5.9: Reference Material
-
- Last updated
- Save as PDF
This chapter was edited by Mary Anne Henderson, with content contributions by Andrew C. Baker, Peter Catapano, Blaine Hamilton, Mary Anne Henderson, Amanda Hughett, Amy Kohout, Maria Montalvo, Brent Ruswick, Philip Luke Sinitiere, Nora Slonimsky, Whitney Stewart, and Brandy Thomas Wells.
Recommended citation: Andrew C. Baker et al., “The Progressive Era,” Mary Anne Henderson, ed., in The American Yawp , eds. Joseph Locke and Ben Wright (Stanford, CA: Stanford University Press, 2018).
Recommended Reading
- Ayers, Edward. The Promise of the New South. New York: Oxford University Press, 1992.
- Bay, Mia. To Tell the Truth Freely: The Life of Ida B. Wells. New York: Hill and Wang, 2010.
- Cott, Nancy. The Grounding of Modern Feminism. New Haven, CT: Yale University Press, 1987.
- Dawley, Alan. Struggles for Justice: Social Responsibility and the Liberal State. Cambridge, MA: Harvard University Press, 1991.
- Dubois, Ellen Carol. Women’s Suffrage and Women’s Rights. New York: New York University Press, 1998.
- Filene, Peter. “An Obituary for ‘The Progressive Movement,’” American Quarterly 22 (Spring 1970): 20–34.
- Flanagan, Maureen. America Reformed: Progressives and Progressivisms, 1890s–1920s. New York: Oxford University Press, 2007.
- Foley, Neil. The White Scourge: Mexicans, Blacks, and Poor Whites in Texas Cotton Culture. Berkeley: University of California Press, 1997.
- Gilmore, Glenda E. Gender and Jim Crow: Women and the Politics of White Supremacy in North Carolina, 1896–1920. Chapel Hill: University of North Carolina Press, 1996.
- Hale, Grace Elizabeth. Making Whiteness: The Culture of Segregation in the South, 1890–1940. New York: Oxford University Press, 1998.
- Hicks, Cheryl. Talk with You Like a Woman: African American Women, Justice, and Reform in New York, 1890–1935. Chapel Hill: University of North Carolina Press, 2010.
- Hofstadter, Richard. The Age of Reform: From Bryan to F.D.R. New York: Knopf, 1955.
- Johnson, Kimberley. Governing the American State: Congress and the New Federalism, 1877–1929. Princeton, NJ: Princeton University Press, 2006.
- Kessler-Harris, Alice. In Pursuit of Equity: Women, Men, and the Quest for Economic Citizenship in 20th-Century America. New York: Oxford University Press, 2001.
- Kloppenberg, James T. Uncertain Victory: Social Democracy and Progressivism in European and American Thought, 1870–1920. New York: Oxford University Press, 1986.
- Kolko, Gabriel. The Triumph of Conservatism. New York: Free Press, 1963.
- Kousser, J. Morgan. The Shaping of Southern Politics: Suffrage Restriction and the Establishment of the One-Party South, 1880–1910 . New Haven, CT: Yale University Press, 1974.
- McGerr, Michael. A Fierce Discontent: The Rise and Fall of the Progressive Movement in America, 1870–1920. New York: Free Press, 2003.
- Molina, Natalia. Fit to Be Citizens?: Public Health and Race in Los Angeles, 1879–1939. Berkeley: University of California Press, 2006.
- Muncy, Robyn. Creating a Female Dominion in American Reform, 1890–1935. New York: Oxford University Press, 1991.
- Rodgers, Daniel T. Atlantic Crossings: Social Politics in a Progressive Age. Cambridge, MA: Harvard University Press, 2000.
- Sanders, Elizabeth. The Roots of Reform: Farmers, Workers, and the American State, 1877–1917. Chicago: University of Chicago Press, 1999.
- Stromquist, Shelton. Re-Inventing “The People”: The Progressive Movement, the Class Problem, and the Origins of Modern Liberalism. Champaign: University of Illinois Press, 2006.
- White, Deborah. Too Heavy a Load: In Defense of Themselves. New York: Norton, 1999.
- Wiebe, Robert. The Search for Order, 1877–1920. New York: Hill and Wang, 1967.
Notes
- Jack London, The Iron Heel (New York: Macmillan, 1908), 104.
- Philip Foner, Women and the American Labor Movement: From Colonial Times to the Eve of World War I (New York: Free Press, 1979.).
- Leon Stein, The Triangle Fire (Ithaca, NY: Syracuse University Press, 1962), 20.
- Ibid., 144
- Ray Stannard Baker, American Chronicle: The Autobiography of Ray Stannard Baker (New York: Scribner, 1945), 183.
- Jacob A. Riis, How the Other Half Lives: Studies Among the Tenements of New York (New York: Scribner, 1890)
- Upton Sinclair, The Jungle (New York: Doubleday, 1906), 40.
- Edward Bellamy, Looking Backward: 2000–1887 (Boston: Ticknor, 1888).
- Ibid., 368.
- Charles M. Sheldon, In His Steps: “What Would Jesus Do?” (Chicago: Advance, 1896), 273.
- Walter Rauschenbusch, A Theology for the Social Gospel (New York: Macmillan, 1917).
- Ibid., 5.
- John Kobler, Ardent Spirits: The Rise and Fall of Prohibition (Boston: Da Capo Press, 1993), 147.
- Toynbee Hall was the first settlement house. It was built in 1884 by Samuel Barnett as a place for Oxford students to live while at the same time working in the house’s poor neighborhood. Daniel Rodgers, Atlantic Crossings: Social Politics in a Progressive Age (Cambridge, MA: Belknap Press, 1998), 64–65; Victoria Bissell Brown, The Education of Jane Addams (Philadelphia: University of Pennsylvania Press, 2004).
- Jane Addams, Twenty Years at Hull House (New York: Macmillan, 1911), 125–126.
- Allen Davis, American Heroine: The Life and Legend of Jane Addams (New York: Oxford University Press, 1979), 77.
- Jane Addams, “The Settlement as a Factor in the Labor Movement,” reprinted in Hull-House Maps and Papers: A Presentation of Nationalities and Wages in a Congested District of Chicago Together with Comments and Essays on Problems Growing out of the Social Conditions (Chicago: University of Illinois Press, 2007), 145, 149.
- Kathryn Kish Sklar, “‘Some of Us Who Deal with the Social Fabric’: Jane Addams Blends Peace and Social Justice, 1907–1919,” Journal of the Gilded Age and Progressive Era 2, no. 1 (January 2003).
- Karen Manners Smith, “New Paths to Power: 1890–1920,” in No Small Courage: A History of Women in the United States , ed. Nancy Cott (New York: Oxford University Press, 2000), 392.
- Sarah Eisenstein, Give Us Bread but Give Us Roses: Working Women’s Consciousness in the United States, 1890 to the First World War (New York: Routledge, 1983), 32.
- Ellen Carol Dubois, Women’s Suffrage and Women’s Rights (New York: New York University Press, 1998).
- Herbert Croly, The Promise of American Life (New York: Macmillan, 1911), 145.
- Kevin P. Phillips, Wealth and Democracy: A Political History of the American Rich (New York: Broadway Books, 2003), 307.
- Munn v. Illinois , 94 U.S. 113 (1877).
- Interstate Commerce Act of 1887.
- The writer Henry Adams said that he “showed the singular primitive quality that belongs to ultimate matter—the quality that medieval theology assigned to God—he was pure act.” Henry Adams, The Education of Henry Adams (New York: Houghton Mifflin, 1918), 413.
- Theodore Roosevelt, Addresses and Presidential Messages of Theodore Roosevelt, 1902–1904 , 15.
- The historiography on American progressive politics is vast. See, for instance, Michael McGerr, A Fierce Discontent: The Rise and Fall of the Progressive Movement in America, 1870–1920 (New York: Free Press, 2003).
- Roderick Nash, Wilderness and the American Mind , 4th ed. (New Haven, CT: Yale University Press, 2001), 167–168, 171, 165.
- John Muir, Our National Parks (Boston: Houghton Mifflin, 1901)
- Gifford Pinchot, The Fight for Conservation (New York: Doubleday Page, 1910), 44.
- Michael Perman, Struggle for Mastery: Disfranchisement in the South, 1888–1908 (Chapel Hill: University of North Carolina Press, 2001).
- Edward Ayers, The Promise of the New South (New York: Oxford University Press, 1992), 147
- Ibid.
- Ibid.
- Neil R. McMillen, Dark Journey: Black Mississippians in the Age of Jim Crow (Champaign: University of Illinois Press, 1990), 43.
- Perman, Struggle for Mastery , 147.
- Plessy v. Ferguson , 163 U.S. 537 (1896).
- Booker T. Washington, Up from Slavery: An Autobiography (New York: Doubleday, 1901), 221–222.
- Kate A. Baldwin, Beyond the Color Line and the Iron Curtain: Reading Encounters Between Black and Red (Durham, NC: Duke University Press, 2002), 297 n. 28.
- W. E. B. DuBois, “Niagara’s Declaration of Principles, 1905,” Gilder Lehrman Center for the Study of Slavery, Resistance, and Abolition, https://glc.yale.edu/niagaras-declar...rinciples-1905 , accessed June 15, 2018.


9.20: Group Web Sharing Calls
- Identify strategies for effective use of web sharing in a group setting
The main difference between one-on-one web-sharing calls and group web sharing is that distraction and inattention increase exponentially with each person on the call. In a one-on-one situation, if the person you’re talking to gets lost or needs more explanation, they’ll just ask. In a group situation, they may be too embarrassed, or they may ask long after you’ve moved on. Thus, it is essential that as a presenter you do the following:
- Use navigation language. It becomes even more important with more distractions on the call.
- Regulate your pace so you don’t speed through your information.
- Pause briefly between slides, worksheets, sections, or pages to help people recognize that you’re moving on.
If you are sharing PowerPoint slides, animations can help people stay engaged. This is not to say that you should have text or images swooping in and dancing on the page. Rather, making your bullet points appear one at a time or having a piece of your pie chart flash when you start to talk about it can help your listeners re-engage with the slides.
Most web-sharing platforms have a chat function that you can choose to use. If you do, it’s good to set some ground rules about how participants should engage with the chat. The chat typically appears as a running bar along the side of the screen that looks a bit like an IM thread. Participants can post questions or comments there. Think about using the chat if you are in any of these situations:
-
You are anticipating a lot of questions, and you want to be sure to get through all your content. Having participants post their comments in the chat rather than asking out loud, can help a presenter accomplish the following:
- Delay answering questions you know will be addressed later in the conversation.
- Weed out duplicate questions.
- Table off-topic questions so you can discuss them with the individual at a later time.
- You have a large group on the call, so even one question per person could really derail the rhythm of the meeting.
- You might want to ask poll-type questions of the group. For example, you might say, “Type in the chat area the number of loss-prevention reports you’ve filed so far this year.” Knowing the answer to a question like this might help you shape and prioritize the rest of the meeting.
Skip the chat if you feel it might invite unwanted comments or side conversations, if the group is too small to need it, or if the purpose of the meeting is to have a discussion equally among all participants.
Contributors and Attributions
- Group Web Sharing Calls. Authored by : Barbara Egel. Provided by : Lumen Learning. License : CC BY: Attribution


12.0: Prelude to Planning for Your Future
selected template will load here
This action is not available.
How ready are you to plan your career? Take this quick survey to figure it out, ranking questions on a scale of 1–4, 1 meaning “least like me” and 4 meaning “most like me.”
You can also take the Chapter 12 survey anonymously online.
STUDENT PROFILE
“A lesson I have learned throughout my college career is that changing majors is okay. Fresh out of high school, I always wanted to be an FBI agent. I wanted to be in those forensic shows—like Criminal Minds —that we all see on TV. So I decided to go to college and major in criminal justice with an emphasis in forensic science. When I started to take criminal justice classes, I noticed that my passion and determination were focused on something else: helping others and changing people’s lives so they do not have to struggle as much as I did. I quickly decided to talk to a career counselor and let him know that I wanted to change majors. We discussed the long-term plans and where I would see myself in the future. I changed my major to a related one, sociology. And with that, I want to be a counselor, as I want to help others achieve their goals.”
—Carlos Espinosa
In earlier chapters of this book (1, 3, 4), you learned more about setting the foundation for college and career success by gaining a deeper understanding of why you are attending college, how to set goals and priorities, and how to begin your academic and life planning.


4.3: Newton’s Second Law of Motion- Concept of a System
Learning Objectives
By the end of this section, you will be able to:
- Define net force, external force, and system.
- Understand Newton’s second law of motion.
- Apply Newton’s second law to determine the weight of an object.
Newton’s second law of motion is closely related to Newton’s first law of motion. It mathematically states the cause and effect relationship between force and changes in motion. Newton’s second law of motion is more quantitative and is used extensively to calculate what happens in situations involving a force. Before we can write down Newton’s second law as a simple equation giving the exact relationship of force, mass, and acceleration, we need to sharpen some ideas that have already been mentioned.
First, what do we mean by a change in motion? The answer is that a change in motion is equivalent to a change in velocity. A change in velocity means, by definition, that there is an acceleration . Newton’s first law says that a net external force causes a change in motion; thus, we see that a net external force causes acceleration .
Another question immediately arises. What do we mean by an external force? An intuitive notion of external is correct — an external force acts from outside the system of interest. For example, in Figure \(\PageIndex{1a}\) the system of interest is the wagon plus the child in it. The two forces exerted by the other children are external forces. An internal force acts between elements of the system. Again looking at Figure \(\PageIndex{1a}\), the force the child in the wagon exerts to hang onto the wagon is an internal force between elements of the system of interest. Only external forces affect the motion of a system, according to Newton’s first law. (The internal forces actually cancel, as we shall see in the next section.) You must define the boundaries of the system before you can determine which forces are external . Sometimes the system is obvious, whereas other times identifying the boundaries of a system is more subtle. The concept of a system is fundamental to many areas of physics, as is the correct application of Newton’s laws. This concept will be revisited many times on our journey through physics.
Now, it seems reasonable that acceleration should be directly proportional to and in the same direction as the net (total) external force acting on a system. This assumption has been verified experimentally and is illustrated in Figure . In part (a), a smaller force causes a smaller acceleration than the larger force illustrated in part (c). For completeness, the vertical forces are also shown; they are assumed to cancel since there is no acceleration in the vertical direction. The vertical forces are the weight \(w\) and the support of the ground \(N\), and the horizontal force \(f\) represents the force of friction. These will be discussed in more detail in later sections. For now, we will define friction as a force that opposes the motion past each other of objects that are touching. Figure \(\PageIndex{1b}\) shows how vectors representing the external forces add together to produce a net force, \(F_{net}\).
To obtain an equation for Newton’s second law, we first write the relationship of acceleration and net external force as the proportionality
\[ a \propto F_{net} \]
where the symbol \( \propto \) means “proportional to,” and \( F_{net} \) is the net external force . (The net external force is the vector sum of all external forces and can be determined graphically, using the head-to-tail method, or analytically, using components. The techniques are the same as for the addition of other vectors, and are covered in the chapter section on Two-Dimensional Kinematics .) This proportionality states what we have said in words— acceleration is directly proportional to the net external force . Once the system of interest is chosen, it is important to identify the external forces and ignore the internal ones. It is a tremendous simplification not to have to consider the numerous internal forces acting between objects within the system, such as muscular forces within the child’s body, let alone the myriad of forces between atoms in the objects, but by doing so, we can easily solve some very complex problems with only minimal error due to our simplification
Now, it also seems reasonable that acceleration should be inversely proportional to the mass of the system. In other words, the larger the mass (the inertia), the smaller the acceleration produced by a given force. And indeed, as illustrated in Figure , the same net external force applied to a car produces a much smaller acceleration than when applied to a basketball. The proportionality is written as
\[ a \propto \dfrac{1}{m}, \]
where \(m\) is the mass of the system. Experiments have shown that acceleration is exactly inversely proportional to mass, just as it is exactly linearly proportional to the net external force.
It has been found that the acceleration of an object depends only on the net external force and the mass of the object. Combining the two proportionalities just given yields Newton's second law of motion.
Newton’s Second Law of Motion
The acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass. In equation form, Newton’s second law of motion is
\[ a = \dfrac{F_{net}}{m}\]
This is often written in the more familiar form
\[ F_{net} = ma. \]
When only the magnitude of force and acceleration are considered, this equation is simply
\[ F_{net} = ma. \]
Although these last two equations are really the same, the first gives more insight into what Newton’s second law means. The law is a cause and effect relationship among three quantities that is not simply based on their definitions. The validity of the second law is completely based on experimental verification.
Units of Force
\( F_{net} = ma \) is used to define the units of force in terms of the three basic units for mass, length, and time. The SI unit of force is called the newton (abbreviated N) and is the force needed to accelerate a 1-kg system at the rate of \( 1m/s^2\) That is, since \( F_{net} = ma, \) \[ 1N = 1 kg \cdot s^2 \]
While almost the entire world uses the newton for the unit of force, in the United States the most familiar unit of force is the pound (lb), where 1 N = 0.225 lb.
Weight and the Gravitational Force
When an object is dropped, it accelerates toward the center of Earth. Newton’s second law states that a net force on an object is responsible for its acceleration. If air resistance is negligible, the net force on a falling object is the gravitational force, commonly called its weight \(w\). Weight can be denoted as a vector \(w\) because it has a direction; down is, by definition, the direction of gravity, and hence weight is a downward force. The magnitude of weight is denoted as \(w\). Galileo was instrumental in showing that, in the absence of air resistance, all objects fall with the same acceleration \(w\). Using Galileo’s result and Newton’s second law, we can derive an equation for weight.
Consider an object with mass \(m\) falling downward toward Earth. It experiences only the downward force of gravity, which has magnitude \(w\). Newton’s second law states that the magnitude of the net external force on an object is \(F_{net} = ma\).
Since the object experiences only the downward force of gravity, \(F_{net} = w\). We know that the acceleration of an object due to gravity is \(g\), or \( a = g\). Substituting these into Newton’s second law gives
WEIGHT
This is the equation for weight - the gravitational force on mass \(m\):
\[ w = mg\]
Since weight \( g = 9.80 m/s^2 \) on Earth, the weight of a 1.0 kg object on Earth is 9.8 N, as we see: \[ w = mg = (1.0 kg) (9.8 m/s^2) = 9.8 N. \]
Recall that \(g\) can take a positive or negative value, depending on the positive direction in the coordinate system. Be sure to take this into consideration when solving problems with weight.
When the net external force on an object is its weight, we say that it is in free-fall . That is, the only force acting on the object is the force of gravity. In the real world, when objects fall downward toward Earth, they are never truly in free-fall because there is always some upward force from the air acting on the object.
The acceleration due to gravity \(g\) varies slightly over the surface of Earth, so that the weight of an object depends on location and is not an intrinsic property of the object. Weight varies dramatically if one leaves Earth’s surface. On the Moon, for example, the acceleration due to gravity is only \(1.67 m/s^2\). A 1.0-kg mass thus has a weight of 9.8 N on Earth and only about 1.7 N on the Moon.
The broadest definition of weight in this sense is that the weight of an object is the gravitational force on it from the nearest large body , such as Earth, the Moon, the Sun, and so on. This is the most common and useful definition of weight in physics. It differs dramatically, however, from the definition of weight used by NASA and the popular media in relation to space travel and exploration. When they speak of “weightlessness” and “microgravity,” they are really referring to the phenomenon we call “free-fall” in physics. We shall use the above definition of weight, and we will make careful distinctions between free-fall and actual weightlessness.
It is important to be aware that weight and mass are very different physical quantities, although they are closely related. Mass is the quantity of matter (how much “stuff”) and does not vary in classical physics, whereas weight is the gravitational force and does vary depending on gravity. It is tempting to equate the two, since most of our examples take place on Earth, where the weight of an object only varies a little with the location of the object. Furthermore, the terms mass and weight are used interchangeably in everyday language; for example, our medical records often show our “weight” in kilograms, but never in the correct units of newtons.
COMMON MISCONCEPTIONS: MASS VS. WEIGHT
Mass and weight are often used interchangeably in everyday language. However, in science, these terms are distinctly different from one another. Mass is a measure of how much matter is in an object. The typical measure of mass is the kilogram (or the “slug” in English units). Weight, on the other hand, is a measure of the force of gravity acting on an object. Weight is equal to the mass of an object \((m)\) multiplied by the acceleration due to gravity \((g)\). Like any other force, weight is measured in terms of newtons (or pounds in English units).
Assuming the mass of an object is kept intact, it will remain the same, regardless of its location. However, because weight depends on the acceleration due to gravity, the weight of an object can change when the object enters into a region with stronger or weaker gravity. For example, the acceleration due to gravity on the Moon is \(1.67 m/s^2 \) (which is much less than the acceleration due to gravity on Earth, \(9.80 m/s^2\) ). If you measured your weight on Earth and then measured your weight on the Moon, you would find that you “weigh” much less, even though you do not look any skinnier. This is because the force of gravity is weaker on the Moon. In fact, when people say that they are “losing weight,” they really mean that they are losing “mass” (which in turn causes them to weigh less)
TAKE-HOME EXPERIMENT: MASS AND WEIGHT
What do bathroom scales measure? When you stand on a bathroom scale, what happens to the scale? It depresses slightly. The scale contains springs that compress in proportion to your weight—similar to rubber bands expanding when pulled. The springs provide a measure of your weight (for an object which is not accelerating). This is a force in newtons (or pounds). In most countries, the measurement is divided by 9.80 to give a reading in mass units of kilograms. The scale measures weight but is calibrated to provide information about mass. While standing on a bathroom scale, push down on a table next to you. What happens to the reading? Why? Would your scale measure the same “mass” on Earth as on the Moon?
Example \(\PageIndex{1}\):What Acceleration Can a Person Produce when pushing a Lawn Mower?
Suppose that the net external force (push minus friction) exerted on a lawn mower is 51 N (about 11 lb) parallel to the ground. The mass of the mower is 24 kg. What is its acceleration?
Strategy
Since \( F_{net} \) and \( m \) are given, the acceleration can be calculated directly from Newton’s second law as stated in \( F_{net} = ma \).
Solution
The magnitude of the acceleration \(a\) is \(a = \frac{F_{net}}{m}\). Entering known values gives \[ a = \dfrac{51 \, N}{24 \, kg} \]
Substituting the units \( kg \cdot m/s^2 \) for N yields \[ a = \dfrac{ 51 \, kg/s^2}{24\space kg} = 2.1\space m/s^2 \]
Discussion
The direction of the acceleration is the same direction as that of the net force, which is parallel to the ground. There is no information given in this example about the individual external forces acting on the system, but we can say something about their relative magnitudes. For example, the force exerted by the person pushing the mower must be greater than the friction opposing the motion (since we know the mower moves forward), and the vertical forces must cancel if there is to be no acceleration in the vertical direction (the mower is moving only horizontally). The acceleration found is small enough to be reasonable for a person pushing a mower. Such an effort would not last too long because the person’s top speed would soon be reached.
Example \(\PageIndex{2}\):What Rocket Thrust Accelerates This Sled?
Prior to manned space flights, rocket sleds were used to test aircraft, missile equipment, and physiological effects on human subjects at high speeds. They consisted of a platform that was mounted on one or two rails and propelled by several rockets. Calculate the magnitude of force exerted by each rocket, called its thrust \(T\) for the four-rocket propulsion system shown in Figure. The sled’s initial acceleration is \(49 m/s^2\) the mass of the system is 2100 kg, and the force of friction opposing the motion is known to be 650 N.
Strategy
Although there are forces acting vertically and horizontally, we assume the vertical forces cancel since there is no vertical acceleration. This leaves us with only horizontal forces and a simpler one-dimensional problem. Directions are indicated with plus or minus signs, with right taken as the positive direction. See the free-body diagram in the figure.
Solution
Since acceleration, mass, and the force of friction are given, we start with Newton’s second law and look for ways to find the thrust of the engines. Since we have defined the direction of the force and acceleration as acting “to the right,” we need to consider only the magnitudes of these quantities in the calculations. Hence we begin with \[ F_{net} = ma. \], where \(F_{net}\) is the net force along the horizontal direction. We can see from Figure that the engine thrusts add, while friction opposes the thrust. In equation form, the net external force is \[ F_{net} = 4T - f. \]
Substituting this into Newton’s second law gives \[ F_{net} = ma = 4T - f.\]
Using a little algebra, we solve for the total thrust 4T: \[ 4T = ma + f. \]
Substituting known values yields \[ 4T = ma + f = (2100 \, kg)(49 \, m/s^2) + 650 \, N \]
So the total thrust is \[ 1 \times 10^5 N, \]
and the individual thrusts are \[ T = \dfrac{1\times 10^5}{4} = 2.6 \times 10^4 \, N \]
Discussion
The numbers are quite large, so the result might surprise you. Experiments such as this were performed in the early 1960s to test the limits of human endurance and the setup designed to protect human subjects in jet fighter emergency ejections. Speeds of 1000 km/h were obtained, with accelerations of 45 \(g\)-s. (Recall that \(g\), the acceleration due gravity is \(9.80 \, m/s^2 \). When we say that an acceleration is 45 \(g\)-s, it is \(45 \times 9.80 m/s^2\), which is approximately \(440 m/s^2\) ). While living subjects are not used any more, land speeds of 10,000 km/h have been obtained with rocket sleds. In this example, as in the preceding one, the system of interest is obvious. We will see in later examples that choosing the system of interest is crucial—and the choice is not always obvious.
Newton’s second law of motion is more than a definition; it is a relationship among acceleration, force, and mass. It can help us make predictions. Each of those physical quantities can be defined independently, so the second law tells us something basic and universal about nature. The next section introduces the third and final law of motion.
Summary
- Acceleration, \(a\), is defined as a change in velocity, meaning a change in its magnitude or direction, or both.
- An external force is one acting on a system from outside the system, as opposed to internal forces, which act between components within the system.
- Newton’s second law of motion states that the acceleration of a system is directly proportional to and in the same direction as the net external force acting on the system, and inversely proportional to its mass.
- In equation form, Newton’s second law of motion is \( a = \frac{F_{net}}{m} \)
- This is often written in the more familiar form: \( F_{net} = ma. \)
- The weight \(w\) of an object is defined as the force of gravity acting on an object of mass \(m.\) The object experiences an acceleration due to gravity \(g\): \[ w = mg. \]
- If the only force acting on an object is due to gravity, the object is in free fall.
- Friction is a force that opposes the motion past each other of objects that are touching.


18.13: Sources of Magnetic Fields (Summary)
-
- Last updated
- Save as PDF
Key Terms
| Ampère’s law | physical law that states that the line integral of the magnetic field around an electric current is proportional to the current |
| Biot-Savart law | an equation giving the magnetic field at a point produced by a current-carrying wire |
| diamagnetic materials | their magnetic dipoles align oppositely to an applied magnetic field; when the field is removed, the material is unmagnetized |
| ferromagnetic materials | contain groups of dipoles, called domains, that align with the applied magnetic field; when this field is removed, the material is still magnetized |
| hysteresis | property of ferromagnets that is seen when a material’s magnetic field is examined versus the applied magnetic field; a loop is created resulting from sweeping the applied field forward and reverse |
| magnetic domains | groups of magnetic dipoles that are all aligned in the same direction and are coupled together quantum mechanically |
| magnetic susceptibility | ratio of the magnetic field in the material over the applied field at that time; positive susceptibilities are either paramagnetic or ferromagnetic (aligned with the field) and negative susceptibilities are diamagnetic (aligned oppositely with the field) |
| paramagnetic materials | their magnetic dipoles align partially in the same direction as the applied magnetic field; when this field is removed, the material is unmagnetized |
| permeability of free space | \(\displaystyle μ_0\), measure of the ability of a material, in this case free space, to support a magnetic field |
| solenoid | thin wire wound into a coil that produces a magnetic field when an electric current is passed through it |
| toroid | donut-shaped coil closely wound around that is one continuous wire |
Key Equations
| Permeability of free space | \(\displaystyle μ_0=4π×10^{−7}T⋅m/A\) |
| Contribution to magnetic field from a current element | \(\displaystyle dB=\frac{μ_0}{4π}\frac{Idlsinθ}{r^2}\) |
| Biot–Savart law | \(\displaystyle \vec{B}=\frac{μ_0}{4π}∫_{wire}\frac{Id\vec{l}×\hat{r}}{r^2}\) |
| Magnetic field due to a long straight wire | \(\displaystyle B=\frac{μ_0I}{2πR}\) |
| Force between two parallel currents | \(\displaystyle \frac{F}{l}=\frac{μ_0I_1I_2}{2πr}\) |
| Magnetic field of a current loop | \(\displaystyle B=\frac{μ_0I}{2R}\) (at center of loop) |
| Ampère’s law | \(\displaystyle ∮\vec{B}⋅d\vec{l}=μ_0I\) |
| Magnetic field strength inside a solenoid | \(\displaystyle B=μ_0nI\) |
| Magnetic field strength inside a toroid | \(\displaystyle B=\frac{μ_oNI}{2πr}\) |
| Magnetic permeability | \(\displaystyle μ=(1+χ)μ_0\) |
| Magnetic field of a solenoid filled with paramagnetic material | \(\displaystyle B=μnI\) |
Summary
12.2 The Biot-Savart Law
- The magnetic field created by a current-carrying wire is found by the Biot-Savart law.
- The current element \(\displaystyle Id\vec{l}\) produces a magnetic field a distance r away.
12.3 Magnetic Field Due to a Thin Straight Wire
- The strength of the magnetic field created by current in a long straight wire is given by \(\displaystyle B=\frac{μ_0I}{2πR}\) (long straight wire) where I is the current, R is the shortest distance to the wire, and the constant \(\displaystyle μ_0=4π×10^{−7}T⋅m/s\) is the permeability of free space.
- The direction of the magnetic field created by a long straight wire is given by right-hand rule 2 (RHR-2): Point the thumb of the right hand in the direction of current, and the fingers curl in the direction of the magnetic field loops created by it.
12.4 Magnetic Force between Two Parallel Currents
- The force between two parallel currents \(\displaystyle I_1\) and \(\displaystyle I_2\), separated by a distance r, has a magnitude per unit length given by \(\displaystyle \frac{F}{l}=\frac{μ_0I_1I_2}{2πr}\).
- The force is attractive if the currents are in the same direction, repulsive if they are in opposite directions.
12.5 Magnetic Field of a Current Loop
- The magnetic field strength at the center of a circular loop is given by \(\displaystyle B=\frac{μ_0I}{2R}\) (at center of loop), , where R is the radius of the loop. RHR-2 gives the direction of the field about the loop.
12.6 Ampère’s Law
- The magnetic field created by current following any path is the sum (or integral) of the fields due to segments along the path (magnitude and direction as for a straight wire), resulting in a general relationship between current and field known as Ampère’s law.
- Ampère’s law can be used to determine the magnetic field from a thin wire or thick wire by a geometrically convenient path of integration. The results are consistent with the Biot-Savart law.
12.7 Solenoids and Toroids
- The magnetic field strength inside a solenoid is
\(\displaystyle B=μ_0nI\) (inside a solenoid)
where n is the number of loops per unit length of the solenoid. The field inside is very uniform in magnitude and direction.
- The magnetic field strength inside a toroid is
\(\displaystyle B=\frac{μ_oNI}{2πr}\) (within the toroid)
where N is the number of windings. The field inside a toroid is not uniform and varies with the distance as 1/r.
12.8 Magnetism in Matter
- Materials are classified as paramagnetic, diamagnetic, or ferromagnetic, depending on how they behave in an applied magnetic field.
- Paramagnetic materials have partial alignment of their magnetic dipoles with an applied magnetic field. This is a positive magnetic susceptibility. Only a surface current remains, creating a solenoid-like magnetic field.
- Diamagnetic materials exhibit induced dipoles opposite to an applied magnetic field. This is a negative magnetic susceptibility.
- Ferromagnetic materials have groups of dipoles, called domains, which align with the applied magnetic field. However, when the field is removed, the ferromagnetic material remains magnetized, unlike paramagnetic materials. This magnetization of the material versus the applied field effect is called hysteresis.


9.10: Music of the 20th Century
Let’s begin the study of our final historical period with an overview of major trends and composers from the era. As you read this page, please pay special attention to the fact that this description focuses on compositional techniques and very little is said about dominant genres. The 20th century was clearly a period of widespread experimentation and many composers wanted the freedom to explore new compositional approaches without the restrictions and expectations that accompany traditional genres. Even when longstanding genres were used, composers felt very comfortable abandoning the traditional structures of those genres.
History
At the turn of the century, music was characteristically late Romantic in style. Composers such as Gustav Mahler, Richard Strauss and Jean Sibelius were pushing the bounds of Post-Romantic Symphonic writing. At the same time, the Impressionist movement, spearheaded by Claude Debussy, was being developed in France. The term was actually loathed by Debussy: “I am trying to do ‘something different—in a way realities—what the imbeciles call ‘impressionism’ is a term which is as poorly used as possible, particularly by art critics”—and Maurice Ravel’s music, also often labelled with this term, explores music in many styles not always related to it (see the discussion on Neoclassicism, below).
Many composers reacted to the Post-Romantic and Impressionist styles and moved in quite different directions. The single most important moment in defining the course of music throughout the century was the widespread break with traditional tonality, effected in diverse ways by different composers in the first decade of the century. From this sprang an unprecedented “linguistic plurality” of styles, techniques, and expression. In Vienna, Arnold Schoenberg developed atonality, out of the expressionism that arose in the early part of the 20th century. He later developed the twelve-tone technique which was developed further by his disciples Alban Berg and Anton Webern; later composers (including Pierre Boulez) developed it further still. Stravinsky (in his last works) explored twelve-tone technique, too, as did many other composers; indeed, even Scott Bradley used the technique in his scores for the Tom and Jerry cartoons.
After the First World War, many composers started returning to the past for inspiration and wrote works that draw elements (form, harmony, melody, structure) from it. This type of music thus became labelled neoclassicism. Igor Stravinsky ( Pulcinella and Symphony of Psalms ), Sergei Prokofiev ( Classical Symphony ), Ravel ( Le tombeau de Couperin ) and Paul Hindemith ( Symphony: Mathis der Maler ) all produced neoclassical works.
Italian composers such as Francesco Balilla Pratella and Luigi Russolo developed musical Futurism. This style often tried to recreate everyday sounds and place them in a “Futurist” context. The “Machine Music” of George Antheil (starting with his Second Sonata, “The Airplane”) and Alexander Mosolov (most notoriously his Iron Foundry ) developed out of this. The process of extending musical vocabulary by exploring all available tones was pushed further by the use of Microtones in works by Charles Ives, Julián Carrillo, Alois Hába, John Foulds, Ivan Wyschnegradsky, and Mildred Couper among many others. Microtones are those intervals that are smaller than asemitone; human voices and unfretted strings can easily produce them by going in between the “normal” notes, but other instruments will have more difficulty—the piano and organ have no way of producing them at all, aside from retuning and/or major reconstruction.
In the 1940s and 50s composers, notably Pierre Schaeffer, started to explore the application of technology to music in musique concrète. The term electroacoustic music was later coined to include all forms of music involving magnetic tape, computers, synthesizers, multimedia, and other electronic devices and techniques. Live electronic music uses live electronic sounds within a performance (as opposed to preprocessed sounds that are overdubbed during a performance), Cage’s Cartridge Music being an early example. Spectral music (Gérard Grisey and Tristan Murail) is a further development of electroacoustic music that uses analyses of sound spectra to create music. Cage, Berio, Boulez, Milton Babbitt, Luigi Nono and Edgard Varèse all wrote electroacoustic music.
From the early 1950s onwards, Cage introduced elements of chance into his music. Process music (Karlheinz Stockhausen Prozession , Aus den sieben Tagen ; and Steve Reich Piano Phase , Clapping Music ) explores a particular process which is essentially laid bare in the work. The termexperimental music was coined by Cage to describe works that produce unpredictable results, according to the definition “an experimental action is one the outcome of which is not foreseen.” The term is also used to describe music within specific genres that pushes against their boundaries or definitions, or else whose approach is a hybrid of disparate styles, or incorporates unorthodox, new, distinctly unique ingredients.
Important cultural trends often informed music of this period, romantic, modernist, neoclassical, postmodernist or otherwise. Igor Stravinsky and Sergei Prokofiev were particularly drawn to primitivism in their early careers, as explored in works such as The Rite of Spring and Chout . Other Russians, notably Dmitri Shostakovich, reflected the social impact of communism and subsequently had to work within the strictures of socialist realism in their music. Other composers, such as Benjamin Britten ( War Requiem ), explored political themes in their works, albeit entirely at their own volition. Nationalism was also an important means of expression in the early part of the century. The culture of the United States of America, especially, began informing an American vernacular style of classical music, notably in the works of Charles Ives, John Alden Carpenter, and (later) George Gershwin. Folk music (Vaughan Williams’ Five Variants of Dives and Lazarus , Gustav Holst’s A Somerset Rhapsody ) and Jazz (Gershwin, Leonard Bernstein, and Darius Milhaud’s La création du monde ) were also influential.
In the latter quarter of the century, eclecticism and polystylism became important. These, as well as minimalism, New Complexity, and New Simplicity, are more fully explored in their respective articles.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- History. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/20th-century_classical_music#History . License : CC BY-SA: Attribution-ShareAlike


4.3: Managing Long Documents in Microsoft Word
By the end of this section, you will be able to:
- Use the tools in the Pages command group
- Utilize Format Painter to apply formatting in a long document
- Use some of the tools on the References tab
- Create bookmarks for document navigation
Long documents underscore the importance of good navigation tools. Imagine working through a 200-page report, only to have to scroll endlessly through chapter headings to find what you are looking for. This is why tools like cross-references, bookmarks, tables of contents, and tables of figures are so important. They allow you to quickly access different parts of your document, as does the document outline, available in the Navigation pane.
In a business setting, these navigation tools are especially important. Think about your market trends report for WorldCorp. The high-level executives reading your report will need to be able to quickly glance at the table of contents and read a few pages of the report, without having to scroll or flip through many pages of text. In fact, they likely don’t need to read the whole report; they will read the executive summary and skip around the report to graphs that interest them. (The executive summary is a short overview at the beginning of the report that highlights the main points of a long report.)
This section discusses the different command groups and tools that make long documents more user-friendly.
The Pages Command Group
The Pages command group is on the Insert tab, and it contains three commands. The most used one is Page Break; this command allows the user to finish writing on the current page and start writing on a new page. (You can also access many of these same tools on the Layout tab in the Page Setup command group.) Adding a page break can be particularly useful at the end of a chapter or section. To insert a page break, place the cursor where the break will be (the start of the new page), and select the Page Break option. You may also just use the keyboard shortcut Ctrl+Enter to place a page break. If you turn on the ability to view hidden characters, you can see exactly where the page break is on your page (see Figure 4.48). You can also view it easily in the Print layout option on the View tab. Here, we can add a page beak at the end of each major section of the market trends report.
There is another command, Blank Page , which is not as commonly used as Page Break, but can be impactful in long documents such as reports and textbooks. These intentionally blank pages are added to provide a sense of closure or finality before a new chapter or section begins; they make a bigger impact than just a page break because they leave a full page empty. This convention is more often used in the publishing world when needing to, say, start a new chapter on a right-hand page.
To insert a blank page, go through the same steps as inserting a page break: Put the cursor where you want the blank page to be, and select the Blank Page icon.
The last command to discuss in the Pages group is Cover Page . Adding a cover page, or title page , is a common practice for long documents. It is an informative page that contains basic information about the report, such as the title, authors, publishing date, and the department or/and division. Some companies or organizations follow specific guidelines about title pages; formally written reports that follow the Chicago Manual of Style , for instance, do not include any graphics, and the components must follow a set alignment and format. Yet many business reports do not strictly follow any set manual of style, or may simply follow their company’s brand guidelines. It is appropriate in many business settings to use graphics, such as a logo or photograph, on a title page, as in Figure 4.49.
Insert a title page to your market trends report. To insert a title page, select the drop-down menu of the Cover Page command, and choose a design. Word offers many different options for layouts, fonts, colors, and styles. Just update the text and add an image, if applicable.
Format Painter
Another tool in Word that helps manage long documents is the Format Painter. Format Painter is a command on the Home tab. It is used to copy formatting from another document or section of the same document. This can be useful when you have formatting in one section of the document that you want to copy over to another section, saving you the time and effort of having to format the new document or section again manually.
To use the Format Painter, select the area of the document from which you want to copy the formatting, then choose the Format Painter command. If you single-click the Format Painter, it will only allow it to be used once. To use it multiple times, double-click the Format Painter. It can then be used until you click the Format Painter icon again or press the Esc key. The mouse pointer will change to a little brush, then with the brush, you select all the areas you want to apply the formatting to. Word will automatically mirror and apply the new formatting. If the document has a lot of chapters, if the document outline is multilevel, or if there is a lot of different elements in the previous document, then the Format Painter will not work perfectly. You might need to fix some pages or areas of the newer document. For example, when you have bullets, the Format Painter will not copy over the bullets, but instead will only copy the formatting of the text.
Format Painter is a useful tool for when you want a whole document to have the same formatting, particularly if that document contains copied and pasted text from multiple sources or collaborators. In Figure 4.50, we selected a phrase from the first paragraph, then chose Format Painter and used the brush to select the entire target paragraph. As you can see, the selected text now has matching formatting to the header.
References Tab
The References tab is for working with citations and the front/back matter of books. This is an important tab for anyone who is creating a document with many sources, such as a journal article or economic report. It is also helpful when dealing with long documents that include many sections such as the market trends report. Managing a table of contents, a citation list, or a bibliography manually can be unwieldy, and, fortunately, Word provides lots of tools to help users keep these organized and looking professional.
Let’s review the different command groups on this tab. The first command group is the Table of Contents group. This contains tools for helping the user insert or modify a table of contents. The Footnotes command group is for managing and inserting footnotes and endnotes. The Research command group has two useful tools. The Search tool can be used to find information, definitions, and images online. The Researcher tool allows you to search for sources for information contained in the document. You can then insert the citation or link to the source using the tool. You simply click on the Research tool and type the search term or phrase into the navigation window. The Citations & Bibliography command group is for inserting, formatting, and managing your references. The Captions command group is for when you are working with figures and tables and want to add captions to later create a summary of the figures in your document. The Index command group is for building an index, like what you would find at the end of a textbook. The Table of Authorities is a command group that lawyers use when they need to cite cases and statutes. This next section walks through some of the commands you might find most helpful in this tab.
Table of Contents
If your document is formatted properly, Word will automatically generate a table of contents for you. You can select from a few different styles, which is useful for ensuring that your table of contents matches the look and feel of the rest of your document. Let’s examine the requirements for a table of contents and what you need to have in your document to create a good one.
Headings for Tables of Contents
In the Creating and Working in Documents chapter, we discussed how to insert headings and why these are important for navigating long documents. We also mentioned how to access your document outline in the Navigation Pane . Recall that as you insert different levels of headings (H1, H2, H3, etc.), the Navigation Pane will read the document automatically and display them for you in the form of a document outline. This is particularly useful for long documents, as you can see the outline of the entire document without having to scroll all the way through hundreds of pages of text.
The reason these headings are relevant to generating a table of contents is that the table of contents uses the headings in the same way as the document outline. When you set up your H1s, H2s, and H3s, Word will automatically use those to generate a multilevel table of contents.
Section Breaks for Tables of Contents
In the chapter on Creating and Working in Documents, we also briefly reviewed how to insert section breaks and why they are important. Recall that there are two types of section breaks : continuous section break s and page break s. Continuous section breaks are used when you want a new section but want to remain on the same page. They can also be used if you want to have different margins on two different areas of the same page, or to add columns for one section of the text. Page breaks, however, begin a new section on the next page. These do everything the continuous section break does, but simply adds a new page. These are useful for when you want to start a new chapter, for example.
Both types of section breaks are important for generating your table of contents. When you create a table of contents in Word, the program automatically generates page numbers based on the page numbers in your sections. You may have continuous page numbering throughout your document, and so in this case, the table of contents will not be concerned with your sections. However, if there is different page numbering in different sections, the table of contents will reflect this.
Inserting a Table of Contents
You need to have added your headings and sections for the table of contents to work. In the market trends report, we formatted the headings for each major section and the subsections in those sections. Recall the headings were formatted using the Style command group on the Home tab. To insert a table of contents using the headings we have defined, go to the first page of the first section. For the report, we want to include the table of contents after the cover page. So we will go to the Introduction/Executive Summary heading. Leave a blank line before the section heading by pressing Enter to move the first section downward as least one line, as shown in Figure 4.51. This will determine the placement of the table of contents. It should appear before the actual report starts, so before the first section of the document. Now, go to the References tab, choose the drop-down menu called Table of Contents , and choose any of the preset formats. Choose Automatic Table 2 for the market trends report; it will look like Figure 4.52. Conveniently, the table of contents has clickable hyperlinks by default, meaning that you can press Ctrl on the keyboard while clicking the mouse on any part of the table of contents and it will take you to that part of the document.
Note that any time you update your document, and add or delete any headings, you will also need to update your table of contents. You can do this simply by putting your cursor anywhere on the table of contents and selecting the Update Table button. This will automatically update your headings and page numbers.
Footnotes and Endnotes
A footnote is a way to add notes to a document, and/or a way to cite a reference . A footnote appears at the bottom of the page where the reference occurs. When you add a footnote, a small superscript number or symbol will appear above your text, and then the footnote will automatically be added to the bottom of the page. This is an easy way to add a reference to a source or other general notes. For example, the author may want to add a related thought or comment about the text and have it appear at the bottom of the page in the form of a footnote, so that the reader can quickly and easily read it.
An endnote is similar to a footnote, except that the note appears on a page at the end of the document. Endnotes are typically part of the back matter. They are useful for when you want to have all of your references and comments in one place. However, having them at the end of the document makes it harder for the reader to read them in the moment.
To insert a footnote or endnote, place your cursor at the end of a sentence, and choose the Insert Footnote or Insert Endnote command on the References tab. Word will automatically take you to the bottom of the page (for a footnote) or the end of the document (for an endnote) and add a short line to indicate where the footnote/endnote space will be.
For the market trends report, we need to add a footnote to provide the reference for the information about laptop sales in the industry. We will place the cursor at the end of the sentence (at the end of the word “billion”) as the place where the footnote notation will be inserted (Figure 4.53). The actual footnote will be inserted at the bottom of the page as a blank footnote. You can then insert the appropriate citation for the footnote. In this case, the website for the reference was copied and pasted into the footnote (Figure 4.54). To format the text in the footnote, click on it twice, and the whole line will be selected, then right-click on it and select Style. You can change the font type if you click on Modify. You can further change the footnote type, as there are many standard layouts of how to present a footnote, as Figure 4.55 shows. Enter the citation/text for the footnote.
An endnote will look exactly the same as a footnote. The only difference is that it is at the end of the document instead of at the bottom of the page. For the market trends report, we could use endnotes to identify the departments that contributed the specific information in the sections or to add additional context to a statement in the report.
For both footnotes and endnotes, you have the option of changing the numbering system. Word will automatically use numbers (for footnotes) or Roman numerals (for endnotes), but you can change this. Double-click to select the endnote or footnote, then right-click it to activate the context menu, and select Note Options where you can change the numbering format. If you want to change the font or layout, select Style from the context menu.
Table of Figures
A table of figures is similar to a table of contents, except that instead of listing your document’s headings and section breaks, it lists the captioned figures, such as tables and images. This is useful in documents with a lot of statistical analyses, for instance, in which readers will need to easily and quickly find specific graphs or charts. It can also be useful in documents like business reports, where you might want to have a list of the different data tables. Like a table of contents, there are a few requirements that must be met before the table of figures can be generated.
Captions
For Word to automatically generate your table of figures, you need to insert captions for each of them. Let’s first insert a summary chart of data using the skills learned earlier in the chapter into the market trends report. To “caption” a figure, select the entire figure, then choose Insert Caption (see Figure 4.56). In the dialog box that appears (see Figure 4.57), the caption is automatically generated to be “Figure 1,” but you can rename it to whatever you like. There are also options where you can choose the placement of the caption. Do the same for all the figures in your document. Once all your figures are captioned, then you can insert your table of figures.
Inserting a Table of Figures
As with inserting a table of contents, first go to the top of your document where you will want your table of figures to appear. Typically, a table of figures is placed right below the table of contents. Now that we have labeled our table with a caption, we can create a table of figures to see how it works. As seen in Figure 4.58, insert a page break, and choose the Insert a Table of Figures command from the References tab. Word will automatically “read” the existing captions, and a table of figures will be generated.
To make your document neat and organized, add the title Table of Figures on top, and insert another page break to separate the Table of Figures page from the beginning of the document.
Linked Bookmarks
Another helpful tool for navigating long documents is the bookmark . You can add bookmark links throughout your document for better navigation. Place the cursor where you want the bookmark to link up to—a section header, for example, or the beginning of an important paragraph—and go to the Insert tab and click Bookmark. In the dialog box, type a name for your bookmark Figure 4.60. In this case, we chose to bookmark the start of the body text. Now, select a word that will link up to the bookmark, right-click on it, and select Hyperlink from the context menu. The same hyperlink dialog box will appear, but this time select Place in This Document. Click on the bookmark you just made, and then click OK; the link will be set Figure 4.61. Conversely, if you want to remove a bookmark from Word, click the Bookmark command in the Insert menu, locate it, and then select Delete.
Another reason to use bookmarks is to help digital readers of your document navigate back to important sections of the document while they read. For example, you can add a link at the end of each chapter that says, “Go to Table of Contents.” To do this, first place a bookmark in front of the table of contents as Figure 4.62 shows. To locate the roster of bookmarks, click the Bookmark command on the Insert tab, and then click Go to. If you want to relocate the bookmark, place the cursor where you want the bookmark to be relocated to, then open the Bookmark dialog box, click the original bookmark’s name, and then click Add. This will update the new location.
Next, move to the area of the document where you want to link the table of contents. Go to the Insert tab and select Link. From the Insert Link window, choose the bookmark you just created. In the text box that appears, write the text you want readers to see (“Go to Table of Contents”), then choose Place in this Document as shown in Figure 4.63 and Figure 4.64. Choose the placement in the document and click OK.
When you are creating documents, especially long documents that might include research or data that has been created by others, it is essential that you give proper credit. You might have encountered citing sources when completing a research report for a class. We have an ethical obligation to cite sources to acknowledge the work of researchers and their findings from the studies. If proper citation is not given to material that is not your own, this is called plagiarism. Plagiarism is part of a broader category called “academic integrity.” Academic integrity goes beyond giving credit for work that is not your own, but it also covers issues such as cheating on a test or paying to have someone create a research report for you.
This is not to say that we cannot use information from others to support our thoughts or to help us develop a new understanding of the material. Instead, it means we have to acknowledge that the information came from another source and we are using their research in our work. From the researcher’s standpoint, a measure of the quality of the research is the number of times the work is cited by others. This is similar to a product online having a large number of five-star reviews. The more the research is cited by others, the higher the credibility of the information.


5.9: Measures of Variability
-
- Last updated
- Save as PDF
Learning Objectives
- Compute the range
- Compute the variance in the population
- Compute the standard deviation from the variance
What is Variability?
Variability refers to how "spread out" a group of scores is. To see what we mean by spread out, consider graphs in Figure \(\PageIndex{1}\). These graphs represent the scores on two quizzes. The mean score for each quiz is \(7.0\). Despite the equality of means, you can see that the distributions are quite different. Specifically, the scores on \(\text{Quiz 1}\) are more densely packed and those on \(\text{Quiz 2}\) are more spread out. The differences among students were much greater on \(\text{Quiz 2}\) than on \(\text{Quiz 1}\).
\(\text{Quiz 1}\)
\(\text{Quiz 1}\)
Figure \(\PageIndex{1}\) : Bar charts of two quizzes
The terms variability, spread, and dispersion are synonyms, and refer to how spread out a distribution is. Just as in the section on central tendency where we discussed measures of the center of a distribution of scores, in this chapter we will discuss measures of the variability of a distribution. There are four frequently used measures of variability: the range, interquartile range, variance, and standard deviation. In the next few paragraphs, we will look at each of these four measures of variability in more detail.
Range
The range is the simplest measure of variability to calculate, and one you have probably encountered many times in your life. The range is simply the highest score minus the lowest score. Let’s take a few examples. What is the range of the following group of numbers: \({10, 2, 5, 6, 7, 3, 4}\)? Well, the highest number is \(10\), and the lowest number is \(2\), so \(10 - 2 = 8\). The range is \(8\). Let’s take another example. Here’s a dataset with \(10\) numbers: \({99, 45, 23, 67, 45, 91, 82, 78, 62, 51}\). What is the range? The highest number is \(99\) and the lowest number is \(23\), so \(99 - 23\) equals \(76\); the range is \(76\). Now consider the two quizzes shown in Figure \(\PageIndex{1}\). On \(\text{Quiz 1}\) , the lowest score is \(5\) and the highest score is \(9\). Therefore, the range is \(4\). The range on \(\text{Quiz 2}\) was larger: the lowest score was \(4\) and the highest score was \(10\). Therefore the range is \(6\).
Interquartile Range
The interquartile range (IQR) is the range of the middle \(50\%\) of the scores in a distribution. It is computed as follows:
\[IQR = 75^{th} percentile - 25^{th} percentile\]
For \(\text{Quiz 1}\) , the \(75^{th}\) percentile is \(8\) and the \(25^{th}\) percentile is \(6\). The interquartile range is therefore \(2\). For \(\text{Quiz 2}\) , which has greater spread, the \(75^{th}\) percentile is \(9\), the \(25^{th}\) percentile is \(5\) , and the interquartile range is \(4\). Recall that in the discussion of box plots, the \(75^{th}\) percentile was called the upper hinge and the \(25^{th}\) percentile was called the lower hinge. Using this terminology, the interquartile range is referred to as the \(H\)-spread.
A related measure of variability is called the semi-interquartile range. The semi-interquartile range is defined simply as the interquartile range divided by \(2\). If a distribution is symmetric, the median plus or minus the semi-interquartile range contains half the scores in the distribution.
Variance
Variability can also be defined in terms of how close the scores in the distribution are to the middle of the distribution. Using the mean as the measure of the middle of the distribution, the variance is defined as the average squared difference of the scores from the mean. The data from \(\text{Quiz 1}\) are shown in Table \(\PageIndex{1}\). The mean score is \(7.0\). Therefore, the column "Deviation from Mean" contains the score minus \(7\). The column "Squared Deviation" is simply the previous column squared.
| Scores | Deviation from Mean | Squared Deviation |
|---|---|---|
| 9 | 2 | 4 |
| 9 | 2 | 4 |
| 9 | 2 | 4 |
| 8 | 1 | 1 |
| 8 | 1 | 1 |
| 8 | 1 | 1 |
| 8 | 1 | 1 |
| 7 | 0 | 0 |
| 7 | 0 | 0 |
| 7 | 0 | 0 |
| 7 | 0 | 0 |
| 7 | 0 | 0 |
| 6 | -1 | 1 |
| 6 | -1 | 1 |
| 6 | -1 | 1 |
| 6 | -1 | 1 |
| 6 | -1 | 1 |
| 6 | -1 | 1 |
| 5 | -2 | 4 |
| 5 | -2 | 4 |
| Means | ||
| 7 | 0 | 1.5 |
One thing that is important to notice is that the mean deviation from the mean is \(0\). This will always be the case. The mean of the squared deviations is \(1.5\). Therefore, the variance is \(1.5\). Analogous calculations with \(\text{Quiz 2}\) show that its variance is \(6.7\). The formula for the variance is:
\[s^2=\frac{\sum (X-\mu )^2}{N}\]
where \(\sigma ^2\) is the variance, \(\mu\) is the mean, and \(N\) is the number of numbers. For \(\text{Quiz 1}\) , \(\mu = 7\) and \(N = 20\).
If the variance in a sample is used to estimate the variance in a population, then the previous formula underestimates the variance and the following formula should be used:
\[s^2=\frac{\sum (X-M)^2}{N-1}\]
where \(s^2\) is the estimate of the variance and \(M\) is the sample mean. Note that \(M\) is the mean of a sample taken from a population with a mean of \(\mu\). Since, in practice, the variance is usually computed in a sample, this formula is most often used. The simulation "estimating variance" illustrates the bias in the formula with \(N\) in the denominator.
Let's take a concrete example. Assume the scores \({1, 2, 4,\: and\; 5 }\) were sampled from a larger population. To estimate the variance in the population you would compute \(s^2\) as follows:
\[M = \dfrac {1 + 2 + 4 + 5}{4} = \dfrac {12}{4} = 3\]
\[\begin{align*} s^2 &= \dfrac{[(1-3)^2 + (2-3)^2 + (4-3)^2 + (5-3)^2]}{(4-1)}\\ &= \dfrac{(4 + 1 + 1 + 4)}{3}\\ &= \dfrac{10}{3}\\ &= 3.333 \end{align*}\]
There are alternate formulas that can be easier to use if you are doing your calculations with a hand calculator. You should note that these formulas are subject to rounding error if your values are very large and/or you have an extremely large number of observations.
\[\sigma ^2=\cfrac{\sum X^2-\cfrac{(\sum X)^2}{N}}{N}\]
and
\[s^2=\cfrac{\sum X^2-\cfrac{(\sum X)^2}{N}}{N-1}\]
For this example,
\[\sum X^2=1^2+2^2+4^2+5^2=46\]
\[\dfrac {(\sum X)^2}{N}=\dfrac {(1+2+4+5)^2}{4}=\dfrac {144}{4}=36\]
\[\sigma ^2=\dfrac {(46-36)}{4}=2.5\]
\[s^2=\dfrac {(46-36)}{3}=3.333\; \; \text{as with the other formula}\]
Standard Deviation
The standard deviation is simply the square root of the variance. This makes the standard deviations of the two quiz distributions \(1.225\) and \(2.588\). The standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal (see Chapter on Normal Distributions) because the proportion of the distribution within a given number of standard deviations from the mean can be calculated. For example, \(68\%\) of the distribution is within one standard deviation of the mean and approximately \(95\%\) of the distribution is within two standard deviations of the mean. Therefore, if you had a normal distribution with a mean of \(50\) and a standard deviation of \(10\), then \(68\%\) of the distribution would be between \(50 - 10 = 40\) and \(50 +10 =60\). Similarly, about \(95\%\) of the distribution would be between \(50 - 2 \times 10 = 30\) and \(50 + 2 \times 10 = 70\). The symbol for the population standard deviation is \(\sigma\); the symbol for an estimate computed in a sample is \(s\). Figure \(\PageIndex{2}\) shows two normal distributions. The red distribution has a mean of \(40\) and a standard deviation of \(5\); the blue distribution has a mean of \(60\) and a standard deviation of \(10\). For the red distribution, \(68\%\) of the distribution is between \(35\) and \(45\); for the blue distribution, \(68\%\) is between \(50\) and \(70\).


2.1: Membrane Fluctuations
In 1972, Singer and Nicolson [1], proposed a fluid mosaic model for the cell membrane, which viewed the entire membrane as a dynamic system constantly in flux. Since then, the dynamics of cell membranes were studied extensively and have shown that the membrane is in even more flux than the fluid mosaic model had claimed. The membrane is inhomogeneous, with some parts changing their position and composition more rapidly than others. Proteins get inserted and reinternalized, rafts form and dissolve, cytoskeleton is remodeled, and the membrane itself changes its thickness and curvature. Together, all of these perturbations are called membrane fluctuations.
Membrane Composition
In order to understand the fluctuations that happen within a membrane, a brief overview of the membrane components is required. The membrane consists of a phospholipid bilayer. This bilayer can be thought of as a sea, in which proteins are floating. Proteins can be inserted in either both or a single layer of the membrane. Also in the membrane, there are sterol molecules, which add rigidity. On the inside, the membrane is supported by a cytoskeletal network, which consists mainly of actin (Figure \(\PageIndex{1}\)).
Lipid Rafts
The membrane is not homogeneous; it is full of rafts, or areas of lipids that are in liquid ordered state, which is more ordered than the surrounding area. These rafts are important for a number of reasons; they are thicker than the lesser ordered membrane, and there are transmembrane proteins that only fit in these thicker rafts [2]. These rafts may act as a means to concentrate proteins to an area of the membrane.
Studies of diffusion rates of raft versus non-raft proteins suggest that the rafts diffuse at a rate similar to that of non-raft transmembrane proteins (Fig 3). This indicates that the rafts are very small, and of similar size to transmembrane proteins.
 These proteins are continuously recycled, with new proteins being delivered by vesicles. Typically, a transmembrane protein would be synthesized in the rough endoplasmic reticulum (rER) and then transported in a clathrin-coated vesicle to the cell membrane. The vesicle fuses with the membrane, and the proteins become a part of it. Vesicle fusion is a major process of membrane remodeling.
Membrane Remodeling Proteins
There is a number of proteins that can remodel the membrane. One of them is known as BAR (Figure 4) This protein has an inherent curvature, which allows it to sense membrane curvature and induce further curvature by binding. When multiple subunits bind to the membrane, they can cause tubulation.
Actin Network
The cell membrane is anchored to the cytoskeleton, which mainly consists of actin [3], (Fig.5, 6). Actin in itself is a dynamic polymer that is both depolymerizing and polymerizing, depending on the concentration of monomers. This activity is tightly controlled and is extremely important for mobility, phagocytosis, and most processes that require large scale membrane remodeling. By growing actin in a certain point in the cell, membrane can be made to grow protrusions, which can be used to engulf foreign particles or move.
Layer to Layer Fluctuations
It is extremely energetically unfavorable for a lipid from one leaflet of the membrane to flip over to the other. However, this is a process necessary for membrane remodeling and it does occur. A group of enzymes, know as flippases, assists this flipping of lipids [4]. A number of flippases have been identified [5], but the mechanism remains unclear.
Protrusions and Bending
Lipids can move laterally within the membrane, or they can protrude from it. If they form a protrusion, it can be detected with neutron scattering (Figure 7) Bending of the membrane has been suggested to be important in clathrin aggregation using computational methods [6].
.
References
- Singer SJ, Nicolson GL (Feb 1972). "The fluid mosaic model of the structure of cell membranes". Science 175 (4023): 720– 31
- Jérémie Rossy, Yuanqing Ma, Katharina Gaus, "The organisation of the cell membrane: do proteins rule lipids?", Current Opinion in Chemical Biology, Volume 20, June 2014, Pages 54-59, ISSN 1367-5931
- Mithilesh Mishra The yeast actin cytoskeleton"
- Bretscher, MS. (1972). "Asymmetrical lipid bilayer structure for biological membranes." Nature New Biology 236:11-12
- Islam, S. T. and Lam, J. S. (2013), " Wzx flippase-mediated membrane translocation of sugar polymer precursors in bacteria. Environmental Microbiology", 15: 1001–1015
- Cordella, Nicholas et al. “Membrane Fluctuations Destabilize Clathrin Protein Lattice Order.” Biophysical Journal 106.7 (2014): 1476–1488


5.2: Ethical Principles
The American Nurses Association (ANA) designates Ethics as the first Standard of Professional Performance in Nursing: Scope and Standards of Practice , stating, “The registered nurse integrates ethics in all aspects of practice.” [1] See the following box for competencies associated with the ANA’s Ethics Standard of Professional Performance.
ANA’s Ethics Competencies [2]
The registered nurse:
- Uses the Code of Ethics for Nurses With Interpretive Statements as a moral foundation to guide nursing practice and decision-making.
- Demonstrates that every person is worthy of nursing care through the provision of respectful, person-centered, compassionate care, regardless of personal history or characteristics (Beneficence).
- Advocates for health care consumer perspectives, preferences, and rights to informed decision-making and self-determination (Respect for autonomy).
- Demonstrates a primary commitment to the recipients of nursing and health care services in all settings and situations (Fidelity).
- Maintains therapeutic relationships and professional boundaries.
- Safeguards sensitive information within ethical, legal, and regulatory parameters (Nonmaleficence).
- Identifies ethics resources within the practice setting to assist and collaborate in addressing ethical issues.
- Integrates principles of social justice in all aspects of nursing practice (Justice).
- Refines ethical competence through continued professional education and personal self-development activities.
- Depicts one’s professional nursing identity through demonstrated values and ethics, knowledge, leadership, and professional comportment.
- Engages in self-care and self-reflection practices to support and preserve personal health, well-being, and integrity.
- Contributes to the establishment and maintenance of an ethical environment that is conducive to safe, quality health care.
- Collaborates with other health professionals and the public to protect human rights, promote health diplomacy, enhance cultural sensitivity and congruence, and reduce health disparities.
- Represents the nursing perspective in clinic, institutional, community, or professional association ethics discussions.
American Nurse Association Code of Ethics
The American Nurses Association Code of Ethics for Nurses With Interpretive Statements is an ethical standard that guides nursing practice and ethical decision-making. It states, “Individuals who become nurses are expected to adhere to the ideals and moral norms of the profession and also to embrace them as a part of what it means to be a nurse. The ethical tradition of nursing is self-reflective, enduring, and distinctive. A code of ethics makes explicit the primary goals, values, and obligations of the profession.” [3]
The Code of Ethics for Nurses contains nine provisions. Each provision contains several clarifying or “interpretive” statements. Read a summary of the nine provisions in the following box.
Nine Provisions of the ANA’s Code of Ethics for Nurses [4]
Provision 1: The nurse practices with compassion and respect for the inherent dignity, worth, and unique attributes of every person.
Provision 2 : The nurse’s primary commitment is to the patient, whether an individual, family, group, community, or population.
Provision 3: The nurse promotes, advocates for, and protects the rights, health, and safety of the patient.
Provision 4: The nurse has authority, accountability, and responsibility for nursing practice; makes decisions; and takes action consistent with the obligation to promote health and to provide optimal care.
Provision 5: The nurse owes the same duties to self as to others, including the responsibility to promote health and safety, preserve wholeness of character and integrity, maintain competence, and continue personal and professional growth.
Provision 6: The nurse, through individual and collective effort, establishes, maintains, and improves the ethical environment of the work setting and conditions of employment that are conducive to safe, quality health care.
Provision 7: The nurse, in all roles and settings, advances the profession through research and scholarly inquiry, professional standards development, and the generation of both nursing and health policy.
Provision 8: The nurse collaborates with other health professionals and the public to protect human rights, promote health diplomacy, and reduce health disparities.
Provision 9: The profession of nursing, collectively through its professional organizations, must articulate nursing values, maintain the integrity of the profession, and integrate principles of social justice into nursing and health policy.
Read the free, online, full version of the Code of Ethics for Nurses With Interpretive Statements .
Ethical Principles
Ethical principles are used to define right from wrong action. Although there are many ethical principles that guide nursing practice, foundational ethical principles include respect for autonomy (self-determination), beneficence (do good), nonmaleficence (do no harm), justice (fairness), fidelity (keep promises), and veracity (tell the truth).
Autonomy
The ethical principle of autonomy recognizes each individual’s right to self-determination and decision-making based on their unique values, beliefs, and preferences. The American Nurses Association (ANA) defines autonomy as, “the capacity to determine one’s own actions through independent choice, including demonstration of competence.” [5] The nurse’s primary ethical obligation is client autonomy. [6] Based on autonomy, clients have the right to refuse nursing care and medical treatment.
Beneficence
Beneficence is defined by the ANA as, “the bioethical principle of benefiting others by preventing harm, removing harmful conditions, or affirmatively acting to benefit another or others, often going beyond what is required by law.” [7] When caring for clients with mental health disorders, nurses implement beneficence when advocating for evidence-based treatment for clients’ mental health.
Nonmaleficence
Nonmaleficence is defined by the ANA as, “the bioethical principle that specifies a duty to do no harm and balances avoidable harm with benefits of good achieved.” [8] A classic example of doing no harm in nursing practice is reflected by nurses checking medication rights three times before administering medications. In this manner, medication errors can be avoided, and the duty to do no harm is met. An example of nurses implementing nonmaleficence in mental health care is ensuring that clients are not harmed by adverse effects of psychotropic medications (such as lithium toxicity or tardive dyskinesia).
Justice
Justice is defined by the ANA as, “a moral obligation to act on the basis of equality and equity and a standard linked to fairness for all in society.” [9] The principle of justice requires health care to be provided in a fair and equitable way. Nurses provide quality care for all individuals with the same level of fairness despite their personalities or characteristics, such as financial status, cultural beliefs, religion, gender, or sexual orientation. Nurses have a social contract to “provide compassionate care that addresses the individual’s needs for protection, advocacy, empowerment, optimization of health, prevention of illness and injury, alleviation of suffering, comfort, and well-being.” [10] An example of a nurse using the principle of justice in mental health care settings is ensuring that quality care is provided to all clients, even those who do not have the cognitive ability to communicate their needs.
Fidelity
Role fidelity is defined as being responsible for providing competent nursing care. An example of role fidelity in nursing is remaining up-to-date with evidence-based practice and implementing effective mental health interventions.
Veracity
Veracity means telling the truth. An example of veracity in health care is informed consent. Nurses ensure that clients have a good understanding of the benefits and risks of a prescribed procedure or psychotropic medication.
Role of Caring
Nurses use a client-centered, care-based, ethical approach to nursing care that focuses on the specific circumstances of each situation. This approach aligns with the foundational nursing concepts of holism and caring in a nurse-client relationship rooted in dignity, respect, kindness, and compassion. [11]
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2015). Code of ethics for nurses with interpretive statements. American Nurses Association. https://www.nursingworld.org/practice-policy/nursing-excellence/ethics/code-of-ethics-for-nurses/coe-view-only/ ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- Legal Information Institute. (n.d.). Welcome to LII. Cornell Law School. https://www.law.cornell.edu ↵


3.1.10: T.S. Eliot (1888-1965)
Selected Poems Tradition and the Individual Talent American/British Modernism Eliot was born in St. Louis, the youngest of seven children. He attended Smith Academy in St. Louis, and went on to study at Harvard. After finishing his bachelor's degree, he began his graduate studies. During this time, he focused on Symbolist poetry. He tried to study abroad in Germany in 1914, but left the country early due to the threat of war. Instead, he went to England, where he met Ezra Pound, who would have a profound influence on Eliot's work. While Eliot did occasionally return to the United States, he settled in England and eventually became a citizen of the country. It was Pound who helped Eliot publish "The Love Song of J. Alfred Prufrock" (1915) in Poetry . The poem established Eliot's reputation as an experimental, intellectual writer. Eliot possessed an amazing versatility. By the time he was 40, he had published over 20 books, which included volumes of poetry, criticism, and plays. His most notable work is The Waste Land (1922), which explores the disenfranchisement and ennui felt by the post-World War I, Lost Generation. The work is experimental in its fracture perspectives, play with tone and language, and disrupted narrative. His criticism, most specifically works from The Sacred Wood , such as "Tradition and the Individual Talent" (1920), constructs a comprehensive literary theory, where the poet is not merely repeating popular ideas, but is interacting with an entire body of literary history, starting with Homer. By the time he won the Nobel Prize in 1948, he was considered one of the most influential writers in the English language. Consider while reading:
- In The Waste Land , how does Eliot portray time in relation to war? What larger point is he trying to make?
- How does Eliot break with traditional forms and content? What is the significance of these breaks?
- How does Eliot reimagine classic stories and figures (such as Tristen and Isolde and Tiresias ) in his work? What is his purpose in this rewriting?
- What does Eliot's work reveal about the effects of historical events, such as World War I, on the poet?
- Eliot uses allusion to enhance his poetry. How do the lines from Dante's Inferno set the tone for "The Love Song of J. Alfred Prufrock? What do the two figures have in common?
- In "The Love Song of J. Alfred Prufrock," Eliot uses a refrain, what is he trying to emphasis?
- Eliot experiments with conceptions of time in both "The Love Song of J. Alfred Prufrock" and The Waste Land . How does he express time in each work? How do these expressions of time underscore his meanings?
- In his essay, "Tradition and Individual Talent," Eliot's conception of art emphasizes the idea of art being impersonal? How does the structure of his argument both uphold and undercut that idea?
- Eliot discusses what a poet must sacrifice in "Tradition and Individual Talent." What does this sacrifice comprise? What does he suggest is the relationship between the poet, history, and society?
Written by Laura Ng


6.9: Proof by Contradiction
For any logically false statement \(e\text{,}\) we have
- Proof
-
First, \(s\rightarrow t\) is false precisely when \(s\) is true and \(t\) is false. On the other hand, \((s \land \neg t) \rightarrow e\) is false precisely when \(s \land \neg t\) is true, and \(s \land \neg t\) is true precisely when \(s, \neg t\) are both true, i.e. when \(s\) is true and \(t\) is false.
- To prove \(P \Rightarrow Q\text{,}\) devise a false statement \(E\) such that \((P \land \neg Q) \Rightarrow E\text{.}\)
- To prove \((\forall x)(P(x) \Rightarrow Q(x))\text{,}\) devise a predicate \(E(x)\) such that \((\forall x)(\neg E(x))\) is true (i.e. \(E(x)\) is false for all \(x\) in the domain), but \((\forall x)\bigl[(P(x) \land \neg Q(x)) \Rightarrow E(x)\bigr]\text{.}\)
Usually \(E\) is taken to be some variation of \(C \land \neg C\text{,}\) for some statement \(C\text{.}\) (See the Law of Contradiction , recorded as Basic Tautology 4 in Example 1.4.1 .)
Prove that \(\sqrt{2}\) is irrational.
Solution
We want to prove the quantified conditional with domain the real numbers: for all \(x\text{,}\) if \(x^2 = 2\) and \(x \gt 0\) then \(x\) is not rational.
Suppose that \(x\) is a real number such that \(x^2 = 2\) and \(x \gt 0\text{.}\) By contradiction, also assume that \(x\) is rational. We want this extra assumption to lead to a false statement. Now, \(x\) rational means \(x = a/b\) for some integers \(a,b\text{.}\) We may assume \(a,b\) are both positive, since \(x \gt 0\text{.}\) We may also assume \(a,b\) have no common factors (i.e. fraction \(a/b\) is in lowest terms). Then,
\begin{align*} x^2 = 2 & \; \Rightarrow \; a^2 = 2 b^2, \\ & \; \Rightarrow \; a^2 \text{ even}, \\ & \; \Rightarrow \; a \text{ even}, \\ & \; \Rightarrow \; a = 2m, \text{ some } m, \\ & \; \Rightarrow \; 2 b^2 = a^2 = 4m^2, \\ & \; \Rightarrow \; b^2 = 2m^2, \\ & \; \Rightarrow \; b^2 \text{ even}, \\ & \; \Rightarrow \; b \text{ even}. \end{align*}
But if both \(a,b\) are even, then \(a\) and \(b\) are both divisible by \(2\text{.}\) We have arrived at our contradiction: \(a,b\) have no common factor but \(a,b\) have a common factor of \(2\text{.}\) That is, we have shown the following.
For all \(x\text{,}\) if \(( (x^2 = 2 \text{ and } x \gt 0) \text{ and } x \text{ not irrational}) \) then (there exist positive integers \(a,b\) such that \(x = a/b\) and \(a,b\) have no common divisor and \(a,b\) have a common divisor).


4.1: Introduction
Apportionment is the problem of dividing up a fixed number of things among groups of different sizes. In politics, this takes the form of allocating a limited number of representatives amongst voters. This problem, presumably, is older than the United States, but the best-known ways to solve it have their origins in the problem of assigning each state an appropriate number of representatives in the new Congress when the country was formed. States also face this apportionment problem in defining how to draw districts for state representatives. The apportionment problem comes up in a variety of non-political areas too, though. We face several restrictions in this process:
- The things being divided up can exist only in whole numbers.
- We must use all of the things being divided up, and we cannot use any more.
- Each group must get at least one of the things being divided up.
- The number of things assigned to each group should be at least approximately proportional to the population of the group. (Exact proportionality isn’t possible because of the whole number requirement, but we should try to be close, and in any case, if Group A is larger than Group B, then Group B shouldn’t get more of the things than Group A does.)
In terms of the apportionment of the United States House of Representatives, these rules imply:
- We can only have whole representatives (a state can’t have 3.4 representatives)
- We can only use the (currently) 435 representatives available. If one state gets another representative, another state has to lose one.
- Every state gets at least one representative
- The number of representatives each state gets should be approximately proportional to the state population. This way, the number of constituents each representative has should be approximately equal.
We will look at four ways of solving the apportionment problem. Three of them (Lowndes’s method is the exception) have been used at various times to apportion the U.S. Congress, although the method currently in use (the Huntington-Hill method) is significantly more complicated.


15.1: Abstract
-
- Last updated
- Save as PDF
During the 18th–20th centuries writing is extremely common on objects made from a wide range of materials that are recovered archaeologically. This evidence is particularly susceptible to nuanced interpretation, as it often forms part of short term deliberate depositional events linked to specific households and consisting of large numbers of items. The nature of the evidence also means that a biographical approach to both individual items and groups of objects can be fruitfully applied. Despite this, such finds have attracted relatively little attention, principally because they are conceived of as part of an unproblematic ‘familiar past’. By looking in detail at six assemblages of material spanning the late 18th to early 20th century recovered during recent excavations at the Grand Arcade site in Cambridge, England, this chapter focusses on how the different materials, sizes, forms, and functions of different types of artefacts affect how writing was employed upon them, as well as why writing does not occur on certain types of artefacts. What was the function of the various types of writing and who was the intended audience? To what extent is some of the writing primarily tactile rather than visual? To what extent was some of the writing meant to be read at all, as some of it was effectively hidden? It also explores the relationship between writing on materials that survive archaeologically and the dominant form of writing on paper that has usually perished. It emerges that much of the writing relates to regulation, although there is also evidence for resistance, as well as a repeated a link to children, and commercial and institutional branding. As writing on objects becomes more common between the late 18th and early 20th centuries and a text-saturated culture develops, the individual texts in later assemblages are often less visible than in earlier groups and are frequently apparently not intended to be read by the consumers of the objects themselves.
How to cite this book chapter:
Cessford, C. 2013. Different Times, Different Materials and Different Purposes: Writing on objects at the Grand Arcade site in Cambridge. In: Piquette, K. E. and Whitehouse, R. D. (eds.)
Writing as Material Practice: Substance, surface and medium.
Pp. 289-317. London: Ubiquity Press. DOI:
http://dx.doi.org/10.5334/bai.o