).
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 (discussed in a previous chapter, Figure 24.3.3). 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 24.3.4). 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 24.3.5) 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.
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 24.3.6). 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.
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
A
Which amendment did Alice Paul promote to end gender discrimination?
- Prohibition Amendment
- Equal Rights Amendment
- Sheppard-Towner Amendment
- Free Exercise Amendment
B
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
D
Why did the prohibition amendment fail after its adoption in 1919?
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.
What was the Harlem Renaissance, and who were some of the most famous participants?
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.
Glossary
- 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


35.13: The Sun and The God
“But at length, a more comfortable view of my condition again presented itself to my mind, and I was consoled: for I again reflected on the great improbability that there was of my finding such a resource in my captivity, as the hermit’s book, and how thankful I ought to be. At this instant, I experienced such an inward persuasion in my mind, that I should escape from this island, that every uneasy thought fled, and left my mind a calm, scarcely to be expressed. I therefore arose, and went cheerfully about my little concerns; but not without having first thanked that God who had given me this consultation.”
Taking religion and spirituality into consideration when analyzing this passage reveals the powers that are pulling Eliza in different directions.
In the passage, we see that Eliza wishes to escape from the island that she has been stranded on. She finds consultation and guidance in the hermit’s book that has been left behind. This text shows her how to live off the land and survive. She also sees it as a crucial aid for her future, as the nourishment from the generous landscape will give her time to formulate a plan of escape.
What’s interesting, however, is that she thanks “God” for leaving this book behind. She does not thank the hermit, or her own good fortune; she attributes the resource to her Christian beliefs.
Th earthly resources bestowed upon her through the guidance of the book, however, seem more aligned with the other God. The god of the indigenous people. The “Sun.”
There are are other points in the novel where the “Sun” directly helps Eliza, like a benevolent force guiding her way. For example, Eliza writes that the hot sun dries her clothes after she is stranded (65). When she discovers the “apartment”, the sun shines brightly through its stones, illuminating the space (66). She also refers to the sun as scorching, but its scorching rays are the cause of the great bounties she harvests for her survival (76).
Eliza praises God rather than the Sun, but the Sun is what seems to be helping her in her time of need. This illuminates the a significant preference in the two beliefs systems at play. The European belief system favors an indirect, illusionary power that dictates morality. The native belief system, however, favors tangible power, like the power of the sun to grow the land.
Eliza seems to prefer the European power, following suit with the culture of her father. But let’s not forget this important detail – Eliza’s father drifted back and forth from Europe to America after the death of his wife, looking to fill the hole that his “native” wife left behind.


4.3: Conditional Probability
Suppose you want to figure out if you should buy a new car. When you first go and look, you find two cars that you like the most. In your mind they are equal, and so each has a 50% chance that you will pick it. Then you start to look at the reviews of the cars and realize that the first car has had 40% of them needing to be repaired in the first year, while the second car only has 10% of the cars needing to be repaired in the first year. You could use this information to help you decide which car you want to actually purchase. Both cars no longer have a 50% chance of being the car you choose. You could actually calculate the probability you will buy each car, which is a conditional probability. You probably wouldn’t do this, but it gives you an example of what a conditional probability is.
Conditional probabilities are probabilities calculated after information is given. This is where you want to find the probability of event A happening after you know that event B has happened. If you know that B has happened, then you don’t need to consider the rest of the sample space. You only need the outcomes that make up event B. Event B becomes the new sample space, which is called the restricted sample space , R. If you always write a restricted sample space when doing conditional probabilities and use this as your sample space, you will have no trouble with conditional probabilities. The notation for conditional probabilities is \(P(A, \text { given } B)=P(A | B)\). The event following the vertical line is always the restricted sample space.
Example \(\PageIndex{1}\) conditional probabilities
- Suppose you roll two dice. What is the probability of getting a sum of 5, given that the first die is a 2?
- Suppose you roll two dice. What is the probability of getting a sum of 7, given the first die is a 4?
- Suppose you roll two dice. What is the probability of getting the second die a 2, given the sum is a 9?
- Suppose you pick a card from a deck. What is the probability of getting a Spade, given that the card is a Jack?
- Suppose you pick a card from a deck. What is the probability of getting an Ace, given the card is a Queen?
Solution
a. Since you know that the first die is a 2, then this is your restricted sample space, so
R = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)}
Out of this restricted sample space, the way to get a sum of 5 is {(2,3)}. Thus
\(P(\text { sum of } 5 | \text { the first die is a } 2)=\dfrac{1}{6}\)
b. Since you know that the first die is a 4, this is your restricted sample space, so
R = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)}
Out of this restricted sample space, the way to get a sum of 7 is {(4,3)}. Thus
\(P(\text { sum of } 7 | \text { the first die is a } 4)=\dfrac{1}{6}\)
c. Since you know the sum is a 9, this is your restricted sample space, so
R = {(3,6), (4,5), (5,4), (6,3)}
Out of this restricted sample space there is no way to get the second die a 2.  Since you know that the card is a Jack, this is your restricted sample space, so
R = {JS, JC, JD, JH}
Out of this restricted sample space, the way to get a Spade is {JS}. Thus
\(P(\text { Spade } | \mathrm{Jack})=\dfrac{1}{4}\)
e. on: Since you know that the card is a Queen, then this is your restricted sample space, so
R = {QS, QC, QD, QH}
Out of this restricted sample space, there is no way to get an Ace, thus
\(P(\text { Ace | Queen })=0\)
If you look at the results of Example \(\PageIndex{7}\) part d and Example \(\PageIndex{1}\) part b, you will notice that you get the same answer. This means that knowing that the first die is a 4 did not change the probability that the sum is a 7. This added knowledge did not help you in any way. It is as if that information was not given at all. However, if you compare Example \(\PageIndex{7}\) part b and Example \(\PageIndex{1}\) part a, you will notice that they are not the same answer. In this case, knowing that the first die is a 2 did change the probability of getting a sum of 5. In the first case, the events sum of 7 and first die is a 4 are called independent events . In the second case, the events sum of 5 and first die is a 2 are called dependent events .
Events A and B are considered independent events if the fact that one event happens does not change the probability of the other event happening. In other words, events A and B are independent if the fact that B has happened does not affect the probability of event A happening and the fact that A has happened does not affect the probability of event B happening. Otherwise, the two events are dependent. In symbols, A and B are independent if
\(P(A | B)=P(A) \text { or } P(B | A)=P(B)\)
Example \(\PageIndex{2}\) independent events
- Suppose you roll two dice.
- Suppose you roll two dice.
- Suppose you pick a card from a deck. Are the events “Jack” and “Spade” independent?
- Suppose you pick a card from a deck. Are the events “Heart” and “Red” card independent?
- Suppose you have two children via separate births. Are the events “the first is a boy” and “the second is a girl” independent?

Solution
a. To determine if they are independent, you need to see if \(P(A | B)=P(A)\). It doesn’t matter which event is A or B, so just assign one as A and one as B.
Let A = sum of 7 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} and B = first die is a 3 = {(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)} \(P(A | B)\) means that you assume that B has happened. The restricted sample space is B,
R = {(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)}
In this restricted sample space, the way for A to happen is {(3,4)}, so
\(P(A | B)=\dfrac{1}{6}\)
The \(P(A)=\dfrac{6}{36}=\dfrac{1}{6}\)
\(P(A | B)=P(A)\) Thus “sum of 7” and “first die is a 3” are independent events.
b. To determine if they are independent, you need to see if \(P(A | B)=P(A)\). It doesn’t matter which event is A or B, so just assign one as A and one as B.
Let A = sum of 6 = {(1,5), (2,4), (3,3), (4,2), (5,1)} and B = first die is a 4 = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)}, so
\(P(A)=\dfrac{5}{36}\)
For \(P(A | B)\), the restricted sample space is B,
R = {(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)}
In this restricted sample space, the way for A to happen is {(4,2)}, so
\(P(A | B)=\dfrac{1}{6}\).
In this case, “sum of 6” and “first die is a 4” are dependent since \(P(A | B) \neq P(A)\).
c. To determine if they are independent, you need to see if \(P(A | B)=P(A)\). It doesn’t matter which event is A or B, so just assign one as A and one as B.
Let A = Jack = {JS, JC, JD, JH} and B = Spade {2S, 3S, 4S, 5S, 6S, 7S, 8S, 9S, 10S, JS, QS, KS, AS}
\(P(A)=\dfrac{4}{52}=\dfrac{1}{13}\)
For \(P(A | B)\), the restricted sample space is B,
R = {2S, 3S, 4S, 5S, 6S, 7S, 8S, 9S, 10S, JS, QS, KS, AS}
In this restricted sample space, the way A happens is {JS}, so
\(P(A | B)=\dfrac{1}{13}\)
In this case, “Jack” and “Spade” are independent since \(P(A | B)=P(A)\).
d. To determine if they are independent, you need to see if \(P(A | B)=P(A)\). It doesn’t matter which event is A or B, so just assign one as A and one as B.
Let A = Heart = {2H, 3H, 4H, 5H, 6H, 7H, 8H, 9H, 10H, JH, QH, KH, AH} and B = Red card = {2D, 3D, 4D, 5D, 6D, 7D, 8D, 9D, 10D, JD, QD, KD, AD, 2H, 3H, 4H, 5H, 6H, 7H, 8H, 9H, 10H, JH, QH, KH, AH}, so
\(P(A)=\dfrac{13}{52}=\dfrac{1}{4}\)
For \(P(A | B)\), the restricted sample space is B,
R = {2D, 3D, 4D, 5D, 6D, 7D, 8D, 9D, 10D, JD, QD, KD, AD, 2H, 3H, 4H, 5H, 6H, 7H, 8H, 9H, 10H, JH, QH, KH, AH}
In this restricted sample space, the way A can happen is 13,
\(P(A | B)=\dfrac{13}{26}=\dfrac{1}{2}\).
In this case, “Heart” and “Red” card are dependent, since \(P(A | B) \neq P(A)\).
e. In this case, you actually don’t need to do any calculations. The gender of one child does not affect the gender of the second child, the events are independent.
f. Since one flip of the coin does not affect the next flip (the coin does not remember what it did the time before), the probability of getting a head on the next flip is still one-half.
Multiplication Rule:
Two more useful formulas: If two events are dependent, then \(P(A \text { and } B)=P(A) * P(B | A)\)
If two events are independent, then \(P(A \text { and } B)=P(A)^{*} P(B)\)
If you solve the first equation for \(P(B | A)\), you obtain \(P(B | A)=\dfrac{P(A \text { and } B)}{P(A)}\), which is a formula to calculate a conditional probability. However, it is easier to find a conditional probability by using the restricted sample space and counting unless the sample space is large.
Example \(\PageIndex{3}\) Multiplication rule
- Suppose you pick three cards from a deck, what is the probability that they are all Queens if the cards are not replaced after they are picked?
- Suppose you pick three cards from a deck, what is the probability that they are all Queens if the cards are replaced after they are picked and before the next card is picked?
Solution
a. This sample space is too large to write out, so using the multiplication rule makes sense. Since the cards are not replaced, then the probability will change for the second and third cards. They are dependent events. This means that on the second draw there is one less Queen and one less card, and on the third draw there are two less Queens and 2 less cards.
P (3 Queens)= P (Q on 1st and Q on 2nd and Q on 3rd)
= P (Q on 1st)* P (Q on 2nd|Q on 1st)* P (Q on 3rd|1st and 2nd Q)
\(=\dfrac{4}{52} * \dfrac{3}{51} * \dfrac{2}{50}\)
\(=\dfrac{24}{132600}\)
b. Again, the sample space is too large to write out, so using the multiplication rule makes sense. Since the cards are put back, one draw has no affect on the next draw and they are all independent.
P (3 Queens)= P (Queen on 1st and Queen on 2nd and Queen on 3rd)
= P (Queen on 1st)* P (Queen on 2nd)* P (Queen on 3rd)
\(=\dfrac{4}{52} * \dfrac{4}{52} * \dfrac{4}{52}\)
\(=\left(\dfrac{4}{52}\right)^{3}\)
\(=\dfrac{64}{140608}\)
Example \(\PageIndex{4}\) application problem
The World Health Organization (WHO) keeps track of how many incidents of leprosy there are in the world. Using the WHO regions and the World Banks income groups, one can ask if an income level and a WHO region are dependent on each other in terms of predicting where the disease is. Data on leprosy cases in different countries was collected for the year 2011 and a summary is presented in Example \(\PageIndex{1}\) ("Leprosy: Number of," 2013).
| WHO Region | World Bank Income Group | Row Total | |||
| High Income | Upper Middle Income | Lower Middle Income | Low Income | ||
| Americas | 174 | 36028 | 615 | 0 | 36817 |
| Eastern Mediterranean | 54 | 6 | 1883 | 604 | 2547 |
| Europe | 10 | 0 | 0 | 0 | 10 |
| Western Pacific | 26 | 216 | 3689 | 1155 | 5086 |
| Africa | 0 | 39 | 1986 | 15928 | 17953 |
| South-East Asia | 0 | 0 | 149896 | 10236 | 160132 |
| Column Total | 264 | 36289 | 158069 | 27923 | 222545 |
- Find the probability that a person with leprosy is from the Americas.
- Find the probability that a person with leprosy is from a high-income country.
- Find the probability that a person with leprosy is from the Americas and a high-income country.
- Find the probability that a person with leprosy is from a high-income country, given they are from the Americas.
- Find the probability that a person with leprosy is from a low-income country.
- Find the probability that a person with leprosy is from Africa.
- Find the probability that a person with leprosy is from Africa and a low-income country.
- Find the probability that a person with leprosy is from Africa, given they are from a low-income country.
- Are the events that a person with leprosy is from “Africa” and “low-income country” independent events? Why or why not?
- Are the events that a person with leprosy is from “Americas” and “high-income country” independent events? Why or why not?
Solution
a. There are 36817 cases of leprosy in the Americas out of 222,545 cases worldwide. So,
\(P(\text { Americas })=\dfrac{36817}{222545} \approx 0.165\)
There is about a 16.5% chance that a person with leprosy lives in a country in the Americas.
b. There are 264 cases of leprosy in high-income countries out of 222,545 cases worldwide. So,
\(P(\text { high-income })=\dfrac{264}{222545} \approx 0.0001\)
There is about a 0.1% chance that a person with leprosy lives in a high-income country.
c. There are 174 cases of leprosy in countries in a high-income country in the Americas out the 222,545 cases worldwide. So,
\(P(\text { Americas and high-income })=\dfrac{174}{222545} 0.0008\)
There is about a 0.08% chance that a person with leprosy lives in a high-income country in the Americas.
d. In this case you know that the person is in the Americas. You don’t need to consider people from Easter Mediterranean, Europe, Western Pacific, Africa, and South-east Asia. You only need to look at the row with Americas at the start. In that row, look to see how many leprosy cases there are from a high-income country. There are 174 countries out of the 36,817 leprosy cases in the Americas. So,
\(P(\text { high-income } | \text { Americas })=\dfrac{174}{36817} \approx 0.0047\)
There is 0.47% chance that a person with leprosy is from a high-income country given that they are from the Americas.
e. There are 27,923 cases of leprosy in low-income countries out of the 222,545 leprosy cases worldwide. So,
\(P(\text { low-income })=\dfrac{27923}{222545} \approx 0.125\)
There is a 12.5% chance that a person with leprosy is from a low-income country.
f. There are 17,953 cases of leprosy in Africa out of 222,545 leprosy cases worldwide. So,
\(P(\text { Africa })=\dfrac{17953}{222545} \approx 0.081\)
There is an 8.1% chance that a person with leprosy is from Africa.
g. There are 15,928 cases of leprosy in low-income countries in Africa out of all the 222,545 leprosy cases worldwide. So,
\(P(\text { Africa and low-income })=\dfrac{15928}{222545} \approx 0.072\)
There is a 7.2% chance that a person with leprosy is from a low-income country in Africa.
h. In this case you know that the person with leprosy is from low-income country. You don’t need to include the high income, upper-middle income, and lowermiddle income country. You only need to consider the column headed by lowincome. In that column, there are 15,928 cases of leprosy in Africa out of the 27,923 cases of leprosy in low-income countries. So,
\(P(\text { Africa | low-income })=\dfrac{15928}{27923} \approx 0.570\)
There is a 57.0% chance that a person with leprosy is from Africa, given that they are from a low-income country.
i. In order for these events to be independent, either \(P(\text { Africa } | \text { low-income })=P(\text { Africa })\) or \(P(\text { low-income } | \text { Africa })=P(\text { low-income })\) have to be true. Part (h) showed \(P(\text { Africa | low-income }) \approx 0.570\) and part (f) showed \(P(\text { Africa }) \approx 0.081\). Since these are not equal, then these two events are dependent.
j. In order for these events to be independent, either \(P(\text { Americas } | \text { high-income })=P(\text { Americas })\) or \(P(\text { high-income |} \text { Americas })=P(\text { high-income })\) have to be true. Part (d) showed \(P(\text { high-income } | \text { Americas }) \approx 0.0047\) and part (b) showed \(P(\text { high-income }) \approx 0.001\). Since these are not equal, then these two events are dependent.
A big deal has been made about the difference between dependent and independent events while calculating the probability of and compound events. You must multiply the probability of the first event with the conditional probability of the second event.
Why do you care? You need to calculate probabilities when you are performing sampling, as you will learn later. But here is a simplification that can make the calculations a lot easier: when the sample size is very small compared to the population size, you can assume that the conditional probabilities just don't change very much over the sample.
For example, consider acceptance sampling. Suppose there is a big population of parts delivered to you factory, say 12,000 parts. Suppose there are 85 defective parts in the population. You decide to randomly select ten parts, and reject the shipment. What is the probability of rejecting the shipment?
There are many different ways you could reject the shipment. For example, maybe the first three parts are good, one is bad, and the rest are good. Or all ten parts could be bad, or maybe the first five. So many ways to reject! But there is only one way that you’d accept the shipment: if all ten parts are good. That would happen if the first part is good, and the second part is good, and the third part is good, and so on. Since the probability of the second part being good is (slightly) dependent on whether the first part was good, technically you should take this into consideration when you calculate the probability that all ten are good.
The probability of getting the first sampled part good is \(\dfrac{12000-85}{12000}=\dfrac{11915}{12000}\). So the probability that all ten being good is \(\dfrac{11915}{12000} * \dfrac{11914}{11999} * \dfrac{11913}{11998} * \ldots * \dfrac{11906}{11991} \approx 93.1357 \%\). If instead you assume that the probability doesn’t change much, you get \(\left(\dfrac{11915}{12000}\right)^{10} \approx 93.1382 \%\). So as you can see, there is not much difference. So here is the rule: if the sample is very small compared to the size of the population, then you can assume that the probabilities are independent, even though they aren’t technically. By the way, the probability of rejecting the shipment is \(1-0.9314=0.0686=6.86 \%\).
Homework
Exercise \(\PageIndex{1}\)
- Are owning a refrigerator and owning a car independent events? Why or why not?
- Are owning a computer or tablet and paying for Internet service independent events? Why or why not?
- Are passing your statistics class and passing your biology class independent events? Why or why not?
- Are owning a bike and owning a car independent events? Why or why not?
-
An experiment is picking a card from a fair deck.
- What is the probability of picking a Jack given that the card is a face card?
- What is the probability of picking a heart given that the card is a three?
- What is the probability of picking a red card given that the card is an ace?
- Are the events Jack and face card independent events? Why or why not?
- Are the events red card and ace independent events? Why or why not?
-
An experiment is rolling two dice.
- What is the probability that the sum is 6 given that the first die is a 5?
- What is the probability that the first die is a 3 given that the sum is 11?
- What is the probability that the sum is 7 given that the fist die is a 2?
- Are the two events sum of 6 and first die is a 5 independent events? Why or why not?
- Are the two events sum of 7 and first die is a 2 independent events? Why or why not?
- You flip a coin four times. What is the probability that all four of them are heads?
- You flip a coin six times. What is the probability that all six of them are heads?
- You pick three cards from a deck with replacing the card each time before picking the next card. What is the probability that all three cards are kings?
- You pick three cards from a deck without replacing a card before picking the next card. What is the probability that all three cards are kings?
-
The number of people who survived the Titanic based on class and sex is in Example \(\PageIndex{2}\) ("Encyclopedia Titanica," 2013). Suppose a person is picked at random from the survivors.
Class Sex Total Female Male 1st 134 59 193 2nd 94 25 119 3rd 80 58 138 Total 308 142 450 Table \(\PageIndex{2}\) : Surviving the Titanic
a. What is the probability that a survivor was female?
b. What is the probability that a survivor was in the 1st class?
c. What is the probability that a survivor was a female given that the person was in 1st class?
d. What is the probability that a survivor was a female and in the 1st class?
e. What is the probability that a survivor was a female or in the 1st class?
f. Are the events survivor is a female and survivor is in 1st class mutually exclusive? Why or why not?
g. Are the events survivor is a female and survivor is in 1st class independent? Why or why not? -
Researchers watched groups of dolphins off the coast of Ireland in 1998 to determine what activities the dolphins partake in at certain times of the day ("Activities of dolphin," 2013). The numbers in Example \(\PageIndex{3}\) represent the number of groups of dolphins that were partaking in an activity at certain times of days.
Activity Period Total Morning Noon Afternoon Evening Travel 6 6 14 13 39 Feed 28 4 0 56 88 Social 38 5 9 10 62 Total 72 15 23 79 189 Table \(\PageIndex{3}\) : Dolphin Activity
a. What is the probability that a dolphin group is partaking in travel?
b. What is the probability that a dolphin group is around in the morning?
c. What is the probability that a dolphin group is partaking in travel given that it is morning?
d. What is the probability that a dolphin group is around in the morning given that it is partaking in socializing?
e. What is the probability that a dolphin group is around in the afternoon given that it is partaking in feeding?
f. What is the probability that a dolphin group is around in the afternoon and is partaking in feeding?
g. What is the probability that a dolphin group is around in the afternoon or is partaking in feeding?
h. Are the events dolphin group around in the afternoon and dolphin group feeding mutually exclusive events? Why or why not?
i. Are the events dolphin group around in the morning and dolphin group partaking in travel independent events? Why or why not?
- Answer
-
1. Independent, see solutions
3. Dependent, see solutions
5. a. P(Jack/face card) = 0.333, b. P(heart/card a 3) = 0.25, c. P(red card/ace) = 0.50, d. not independent, see solutions, e. independent, see solutions
7. 0.0625
9. \(4.55 \times 10^{-4}\)
11. a. P(female) = 0.684, b. P(1st class) = 0.429, c. P(female/1st class) = 0.694, d. P(female and 1st class) = 0.298, e. P(female or 1st class) = 0.816, f. No, see solutions, g. Dependent, see solutions


13.4: Gravitational Potential Energy and Total Energy
- Determine changes in gravitational potential energy over great distances
- Apply conservation of energy to determine escape velocity
- Determine whether astronomical bodies are gravitationally bound
We studied gravitational potential energy in Potential Energy and Conservation of Energy , where the value of \(g\) remained constant. We now develop an expression that works over distances such that g is not constant. This is necessary to correctly calculate the energy needed to place satellites in orbit or to send them on missions in space.
Gravitational Potential Energy beyond Earth
We defined work and potential energy , previously. The usefulness of those definitions is the ease with which we can solve many problems using conservation of energy. Potential energy is particularly useful for forces that change with position, as the gravitational force does over large distances. In Potential Energy and Conservation of Energy , we showed that the change in gravitational potential energy near Earth’s surface is
\[ \Delta U = mg(y_2− y_1) \label{simple}\]
This works very well if \(g\) does not change significantly between y 1 and y 2 . We return to the definition of work and potential energy to derive an expression that is correct over larger distances. Recall that work (W) is the integral of the dot product between force and distance. Essentially, it is the product of the component of a force along a displacement times that displacement. We define \(\Delta u\) as the negative of the work done by the force we associate with the potential energy. For clarity, we derive an expression for moving a mass m from distance r 1 from the center of Earth to distance r 2 . However, the result can easily be generalized to any two objects changing their separation from one value to another.
Consider Figure \(\PageIndex{1}\), in which we take m from a distance r 1 from Earth’s center to a distance that is r 2 from the center. Gravity is a conservative force (its magnitude and direction are functions of location only), so we can take any path we wish, and the result for the calculation of work is the same. We take the path shown, as it greatly simplifies the integration. We first move radially outward from distance r 1 to distance r 2 , and then move along the arc of a circle until we reach the final position. During the radial portion, \(\vec{F}\) is opposite to the direction we travel along d\(\vec{r}\), so
\[E = K_1 + U_1 = K_2 + U_2\]
Along the arc, \(\vec{F}\) is perpendicular to d\(\vec{r}\), so \(\vec{F}\; \cdotp d \vec{r}\) = 0. No work is done as we move along the arc. Using the expression for the gravitational force and noting the values for \(\vec{F}\; \cdotp d \vec{r}\) along the two segments of our path, we have
\[ \begin{align} \Delta U &= - \int_{r_{1}}^{r_{2}} \vec{F}\; \cdotp d \vec{r} \\[4pt] &= GM_{E} m \int_{r_{1}}^{r_{2}} \frac{dr}{r^{2}} \\[4pt] &= GM_{E} m \left(\dfrac{1}{r_{1}} - \dfrac{1}{r_{2}}\right) \ldotp \label{eq13.3} \end{align} \]
Since \(\Delta U = U_2 − U_1\) we can adopt a simple expression for \(U\):
\[U = - \frac{GM_{E} m}{r} \ldotp \label{13.4}\]
Note two important items with this definition. First, \(U → 0\) as \(r → \infty\). The potential energy is zero when the two masses are infinitely far apart. Only the difference in \(U\) is important, so the choice of \(U = 0\) for \(r = \infty\) is merely one of convenience. (Recall that in earlier gravity problems, you were free to take \(U = 0\) at the top or bottom of a building, or anywhere.) Second, note that \(U\) becomes increasingly more negative as the masses get closer. That is consistent with what you learned about potential energy in Potential Energy and Conservation of Energy . As the two masses are separated, positive work must be done against the force of gravity, and hence, \(U\) increases (becomes less negative). All masses naturally fall together under the influence of gravity, falling from a higher to a lower potential energy.
How much energy is required to lift the 9000-kg Soyuz vehicle from Earth’s surface to the height of the ISS, 400 km above the surface?
Strategy
Use Equation \ref{eq13.3} to find the change in potential energy of the payload. That amount of work or energy must be supplied to lift the payload.
Solution
Paying attention to the fact that we start at Earth’s surface and end at 400 km above the surface, the change in \(U\) is
\[ \begin {align*} \Delta U &= U_{orbit} - U_{Earth} \\[4pt] &= - \dfrac{GM_{E} m}{R_{E} + 400\; km} - \left(- \dfrac{GM_{E} m}{R_{E}}\right) \ldotp \end{align*}\]
We insert the values
- \(m = 9000\; kg\)
- \(M_{E} = 5.96 \times 10^{24}\; kg\)
- \(R_{E} = 6.37 \times 10^{6}\; m\)
and convert 400 km into 4.00 x 10 5 m. We find \(\Delta U = 3.32 \times 10^{10} J\). It is positive, indicating an increase in potential energy, as we would expect.
Significance
For perspective, consider that the average US household energy use in 2013 was 909 kWh per month. That is energy of
\[909\; kWh \times 1000\; W/kW \times 3600\; s/h = 3.27 \times 10^{9}\; J\; per\; month \ldotp \nonumber\]
So our result is an energy expenditure equivalent to 10 months. However, this is just the energy needed to raise the payload 400 km. If we want the Soyuz to be in orbit so it can rendezvous with the ISS and not just fall back to Earth, it needs a lot of kinetic energy. As we see in the next section, that kinetic energy is about five times that of \(\Delta\)U. In addition, far more energy is expended lifting the propulsion system itself. Space travel is not cheap.
Why not use the simpler expression in Equation \ref{simple} instead? How significant would the error be? (The value \(g\) at 400 km above the Earth is 8.67 m/s 2 .)
Conservation of Energy
In Potential Energy and Conservation of Energy , we described how to apply conservation of energy for systems with conservative forces . We were able to solve many problems, particularly those involving gravity, more simply using conservation of energy. Those principles and problem-solving strategies apply equally well here. The only change is to place the new expression for potential energy into the conservation of energy equation,
\[E_{tot} = K_1 + U_1 = K_2 + U_2.\]
\[\frac{1}{2} mv_{1}^{2} - \frac{GMm}{r_{1}} = \frac{1}{2} mv_{2}^{2} - \frac{GMm}{r_{2}} \label{13.5}\]
Note that we use M, rather than M E , as a reminder that we are not restricted to problems involving Earth. However, we still assume that m << M. (For problems in which this is not true, we need to include the kinetic energy of both masses and use conservation of momentum to relate the velocities to each other. But the principle remains the same.)
Escape velocity
Escape velocity is often defined to be the minimum initial velocity of an object that is required to escape the surface of a planet (or any large body like a moon) and never return. As usual, we assume no energy lost to an atmosphere, should there be any.
Consider the case where an object is launched from the surface of a planet with an initial velocity directed away from the planet. With the minimum velocity needed to escape, the object would just come to rest infinitely far away, that is, the object gives up the last of its kinetic energy just as it reaches infinity, where the force of gravity becomes zero. Since U → 0 as r → \(\infty\), this means the total energy is zero. Thus, we find the escape velocity from the surface of an astronomical body of mass M and radius R by setting the total energy equal to zero. At the surface of the body, the object is located at \(r_1 = R\) and it has escape velocity \(v_1 = v_{esc}\). It reaches \(r_2 = \infty\) with velocity \(v_2 = 0\). Substituting into Equation \ref{13.5}, we have
\[\frac{1}{2} mv_{esc}^{2} - \frac{GMm}{R} = \frac{1}{2} m0^{2} - \frac{GMm}{\infty} = 0 \ldotp\]
Solving for the escape velocity,
\[v_{esc} = \sqrt{\frac{2GM}{R}} \ldotp \label{13.6}\]
Notice that \(m\) has canceled out of the equation. The escape velocity is the same for all objects, regardless of mass . Also, we are not restricted to the surface of the planet; R can be any starting point beyond the surface of the planet.
What is the escape speed from the surface of Earth? Assume there is no energy loss from air resistance. Compare this to the escape speed from the Sun, starting from Earth’s orbit.
Strategy
We use Equation 13.6, clearly defining the values of R and M. To escape Earth, we need the mass and radius of Earth. For escaping the Sun, we need the mass of the Sun, and the orbital distance between Earth and the Sun.
Solution
Substituting the values for Earth’s mass and radius directly into Equation 13.6, we obtain
\[ \begin {align*} v_{esc} &= \sqrt{\frac{2GM}{R}} \\[4pt] &= \sqrt{\frac{2 (6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2})(5.96 \times 10^{24}\; kg)}{6.37 \times 10^{6}\; m}} \\[4pt] &= 1.12 \times 10^{4}\; m/s \ldotp \end{align*}\]
That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use R = R ES = 1.50 x 10 11 m and M Sun = 1.99 x 10 30 kg. The result is v esc = 4.21 x 10 4 m/s or about 42 km/s.
Significance
The speed needed to escape the Sun (leave the solar system) is nearly four times the escape speed from Earth’s surface. But there is help in both cases. Earth is rotating, at a speed of nearly 1.7 km/s at the equator, and we can use that velocity to help escape, or to achieve orbit. For this reason, many commercial space companies maintain launch facilities near the equator. To escape the Sun, there is even more help. Earth revolves about the Sun at a speed of approximately 30 km/s. By launching in the direction that Earth is moving, we need only an additional 12 km/s. The use of gravitational assist from other planets, essentially a gravity slingshot technique, allows space probes to reach even greater speeds. In this slingshot technique, the vehicle approaches the planet and is accelerated by the planet’s gravitational attraction. It has its greatest speed at the closest point of approach, although it decelerates in equal measure as it moves away. But relative to the planet, the vehicle’s speed far before the approach, and long after, are the same. If the directions are chosen correctly, that can result in a significant increase (or decrease if needed) in the vehicle’s speed relative to the rest of the solar system.
Energy and gravitationally bound objects
As stated previously, escape velocity can be defined as the initial velocity of an object that can escape the surface of a moon or planet. More generally, it is the speed at any position such that the total energy is zero. If the total energy is zero or greater, the object escapes. If the total energy is negative, the object cannot escape. Let’s see why that is the case.
As noted earlier, we see that \(U → 0\) as \(r → \infty\). If the total energy is zero, then as m reaches a value of r that approaches infinity, U becomes zero and so must the kinetic energy. Hence, m comes to rest infinitely far away from M. It has “just escaped” M. If the total energy is positive, then kinetic energy remains at \(r = \infty\) and certainly m does not return. When the total energy is zero or greater, then we say that m is not gravitationally bound to M.
On the other hand, if the total energy is negative, then the kinetic energy must reach zero at some finite value of r, where U is negative and equal to the total energy. The object can never exceed this finite distance from M, since to do so would require the kinetic energy to become negative, which is not possible. We say m is gravitationally bound to M.
We have simplified this discussion by assuming that the object was headed directly away from the planet. What is remarkable is that the result applies for any velocity. Energy is a scalar quantity and hence Equation \ref{13.5} is a scalar equation—the direction of the velocity plays no role in conservation of energy. It is possible to have a gravitationally bound system where the masses do not “fall together,” but maintain an orbital motion about each other.
We have one important final observation. Earlier we stated that if the total energy is zero or greater, the object escapes. Strictly speaking, Equation \ref{13.5} and Equation \ref{13.6} apply for point objects. They apply to finite-sized, spherically symmetric objects as well, provided that the value for \(r\) in Equation \ref{13.5} is always greater than the sum of the radii of the two objects. If r becomes less than this sum, then the objects collide. (Even for greater values of r, but near the sum of the radii, gravitational tidal forces could create significant effects if both objects are planet sized. We examine tidal effects in Tidal Forces .) Neither positive nor negative total energy precludes finite-sized masses from colliding. For real objects, direction is important.
Let’s consider the preceding example again, where we calculated the escape speed from Earth and the Sun, starting from Earth’s orbit. We noted that Earth already has an orbital speed of 30 km/s. As we see in the next section, that is the tangential speed needed to stay in circular orbit. If an object had this speed at the distance of Earth’s orbit, but was headed directly away from the Sun, how far would it travel before coming to rest? Ignore the gravitational effects of any other bodies.
Strategy
The object has initial kinetic and potential energies that we can calculate. When its speed reaches zero, it is at its maximum distance from the Sun. We use Equation 13.5, conservation of energy, to find the distance at which kinetic energy is zero.
Solution
The initial position of the object is Earth’s radius of orbit and the initial speed is given as 30 km/s. The final velocity is zero, so we can solve for the distance at that point from the conservation of energy equation. Using R ES = 1.50 x 10 11 m and M Sun = 1.99 x 10 30 kg, we have
\[\begin{split} \frac{1}{2} mv_{1}^{2} - \frac{GMm}{r_{1}} & = \frac{1}{2} mv_{2}^{2} - \frac{GMm}{r_{2}} \\ \frac{1}{2} \cancel{m} (30\; km/s)^{2} - \frac{(6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2})(1.99 \times 10^{30}\; kg) \cancel{m}}{1.50 \times 10^{11}\; m} & = \frac{1}{2} m(0)^{2} - \frac{(6.67 \times 10^{-11}\; N\; \cdotp m^{2}/kg^{2})(1.99 \times 10^{30}\; kg) \cancel{m}}{r_{2}} \end{split}\]
where the mass m cancels. Solving for r 2 we get r 2 = 3.0 x 10 11 m. Note that this is twice the initial distance from the Sun and takes us past Mars’s orbit, but not quite to the asteroid belt.
Significance
The object in this case reached a distance exactly twice the initial orbital distance. We will see the reason for this in the next section when we calculate the speed for circular orbits.
Assume you are in a spacecraft in orbit about the Sun at Earth’s orbit, but far away from Earth (so that it can be ignored). How could you redirect your tangential velocity to the radial direction such that you could then pass by Mars’s orbit? What would be required to change just the direction of the velocity?


20.6: Area of solid parallelograms
Let \(\square ABCD\) be a parallelogram in the Euclidean plane, \(a=AB\) and \(h\) be the distance between the lines \((AB)\) and \((CD)\) . Then
\(\text{area }(\blacksquare ABCD)=a\cdot h.\)
- Proof
-
Let \(A'\) and \(B'\) denote the foot points of \(A\) and \(B\) on the line \((CD)\) .
Note that \(ABB'A'\) is a rectangle with sides \(a\) and \(h\) . By Theorem 20.5.1 ,
\[\text{area }(\blacksquare ABB'A')=h\cdot a.\]
Without loss of generality, we may assume that \(\ blacksquare ABCA'\) contains \(\ blacksquare ABCD\) and \(\ blacksquare ABB'A'\) . In this case \(\ blacksquare ABCA'\) admits two subdivisions:
\(\blacksquare ABCA'=\blacksquare ABCD\cup\blacktriangle AA'D=\blacksquare ABB'A'\cup\blacksquare BB'C.\)
By Proposition 20.4.2 ,
\[\begin{aligned} \text{area }( \blacksquare ABCD)&+\text{area }(\blacktriangle AA'D)= \\ &= \text{area }(\blacksquare ABB'A')+ \text{area } (\blacktriangle BB'C). \end{aligned}\]
Note that
Indeed, since the quadrangles \(ABB'A'\) and \(ABCD\) are parallelograms, by Lemma 7.5.1 , we have that \(AA'=BB'\) , \(AD=BC\) , and \(DC=AB=A'B'\) . It follows that \(A'D=B'C\) . Applying the SSS congruence condition, we get 20.6.3.
In particular,
\[\text{area }(\blacktriangle BB'C)=\text{area } (\blacktriangle AA'D). \]
Subtracting 20.6.4 from 20.4.2, we get that
\[\text{area } (\blacksquare ABCD)=\text{area }(\blacksquare ABB'D).\]
It remains to apply 20.6.1.
Assume \(\square ABCD\) and \(\square AB'C'D'\) are two parallelograms such that \(B'\in[BC]\) and \(D\in [C'D']\) . Show that
\(\text{area }(\blacksquare ABCD)=\text{area }(\blacksquare AB'C'D').\)
- Hint
-
Suppose that \(E\) denotes the point of intersection of the lines \((BC)\) and \((C'D')\).
Use Proposition \(\PageIndex{1}\) to prove the following two identities:
\(\begin{array} {l} {\text{area } (\blacksquare AB'ED) = \text{area } (\blacksquare ABCD),} \\ {\text{area } (\blacksquare AB'ED) = \text{area } (\blacksquare AB'C'D')} \end{array}\)


4.14: Conclusion
We have now passed through nearly four millennia of East Asian history. We prefaced that history with the development of Neolithic cultures leading up to the second millennium BCE, at which point the first kingdoms in Chinese history emerged. Longshan culture (3000 – 1900 BCE) consisted of competing chiefdoms that laid the foundations for the emergence of the first major states.
The first major state was the Xia Dynasty (c. 1900 – 1600 BCE). Although historians debate whether or not it was legendary and lack written sources from that time, the archaeological evidence points to a Bronze Age civilization with a capital city at Erlitou. The next dynasty, the Shang (c. 1600 – 1046 BCE), brings us into history proper, because we have written evidence in the form of oracle bones and bronze inscriptions. The last Shang capital was located at Anyang on the North China Plain near the Yellow River. The dynasty’s most important legacies were the earliest form of Chinese writing and ancestor worship. The oracle bones were used to divine the will of higher powers and the spirits of deceased Shang kings.
The next dynasty, the Zhou Dynasty (1045 – 256 BCE), began when chieftains to the west of the Shang declared themselves kings and overthrew it. They justified doing so with the Mandate of Heaven, which states that a higher power withdraws its support from corrupt ruling families and transfers it to righteous ones. Because it lasted eight hundred years and saw much change, the Zhou is divided into two major periods, the Western (1045 – 771 BCE) and Eastern Zhou (770 – 256 BCE). During the Western Zhou, Zhou kings ruled by granting land and noble titles to kinsmen and allies in exchange for loyalty and service. The resulting political system, which saw China divided up by over one hundred feudal states, is known as Zhou feudalism. During the Eastern Zhou, this nobility became increasingly independent and fought with each other for power and territory. A few declared themselves kings and forged powerful militaries by gearing their kingdoms for war. In the end, in 221 BCE, the state of Qin prevailed, initiating China’s imperial era. Also, in the midst of the dislocation caused by Eastern Zhou instability, three of China’s major philosophical traditions emerged: Confucianism, Legalism, and Daoism.
The short-lived Qin Dynasty was followed by the much longer lasting Han Dynasty (202 BCE – 220 CE). During the Han Dynasty, Confucianism was adopted as the governing philosophy and also penetrated society, shaping the Chinese worldview. The Han also became an empire, as borders were expanded in all directions. During the reign of Emperor Wu (141 – 87 BCE), China extended control over parts of Korea, northern Vietnam, and also much of Central Asia. The resulting stability and the productivity of the economy spurred the development of the Silk Routes.
After the Han dynasty fell, China was divided up by independent, short-lived kingdoms until 589 CE, when the Sui Dynasty reunited most of the territory once controlled by the Han. Thus, for four centuries, during what is known as the Period of Division (220 – 589 CE), China was politically unstable and racked by endemic warfare. Yet, in spite of the violence, these centuries also saw vibrant cultural developments, as Buddhism became an organized institutional religion reshaping the spiritual landscape.
The Tang Dynasty (618 – 907 CE) was yet another long-lived one in Chinese history. Tang rulers built an empire on the foundations of solid political and legal institutions, agricultural policy, and a formidable military. Also during the Tang Dynasty, East Asia first emerged as an identifiable cultural sphere. By Tang times, kingdoms had already formed on the Korean Peninsula and the main islands of Japan, but it was during the Tang that ruling elites in both of these states made extensive efforts to adapt components of the Chinese political, legal, and writing system, as well as of Chinese culture, to their own societies. During the seventh century CE, the Silla Dynasty unified the Korean Peninsula, and the Yamato heavenly sovereigns unified much of Japan. Two other unique East Asian civilizations had taken shape.
The last major dynasty surveyed was the Song Dynasty (960 – 1279 CE), one that is notable for the challenges it faced from northern conquest dynasties, its economic prosperity, and the civil service examination system and the educated elite of scholar-officials it created. Also, since footbinding developed during the Song, we considered the status of women in Chinese society, where gender hierarchy was the norm.
The Song Dynasty ended with Mongol conquests in 1279 CE. The Mongol Yuan Dynasty made China one part of a much larger Eurasian territorial empire. Mongol ruled lasted until 1368 CE, when native rebellions overthrew a faltering Yuan state, initiating a new period in Chinese history: the Ming Dynasty. This dynasty properly belongs to early modern history.


12.1: The Economics of Cotton
In the antebellum era—that is, in the years before the Civil War—American planters in the South continued to grow Chesapeake tobacco and Carolina rice as they had in the colonial era. Cotton, however, emerged as the antebellum South’s major commercial crop, eclipsing tobacco, rice, and sugar in economic importance. By 1860, the region was producing two-thirds of the world’s cotton. In 1793, Eli Whitney revolutionized the production of cotton when he invented the cotton gin, a device that separated the seeds from raw cotton. Suddenly, a process that was extraordinarily labor-intensive when done by hand could be completed quickly and easily. American plantation owners, who were searching for a successful staple crop to compete on the world market, found it in cotton.
As a commodity, cotton had the advantage of being easily stored and transported. A demand for it already existed in the industrial textile mills in Great Britain, and in time, a steady stream of slave-grown American cotton would also supply northern textile mills. Southern cotton, picked and processed by American slaves, helped fuel the nineteenth-century Industrial Revolution in both the United States and Great Britain.
KING COTTON
Almost no cotton was grown in the United States in 1787, the year the federal constitution was written. However, following the War of 1812, a huge increase in production resulted in the so-called cotton boom, and by midcentury, cotton became the key cash crop (a crop grown to sell rather than for the farmer’s sole use) of the southern economy and the most important American commodity. By 1850, of the 3.2 million slaves in the country’s fifteen slave states, 1.8 million were producing cotton; by 1860, slave labor was producing over two billion pounds of cotton per year. Indeed, American cotton soon made up two-thirds of the global supply, and production continued to soar. By the time of the Civil War, South Carolina politician James Hammond confidently proclaimed that the North could never threaten the South because “cotton is king.”
The crop grown in the South was a hybrid: Gossypium barbadense , known as Petit Gulf cotton, a mix of Mexican, Georgia, and Siamese strains. Petit Gulf cotton grew extremely well in different soils and climates. It dominated cotton production in the Mississippi River Valley—home of the new slave states of Louisiana, Mississippi, Arkansas, Tennessee, Kentucky, and Missouri—as well as in other states like Texas. Whenever new slave states entered the Union, white slaveholders sent armies of slaves to clear the land in order to grow and pick the lucrative crop. The phrase “to be sold down the river,” used by Harriet Beecher Stowe in her 1852 novel Uncle Tom’s Cabin , refers to this forced migration from the upper southern states to the Deep South, lower on the Mississippi, to grow cotton.
The slaves who built this cotton kingdom with their labor started by clearing the land. Although the Jeffersonian vision of the settlement of new U.S. territories entailed white yeoman farmers single-handedly carving out small independent farms, the reality proved quite different. Entire old-growth forests and cypress swamps fell to the axe as slaves labored to strip the vegetation to make way for cotton. With the land cleared, slaves readied the earth by plowing and planting. To ambitious white planters, the extent of new land available for cotton production seemed almost limitless, and many planters simply leapfrogged from one area to the next, abandoning their fields every ten to fifteen years after the soil became exhausted. Theirs was a world of mobility and restlessness, a constant search for the next area to grow the valuable crop. Slaves composed the vanguard of this American expansion to the West.
Cotton planting took place in March and April, when slaves planted seeds in rows around three to five feet apart. Over the next several months, from April to August, they carefully tended the plants. Weeding the cotton rows took significant energy and time. In August, after the cotton plants had flowered and the flowers had begun to give way to cotton bolls (the seed-bearing capsule that contains the cotton fiber), all the plantation’s slaves—men, women, and children—worked together to pick the crop (Figure 12.1.2). On each day of cotton picking, slaves went to the fields with sacks, which they would fill as many times as they could. The effort was laborious, and a white “driver” employed the lash to make slaves work as quickly as possible.
Cotton planters projected the amount of cotton they could harvest based on the number of slaves under their control. In general, planters expected a good “hand,” or slave, to work ten acres of land and pick two hundred pounds of cotton a day. An overseer or master measured each individual slave’s daily yield. Great pressure existed to meet the expected daily amount, and some masters whipped slaves who picked less than expected.
Cotton picking occurred as many as seven times a season as the plant grew and continued to produce bolls through the fall and early winter. During the picking season, slaves worked from sunrise to sunset with a ten-minute break at lunch; many slaveholders tended to give them little to eat, since spending on food would cut into their profits. Other slaveholders knew that feeding slaves could increase productivity and therefore provided what they thought would help ensure a profitable crop. The slaves’ day didn’t end after they picked the cotton; once they had brought it to the gin house to be weighed, they then had to care for the animals and perform other chores. Indeed, slaves often maintained their own gardens and livestock, which they tended after working the cotton fields, in order to supplement their supply of food.
Sometimes the cotton was dried before it was ginned (put through the process of separating the seeds from the cotton fiber). The cotton gin allowed a slave to remove the seeds from fifty pounds of cotton a day, compared to one pound if done by hand. After the seeds had been removed, the cotton was pressed into bales. These bales, weighing about four hundred to five hundred pounds, were wrapped in burlap cloth and sent down the Mississippi River.
Visit the Internet Archive to watch a 1937 WPA film showing cotton bales being loaded onto a steamboat.
As the cotton industry boomed in the South, the Mississippi River quickly became the essential water highway in the United States. Steamboats, a crucial part of the transportation revolution thanks to their enormous freight-carrying capacity and ability to navigate shallow waterways, became a defining component of the cotton kingdom. Steamboats also illustrated the class and social distinctions of the antebellum age. While the decks carried precious cargo, ornate rooms graced the interior. In these spaces, whites socialized in the ship’s saloons and dining halls while black slaves served them (Figure 12.1.3).
Investors poured huge sums into steamships. In 1817, only seventeen plied the waters of western rivers, but by 1837, there were over seven hundred steamships in operation. Major new ports developed at St. Louis, Missouri; Memphis, Tennessee; and other locations. By 1860, some thirty-five hundred vessels were steaming in and out of New Orleans, carrying an annual cargo made up primarily of cotton that amounted to $220 million worth of goods (approximately $6.5 billion in 2014 dollars).
New Orleans had been part of the French empire before the United States purchased it, along with the rest of the Louisiana Territory, in 1803. In the first half of the nineteenth century, it rose in prominence and importance largely because of the cotton boom, steam-powered river traffic, and its strategic position near the mouth of the Mississippi River. Steamboats moved down the river transporting cotton grown on plantations along the river and throughout the South to the port at New Orleans. From there, the bulk of American cotton went to Liverpool, England, where it was sold to British manufacturers who ran the cotton mills in Manchester and elsewhere. This lucrative international trade brought new wealth and new residents to the city. By 1840, New Orleans alone had 12 percent of the nation’s total banking capital, and visitors often commented on the great cultural diversity of the city. In 1835, Joseph Holt Ingraham wrote: “Truly does New-Orleans represent every other city and nation upon earth. I know of none where is congregated so great a variety of the human species.” Slaves, cotton, and the steamship transformed the city from a relatively isolated corner of North America in the eighteenth century to a thriving metropolis that rivaled New York in importance (Figure 12.1.4).
THE DOMESTIC SLAVE TRADE
The South’s dependence on cotton was matched by its dependence on slaves to harvest the cotton. Despite the rhetoric of the Revolution that “all men are created equal,” slavery not only endured in the American republic but formed the very foundation of the country’s economic success. Cotton and slavery occupied a central—and intertwined—place in the nineteenth-century economy.
In 1807, the U.S. Congress abolished the foreign slave trade, a ban that went into effect on January 1, 1808. After this date, importing slaves from Africa became illegal in the United States. While smuggling continued to occur, the end of the international slave trade meant that domestic slaves were in very high demand. Fortunately for Americans whose wealth depended upon the exploitation of slave labor, a fall in the price of tobacco had caused landowners in the Upper South to reduce their production of this crop and use more of their land to grow wheat, which was far more profitable. While tobacco was a labor-intensive crop that required many people to cultivate it, wheat was not. Former tobacco farmers in the older states of Virginia and Maryland found themselves with “surplus” slaves whom they were obligated to feed, clothe, and shelter. Some slaveholders responded to this situation by freeing slaves; far more decided to sell their excess bondsmen. Virginia and Maryland therefore took the lead in the domestic slave trade, the trading of slaves within the borders of the United States.
The domestic slave trade offered many economic opportunities for white men. Those who sold their slaves could realize great profits, as could the slave brokers who served as middlemen between sellers and buyers. Other white men could benefit from the trade as owners of warehouses and pens in which slaves were held, or as suppliers of clothing and food for slaves on the move. Between 1790 and 1859, slaveholders in Virginia sold more than half a million slaves. In the early part of this period, many of these slaves were sold to people living in Kentucky, Tennessee, and North and South Carolina. By the 1820s, however, people in Kentucky and the Carolinas had begun to sell many of their slaves as well. Maryland slave dealers sold at least 185,000 slaves. Kentucky slaveholders sold some seventy-one thousand individuals. Most of the slave traders carried these slaves further south to Alabama, Louisiana, and Mississippi. New Orleans, the hub of commerce, boasted the largest slave market in the United States and grew to become the nation’s fourth-largest city as a result. Natchez, Mississippi, had the second-largest market. In Virginia, Maryland, the Carolinas, and elsewhere in the South, slave auctions happened every day.
All told, the movement of slaves in the South made up one of the largest forced internal migrations in the United States. In each of the decades between 1820 and 1860, about 200,000 people were sold and relocated. The 1800 census recorded over one million African Americans, of which nearly 900,000 were slaves. By 1860, the total number of African Americans increased to 4.4 million, and of that number, 3.95 million were held in bondage. For many slaves, the domestic slave trade incited the terror of being sold away from family and friends.
Solomon Northup was a free black man living in Saratoga, New York, when he was kidnapped and sold into slavery in 1841. He later escaped and wrote a book about his experiences: Twelve Years a Slave. Narrative of Solomon Northup, a Citizen of New-York, Kidnapped in Washington City in 1841 and Rescued in 1853 (the basis of a 2013 Academy Award–winning film) . This excerpt derives from Northup’s description of being sold in New Orleans, along with fellow slave Eliza and her children Randall and Emily.
One old gentleman, who said he wanted a coachman, appeared to take a fancy to me. . . .
The same man also purchased Randall. The little fellow was made to jump, and run across the floor, and perform many other feats, exhibiting his activity and condition. All the time the trade was going on, Eliza was crying aloud, and wringing her hands. She besought the man not to buy him, unless he also bought her self and Emily. . . .  He would not have such work—such snivelling; and unless she ceased that minute, he would take her to the yard and give her a hundred lashes. . . . Eliza shrunk before him, and tried to wipe away her tears, but it was all in vain. She wanted to be with her children, she said, the little time she had to live. All the frowns and threats of Freeman, could not wholly silence the afflicted mother.
What does Northup’s narrative tell you about the experience of being a slave? How does he characterize Freeman, the slave trader? How does he characterize Eliza?
THE SOUTH IN THE AMERICAN AND WORLD MARKETS
The first half of the nineteenth century saw a market revolution in the United States, one in which industrialization brought changes to both the production and the consumption of goods. Some southerners of the time believed that their region’s reliance on a single cash crop and its use of slaves to produce it gave the South economic independence and made it immune from the effects of these changes, but this was far from the truth. Indeed, the production of cotton brought the South more firmly into the larger American and Atlantic markets. Northern mills depended on the South for supplies of raw cotton that was then converted into textiles. But this domestic cotton market paled in comparison to the Atlantic market. About 75 percent of the cotton produced in the United States was eventually exported abroad. Exporting at such high volumes made the United States the undisputed world leader in cotton production. Between the years 1820 and 1860, approximately 80 percent of the global cotton supply was produced in the United States. Nearly all the exported cotton was shipped to Great Britain, fueling its burgeoning textile industry and making the powerful British Empire increasingly dependent on American cotton and southern slavery.
The power of cotton on the world market may have brought wealth to the South, but it also increased its economic dependence on other countries and other parts of the United States. Much of the corn and pork that slaves consumed came from farms in the West. Some of the inexpensive clothing, called “slops,” and shoes worn by slaves were manufactured in the North. The North also supplied the furnishings found in the homes of both wealthy planters and members of the middle class. Many of the trappings of domestic life, such as carpets, lamps, dinnerware, upholstered furniture, books, and musical instruments—all the accoutrements of comfortable living for southern whites—were made in either the North or Europe. Southern planters also borrowed money from banks in northern cities, and in the southern summers, took advantage of the developments in transportation to travel to resorts at Saratoga, New York; Litchfield, Connecticut; and Newport, Rhode Island.
Section Summary
In the years before the Civil War, the South produced the bulk of the world’s supply of cotton. The Mississippi River Valley slave states became the epicenter of cotton production, an area of frantic economic activity where the landscape changed dramatically as land was transformed from pinewoods and swamps into cotton fields. Cotton’s profitability relied on the institution of slavery, which generated the product that fueled cotton mill profits in the North. When the international slave trade was outlawed in 1808, the domestic slave trade exploded, providing economic opportunities for whites involved in many aspects of the trade and increasing the possibility of slaves’ dislocation and separation from kin and friends. Although the larger American and Atlantic markets relied on southern cotton in this era, the South depended on these other markets for food, manufactured goods, and loans. Thus, the market revolution transformed the South just as it had other regions.
Review Questions
Which of the following was not one of the effects of the cotton boom?
- U.S. trade increased with France and Spain.
- Northern manufacturing expanded.
- The need for slave labor grew.
- Port cities like New Orleans expanded.
A
The abolition of the foreign slave trade in 1807 led to _______.
- a dramatic decrease in the price and demand for slaves
- the rise of a thriving domestic slave trade
- a reform movement calling for the complete end to slavery in the United States
- the decline of cotton production
B
Why did some southerners believe their region was immune to the effects of the market revolution? Why was this thinking misguided?
Some southerners believed that their region’s monopoly over the lucrative cotton crop—on which both the larger American and Atlantic markets depended—and their possession of a slave labor force allowed the South to remain independent from the market revolution. However, the very cotton that provided the South with such economic potency also increased its reliance on the larger U.S. and world markets, which supplied—among other things—the food and clothes slaves needed, the furniture and other manufactured goods that defined the southern standard of comfortable living, and the banks from which southerners borrowed needed funds.
Glossary
- antebellum
- a term meaning “before the war” and used to describe the decades before the American Civil War began in 1861
- cash crop
- a crop grown to be sold for profit instead of consumption by the farmer’s family
- cotton boom
- the upswing in American cotton production during the nineteenth century
- cotton gin
- a device, patented by Eli Whitney in 1794, that separated the seeds from raw cotton quickly and easily
- domestic slave trade
- the trading of slaves within the borders of the United States


16.3: Organizational Change
-
- Last updated
- Save as PDF
- What are the fundamental dimensions of change?
Our discussion about organizational structure to this point has focused on the forms that an organization might take and the options that are available to managers as they design structures for their organizations. However, organizations are constantly evolving. One common refrain is that "there is nothing so constant as change." Because of this, there is no one best way to organize in all circumstances. Effective managers need to be aware of the various factors that drive the need for change. There advantages and disadvantages of each the various forms of organizing we have discussed. Managers need to adapt the organization so that it is ideally situated to accomplish current organizational goals. Thus, effective managers need to know how to plan and implement change to achieve organizational success.
We will begin this section by reviewing the types of changes that may occur in an organization. Then we will explore the organizational life cycle model, which explains how the structural needs of an organization evolve over time.
Types of Change
There are many different types of changes in organizations. The first, consistent with what we talked about so far in this chapter, is structural change . This has to do with the changes in the overall formal relationships within an organization. Examples of structural change include reorganizing departments or business units, adding employee positions, or revising job roles and assignments. These changes should be made to support broader objectives such as to centralize or decentralize operations, empower employees, or find greater efficiencies.
Another common type of change is technological change . Implementation of new technologies is often forced upon an organization as the environment shifts. For example, an industry upgrade in a commonly used software platform may require that employees learn new ways of working. Upgraded machinery or hardware may require employees to learn new procedures or restructure the way that they interact with one another. The advent of web-based cloud technologies is an example from the last decade and an example of ways which new forms of collaboration are becoming more available. Technological change often induces structural change because it requires different ways of connecting across an organizational system.
A third type of organizational change is culture change . Organizational culture refers to the common patterns of thinking and behaving within an organization. Culture is rooted in the underlying beliefs and assumptions that people hold of themselves and of the organization. These beliefs and assumptions create mindsets that shape the culture. Culture change is among the most difficult kinds of changes to create within an organizational system. It often involves reshaping and reimagining the core identity of the organization. A typical culture change process, if it is successful, requires many years to achieve.
The Organizational Life Cycle
Most organizations begin as very small systems that feature very loose structures. In a new venture, nearly every employee might contribute to many aspects of an organization’s work. As the business grows, the workload increases, and more workers are needed. Naturally, as the organization hires more and more people, employees being to specialize. Over time, these areas of specialization mature through differentiation , the process of organizing employees into groups that focus on specific functions in the organization. Usually, differentiated tasks should be organized in a way that makes them complementary, where each employee contributes an essential activity that supports the work and outputs of others in the organization.
The patterns and structures that appear in an organization need to evolve over time as an organization grows or declines, through four predictable phases (see Figure \(\PageIndex{1}\)). In the entrepreneurship phase, the organization is usually very small and agile, focusing on new products and markets. The founders typically focus on a variety of responsibilities, and they often share frequent and informal communication with all employees in the new company. Employees enjoy a very informal relationship, and the work assignments are very flexible. Usually, there is a loose, organic organizational structure in this phase.
The second phase, survival and early success , occurs as an organization begins to scale up and find continuing success. The organization develops more formal structures around more specialized job assignments. Incentives and work standards are adopted. The communication shifts to a more formal tone with the introduction of hierarchy with upper- and lower-level managers. It becomes impossible for every employee to have personal relationships with every other employee in the organization. At this stage, it becomes appropriate for introduce mechanistic structures that support the standardization and formalization required to create effective coordination across the organization.
In a third phase, sustained success or maturity , the organization expands and the hierarchy deepens, now with multiple levels of employees. Lower-level managers are given greater responsibility, and managers for significant areas of responsibility may be identified. Top executives begin to rely almost exclusively on lower-level leaders to handle administrative issues so that they can focus on strategic decisions that affect the overall organization. At this stage, the mechanistic structures of the organization are strengthened, and functional structures may be introduced. Often, tension emerges over how to find balance in the structure. Most organizations at this stage of development need to have elements of a mechanistic bureaucracy while maintaining an environment that allows for the innovation and flexibility that is a feature of an organic structure.
A transition to the fourth phase, renewal or decline , occurs when an organization expands to the point that its operations are far-flung and need to operate somewhat autonomously. Functional structures become almost essential, and subunits may begin to operate as independent businesses. Often, the tensions in the company between mechanistic and organic inclinations may be out of balance. To address these issues, the organization has to be reorganized or restructured to achieve higher levels of coordination between and among different groups or subunits. Managers may need to address fundamental questions about the overall direction and administration of the organization.
To summarize, the key insight about the organizational life cycle is that the needs of an organization will evolve over time. Different structures are needed at different stages as an organization develops. The needs of employees will also change. An understanding of the organizational life cycle provides a framework for thinking about changes that may be needed over time.
Dimensions of Change
When considering how to assess the need for change in an organization, it can be helpful to think of three dimensions: the scope of change, the level of change, and the intentionality of change.
The first, the scope of change refers to the degree to which the required change will disrupt current patterns and routines. Incremental change refers to small refinements in current organizational practices or routines that do not challenge, but rather build on or improve, existing aspects and practices within the organization. Common incremental change practices are LEAN and Six Sigma, which are used to find relatively small changes that can generate greater efficiencies in a process. An organization can improve its product-line efficiencies by identifying small discrepancies in process, then fixing them in a systematic way. Incremental change does not typically challenge people to be at the edge of their comfort zone.
In contrast, transformational change refers to significant shifts in an organizational system that may cause significant disruption to some underlying aspect of the organization, its processes, or structures. Transformational change can be invigorating for some employees, but also highly disruptive and stressful for others. Examples of transformational change include large systems changes and organizational restructuring. Culture change often requires transformational change to be successful. 7
Finally, a strategic change is a change, either incremental or transformational, that helps align an organization’s operations with its strategic mission and objectives. This kind of change is necessary for an organization to achieve the focus it needs to make needed transfer missions and work it does feel to stay competitive in the current or larger organization, larger market environment, or societal environment.
The level of change refers to the breadth of the systems that need to be changed within an organization. Individual-level change focuses on how to help employees to improve some active aspect of their performance or the knowledge they need to continue to contribute to the organization in an effective manner. Individual-level change programs include leadership development, training, and performance management. Group-level change centers on the relationships between people and usually focuses on helping people to work more effectively together. Team development, or teambuilding, is one of the most common forms of a team change process. Organization-level change is a change that affects an entire organizational system or several of its units. Strategic planning and implementation is perhaps the most common type of organization-level change. Higher-level change programs usually require changes at lower levels—an organization-level change may require change at both team and individual levels as well.
Intentionality is the final dimension of change and refers to the degree to which the change is intentionally designed or purposefully implemented. Planned change is an intentional activity or set of intentional activities that are designed to create movement toward a specific goal or end. Planned change processes often involve large groups of people and step-by-step or phase-by-phase activities that unfold over a period of time. Usually, effective leaders identify clear objectives for the change, the specific activities that will achieve those objectives, and the indicators of success.
In contrast, unplanned change is unintentional and is usually the result of informal organizing. It may or may not serve the aims of the organization as a whole. Unplanned change may be completely spontaneous, occurring simply because employees in some part of an organization want to initiate change. But sometimes it occurs as a byproduct of a planned change process. This is because it is difficult for leaders to anticipate all the consequences of a planned change effort. Employees react in unpredictable ways, technologies don't work as expected, changes in the marketplace don't happen as expected, or other actors may react in unanticipated ways.
As we will discuss below, some change models are designed to take advantage of the potential for spontaneous organizing among employees. Unplanned change can be harnessed as a positive force when employees are invited to be proactive about working toward common organizational goals.
concept check
- What are the fundamental dimensions of change?