20.5: Writing Process- Looking Back, Looking Forward
By the end of this section, you will be able to:
- Apply composing processes and tools as a means to discover and reconsider ideas.
- Reflect on the development and insights of composing processes and how they affect your work.
- Adapt and apply composing processes for a variety of technologies and modalities.
One of the best ways to reveal who you are as a writer is to show yourself becoming aware of your strengths and weakness. This awareness can help you discover not only new ways of seeing the world but also new insights into yourself. Although such awareness can occur for unexplainable reasons, it usually happens when you encounter new ideas or have experiences that change you in some way. Reflection allows you to begin this journey. To grow as a writer, look back at your previous writing. If you look back at a drawing you did in first grade, you might find it funny or cute. Additionally, and more likely than not, you could do that same drawing now with a lot more detail and skill than you did back then. Think about writing in the same way: as you add to your writing skills and abilities, you become more proficient and can take on more challenging writing tasks. In this section, as you reflect on your writing development during the course, you will find areas of strength and weakness. The weaker areas are the ones you will want to improve.
Prewriting
Before beginning your reflective essay, take some time to review your work from the course. Write a few sentences or paragraphs about specific aspects of each assignment, such as its purpose, your feelings, what you learned, what you did well (and not so well), and where you think you can do significantly better. This prewriting work will be useful later.
Summary of Assignment: Portfolio Reflection and Self-Evaluation
In the form of a letter (e.g., “Dear Reader”), respond to several questions and discuss various topics related to your writing development in this course. For example, you might be asked to identify and discuss your strongest piece of writing. For each claim you make about your strongest assignment, provide reasoning and evidence from your portfolio to support the statement. When you quote directly from your own writing, be sure to state which assignment or draft you are quoting. Within the context of your responses, include commentary on most of the following course topics as well as others that have been significant:
- Writing processes (organizing graphically, outlining, drafting, conferencing, revising, editing, publishing, recursivity)
- Rhetorical situation, rhetoric, and persuasion
- Reasoning strategies, textual and rhetorical analysis
- Evidentiary strategies: evaluation, research
- Word choice, leads, transitions
- Thesis statement, structure and organization, introductions, conclusions
- Showing, not telling; descriptive writing
- Voice; feelings, as hindsight or in process
Depending on the nature of your portfolio, you may be able to create a digital or multimodal reflective letter, as mentioned in Glance at Genre: Purpose and Structure.
Another Lens. Using reflective organization and strategies, create a fictionalized story for readers. Some fiction writers base their stories on real events, adding material or characters to help readers connect the plot points and make the story more memorable and engaging. For example, consider a school-like setting and a host of characters. Incorporate dialogue, details of setting, and other story elements to develop characters and create tension.
Once Upon a Time
James sat silently across the room as Rafael read the paper James had worked so hard to write. He could not have been more nervous watching Rafael, the best writer in the class, review his work. No one had ever read James’s work other than a teacher. His heart was racing, and beads of sweat formed on his forehead.
“What do you think?” James asked. Rafael rolled his eyes before locking eyes with James.
“I’m not finished yet,” Rafael answered and returned to the paper, ignoring James.
James stared at him, contemplating the meaning of every facial wrinkle and twitch of a finger: what did they mean? After several aching minutes, Rafael picked up a pen and wrote for several minutes with a slight smile on his face. He took a long breath, walked over to James, and handed the paper back with a sheet full of notes.
He smiled and said, “It’s a strong paper. I made some notes I hope you find useful. I was confused only a few times, so you could look at where I made suggestions for when you revise.” James smiled back, and the look on his face showed surprise and relief.
After quickly reviewing the comments, James turned to his close friend Jess and said, “He didn’t destroy my paper and actually gave me some good suggestions.”
In this example, James, Rafael, and Jess are not real people, but the characters show how students may react during a peer-review workshop. Of course, you might decide to write about a less-than-ideal experience in which Rafael laughs at James’s work and Jess steps in to help him revise. Or you might set the story in a different time or place and create an entirely different situation. Whatever you decide, use your course experience and some creativity to create scenarios in which a character reflects on their writing in ways that are meaningful and useful not only for you but also for your readers. Then weave these characters into a larger narrative. It might end in a published class book or website of student writing and require James to give a speech about one of his papers, or it might end in another scenario that follows logically from the narrative you have created. Regardless of where you take the story, include realistic elements of reflection as well as how your main character develops across the story, faces a challenge, and finds a way to overcome it.
You can show character development in several different ways. One way is to be inside a character’s mind. To portray a character thinking rather than talking, employ internal monologue by using sentence fragments and other nonacademic writing conventions to show that a person’s thought process doesn’t follow conventional rules of language. For example, a character named Bethany describes her thoughts when revising her first paper. She writes an internal monologue—readers hear her talking to herself while she tries to focus on revision. Note how she provides clues for readers to understand what is going on around her.
OMG, I cannot believe I wrote that! How could I write about a calligraphy pen when I don’t even own a calligraphy pen? I’m not even sure what one looks like! Bonkers! I wonder if the other people in my class are staring at me right now. I’m afraid to look up. I casually tilt my head up and see no one paying any attention to me whatsoever. Wonderful! I’m just another writer in a writing class.
In this example, the character works through a process of reflection based on her experience. She cannot believe she wrote about a calligraphy pen, but perhaps because of her nervousness with writing, this quirk has become a unique aspect of her character. You might create other such quirks in one of your characters, such as a character who always reads aloud, even in the middle of class, or one who taps a pen on their forehead loudly as they read. Then you can use that as a tool to point to another aspect of what you learned in your actual course. When writing this fictionalized piece, be mindful of your focus, which is to reflect on your development as a writer during this course.
Quick Launch: Establishing Criteria for Growth
To get started, you will need to organize your thoughts. After you have reviewed each chapter and its related assignment, reflect on your successes and challenges. Use a graphic organizer similar to Table \(20.1\) to get started. If the information already filled in for Chapter 1 works for you, use it. If it doesn’t, change it accordingly. If you skipped the suggested review of your assignments, do it now. Otherwise, use your notes as you complete the chapter reflection table below. Skip any rows related to chapters that you did not cover in class.
| Chapter Number | Key Skills Learned | Successes | Challenges |
| 1 | Summarize, Analyze, Evaluate | Analyzing wasn’t hard, but I learned to do it better because I really had to think about what the sentence meant and explain it in my own way. | I had a hard time with evaluation because I wasn’t comfortable giving my opinion since I’m just a student. However, I got more comfortable writing about complex topics and ideas. |
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Portfolio Reflection Template Dear Reader, Welcome to my English Composition portfolio. Here, you will find ________. Complete this statement: This semester, I learned that I am (not) a writer because ________ . |
| Answer these questions in paragraph form. For each claim you make, show your reasoning and provide quoted evidence from your portfolio: |
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Sincerely, Your Name |
Structuring Your Responses
As you respond to each of the questions above, use a paragraph planner such as this one.
| Topic Sentence/Claim | Reasoning Strategy (Circle All Used) | Quoted Evidence from Portfolio |
| Revisit Reasoning Strategies: Improving Critical Thinking to find frames and word banks that will help you employ these strategies. | ||
| The strongest piece of writing in my portfolio is ________ because ________. |
Analogy Cause and effect Classification and division Comparison and contrast Problem and solution Definition |
Shows my use of descriptive writing/figurative language: “Learning a foreign language is like learning to ride a bicycle: you must learn to perform multiple tasks at the same time.” |
| The strengths of this essay are ________, ________, and ________. |
Analogy Cause and effect Classification and division Comparison and contrast Problem and solution Definition |
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| This semester, I improved my skills as a writer in the following ways: ________ and ________. |
Analogy Cause and effect Classification and division Comparison and contrast Problem and solution Definition |
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| This essay demonstrates these skills in that it ________. |
Analogy Cause and effect Classification and division Comparison and contrast Problem and solution Definition |


1.1: Basic Set Concepts
After completing this section, you should be able to:
- Represent sets in a variety of ways.
- Represent well-defined sets and the empty set with proper set notation.
- Compute the cardinal value of a set.
- Differentiate between finite and infinite sets.
- Differentiate between equal and equivalent sets.
Sets and Ways to Represent Them
Think back to your kitchen organization. If the drawer is the set , then the forks and knives are elements in the set. Sets can be described in a number of different ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and by Venn diagrams. Sets are typically designated with capital letters. The simplest way to represent a set with only a few members is the roster (or listing ) method , in which the elements in a set are listed, enclosed by curly braces and separated by commas. For example, if represents our set of flatware, we can represent by using the following set notation with the roster method:
Write a set consisting of your three favorite sports and label it with a capital .
- Answer
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There are multiple possible answers depending on what your three favorite sports are, but any answer must list three different sports separated by commas, such as the following:
All the sets we have considered so far have been well-defined sets . A well-defined set clearly communicates whether an element is a member of the set or not. The members of a well-defined set are fixed and do not change over time. Consider the following question. What are your top 10 songs of 2021? You could create a list of your top 10 favorite songs from 2021, but the list your friend creates will not necessarily contain the same 10 songs. So, the set of your top 10 songs of 2021 is not a well-defined set. On the other hand, the set of the letters in your name is a well-defined set because it does not vary (unless of course you change your name). The NFL wide receiver, Chad Johnson, famously changed his name to Chad Ochocinco to match his jersey number of 85.
For each of the following collections, determine if it represents a well-defined set.
- The group of all past vice presidents of the United States.
- A group of old cats.
- Answer
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- The group of all past vice presidents of the United States is a well-defined set, because you can clearly identify if any individual was or was not a member of that group. For example, Britney Spears is not a member of this set, but Joe Biden is a member of this set.
- A group of old cats is not a well-defined set because the word old is ambiguous. Some people might consider a seven-year-old cat to be old, while others might think a cat is not old until it is 13 years old. Because people can disagree on what is and what is not a member of this group, the set is not well-defined.
For each of the following collections, determine if it represents a well-defined set.
A collection of medium-sized potatoes.
The original members of the Black Eyed Peas musical group.
On January 20, 2021, Kamala Harris was sworn in as the first woman vice president of the United States of America. If we were to consider the set of all women vice presidents of the United States of America prior to January 20, 2021, this set would be known as an empty set ; the number of people in this set is 0, since there were no women vice presidents before Harris. The empty set, also called the null set, is written symbolically using a pair of braces, , or a zero with a slash through it, .
The set containing the number , is a set with one element in it. It is not the same as the empty set , , which does not have any elements in it. Symbolically: .
Represent each of the following sets symbolically.
- The set of prime numbers less than 2.
- The set of birds that are also mammals.
- Answer
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- A prime number is a natural number greater than 1 that is only divisible by one and itself. Since there are no prime numbers less than 2, this set is empty, and we can represent it symbolically as follows: These two different symbols for the empty set can be used interchangeably.
- The set of birds and the set of mammals do not intersect, so the set of birds that are also mammals is empty, and we can represent it symbolically as
Represent the set of all numbers divisible by 0 symbolically.
We use the number zero to represent the concept of nothing every day. The machine language of computers is binary, consisting only of zeros and ones, and even way before that, the number zero was a powerful invention that allowed our understanding of mathematics and science to develop. The historical record shows the Babylonians first used zeros around 300 B.C., while the Mayans developed and began using zero separately around 350 A.D. What is considered the first formal use of zero in arithmetic operations was developed by the Indian mathematician Brahmagupta around 650 A.D.
Brahmagupta, Mathematician and Astronomer
Another interesting feature of the number zero is that although it is an even number, it is the only number that is neither negative nor positive.
For larger sets that have a natural ordering, sometimes an ellipsis is used to indicate that the pattern continues. It is common practice to list the first three elements of a set to establish a pattern, write the ellipsis, and then provide the last element. Consider the set of all lowercase letters of the English alphabet, . This set can be written symbolically as .
The sets we have been discussing so far are finite sets . They all have a limited or fixed number of elements. We also use an ellipsis for infinite sets , which have an unlimited number of elements, to indicate that the pattern continues. For example, in set theory, the set of natural numbers , which is the set of all positive counting numbers, is represented as .
Notice that for this set, there is no element following the ellipsis. This is because there is no largest natural number; you can always add one more to get to the next natural number. Because the set of natural numbers grows without bound, it is an infinite set.
Write the set of even natural numbers including and between 2 and 100, and label it with a capital . Include an ellipsis.
- Answer
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Write the label, , followed by an equal sign and then a bracket. Write the first three even numbers separated by commas, beginning with the number two to establish a pattern. Next, write the ellipsis followed by a comma and the last number in the list, 100. Finally, write the closing bracket to complete the set.
Write the label, , followed by an equal sign and then a bracket.
Write the first three even numbers separated by commas, beginning with the number 2 to establish a pattern.
Next, write the ellipsis followed by a comma and the last number in the list, 100.
Finally, write the closing bracket to complete the set.
Use an ellipsis to write the set of single digit numbers greater than or equal to zero and label it with a capital\(\(D\).
Our number system is made up of several different infinite sets of numbers. The set of integers , is another infinite set of numbers. It includes all the positive and negative counting numbers and the number zero. There is no largest or smallest integer.
Write the set of integers using the roster method, and label it with a .
- Answer
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Step 1: As always, we write the label and then the opening bracket. Because the negative counting numbers are infinite, to represent that the pattern continues without bound to the left, we must use an ellipsis as the first element in our list.
Step 2: We place a comma and follow it with at least three consecutive integers separated by commas to establish a pattern.
Step 3: Add an ellipsis to the end of the list to show that the set of integers continues without bound to the right.
Complete the list with a closing bracket. The set of integers may be represented as follows:
Write the set of odd numbers greater than 0 and label it with a capital\(\(M\).
A shorthand way to write sets is with the use of set builder notation , which is a verbal description or formula for the set. For example, the set of all lowercase letters of the English alphabet, , written in set builder notation is:
This is read as, “Set is the set of all elements \(x\) such that \(x\) is a lowercase letter of the English alphabet.”
Using set builder notation, write the set of all types of balls. Explain what the notation means.
- Answer
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The verbal description of the set is, “Set is the set of all elements such that is a ball.” This set can be written in set builder notation as follows:
Using set builder notation, write the set\(\(C\) of all types of cars.
Consider the set of letters in the word “happy.” Determine the best way to represent this set, and then write the set using either the roster method or set builder notation, whichever is more appropriate.
- Answer
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Because the letters in the word “happy” consist of a small finite set, the best way to represent this set is with the roster method. Choose a label to represent the set, such as .
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Notice that the letter “p” is only represented one time. This occurs because when representing members of a set, each unique element is only listed once no matter how many times it occurs. Duplicate elements are never repeated when representing members of a set.
Use the roster method or set builder notation to represent the collection of all musical instruments.
Computing the Cardinal Value of a Set
Almost all the sets most people work with outside of pure mathematics are finite sets. For these sets, the cardinal value or cardinality of the set is the number of elements in the set. For finite set , the cardinality is denoted symbolically as . For example, a set that contains four elements has a cardinality of 4.
How do we measure the cardinality of infinite sets? The ‘smallest’ infinite set is the set of natural numbers, or counting numbers, . This set has a cardinality of (pronounced "aleph-null"). All sets that have the same cardinality as the set of natural numbers are countably infinite . This concept, as well as notation using aleph, was introduced by mathematician Georg Cantor who once said, “A set is a Many that allows itself to be thought of as a One.”
Write the cardinal value of each of the following sets in symbolic form.
- The empty set.
- Answer
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- There are 5 distinct elements in set : a fork, a spoon, a knife, a meat thermometer, and a can opener. Therefore, the cardinal value of set is 5 and written symbolically as
- Because the empty set does not have any elements in it, the cardinality of the empty set is zero. Symbolically we write this as:
Write the cardinal value of each of the following sets in symbolic form.
Set \(P\) is the set of prime numbers less than 2 .
2. Set \(A\) is the set of lowercase letters of the English alphabet, \(A=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \ldots, \mathrm{z}\}\).
Now that we have learned to represent finite and infinite sets using both the roster method and set builder notation, we should also be able to determine if a set is finite or infinite based on its verbal or symbolic description. One way to determine if a set is finite or not is to determine the cardinality of the set. If the cardinality of a set is a natural number, then the set is finite.
Classify each of the following sets as infinite or finite.
- is the set of lowercase letters of the English Alphabet, .
- Answer
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- . Since 5 is a natural number, the set is finite.
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26 n . Since 26 is a natural number, the set is finite.( A ) = 26 - Set is the set of rational numbers or fractions. Because the set of integers is a subset of the set of rational numbers, and the set of integers is infinite, the set of rational numbers is also infinite. There is no smallest or largest rational number.
Classify each of the following sets as infinite or finite.
\(B=\{b, a, k, e\}\)
2. \(\mathbb{R}=\{x \mid x\) is a real number \(\}\)
Equal versus Equivalent Sets
When speaking or writing we tend to use equal and equivalent interchangeably, but there is an important distinction between their meanings. Consider a new Ford Escape Hybrid and a new Toyota Rav4 Hybrid. Both cars are hybrid electric sport utility vehicles; in that sense, they are equivalent . They will both get you from place to place in a relatively fuel-efficient way. In this example we are comparing the single member set {Toyota Rav4 Hybrid} to the single member set {Ford Escape Hybrid}. Since these two sets have the same number of elements, they are also equivalent mathematically, meaning they have the same cardinality. But they are not equal, because the two cars have different looks and features, and probably even handle differently. Each manufacturer will emphasize the features unique to their vehicle to persuade you to buy it; if the SUVs were truly equal, there would be no reason to choose one over the other.
Now consider two Honda CR-Vs that are made with exactly the same parts, on the same assembly line within a few minutes of each other—these SUVs are equal . They are identical to each other, containing the same elements without regard to order, and the only differentiator when making a purchasing decision would be varied pricing at different dealerships. The set {Honda CR-V} is equal to the set {Honda CR-V}. Symbolically, we represent equal sets as and equivalent sets as .
Now, let us consider a Toyota dealership that has 10 RAV4s on the lot, 8 Prii, 7 Highlanders, and 12 Camrys. There is a one-to-one relationship between the set of vehicles on the lot and the set consisting of the number of each type of vehicle on the lot. Therefore, these two sets are equivalent, but not equal. The set {RAV4, Prius, Highlander, Camry} is equivalent to the set {10, 8, 7, 12} because they have the same number of elements.
If two sets are equal, they are also equivalent, because equal sets also have the same cardinality.
Determine if the following pairs of sets are equal, equivalent, or neither.
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- The empty set and the set of prime numbers less than 2.
- The set of vowels in the word happiness and the set of consonants in the word happiness.
- Answer
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- Sets E and F both have a cardinal value of 5, but the elements in these sets are different. So, the two sets are equivalent, but they are not equal: .
- The set of prime numbers consists of the set of counting numbers greater than one that can only be divided evenly by one and itself. The set of prime numbers less than 2 is an empty set, since there are no prime numbers less than 2. Therefore, these two sets are equal (and equivalent).
- The set of vowels in the word happiness is and the set of consonants in the word happiness is The cardinal value of these sets two sets is and respectively. Because the cardinality of the two sets differs, they are not equivalent. Further, their elements are not identical, so they are also not equal.
Determine if the following pairs of sets are equal, equivalent, or neither.
Set \(B=\{b, a, k, e\}\) and set \(A=\{a, b, e, k\}\)
2. Set \(B=\{b, a, k, e\}\) and set \(F=\{f, l, a, k, e\}\)
3. Set \(B=\{b, a, k, e\}\) and set \(C=\{c, a, k, e\}\)
Georg Cantor, the father of modern set theory, was born during the year 1845 in Saint Petersburg, Russa and later moved to Germany as a youth. Besides being an accomplished mathematician, he also played the violin. Cantor received his doctoral degree in Mathematics at the age of 22.
In 1870, at the age of 25 he established the uniqueness theorem for trigonometric series. His most significant work happened between 1874 and 1884, when he established the existence of transcendental numbers (also called irrational numbers) and proved that the set of real numbers are uncountably infinite—despite the objections of his former professor Leopold Kronecker.
Cantor published his final treatise on set theory in 1897 at the age of 52, and was awarded the Sylvester Medial from the Royal Society of London in 1904 for his contributions to the field. At the heart of Cantor’s work was his goal to solve the continuum problem, which later influenced the works of David Hilbert and Ernst Zermelo.
References:
Wikipedia contributors. “Cantor.” Wikipedia, The Free Encyclopedia, 23 Mar. 2021. Web. 20 Jul. 2021.
Akihiro Kanamori, “Set Theory from Cantor to Cohen,” Editor(s): Dov M. Gabbay, Akihiro Kanamori, John Woods, Handbook of the History of Logic , North-Holland, Volume 6, 2012.
Check Your Understanding
- A _____________ is a well-defined collection of objects.
- The ___________ of a finite set \(A\), denoted \(n(A)\), is the number of elements in set \(A\).
- Determine if the following description describes a well-defined set: “The top 5 pizza restaurants in Chicago.”
- The United States is the only country to have landed people on the moon as of March 21, 2021. What is the cardinality of the set of all people who have walked on the moon prior to this date?
- Set \(A\) is a set of a dozen distinct donuts, and set \(B\) is a set of a dozen different types of apples. Is set \(A\) equal to set \(B\), equivalent to set \(B\), or neither?
- Is the set of all butterflies in the world a finite set or an infinite set?
- Represent the set of all upper-case letters of the English alphabet using both the roster method and set builder notation.


4.7: Ocean Tides and the Moon
By the end of this section, you will be able to:
- Describe what causes tides on Earth
- Explain why the amplitude of tides changes during the course of a month
Anyone living near the sea is familiar with the twice-daily rising and falling of the tides . Early in history, it was clear that tides must be related to the Moon because the daily delay in high tide is the same as the daily delay in the Moon’s rising. A satisfactory explanation of the tides, however, awaited the theory of gravity, supplied by Newton.
The Pull of the Moon on Earth
The gravitational forces exerted by the Moon at several points on Earth are illustrated in Figure 4.16. These forces differ slightly from one another because Earth is not a point, but has a certain size: all parts are not equally distant from the Moon, nor are they all in exactly the same direction from the Moon. Moreover, Earth is not perfectly rigid. As a result, the differences among the forces of the Moon’s attraction on different parts of Earth (called differential forces ) cause Earth to distort slightly. The side of Earth nearest the Moon is attracted toward the Moon more strongly than is the center of Earth, which in turn is attracted more strongly than is the side opposite the Moon. Thus, the differential forces tend to stretch Earth slightly into an oblate spheroid (which is something like an American football shape), with its long diameter pointed toward the Moon.
If Earth were made of water, it would distort until the Moon’s differential forces over different parts of its surface came into balance with Earth’s own gravitational forces pulling it together. Calculations show that in this case, Earth would distort from a sphere by amounts ranging up to nearly 1 meter. Measurements of the actual deformation of Earth show that the solid Earth does distort, but only about one-third as much as water would, because of the greater rigidity of Earth’s interior.
Because the tidal distortion of the solid Earth amounts—at its greatest—to only about 20 centimeters, Earth does not distort enough to balance the Moon’s differential forces with its own gravity. Hence, objects at Earth’s surface experience tiny horizontal tugs, tending to make them slide about. These tide-raising forces are too insignificant to affect solid objects like astronomy students or rocks in Earth’s crust, but they do affect the waters in the oceans.
The Formation of Tides
The tide-raising forces, acting over a number of hours, produce motions of the water that result in measurable tidal bulges in the oceans. Water on the side of Earth facing the Moon flows toward it, with the greatest depths roughly at the point below the Moon. On the side of Earth opposite the Moon, water also flows to produce a tidal bulge (Figure 4.17).
Note that the tidal bulges in the oceans do not result from the Moon’s compressing or expanding the water, nor from the Moon’s lifting the water “away from Earth.” Rather, they result from an actual flow of water over Earth’s surface toward the two regions below and opposite the Moon, causing the water to pile up to greater depths at those places (Figure 4.18).
In the idealized (and, as we shall see, oversimplified) model just described, the height of the tides would be only a few feet. The rotation of Earth would carry an observer at any given place alternately into regions of deeper and shallower water. An observer being carried toward the regions under or opposite the Moon, where the water was deepest, would say, “The tide is coming in”; when carried away from those regions, the observer would say, “The tide is going out.” During a day, the observer would be carried through two tidal bulges (one on each side of Earth) and so would experience two high tides and two low tides.
The Sun also produces tides on Earth, although it is less than half as effective as the Moon at tide raising. The actual tides we experience are a combination of the larger effect of the Moon and the smaller effect of the Sun. When the Sun and Moon are lined up (at new moon or full moon), the tides produced reinforce each other and so are greater than normal (Figure 4.19). These are called spring tides (the name is connected not to the season but to the idea that higher tides “spring up”). Spring tides are approximately the same, whether the Sun and Moon are on the same or opposite sides of Earth, because tidal bulges occur on both sides. When the Moon is at first quarter or last quarter (at right angles to the Sun’s direction), the tides produced by the Sun partially cancel the tides of the Moon, making them lower than usual. These are called neap tides.
The “simple” theory of tides, described in the preceding paragraphs, would be sufficient if Earth rotated very slowly and were completely surrounded by very deep oceans. However, the presence of land masses stopping the flow of water, the friction in the oceans and between oceans and the ocean floors, the rotation of Earth, the wind, the variable depth of the ocean, and other factors all complicate the picture. This is why, in the real world, some places have very small tides while in other places huge tides become tourist attractions. If you have been in such places, you may know that “tide tables” need to be computed and published for each location; one set of tide predictions doesn’t work for the whole planet. In this introductory chapter, we won’t delve further into these complexities.
The rubbing of water over the face of Earth involves an enormous amount of energy. Over long periods of time, the friction of the tides is slowing down the rotation of Earth. Our day gets longer by about 0.002 second each century. That seems very small, but such tiny changes can add up over millions and billions of years.
Although Earth’s spin is slowing down, the angular momentum (see Orbits and Gravity) in a system such as the Earth-Moon system cannot change. Thus, some other spin motion must speed up to take the extra angular momentum. The details of what happens were worked out over a century ago by George Darwin , the son of naturalist Charles Darwin. George Darwin (see Figure 4.20) had a strong interest in science but studied law for six years and was admitted to the bar. However, he never practiced law, returning to science instead and eventually becoming a professor at Cambridge University. He was a protégé of Lord Kelvin, one of the great physicists of the nineteenth century, and he became interested in the long-term evolution of the solar system. He specialized in making detailed (and difficult) mathematical calculations of how orbits and motions change over geologic time.
What Darwin calculated for the Earth-Moon system was that the Moon will slowly spiral outward, away from Earth. As it moves farther away, it will orbit less quickly (just as planets farther from the Sun move more slowly in their orbits). Thus, the month will get longer. Also, because the Moon will be more distant, total eclipses of the Sun will no longer be visible from Earth.
Both the day and the month will continue to get longer, although bear in mind that the effects are very gradual. Darwin’s calculations were confirmed by mirrors placed on the Moon by Apollo 11 astronauts. These show that the Moon is moving away by 3.8 centimeters per year, and that ultimately—billions of years in the future—the day and the month will be the same length (about 47 of our present days). At this point the Moon will be stationary in the sky over the same spot on Earth, meaning some parts of Earth will see the Moon and its phases and other parts will never see them. This kind of alignment is already true for Pluto’s moon Charon (among others). Its rotation and orbital period are the same length as a day on Pluto.


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The gastrointestinal system has four main functions: ingestion, digestion, absorption, and metabolization.
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The small intestine handles 90% of nutrient and water absorption.
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The duodenum is the site for iron absorption and some vitamin B
9
(folate) absorption, but the major site for vitamin B
9
absorption is the jejunum.
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Minerals and vitamins A, D, E, and K are absorbed by the small intestine.
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The last segment of the small intestine, the ileum, is responsible for the absorption of bile salts and vitamin B
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(cobalamin).
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The large intestine has three main functions in digestion overall: absorption of the remaining water and electrolytes, absorption and production of vitamins, and formation and movement of feces to the rectum for elimination.
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The first portion of the large intestine, the ascending colon, removes any remaining water and key nutrients from the indigestible material that passes through it, while solidifying this waste to form stool.
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The nutritional plan for clients with gastrointestinal diseases will include supplementation and higher nutritional intake levels than for a healthy client.
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Challenges to nutritional goals include malabsorption, food allergies, comorbidities, and side effects of the gastrointestinal disease processes.


8.4: Starting Tips for Practice
Above all, avoid the practice of “cramming” when deciding on a practice strategy. While this strategy may work on a history exam when trying to memorize dates and facts at the last minute to regurgitate them before forgetting them, this strategy does not work as well for communicators. Remember, public speaking acts like any other learned skill that individuals develop and refine, like performing a saxophone solo, shooting a free-throw shot, or learning a new language. Practice should begin well in advance of the speech and spread out over time, rather than concentrated all in one or two sessions. Brief yet frequent practice sessions held over a period of time works best.
When beginning to practice, understand the vast, yet subtle difference between memorization and learning. For example, people asked to tell the story of their first kiss, the time they learned to ride a bike, or the first time they flew in an airplane could most likely recall and tell others that story with relative ease. They did not need to memorize these stories or write down lines, study them, and then repeat them until they could recite them perfectly each time. People find these stories easy to tell without much advance preparation because they represent learned experiences. The speaker knows the material; telling the story involves putting the concepts into words. At most, notes for telling these types of stories would consist of a few keywords and phrases to jog the memory and keep it on track.
Home Memorization , on the other hand, refers to the process of storing and recalling exact phrases and words. Actors use this process to recall exact lines in prewritten scripts, but for a public speaker, whose primary goal consists of having a fluid conversation with an audience, it will not suffice. Audiences, first of all, can easily see through a speaker with a scripted speech. From eye contact to vocal tone, rate, and variety, everything about that speaker’s delivery will seem canned or manufactured, anything but genuine and authentic. Additionally, what would happen if the speaker forgot a line? Potentially, that could cause what some might refer to as a “speech fatality”—an unrecoverable error that causes extreme anxiety, resulting in panic and possibly failure. As opposed to memorized speeches, learned speeches offer speakers a backup plan, because they know the concepts they are trying to convey, so the words they choose to convey those concepts may vary, but the essence of the message remains the same. For that reason, avoid rote memorization at all costs, and instead, get to know the material and use notes to help provide guidance through that material.
When practicing, remember to recite certain phrases differently each time. Not only does this prevent memorization as a result of repeated practice, but it also effectively “trains” speakers to deliver their message in a variety of ways, developing multiple phrases for saying the same thing. That way, if they get stuck trying to remember how they planned on phrasing a certain point, they will have numerous options from which to draw, keeping the speech fluid and conversational.
Home Remember that, during practice sessions, speakers should practice in the exact same way they intend to present when in front of the audience. This means that, if they plan to stand and move around, they should not practice from a seated position, staring at a computer screen, and going through the lines in their head. They will want to stand up, move around, gesture, and practice saying phrases out loud, projecting their voice just as they will in front of the audience. This practice method trains the body (the delivery vehicle) via muscle memory . In the same way a professional golfer practices hitting a drive hundreds of times to develop and refine the swing, a public speaker needs to train the muscles of the body like the neck so he or she remembers to move the head around the room, effectively maintaining eye contact with the entire room. This also includes the diaphragm, which is needed to expel enough air through the vocal cords to project the voice to the back of the room.
For instance, what would happen if a speaker only practiced from a seated position? That person will likely stand to deliver the message feeling uncomfortable due to the unfamiliar method of delivery, which leads to uncertainty and generates natural anxiety. In the same way, practicing phrases internally instead of speaking them out loud only teaches a person to remain quiet and reserved.
Lastly, consider conducting at least one practice session as a full-dress rehearsal.
 This technique involves engaging one’s imagination to visualize a positive outcome from the upcoming performance. For example, a basketball player preparing to take a crucial free-throw shot to win the game at the last minute might dribble the ball a few times, then pause and visualize the shot before taking it, imagining the ball going through the air in a perfect arc and sinking through the hoop, touching nothing but the net. Only after that visualization will the player physically take the shot. Public speakers should imagine themselves successfully delivering the speech as close to an ideal reality as possible. One of the worst things a speaker can do prior to the speech is engage in negative self-talk, such as thinking “I’m not ready for this” or “This is going to stink!”
While everyone will have a different “best practice” number of practice sessions to effectively learn a speech, it requires some trial and error to find out what will work best. Some people have the good fortune to possess photographic memories and can recall written information effortlessly, and for them, perhaps one practice session will work fine. Others may end up on the opposite end of that spectrum, requiring 30 or more practice run-throughs before they feel comfortable enough to deliver their speech. Despite most people falling anywhere along this spectrum, many novice public speaking students suggest that the following three practice “sessions” outlined in this chapter provide a recipe for success.


4.3: Measuring Your Success
When is this going to make money?
In the typical business world, profits are the ultimate measure of success. Investors look to the financial statements to find the story of how the company performed. Projected financials are also important. If you don’t know how to create financial statements, ask a business major or accountant to help you.
Cash is king
For a start-up business, or any business, having cash when you need it is crucial. In fact, many profitable businesses fail because they grow too quickly and run out of cash along the way. You could face this problem too if the payments to your suppliers are due before you get paid by your customers. Predict the timing of your cash inflow and outflow, and make sure you have the cash to cover yourself when you make decisions.
In operating your company, keep close track of your cash position. While you’re working to make your venture viable, you should probably estimate your cash flow on a daily basis. The term burn rate refers to how quickly a company is using up its available cash. If you don’t have a perfect picture of how you are using money, estimate your burn rate to determine how many days you have until you need more funding or have to cease operations.
Your original business plan might include a rough cash flow estimation. Your financiers will use it to determine when they will begin getting a return on their investment, and how high the return will be.
The balance sheet reports the position of the company at a point in time. It is divided into three parts: assets, liabilities, and owner’s equity. Assets represent everything of value in the firm, such as property, inventory, cash, and payments owed by customers. Typically the assets are listed in order of decreasing liquidity, meaning that the ones easiest to convert cash are first. Liabilities represent debts owed to customers, employees, and banks. These are listed roughly in order of when they come due. Owner’s equity is what’s left over when liabilities are subtracted from assets. It’s the value of the owner’s (or stockholder’s) share of the company if it were to cease operations. The actual value of the company is based on its future potential (and should be higher than owner’s equity on the balance sheet).
The income statement reports the revenues and expenses of the company over a specified time period, culminating in a profit or loss. Income is reported based on accounting rules and often does not reflect cash changing hands. Revenues can be reported when a product is sold and delivered to the buyer even if they have not paid (but will pay in the future). The income statement also follows the “matching principle,” which requires expenses to be matched to revenues and reported at the same time. This means that expenses incurred to produce a product are not reported in the income statement until that product is sold.
The cash flow statement documents the sources and uses of cash over a specified time period. Each activity where cash is gained or lost is listed, along with the amount. The activities are typically divided into operations, investment, and financing. Positive cash flow from operations is a sign that the business is making money from the goods and services it’s selling. The investment section documents cash used to purchase assets or gained in the sale of assets. The financing section represents cash borrowed from banks or gained through the sale of stock, and cash used to pay back loans or repurchase stock.
More than one bottom line
You and your investors may want to achieve other returns on your venture. Again, we’re referring to the triple bottom line. While the first bottom line, profit, is shown in your financial statements, you must create metrics to report your other bottom lines (see the sidebar on the bottom left). Committing to goals that improve the environment or the human condition should be coupled with establishing a system to track your progress. It’s difficult to see how far you have come if you don’t have consistent ways to measure your impact.
The National Social Venture Competition recommends two different options for quantifying social impact in dollar terms. One, based on project valuation techniques at the International Finance Corporation, involves thinking of the best alternative solution that exists to meeting the social need you are tackling. To find the monetary benefit of your method, subtract your expenses to meet this need from the expenses that would be incurred if the alternative solution was used instead. The second is based on the approach used by Roberts Enterprise Development Fund to value its social venture returns. The monetary return is calculated by adding the savings to society (in governmental program costs that do not have to be spent) and the gains to society (in tax revenues) from your work, and subtracting your business expenses that are related to implementing the social purpose (and would not be otherwise incurred).
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The Inexact Science of Social Return on Investment (SROI)
Social Return on Investment (SROI) measures social impact in dollar terms. Whereas financial return is relatively straightforward, differences in personal values and definitions of “social impact” require varying measures of social return. In fact, many social returns are impossible to quantify, making this a necessarily inexact science. SROI can be thought of in terms of jobs created, houses built, or lives saved. But to start a social venture, you need to come up with a model to measure your particular social impact. A working SROI model helps investors determine the attractiveness of a potential investment. Documenting meaningful social returns is crucial; without an understanding of the tangible and measurable effect created through an investor’s activity, the reasons to engage in this critical work are merely anecdotal.
Developing a cash flow budget
Here are some steps you can take to create a cash flow budget you can rely on:
From “Financial Troubleshooting,” Inc. Magazine, August 2000, by Michael Pellecchia |


10.4: Treatments and Nutrition
By the end of this section, you should be able to:
- 10.3.1 Identify drug–food interactions for their impact on treatments used for hematologic disorders.
- 10.3.2 Identify treatments and medications that can cause nutritional deficiencies in clients with hematologic disorders.
Drug–Food Interactions with Treatments for Hematologic Disorders
Some treatments for hematologic disorders will not work as intended if taken with certain foods. Limiting groups of foods during an already vulnerable state of nutritional need can lead to additional micronutrient deficiencies. For example, individuals requiring iron replacement are instructed to avoid cereals, dietary fiber, tea, coffee, eggs, and milk while taking iron supplement s because these foods are known to decrease iron absorption (Moustarah, 2022).
Unlike individuals with other hematologic illnesses previously discussed, individuals with SCD and thalassemia are at risk for iron overload rather than iron deficiency. The primary cause of iron overload is red blood cell transfusion therapy. However, iron overload can occur in people with and without transfusion-dependent thalassemia because of intestinal absorption of iron secondary to ineffective erythropoiesis (Srichairatanakool et al., 2020). These individuals are therefore counseled to avoid iron-rich foods. The rate of iron accumulation varies and correlates with the number of red blood cell transfusions received. Each unit of packed red blood cells contains 200–250 mg of elemental iron, equating to 0.3–0.6 mg/kg/day of elemental iron , and most transfusion-dependent older children and adults will receive 2–4 units of blood monthly. The symptoms and associated health risks from iron overload depend on where the iron accumulates. The liver, heart, and endocrine glands are the most common initial sites (Srichairatanakool et al., 2020). Frequent complications of iron overload in people living with hemoglobinopathies include:
- Increased risk for thrombosis
- Pulmonary hypertension
- Hypothyroidism
- Hypogonadism
- Osteoporosis
Nurses should ensure client rights regarding medication administration, which includes understanding any contraindications and drug–food interactions. Equally important is a nutritional evaluation prior to the start of treatment for hematologic disorders and the development of a plan to avoid further nutritional deficiencies that can impact the desired response of treatment.
Read the following clinical scenario and then answer the questions that follow. This case study is a follow-up to Case Study Part A.
Catalina’s health care provider confirms that her physical exam is normal other than signs of pallor. A complete blood count (CBC) and ferritin level are ordered to evaluate for ID and anemia. Catalina is also referred to a gynecologist for further evaluation of her heavy menstrual bleeding.
Catalina’s laboratory test results reveal a hemoglobin level of 7 g/dL (expected range: 12–15 g/dL) and a ferritin level of 9 ng/mL (expected range: 12–150 ng/mL), confirming IDA. She is started on oral iron replacement and counseled on foods to avoid while taking oral iron for better absorption. She is also encouraged to add iron-rich foods to her diet daily and to return in 6 weeks.
At her 6-week follow-up visit, Catalina reports that she is feeling less tired, and her shortness of breath has resolved. Her hemoglobin level at this visit is 10 g/dL. She has also been evaluated by a gynecologist and started on oral birth control pills to decrease her heavy menstrual bleeding. Catalina is instructed to continue oral iron therapy for at least 2 more months and then return for follow-up.
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Constipation
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Increased appetite
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Weakness and dizziness
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Increased menstrual bleeding
Treatments and Medications That Can Cause Nutritional Deficiencies
Treatments for hematologic malignancies can lead to nutritional deficiencies due to decreased food intake that is secondary to adverse treatment effects such as nausea, vomiting, diarrhea, mouth sores, and loss of appetite. Traditional chemotherapy and novel targeted therapy can affect taste and saliva production, further affecting the individual’s interest in food.
Several of the most frequent hematologic malignancies, such as myelodysplasia and multiple myeloma (a blood cancer affecting plasma cells found in bone marrow), occur in older adults. The average age at diagnosis for hematologic malignancies in adults is 70 years, and 40% already have reduced food intake. The prevalence of malnutrition in these individuals is approximately 30% and is more common in male clients (Stauder et al., 2020). Associated symptoms include a high systemic inflammatory score, alterations in mood and cognition, and moderate to severe fatigue. Decreased food intake, weight loss, low body mass index, and laboratory evidence of malnourishment have been associated with decreased survival rates in older adults with hematologic malignancies (Stauder et al., 2020). A complete nutritional assessment, timely intervention with micronutrient and macronutrient replacement, and dietary referral are essential.


3.2: Numerical Techniques
Engineering majors are a majority of the students in the kind of physics course for which this book is designed, so most likely you fall into that category. Although you surely recognize that physics is an important part of your training, if you've had any exposure to how engineers really work, you're probably skeptical about the flavor of problem-solving taught in most science courses. You realize that not very many practical engineering calculations fall into the narrow range of problems for which an exact solution can be calculated with a piece of paper and a sharp pencil. Real-life problems are usually complicated, and typically they need to be solved by number-crunching on a computer, although we can often gain insight by working simple approximations that have algebraic solutions. Not only is numerical problem-solving more useful in real life, it's also educational; as a beginning physics student, I really only felt like I understood projectile motion after I had worked it both ways, using algebra and then a computer program. (This was back in the days when 64 kilobytes of memory was considered a lot.)
In this section, we'll start by seeing how to apply numerical techniques to some simple problems for which we know the answer in “closed form,” i.e., a single algebraic expression without any calculus or infinite sums. After that, we'll solve a problem that would have made you world-famous if you could have done it in the seventeenth century using paper and a quill pen! Before you continue, you should read Appendix 1 on page 908 that introduces you to the Python programming language.
First let's solve the trivial problem of finding how much time it takes an object moving at speed v to travel a straight-line distance dist . This closed-form answer is, of course, dist/v , but the point is to introduce the techniques we can use to solve other problems of this type. The basic idea is to divide the distance up into n equal parts, and add up the times required to traverse all the parts. The following Python function does the job. Note that you shouldn't type in the line numbers on the left, and you don't need to type in the comments, either. I've omitted the prompts >>> and ... in order to save space.
import math def time1(dist,v,n): x=0 # Initialize the position. dx = dist/n # Divide dist into n equal parts. t=0 # Initialize the time. for i in range(n): x = x+dx # Change x. dt=dx/v # time=distance/speed t=t+dt # Keep track of elapsed time. return t
How long does it take to move 1 meter at a constant speed of 1 m/s? If we do this,
>>> print(time1(1.0,1.0,10)) # dist, v, n 0.99999999999999989
Python produces the expected answer by dividing the distance into ten equal 0.1-meter segments, and adding up the ten 0.1-second times required to traverse each one. Since the object moves at constant speed, it doesn't even matter whether we set n to 10, 1, or a million:
>>> print(time1(1.0,1.0,1)) # dist, v, n 1.0
Now let's do an example where the answer isn't obvious to people who don't know calculus: how long does it take an object to fall through a height h , starting from rest? We know from example 8 on page 83 that the exact answer, found using calculus, is \(\sqrt{2h/g}\). Let's see if we can reproduce that answer numerically. The main difference between this program and the previous one is that now the velocity isn't constant, so we need to update it as we go along. Conservation of energy gives \(mgh=(1/2)mv^2+mgy\) for the velocity \(v\) at height \(y\), so \(v=-\sqrt{2g(h-y)}\). (We choose the negative root because the object is moving down, and our coordinate system has the positive \(y\) axis pointing up.)
import math def time2(h,n): g=9.8 # gravitational field y=h # Initialize the height. v=0 # Initialize the velocity. dy = -h/n # Divide h into n equal parts. t=0 # Initialize the time. for i in range(n): y = y+dy # Change y. (Note dy<0.) v = -math.sqrt(2*g*(h-y)) # from cons. of energy dt=dy/v # dy and v are <0, so dt is >0 t=t+dt # Keep track of elapsed time. return t
For h =1.0 m, the closed-form result is \(\sqrt{2\cdot1.0\ \text{m}/9.8\ \text{m}/\text{s}^2}=0.45\ \text{s}\). With the drop split up into only 10 equal height intervals, the numerical technique provides a pretty lousy approximation:
>>> print(time2(1.0,10)) # h, n 0.35864270709233342
But by increasing n to ten thousand, we get an answer that's as close as we need, given the limited accuracy of the raw data:
>>> print(time2(1.0,10000)) # h, n 0.44846664060793945
A subtle point is that we changed y in line 9, and then on line 10 we calculated v , which depends on y . Since y is only changing by a ten-thousandth of a meter with each step, you might think this wouldn't make much of a difference, and you'd be almost right, except for one small problem: if we swapped lines 9 and 10, then the very first time through the loop, we'd have v =0, which would produce a division-by-zero error when we calculated dt ! Actually what would make the most sense would be to calculate the velocity at height y and the velocity at height y+dy (recalling that dy is negative), average them together, and use that value of y to calculate the best estimate of the velocity between those two points. Since the acceleration is constant in the present example, this modification results in a program that gives an exact result even for n =1:
import math def time3(h,n): g=9.8 y=h v=0 dy = -h/n t=0 for i in range(n): y_old = y y = y+dy v_old = math.sqrt(2*g*(h-y_old)) v = math.sqrt(2*g*(h-y)) v_avg = -(v_old+v)/2. dt=dy/v_avg t=t+dt return t
>>> print(time3(1.0,1)) # h, n 0.45175395145262565
Now we're ready to attack a problem that challenged the best minds of Europe back in the days when there were no computers. In 1696, the mathematician Johann Bernoulli posed the following famous question. Starting from rest, an object slides frictionlessly over a curve joining the point \((a,b)\) to the point \((0,0)\). Of all the possible shapes that such a curve could have, which one gets the object to its destination in the least possible time, and how much time does it take? The optimal curve is called the brachistochrone , from the Greek “short time.” The solution to the brachistochrone problem evaded Bernoulli himself, as well as Leibniz, who had been one of the inventors of calculus. The English physicist Isaac Newton, however, stayed up late one night after a day's work running the royal mint, and, according to legend, produced an algebraic solution at four in the morning. He then published it anonymously, but Bernoulli is said to have remarked that when he read it, he knew instantly from the style that it was Newton --- he could “tell the lion from the mark of his claw.”
Rather than attempting an exact algebraic solution, as Newton did, we'll produce a numerical result for the shape of the curve and the minimum time, in the special case of \(a\)=1.0 m and \(b\)=1.0 m. Intuitively, we want to start with a fairly steep drop, since any speed we can build up at the start will help us throughout the rest of the motion. On the other hand, it's possible to go too far with this idea: if we drop straight down for the whole vertical distance, and then do a right-angle turn to cover the horizontal distance, the resulting time of 0.68 s is quite a bit longer than the optimal result, the reason being that the path is unnecessarily long. There are infinitely many possible curves for which we could calculate the time, but let's look at third-order polynomials,
where we require \(c_3=(b-c_1a-c_2a^2)/a^3\) in order to make the curve pass through the point \((a,b)\). The Python program, below, is not much different from what we've done before. The function only asks for \(c_1\) and \(c_2\), and calculates \(c_3\) internally at line 4. Since the motion is two-dimensional, we have to calculate the distance between one point and the next using the Pythagorean theorem, at line 16.
import math def timeb(a,b,c1,c2,n): g=9.8 c3 = (b-c1*a-c2*a**2)/(a**3) x=a y=b dx = -a/n t=0 for i in range(n): y_old = y x = x+dx y = c1*x+c2*x**2+c3*x**3 dy = y-y_old v_old = math.sqrt(2*g*(b-y_old)) v = math.sqrt(2*g*(b-y)) v_avg = (v_old+v)/2. ds = math.sqrt(dx**2+dy**2) # Pythagorean thm. dt=ds/v_avg t=t+dt return t
As a first guess, we could try a straight diagonal line, \(y=x\), which corresponds to setting \(c_1=1\), and all the other coefficients to zero. The result is a fairly long time:
>>> a=1. >>> b=1. >>> n=10000 >>> c1=1. >>> c2=0. >>> print(timeb(a,b,c1,c2,n)) 0.63887656499994161
What we really need is a curve that's very steep on the right, and flatter on the left, so it would actually make more sense to try \(y=x^3\):
>>> c1=0. >>> c2=0. >>> print(timeb(a,b,c1,c2,n)) 0.59458339947087069
This is a significant improvement, and turns out to be only a hundredth of a second off of the shortest possible time! It's possible, although not very educational or entertaining, to find better approximations to the brachistochrone curve by fiddling around with the coefficients of the polynomial by hand. The real point of this discussion was to give an example of a nontrivial problem that can be attacked successfully with numerical techniques. I found the first approximation shown in figure a ,
by using the program listed in appendix 2 on page 911 to search automatically for the optimal curve. The seventh-order approximation shown in the figure came from a straightforward extension of the same program.
Contributors and Attributions
Benjamin Crowell (Fullerton College). Conceptual Physics is copyrighted with a CC-BY-SA license.


21.6: Chemistry of Acid Anhydrides
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Objectives
After completing this section, you should be able to
- write an equation to illustrate the preparation of an acid anhydride from an acid halide and the sodium salt of a carboxylic acid.
- identify the product formed from the reaction of a given acid halide with the sodium salt of a given carboxylic acid.
- identify the acid halide, carboxylate salt, or both, required to prepare a given acid anhydride.
- write an equation to describe the reaction of an acid anhydride with each of the following: water, alcohol, ammonia, a primary or secondary amine, lithium aluminum hydride.
- identify the product formed when a given acid anhydride is reacted with any of the reagents listed in Objective 4, above.
- write a detailed mechanism for the reaction of an acid anhydride with any of the reagents listed in Objective 4, above.
- identify the acid anhydride, the nucleophilic reagent, or both, needed to prepare a specified carboxylic acid, ester, amide, or primary alcohol.
Study Notes
The reactions described in this section are, in principle, identical to those discussed in Section 21.4. Once you have understood the mechanism of nucleophilic acyl substitution, these reactions should not present you with any great difficulty, and memorization can be kept to a minimum.
This figure provides a convenient general summary of a few of the reactions described in Section 21.5. Note that from a synthetic perspective the ester‑ and amide‑forming reactions are the most common, so they are the focus of this section.
Preparation of Acid Anhydrides
As show in Section 21.4, acid anhydrides are generally made using a nucleophilic acyl substitution reaction of an acid chloride with a carboxylic acid or a carboxylate anion.
Chemistry of Anhydrides
Anhydrides are highly reactive to nucleophilic attack and undergo many of the same reactions as acid chlorides that were explored in section 21.4. Although slower reacting than acid chlorides, anhydrides react with water to form carboxylic acids, with alcohols to form esters, and with amines to form amides. Anhydrides can also be reduced to 1 o alcohols by hydride reduction. Because many anhydrides are made from the coupling of two carboxylic acids, reactions using anyhydrides waste one equivalent of the carboxylic acid as a leaving group. As a consequence, reactions are only commonly performed with inexpensive, readily available anhydrides, such as acetic anhydride or benzoic anhydride. Anydrides have the advantage of being easier to work with than acid chlorides.
Generic Reaction with an Anhydride
Generic Mechanism of Nucleophilic Acyl Substitution using an Anhydride
The following nucleophilic acyl substitution reactions are all similar and can be represented by one generic mechanism. The nuceophile (water, ammonia, amine, or alcohol) adds to one of the carbonyl carbons in the anhydride forming a tetrahedral alkoxide intermediate. The reforming of the carbonyl C=O bond eliminates a carboxylate leaving group. Because the nucleophile is neutral, a protonated intermediate is formed. A base deprotonates the intermediate to form the neutral carboxylic acid derivative product (carboxylic acid, ester, amide).
1) Nucleophilic attack
2) Deprotonation
3) Leaving group removal
Predicting the Products of an Anhydride Reaction
An anhydride and a neutral nucleophile react to form a new acyl compound through nucleophilic acyl substitution. During this reaction there are three changes in bonding. The leaving group is removed from the anhydride. The neutral nucleophile loses a hydrogen. A C-Nuc bond is formed between the nucleophile and the anhydride’s electrophilic carbonyl carbon.
Reactions with Asymmetrical Anhydrides
Asymmetrical anhydrides are typically not used in the formation of esters and amides because they have the possibility of forming two different products. This is not the case when symmetrical anhydrides are used.
Conversion of Acid Anhydrides to Carboxylic Acids: Hydrolyisis
Anhydrides react rapidly with water to form two carboxylic acids compounds. Because anhydrides are often prepared from carboxylic acids this reaction serves little synthetic value. However, this reaction serves as a reminder to prevent the exposure of anhydrides to moisture because they will become contaminated with the corresponding carboxylic acids.
General Reaction
Example
Conversion of Acid Anhydrides to Esters: Alcoholyisis
Anhydrides react with alcohols to form esters as the main product and a carboxylate as a side product. The reaction is typically run with a base, such as NaOH or pyridine, to remove any acid produced. Notice that one acyl group from the anhydride is incorporated into the ester and the other acyl group forms a carboxylate.
General Reaction
Example
Conversion of Acid Anhydrides to Amides: Aminolysis
Acid anhydrides react with ammonia, 1 o , or 2 o amines to form the corresponding amides. Two molar equivalents of amine are required. It is important to run this reaction with a base to neutralize the acid produced otherwise the amine reactant would be become protonated to form a non-nucleophilic ammonium compound. During the reaction, one acyl group from the anhydride becomes part of the amide while the other acyl group becomes a carboxylate.
Example \(\PageIndex{1}\)
Draw the products of the following reaction:
Solution
The reaction of an anhydride and an amine form an amide through nucleophilic acyl substitution. During this reaction three changes in bonding occur. The leaving group is removed from the anhydride. The amine loses a hydrogen. A C-N bond is formed between the amine and the electrophilic carbonyl carbon.
Acid Anhydrides react with alcohols to form esters
Reactions of anhydrides use Pyridine as a solvent
Example 21.5.1
Mechanism
1) Nucleophilic Attack by the Alcohol
2) Deprotonation by pyridine
3) Leaving group removal
4) Protonation of the carboxylate
Acid Anhydrides react with amines to form amides
Example 21.5.2
Mechanism
1) Nucleophilic Attack by the Amine
2) Deprotonation by the amine
3) Leaving group removal


2.29: Escritura
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- Erica Brown, Alejandra Escudero, María Cristina Montoya, & Elizabeth Small
- SUNY Oneonta via OER SUNY
Course Objectives.
At the end of this module the students will be able to:
1. Identify the questions in Spanish
2. Answer questions in the target language
3. Write a short paragraph or sentences in Spanish
This assignment aims to help you prepare to write an assignment where everything is part of your final grade for Escritura 1. What it means is you won't receive any points for this task.
Earlier in this chapter Manuel wrote you a letter introducing himself. Now you are going to write a paragraph-length (60 – 75 words) letter back to him.
Paso 1 : Re-read the letter from Manuel. (From the “En contexto” pages in this chapter, reprinted below).
La fecha: Hoy es el 11 de febrero
¡Hola! Me llamo Manuel y soy de Perú. Mi familia no es muy grande. Mi mamá se llama Teresa y ella es de Lima, la capital de Perú y una ciudad grande. Mi papá se llama Roberto y es de Chiclayo, una ciudad pequeña al norte de Lima. Mi mamá es amable y mi papá es generoso. Mis hermanos son Lucas y Marisol, ellos son inteligentes. Somos una familia muy unida (close-knit). Mis mejores amigos son Juan y Rodolfo, ellos son chicos tontos. Ellos son de Cuzco, cerca de Machu Picchu. Pero ahora, nosotros somos estudiantes en la Universidad de Lima. Las clases son difíciles. ¿Cómo es tu familia? ¿De dónde son tus amigos? ¿Eres estudiante también? ¿Cómo son tus clases?
As you are reading his letter, write down some notes:
- How does he introduce himself?
- What information does he include about himself?
- What type of questions did he ask you?
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What verb did he use to:
- describe something/someone
- identify someone/something
- ask or say where someone is from
- ask and say his or someone’s profession
Paso 2: Write the letter. Be sure to include the date at the top of your letter (¿Qué día es hoy?). Write an introduction sentence similar to Manuel’s (¿Cómo te llamas?). Develop information about yourself and your friends and classmates (¿De dónde eres? ¿Cómo eres tú? ¿Y tus amigos? etc.). Lastly, your conclusion sentences should have new questions for Manuel.
¡A editar!
- Do you properly introduce yourself?
- Did you use the correct form of the verb “ser” with its respective subjects?
- Do your adjectives and articles coordinate with your nouns? (masculine and feminine)
- Have you looked at combining short sentences together with “y” ( and ) or “pero” ( but )?
- Did you check your punctuation and accent marks?
- Did you write a paragraph with 60-75 words?
Contributors and Attributions
- Assignment: Escribir. Authored by : SUNY Oneonta with Lumen Learning. Provided by : SUNY Oneonta. License : CC BY: Attribution


8.1.5D: Family Life
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Family life, including marriage, childbearing and household composition are strongly influenced by social class.
Learning Objectives
- Give examples for effects of social class on marriage, birth rates, and family composition
Key Points
- In the United States, the probability of a first marriage ending is substantially higher for couples with low socioeconomic statuses than for those in the middle or upper class.
- Globally, the birth rate in countries with large impoverished populations is much higher than in wealthier countries.
- In nations with high levels of fertility, upper class individuals tend to have more children than their lower class peers, while in nations with low levels of fertility, upper class families exhibit even lower fertility than average.
- Social class has both a cause and an effect relationship with family composition, and lower social class is often correlated with one-parent households.
Key Terms
- birth rate : The birth rate is typically the rate of births in a population over time.
- Family Life : A general term that refers to marriage and childbearing patterns, household composition and home stability.
- overpopulation : A situation which occurs when the number of occupants of an area exceeds the ability of that area to provide for those occupants.
Family life – marriage and childbearing patterns, household composition, and home stability – are strongly influenced by social class. In the United States, the probability of a first marriage ending is substantially higher for couples with low socioeconomic statuses than for those in the middle or upper class. Research shows that the higher rates of divorce for individuals in lower social classes can often be attributed to the greater financial stress these couples face, though factors like class expectations can also play a role.
Globally, the birth rate in countries with large impoverished populations is much higher than in wealthier countries, indicating that income and wealth play a role in shaping family structures. Demographers have identified a direct relationship between average number of children per household and the economic development of a nation. Today, less developed countries struggle with overpopulation while many governments in developed countries are instituting policies to deal with low birth rates. In nations with high levels of fertility, upper class individuals tend to have more children than their lower class peers. In nations with low levels of fertility, upper class families exhibit even lower fertility than average.
Social class has both a cause and an effect relationship with family composition. For example, single-parent households are likely to have a lower social class because they violate social norms. At the same time, single-parent families can contribute to financial and social instability. A single parent will often face higher costs (in the form of paid childcare), lower earnings (loss of the second parent’s income or loss of time spent at work), or both.


12.3.5: Procedure- Back Rub
Back rubs can be given after baths, before bed time, after repositioning, and to help the patient relax. Back rubs help to relax muscles, stimulate circulation, and promote sleep. Back rubs last about 5 minutes. Always ensure that it is okay to provide a back rub. Home Health Aides/Personal Care Aides must check with their supervisor and in the Care Plan. Observe skin condition during back rubs for bruises, red, white, or open areas, and other signs of skin breakdown. Always report and record these observations and any changes in skin condition. It is helpful to use lotion during back rubs to reduce friction and moisturize skin. Lotion on skin also helps to prevent skin breakdown.
- Explain the procedure to patient.
- Wash and dry your hands. Apply gloves as appropriate.
- Assemble equipment (e.g. lotion) per patient preferences.
- Warm lotion in basin of warm water or by running bottle under warm water for a few minutes. Do not use very hot water. To apply lotion to the patient’s skin, rub some between your hands rather than pouring it directly onto their skin. This serves to lubricate your hands during the back rub and to help warm the lotion so the patient does not feel too cold.
- Position the bed to safe working level at waist height. Lower the head of the bed. Lock the brakes. Lower the bed rail nearest you.
- Provide for privacy. Remove patient clothing from only the area you are working on to minimize exposure. Keep blankets on the patient, exposing only their back to provide warmth.
- When working on the patient’s back position the patient on their side or stomach, according to their preferences and comfort level.
- Keep the patient covered, exposing their back only to the top of their buttocks.
- Place a small amount of lotion in your hand and rub your palms together to warm lotion and lubricate hands. Caution the patient that the lotion may still feel cool.
- Rub the patient’s back with the appropriate pressure, according to their preferences. Rub back in a rhythmic motion in the correct direction.
- Start at the top of the buttocks, near the lower back, and move in long, smooth upward strokes with both hands, using the palms of your hands to create gentle pressure. Move your hands upward along the spine, and across the patient’s shoulders and then down their upper arms.
- Move your hands up over upper arms, back across the shoulders, and then down along the back toward the buttocks.
- Keep your hands in contact with the patient’s skin during movement to prevent friction against the patient’s skin.
- Repeat for 3-5 minutes, or according to patient preferences. Stop the back rub if at any time the patient reports discomfort or no longer wishes you to continue.
- If the patient wishes, you can gently knead the patient’s skin or apply a gentle tapping motion upward along the patient’s back to their shoulders and arms, and then back down again. To knead skin, gently grasp skin between your fingers and thumb. Move from buttocks upward, along the length of their back, across their shoulders, and then down over upper arms. Work your way back down to the buttocks using the same motion. Depending on patient preference, you can speed up or slow down your movements and pressure.
- Be careful to only lightly massage over bony areas so as not to cause pain to the patient. Do not massage areas that are red or that have broken or irritated skin. Do not apply lotion to areas with a rash. Do not remove dressings to apply lotion. Do not massage these areas.
- When you are almost done with the back rub, let the patient know so they are prepared for the ending of their back rub.
- Wipe away any excess lotion. Assist the patient with dressing and position the patient for comfort.
- Lower the bed to its lowest setting. Ensure the side rails are up.
- Put away equipment and supplies. Dispose of any dirty linens.
- Remove your gloves. Wash and dry your hands.
- Document completion of the task and record any changes in condition or behavior. Report any skin conditions, red, white, or open areas, or pressure sores.


3.E: Kinematics (Exercises)
Q1
Fred buys a ticket on a spaceship that will accelerate to an ultrarelativistic speed \(v\) such that \(c - v\) is only \(6\: m/s\). Fred was on the track team in high school, so he knows he can run about \(8\: m/s\). Once the ship is up to speed, Fred plans to run in the forward direction, thereby becoming the first human to exceed the speed of light. Other than the possible lack of gravity to allow running, what is wrong with Fred’s plan?
Q2
- In the equation \(v_c = \tfrac{v_1 +v_2}{1+v_1v_2}\) for combination of velocities, interpret the case where one of the velocities (but not the other) equals the speed of light.
- Interpret the case where the denominator goes to zero.
- Use the geometric series to rewrite the factor \[(tfrac{1}{1+v_1v_2}\), and then expand the expression for \(v_c\) as a series in \(v_1\) and \(v_2\), retaining terms up to third order in velocity. How does this relate to the correspondence principle?
Q3
Determine which of the identities in section 3.6 need to be modified in order to be valid in units with \(c \neq 1\), and describe how they should be modified.
Q4
The Large Hadron Collider accelerates counterrotating beams of protons and collides them head-on. The beam energy has been gradually increased, and the accelerator is designed to reach a maximum energy of \(14\: TeV\), corresponding to a rapidity of \(10.3\).
- Find the velocity of the beam.
- In any collision, the kinetic energy available to do something inelastic (smash up your car, produce nuclear reactions, ...) is the energy in the center of mass frame; in any other frame, there is initial kinetic energy that must also be present in the final state due to conservation of momentum. Suppose that a particular proton in the LHC beam never undergoes a collision with a proton from the opposite beam, and instead is wasted by being dumped into a beamstop. Let’s say that this collision is with a proton in a hydrogen atom left behind by someone’s fingerprint. Find the velocities of the two protons in their common center of mass frame.
Q5
Each GPS satellite is in an orbit with a radius of \(26,600\: km\), with an orbital period of half a sidereal day, giving it a velocity of \(3.88\: km/s\). The atomic clock aboard such a satellite is tuned to \(10.22999999543\: MHz\), which is chosen so that when the satellite is directly overhead, the effect of time dilation (transverse Doppler shift), combined with a general-relativistic effect due to gravity, results in a frequency of exactly \(10.23\: MHz\). (GPS started out as a military project, and legend has it that the top brass, suspicious of the crazy relativity stuff, demanded that the satellites be equipped with a software switch to turn off the correction, just in case the physicists were wrong.) There are oscillations superimposed onto these static effects due to the longitudinal Doppler shifts as the satellites approach and recede from a given observer on the ground.
- Calculate the maximum Doppler-shifted frequency for a hypothetical observer in outer space who is being directly approached by the satellite in its orbit.
- In reality, the greatest possible longitudinal component of the velocity is considerably smaller than this due to the geometry. Use the size of the earth to determine this velocity and the corresponding maximum frequency.
Q6
Verify directly, using the geometry of figure below, that for \(v = 3/5\), the Doppler shift factor is \(D = 2\). (Do not simply plug \(v = 3/5\) into the formula \(D =\sqrt{\frac{1 + v}{1-v}}\).)
Q7
Generalize the numerical calculation of problem Q6 to prove the general result \(D =\sqrt{\frac{1 + v}{1-v}}\).
Q8
Expand the relativistic equation for the longitudinal Doppler shift of light \(D(v)\) in a Taylor series, and find the first two nonvanishing terms. Show that these two terms agree with the nonrelativistic expression, so that any relativistic effect is of higher order in \(v\).
Q9
Prove, as claimed in
section 3.5, that we must have \(a \cdot v = 0\) if the velocity four-vector is to remain properly normalized.Q10
Example 3.5.1 described the motion of an object having constant proper acceleration \(a\), the world-line being \(t = \frac{1}{a} \sinh a\tau\) and \(x = \frac{1}{a} \cosh a\tau\) in a particular observer’s Minkowski coordinates.
- Prove the following results for \(γ\) and for the (three-)velocity and (three-)acceleration measured by this observer. \[\gamma =\cosh a\tau \\ v = \tanh a\tau \\ \text{acceleration} = a\cosh ^{-3}a\tau\] Do the calculations simply by taking the first and second derivatives of position with respect to time. You will find the following facts helpful: \[1-\tanh ^2 = \cosh ^{-2}\\ \frac{\mathrm{d} }{\mathrm{d} x} \tanh = \cosh ^{-2}x\]
- Interpret the results in the limit of large \(τ\).
Q11
Example 3.5.1 described the motion of an object having constant proper acceleration \(a\), the world-line being \(t = \frac{1}{a} \sinh a\tau\) and \(x = \frac{1}{a} \cosh a\tau\) in a particular observer’s Minkowski coordinates. Find the corresponding velocity and acceleration four-vectors.
Q12
Starting from the results of problem Q11, repeat problem Q10a using the techniques of section 3.7. You will find it helpful to know that \(1-\tanh ^2 = \cosh ^{-2}\).
Q13
Let \(v\) be a future-directed, properly normalized velocity vector. Compare the value of \(v\cdot v\) in the \(+---\) signature used in this book with its value in the signature \(-+ ++\).
Q14
- Prove the relation \(\frac{\mathrm{d} \gamma }{\mathrm{d} t} = \gamma ^3 a_o\cdot v_o\), in the special case where the motion is linear.
- Generalize the result to \(3 + 1\) dimensions.
Q15
Derive the identity \(a_o = \frac{1}{(o\cdot v)^2}\left [ P_o a - (o\cdot a)v_a \right ]\).
Q16
Recapitulating the geometry in figure below, let \(W\) be a smooth, timelike world line, \(E\) an event not on \(W\), and \(r\) the vector from \(E\) to a point on \(W\), parametrized by proper time \(τ\). Define the proper distance \(l\) between \(E\) and a point on \(W\) as \(l^2 = -(P_v r)^2\), where the square indicates an inner product of the vector with itself, and the minus sign is because we use the \(+---\) signature. Show that \(\frac{\mathrm{d} l^2}{\mathrm{d} \tau } = 2(r\cdot v)(r\cdot a)(v\cdot v)\), where the final factor is just a signature-dependent sign. Does this make sense when \(W\) is inertial? Give an example where the derivative vanishes because the first factor is zero, and another example where the second factor is the one that vanishes (but \(a \neq 0\)).
Q17
Consider an observer \(O\) moving along a world-line \(W\) with the constant-acceleration motion defined in Example 3.5.1. In
section 3.9, we gave the coordinates of a certain event \(E\) that was never “now” as described by our observer. The purpose of this problem is to analyze this is a more elegant and coordinate-invariant way. Let \(P\) be a point on \(W\), let \(B\) be the event described in section 3.9, and let \(x = \overrightarrow{BP}\), \(h = \overrightarrow{BE}\), and \(r = \overrightarrow{EP}\).- Show that \(W\), which was originally described in a certain set of coordinates, can instead by defined by the fact that \(x\cdot v = 0\) for every point on \(W\).
- Show that if \(h\) is timelike, then \(r\cdot v\) is never zero.


38.3: Types of Skeletal Systems - Human Axial Skeleton
- Describe the bones and function of the human axial skeleton
Human Axial Skeleton
The axial skeleton forms the central axis of the human body and includes the bones of the skull, the ossicles of the middle ear, the hyoid bone of the throat, the vertebral column, and the thoracic cage (ribcage). The function of the axial skeleton is to provide support and protection for the brain, spinal cord, and organs in the ventral body cavity. It also provides a surface for the attachment of muscles that move the head, neck, and trunk; performs respiratory movements; and stabilizes parts of the appendicular skeleton, which will be discussed later.
The Skull
The bones of the skull support the structures of the face and protect the brain. The skull consists of 22 bones, which are divided into two categories: cranial bones and facial bones. The cranial bones are eight bones that form the cranial cavity, which encloses the brain and serves as an attachment site for the muscles of the head and neck. The eight cranial bones include the frontal bone, two parietal bones, two temporal bones, the occipital bone, the sphenoid bone, and the ethmoid bone.
Fourteen facial bones form the face, provide cavities for the sense organs (eyes, mouth, and nose), protect the entrances to the digestive and respiratory tracts, and serve as attachment points for facial muscles. The 14 facial bones are the nasal bones, maxillary bones, zygomatic bones, palatine, vomer, lacrimal bones, inferior nasal conchae, and mandible.
The auditory ossicles of the middle ear transmit sounds from the air as vibrations to the fluid-filled cochlea. The auditory ossicles consist of six bones: two malleus bones, two incus bones, and two stapes, one of each on each side. These bones are unique to mammals.
The hyoid bone lies below the mandible in the front of the neck. It acts as a movable base for the tongue and is connected to muscles of the jaw, larynx, and tongue. The mandible articulates with the base of the skull, controling the opening to the airway and gut. In animals with teeth, the mandible brings the surfaces of the teeth in contact with the maxillary teeth.
The Vertebral Column
The vertebral column, or spinal column, surrounds and protects the spinal cord, supports the head, and acts as an attachment point for the ribs and muscles of the back and neck. The adult vertebral column is comprised of 26 bones: the 24 vertebrae, the sacrum, and the coccyx bones. In the adult, the sacrum is typically composed of five vertebrae that fuse into one. We begin life with approximately 33 vertebrae, but as we grow, several vertebrae fuse together. The adult vertebrae are further divided into the 7 cervical vertebrae, 12 thoracic vertebrae, and 5 lumbar vertebrae.
Each vertebral body has a large hole in the center through which the nerves of the spinal cord pass. There is also a notch on each side through which the spinal nerves, which serve the body at that level, can exit from the spinal cord. The names of the spinal curves correspond to the region of the spine in which they occur. The thoracic and sacral curves are concave, while the cervical and lumbar curves are convex. The arched curvature of the vertebral column increases its strength and flexibility, allowing it to absorb shocks like a spring.
Intervertebral discs composed of fibrous cartilage lie between adjacent vertebral bodies from the second cervical vertebra to the sacrum. Each disc is part of a joint that allows for some movement of the spine, acting as a cushion to absorb shocks from movements, such as walking and running. Intervertebral discs also act as ligaments to bind vertebrae together. The inner part of discs, the nucleus pulposus, hardens as people age, becoming less elastic. This loss of elasticity diminishes its ability to absorb shocks.
The Thoracic Cage
The thoracic cage, also known as the ribcage, is the skeleton of the chest. It consists of the ribs, sternum, thoracic vertebrae, and costal cartilages. The thoracic cage encloses and protects the organs of the thoracic cavity, including the heart and lungs. It also provides support for the shoulder girdles and upper limbs, and serves as the attachment point for the diaphragm, muscles of the back, chest, neck, and shoulders. Changes in the volume of the thorax enable breathing.
The sternum, or breastbone, is a long, flat bone located at the anterior of the chest. It is formed from three bones that fuse in the adult. The ribs are 12 pairs of long, curved bones that attach to the thoracic vertebrae and curve toward the front of the body, forming the ribcage. Costal cartilages connect the anterior ends of the ribs to the sternum, with the exception of rib pairs 11 and 12, which are free-floating ribs.
Key Points
- The axial skeleton provides support and protection for the brain, spinal cord, and the organs in the ventral body cavity; it also provides a surface for the attachment of muscles, directs respiratory movements, and stabilizes portions of the appendicular skeleton.
- The bones of the skull are divided into cranial bones and facial bones; their main roles consist of supporting the structures of the face and protecting the brain.
- The vertebral column protects the spinal cord, supports the head, and acts as an attachment point for the ribs and muscles of the back and neck.
- The thoracic cage’s most notable role is in breathing; however, it also protects the organs of the thoracic cavity, provides support for the shoulder girdles and upper limbs, and functions as the attachment point for the diaphragm, muscles of the back, chest, neck, and shoulders.
Key Terms
- intervertebral disc : a disc between the vertebra in the spine
- ossicle : a small bone (or bony structure), especially one of the three of the middle ear
- convex : curved or bowed outward like the outside of a bowl, sphere or circle
- vertebral column : the series of vertebrae that protect the spinal cord; the spinal column
- concave : curved or bowed inward like the inner surface of a sphere or bowl