2.5: Gilman, Charlotte Perkins "The Yellow Wallpaper" (1892)
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- Heather Ringo & Athena Kashyap
- City College of San Francisco via ASCCC Open Educational Resources Initiative
The Yelllow Wallpaper
by Charlotte Perkins Gilman
1
It is very seldom that mere ordinary people like John and myself secure ancestral halls for the summer.
A colonial mansion, a hereditary estate, I would say a haunted house, and reach the height of romantic felicity—but that would be asking too much of fate!
Still I will proudly declare that there is something queer about it.
Else, why should it be let so cheaply? And why have stood so long untenanted?
John laughs at me, of course, but one expects that in marriage.
John is practical in the extreme. He has no patience with faith, an intense horror of superstition, and he scoffs openly at any talk of things not to be felt and seen and put down in figures.
John is a physician, and perhaps —(I would not say it to a living soul, of course, but this is dead paper and a great relief to my mind)— perhaps that is one reason I do not get well faster.
You see, he does not believe I am sick!
And what can one do?
If a physician of high standing, and one’s own husband, assures friends and relatives that there is really nothing the matter with one but temporary nervous depression—a slight hysterical tendency—what is one to do?
My brother is also a physician, and also of high standing, and he says the same thing.
So I take phosphates or phosphites—whichever it is, and tonics, and journeys, and air, and exercise, and am absolutely forbidden to “work” until I am well again.
Personally, I disagree with their ideas.
Personally, I believe that congenial work, with excitement and change, would do me good.
But what is one to do?
I did write for a while in spite of them; but it does exhaust me a good deal—having to be so sly about it, or else meet with heavy opposition.
I sometimes fancy that in my condition if I had less opposition and more society and stimulus—but John says the very worst thing I can do is to think about my condition, and I confess it always makes me feel bad.
So I will let it alone and talk about the house.
The most beautiful place! It is quite alone, standing well back from the road, quite three miles from the village. It makes me think of English places that you read about, for there are hedges and walls and gates that lock, and lots of separate little houses for the gardeners and people.
There is a delicious garden! I never saw such a garden—large and shady, full of box-bordered paths, and lined with long grape-covered arbors with seats under them.
There were greenhouses, too, but they are all broken now.
There was some legal trouble, I believe, something about the heirs and co-heirs; anyhow, the place has been empty for years.
That spoils my ghostliness, I am afraid; but I don’t care—there is something strange about the house—I can feel it.
I even said so to John one moonlight evening, but he said what I felt was a draught , and shut the window.
I get unreasonably angry with John sometimes. I’m sure I never used to be so sensitive. I think it is due to this nervous condition.
But John says if I feel so I shall neglect proper self-control; so I take pains to control myself,—before him, at least,—and that makes me very tired.
I don’t like our room a bit. I wanted one downstairs that opened on the piazza and had roses all over the window, and such pretty old-fashioned chintz hangings! But John would not hear of it.
He said there was only one window and not room for two beds, and no near room for him if he took another.
He is very careful and loving, and hardly lets me stir without special direction.
I have a schedule prescription for each hour in the day; he takes all care from me, and so I feel basely ungrateful not to value it more.
He said we came here solely on my account, that I was to have perfect rest and all the air I could get. “Your exercise depends on your strength, my dear,” said he, “and your food somewhat on your appetite; but air you can absorb all the time.” So we took the nursery, at the top of the house.
It is a big, airy room, the whole floor nearly, with windows that look all ways, and air and sunshine galore. It was a nursery first and then playroom and gymnasium, I should judge; for the windows are barred for little children, and there are rings and things in the walls.
The paint and paper look as if a boys’ school had used it. It is stripped off—the paper—in great patches all around the head of my bed, about as far as I can reach, and in a great place on the other side of the room low down. I never saw a worse paper in my life.
One of those sprawling flamboyant patterns committing every artistic sin.

The color is repellant, almost revolting; a smouldering, unclean yellow, strangely faded by the slow-turning sunlight.
It is a dull yet lurid orange in some places, a sickly sulphur tint in others.
No wonder the children hated it! I should hate it myself if I had to live in this room long.
There comes John, and I must put this away,—he hates to have me write a word.
2
We have been here two weeks, and I haven’t felt like writing before, since that first day.
I am sitting by the window now, up in this atrocious nursery, and there is nothing to hinder my writing as much as I please, save lack of strength.

I am glad my case is not serious!
But these nervous troubles are dreadfully depressing.
John does not know how much I really suffer. He knows there is no reason to suffer, and that satisfies him.
Of course it is only nervousness. It does weigh on me so not to do my duty in any way!
I meant to be such a help to John, such a real rest and comfort, and here I am a comparative burden already!
Nobody would believe what an effort it is to do what little I am able—to dress and entertain, and order things.
It is fortunate Mary is so good with the baby. Such a dear baby!
And yet I cannot be with him, it makes me so nervous.
I suppose John never was nervous in his life. He laughs at me so about this wallpaper!
At first he meant to repaper the room, but afterwards he said that I was letting it get the better of me, and that nothing was worse for a nervous patient than to give way to such fancies.
He said that after the wallpaper was changed it would be the heavy bedstead, and then the barred windows, and then that gate at the head of the stairs, and so on.

Then he took me in his arms and called me a blessed little goose, and said he would go down cellar if I wished, and have it whitewashed into the bargain.
But he is right enough about the beds and windows and things.
It is as airy and comfortable a room as any one need wish, and, of course, I would not be so silly as to make him uncomfortable just for a whim.
I’m really getting quite fond of the big room, all but that horrid paper.
Out of one window I can see the garden, those mysterious deep-shaded arbors, the riotous old-fashioned flowers, and bushes and gnarly trees.
Out of another I get a lovely view of the bay and a little private wharf belonging to the estate. There is a beautiful shaded lane that runs down there from the house. I always fancy I see people walking in these numerous paths and arbors, but John has cautioned me not to give way to fancy in the least. He says that with my imaginative power and habit of story-making a nervous weakness like mine is sure to lead to all manner of excited fancies, and that I ought to use my will and good sense to check the tendency. So I try.
I think sometimes that if I were only well enough to write a little it would relieve the press of ideas and rest me.
But I find I get pretty tired when I try.
It is so discouraging not to have any advice and companionship about my work. When I get really well John says we will ask Cousin Henry and Julia down for a long visit; but he says he would as soon put fire-works in my pillow-case as to let me have those stimulating people about now.
I wish I could get well faster.
But I must not think about that. This paper looks to me as if it knew what a vicious influence it had!
There is a recurrent spot where the pattern lolls like a broken neck and two bulbous eyes stare at you upside-down.
I get positively angry with the impertinence of it and the everlastingness. Up and down and sideways they crawl, and those absurd, unblinking eyes are everywhere. There is one place where two breadths didn’t match, and the eyes go all up and down the line, one a little higher than the other.
I never saw so much expression in an inanimate thing before, and we all know how much expression they have! I used to lie awake as a child and get more entertainment and terror out of blank walls and plain furniture than most children could find in a toy-store.
I remember what a kindly wink the knobs of our big old bureau used to have, and there was one chair that always seemed like a strong friend.
I used to feel that if any of the other things looked too fierce I could always hop into that chair and be safe.
The furniture in this room is no worse than inharmonious, however, for we had to bring it all from downstairs. I suppose when this was used as a playroom they had to take the nursery things out, and no wonder! I never saw such ravages as the children have made here.
The wallpaper, as I said before, is torn off in spots, and it sticketh closer than a brother—they must have had perseverance as well as hatred.
Then the floor is scratched and gouged and splintered, the plaster itself is dug out here and there, and this great heavy bed, which is all we found in the room, looks as if it had been through the wars.
But I don’t mind it a bit—only the paper.
There comes John’s sister. Such a dear girl as she is, and so careful of me! I must not let her find me writing.
She is a perfect, and enthusiastic housekeeper, and hopes for no better profession. I verily believe she thinks it is the writing which made me sick!
But I can write when she is out, and see her a long way off from these windows.
There is one that commands the road, a lovely, shaded, winding road, and one that just looks off over the country. A lovely country, too, full of great elms and velvet meadows.
This wallpaper has a kind of sub-pattern in a different shade, a particularly irritating one, for you can only see it in certain lights, and not clearly then.
But in the places where it isn’t faded, and where the sun is just so, I can see a strange, provoking, formless sort of figure, that seems to skulk about behind that silly and conspicuous front design.
There’s sister on the stairs!
3
Well, the Fourth of July is over! The people are gone and I am tired out. John thought it might do me good to see a little company, so we just had mother and Nellie and the children down for a week.
Of course I didn’t do a thing. Jennie sees to everything now.
But it tired me all the same.
John says if I don’t pick up faster he shall send me to Weir Mitchell in the fall.
But I don’t want to go there at all. I had a friend who was in his hands once, and she says he is just like John and my brother, only more so!
Besides, it is such an undertaking to go so far.
I don’t feel as if it was worth while to turn my hand over for anything, and I’m getting dreadfully fretful and querulous.

Of course I don’t when John is here, or anybody else, but when I am alone.
And I am alone a good deal just now. John is kept in town very often by serious cases, and Jennie is good and lets me alone when I want her to.
So I walk a little in the garden or down that lovely lane, sit on the porch under the roses, and lie down up here a good deal.
I’m getting really fond of the room in spite of the wallpaper. Perhaps because of the wallpaper.
It dwells in my mind so!
I lie here on this great immovable bed—it is nailed down, I believe—and follow that pattern about by the hour. It is as good as gymnastics, I assure you. I start, we’ll say, at the bottom, down in the corner over there where it has not been touched, and I determine for the thousandth time that I will follow that pointless pattern to some sort of a conclusion.
I know a little of the principle of design, and I know this thing was not arranged on any laws of radiation, or alternation, or repetition, or symmetry, or anything else that I ever heard of.
It is repeated, of course, by the breadths, but not otherwise.
Looked at in one way each breadth stands alone, the bloated curves and flourishes—a kind of “debased Romanesque” with delirium tremens —go waddling up and down in isolated columns of fatuity.
But, on the other hand, they connect diagonally, and the sprawling outlines run off in great slanting waves of optic horror, like a lot of wallowing seaweeds in full chase.
The whole thing goes horizontally, too, at least it seems so, and I exhaust myself in trying to distinguish the order of its going in that direction.
They have used a horizontal breadth for a frieze, and that adds wonderfully to the confusion.
There is one end of the room where it is almost intact, and there, when the cross-lights fade and the low sun shines directly upon it, I can almost fancy radiation after all,—the interminable grotesques seem to form around a common center and rush off in headlong plunges of equal distraction.
It makes me tired to follow it. I will take a nap, I guess.
4
I don’t know why I should write this.
I don’t want to.
I don’t feel able.
And I know John would think it absurd. But I must say what I feel and think in some way—it is such a relief!
But the effort is getting to be greater than the relief.
Half the time now I am awfully lazy, and lie down ever so much.
John says I musn’t lose my strength, and has me take cod-liver oil and lots of tonics and things, to say nothing of ale and wine and rare meat.
Dear John! He loves me very dearly, and hates to have me sick. I tried to have a real earnest reasonable talk with him the other day, and tell him how I wish he would let me go and make a visit to Cousin Henry and Julia.
But he said I wasn’t able to go, nor able to stand it after I got there; and I did not make out a very good case for myself, for I was crying before I had finished.
It is getting to be a great effort for me to think straight. Just this nervous weakness, I suppose.
And dear John gathered me up in his arms, and just carried me upstairs and laid me on the bed, and sat by me and read to me till it tired my head.
He said I was his darling and his comfort and all he had, and that I must take care of myself for his sake, and keep well.
He says no one but myself can help me out of it, that I must use my will and self-control and not let any silly fancies run away with me.
There’s one comfort, the baby is well and happy, and does not have to occupy this nursery with the horrid wallpaper.
If we had not used it that blessed child would have! What a fortunate escape! Why, I wouldn’t have a child of mine, an impressionable little thing, live in such a room for worlds.
I never thought of it before, but it is lucky that John kept me here after all.
Of course I never mention it to them any more,—I am too wise,—but I keep watch of it all the same.
There are things in that paper that nobody knows but me, or ever will.
Behind that outside pattern the dim shapes get clearer every day.
It is always the same shape, only very numerous.
And it is like a woman stooping down and creeping about behind that pattern. I don’t like it a bit. I wonder—I begin to think—I wish John would take me away from here!
5
It is so hard to talk with John about my case, because he is so wise, and because he loves me so.
But I tried it last night.
It was moonlight. The moon shines in all around, just as the sun does.
I hate to see it sometimes, it creeps so slowly, and always comes in by one window or another.
John was asleep and I hated to waken him, so I kept still and watched the moonlight on that undulating wallpaper till I felt creepy.
The faint figure behind seemed to shake the pattern, just as if she wanted to get out.
I got up softly and went to feel and see if the paper did move, and when I came back John was awake.
“What is it, little girl?” he said. “Don’t go walking about like that—you’ll get cold.”
I though it was a good time to talk, so I told him that I really was not gaining here, and that I wished he would take me away.
“Why darling!” said he, “our lease will be up in three weeks, and I can’t see how to leave before.
“The repairs are not done at home, and I cannot possibly leave town just now. Of course if you were in any danger I could and would, but you really are better, dear, whether you can see it or not. I am a doctor, dear, and I know. You are gaining flesh and color, your appetite is better. I feel really much easier about you.”
“I don’t weigh a bit more,” said I, “nor as much; and my appetite may be better in the evening, when you are here, but it is worse in the morning when you are away.”
“Bless her little heart!” said he with a big hug; “she shall be as sick as she pleases! But now let’s improve the shining hours by going to sleep, and talk about it in the morning!”
“And you won’t go away?” I asked gloomily.
“Why, how can I, dear? It is only three weeks more and then we will take a nice little trip of a few days while Jennie is getting the house ready. Really, dear, you are better!”
“Better in body perhaps”—I began, and stopped short, for he sat up straight and looked at me with such a stern, reproachful look that I could not say another word.
“My darling,” said he, “I beg of you, for my sake and for our child’s sake, as well as for your own, that you will never for one instant let that idea enter your mind! There is nothing so dangerous, so fascinating, to a temperament like yours. It is a false and foolish fancy. Can you not trust me as a physician when I tell you so?”
So of course I said no more on that score, and we went to sleep before long. He thought I was asleep first, but I wasn’t,—I lay there for hours trying to decide whether that front pattern and the back pattern really did move together or separately.
On a pattern like this, by daylight, there is a lack of sequence, a defiance of law, that is a constant irritant to a normal mind.
The color is hideous enough, and unreliable enough, and infuriating enough, but the pattern is torturing.
You think you have mastered it, but just as you get well under way in following, it turns a back somersault and there you are.  It is like a bad dream.
The outside pattern is a florid arabesque, reminding one of a fungus. If you can imagine a toadstool in joints, an interminable string of toadstools, budding and sprouting in endless convolutions,—why, that is something like it.
That is, sometimes!
There is one marked peculiarity about this paper, a thing nobody seems to notice but myself, and that is that it changes as the light changes.
When the sun shoots in through the east window—I always watch for that first long, straight ray—it changes so quickly that I never can quite believe it.
That is why I watch it always.
By moonlight—the moon shines in all night when there is a moon—I wouldn’t know it was the same paper.
At night in any kind of light, in twilight, candlelight, lamplight, and worst of all by moonlight, it becomes bars! The outside pattern I mean, and the woman behind it is as plain as can be.
I didn’t realize for a long time what the thing was that showed behind,—that dim sub-pattern,—but now I am quite sure it is a woman.
By daylight she is subdued, quiet. I fancy it is the pattern that keeps her so still. It is so puzzling. It keeps me quiet by the hour.
I lie down ever so much now. John says it is good for me, and to sleep all I can.
Indeed, he started the habit by making me lie down for an hour after each meal.
It is a very bad habit, I am convinced, for, you see, I don’t sleep.
And that cultivates deceit, for I don’t tell them I’m awake,—oh, no!
The fact is, I am getting a little afraid of John.
He seems very queer sometimes, and even Jennie has an inexplicable look.
It strikes me occasionally, just as a scientific hypothesis, that perhaps it is the paper!
I have watched John when he did not know I was looking, and come into the room suddenly on the most innocent excuses, and I’ve caught him several times looking at the paper! And Jennie too. I caught Jennie with her hand on it once.
She didn’t know I was in the room, and when I asked her in a quiet, a very quiet voice, with the most restrained manner possible, what she was doing with the paper she turned around as if she had been caught stealing, and looked quite angry—asked me why I should frighten her so!
Then she said that the paper stained everything it touched, that she had found yellow smooches on all my clothes and John’s, and she wished we would be more careful!
Did not that sound innocent? But I know she was studying that pattern, and I am determined that nobody shall find it out but myself!
6
Life is very much more exciting now than it used to be. You see I have something more to expect, to look forward to, to watch. I really do eat better, and am more quiet than I was.
John is so pleased to see me improve! He laughed a little the other day, and said I seemed to be flourishing in spite of my wallpaper.
I turned it off with a laugh. I had no intention of telling him it was because of the wallpaper—he would make fun of me. He might even want to take me away.
I don’t want to leave now until I have found it out. There is a week more, and I think that will be enough.
7
I’m feeling ever so much better! I don’t sleep much at night, for it is so interesting to watch developments; but I sleep a good deal in the daytime.
In the daytime it is tiresome and perplexing.
There are always new shoots on the fungus, and new shades of yellow all over it. I cannot keep count of them, though I have tried conscientiously.
It is the strangest yellow, that wallpaper! It makes me think of all the yellow things I ever saw—not beautiful ones like buttercups, but old foul, bad yellow things.
But there is something else about that paper—the smell! I noticed it the moment we came into the room, but with so much air and sun it was not bad. Now we have had a week of fog and rain, and whether the windows are open or not, the smell is here.
It creeps all over the house.
I find it hovering in the dining-room, skulking in the parlor, hiding in the hall, lying in wait for me on the stairs.
It gets into my hair.
Even when I go to ride, if I turn my head suddenly and surprise it—there is that smell!
Such a peculiar odor, too! I have spent hours in trying to analyze it, to find what it smelled like.
It is not bad—at first, and very gentle, but quite the subtlest, most enduring odor I ever met.
In this damp weather it is awful. I wake up in the night and find it hanging over me.
It used to disturb me at first. I thought seriously of burning the house—to reach the smell.
But now I am used to it. The only thing I can think of that it is like is the color of the paper! A yellow smell.
There is a very funny mark on this wall, low down, near the mopboard. A streak that runs round the room. It goes behind every piece of furniture, except the bed, a long, straight, even smooch , as if it had been rubbed over and over.
I wonder how it was done and who did it, and what they did it for. Round and round and round—round and round and round—it makes me dizzy!
8
I really have discovered something at last.
Through watching so much at night, when it changes so, I have finally found out.
The front pattern does move—and no wonder! The woman behind shakes it!
Sometimes I think there are a great many women behind, and sometimes only one, and she crawls around fast, and her crawling shakes it all over.
Then in the very bright spots she keeps still, and in the very shady spots she just takes hold of the bars and shakes them hard.
And she is all the time trying to climb through. But nobody could climb through that pattern—it strangles so; I think that is why it has so many heads.
They get through, and then the pattern strangles them off and turns them upside-down, and makes their eyes white!
If those heads were covered or taken off it would not be half so bad.
9
I think that woman gets out in the daytime!

I can see her out of every one of my windows!
It is the same woman, I know, for she is always creeping, and most women do not creep by daylight.
I see her on that long shaded lane, creeping up and down. I see her in those dark grape arbors, creeping all around the garden.
I see her on that long road under the trees, creeping along, and when a carriage comes she hides under the blackberry vines.
I don’t blame her a bit. It must be very humiliating to be caught creeping by daylight!
I always lock the door when I creep by daylight. I can’t do it at night, for I know John would suspect something at once.
And John is so queer now, that I don’t want to irritate him. I wish he would take another room! Besides, I don’t want anybody to get that woman out at night but myself.
I often wonder if I could see her out of all the windows at once.
But, turn as fast as I can, I can only see out of one at one time.
And though I always see her she may be able to creep faster than I can turn!
I have watched her sometimes away off in the open country, creeping as fast as a cloud shadow in a high wind.
10
If only that top pattern could be gotten off from the under one! I mean to try it, little by little.
I have found out another funny thing, but I shan’t tell it this time! It does not do to trust people too much.
There are only two more days to get this paper off, and I believe John is beginning to notice.
And I heard him ask Jennie a lot of professional questions about me. She had a very good report to give.
She said I slept a good deal in the daytime.
John knows I don’t sleep very well at night, for all I’m so quiet!
He asked me all sorts of questions, too, and pretended to be very loving and kind.
As if I couldn’t see through him!
Still, I don’t wonder he acts so, sleeping under this paper for three months.
It only interests me, but I feel sure John and Jennie are secretly affected by it.
11
Hurrah! This is the last day, but it is enough. John is to stay in town over night, and won’t be out until this evening.
Jennie wanted to sleep with me—the sly thing! but I told her I should undoubtedly rest better for a night all alone.
That was clever, for really I wasn’t alone a bit! As soon as it was moonlight, and that poor thing began to crawl and shake the pattern, I got up and ran to help her.
I pulled and she shook, I shook and she pulled, and before morning we had peeled off yards of that paper.
A strip about as high as my head and half around the room.
And then when the sun came and that awful pattern began to laugh at me I declared I would finish it to-day!
We go away to-morrow, and they are moving all my furniture down again to leave things as they were before.
Jennie looked at the wall in amazement, but I told her merrily that I did it out of pure spite at the vicious thing.
She laughed and said she wouldn’t mind doing it herself, but I must not get tired.
How she betrayed herself that time!
But I am here, and no person touches this paper but me—not alive!
She tried to get me out of the room—it was too patent! But I said it was so quiet and empty and clean now that I believed I would lie down again and sleep all I could; and not to wake me even for dinner—I would call when I woke.
So now she is gone, and the servants are gone, and the things are gone, and there is nothing left but that great bedstead nailed down, with the canvas mattress we found on it.
We shall sleep downstairs to-night, and take the boat home to-morrow.
I quite enjoy the room, now it is bare again.
How those children did tear about here!
This bedstead is fairly gnawed!
But I must get to work.
I have locked the door and thrown the key down into the front path.
I don’t want to go out, and I don’t want to have anybody come in, till John comes.
I want to astonish him.
I’ve got a rope up here that even Jennie did not find. If that woman does get out, and tries to get away, I can tie her!
But I forgot I could not reach far without anything to stand on!
This bed will not move!
I tried to lift and push it until I was lame, and then I got so angry I bit off a little piece at one corner—but it hurt my teeth.
Then I peeled off all the paper I could reach standing on the floor. It sticks horribly and the pattern just enjoys it! All those strangled heads and bulbous eyes and waddling fungus growths just shriek with derision!
I am getting angry enough to do something desperate. To jump out of the window would be admirable exercise, but the bars are too strong even to try.
Besides I wouldn’t do it. Of course not. I know well enough that a step like that is improper and might be misconstrued.
I don’t like to look out of the windows even—there are so many of those creeping women, and they creep so fast.
I wonder if they all come out of that wallpaper as I did?
But I am securely fastened now by my well-hidden rope—you don’t get me out in the road there!
I suppose I shall have to get back behind the pattern when it comes night, and that is hard!
It is so pleasant to be out in this great room and creep around as I please!
I don’t want to go outside. I won’t, even if Jennie asks me to.
For outside you have to creep on the ground, and everything is green instead of yellow.
But here I can creep smoothly on the floor, and my shoulder just fits in that long smooch around the wall, so I cannot lose my way.
Why, there’s John at the door!
It is no use, young man, you can’t open it!
How he does call and pound!
Now he’s crying for an axe.
It would be a shame to break down that beautiful door!
“John dear!” said I in the gentlest voice, “the key is down by the front steps, under a plantain leaf!”
That silenced him for a few moments.
Then he said—very quietly indeed, “Open the door, my darling!”
“I can’t,” said I. “The key is down by the front door under a plantain leaf!”
And then I said it again, several times, very gently and slowly, and said it so often that he had to go and see, and he got it, of course, and came in. He stopped short by the door.
“What is the matter?” he cried. “For God’s sake, what are you doing!”
I kept on creeping just the same, but I looked at him over my shoulder.
“I’ve got out at last,” said I, “in spite of you and Jane! And I’ve pulled off most of the paper, so you can’t put me back!”
Now why should that man have fainted? But he did, and right across my path by the wall, so that I had to creep over him every time!
Exercise 5.3.1
Questions
1. What is the meaning of the title? Why did Gilman use first person narration? Why is the female protagonist unnamed throughout the text?
2. How would you describe the narrator's voice in this text? How does that voice evolve over the course of the novella? How does the fragmented style of the narrative influence you as a reader?
3. How does Gilman engage with traditional ideas of femininity in this story? What do the house and wallpaper symbolize? What does the woman/women in the wallpaper represent?
4. What is the meaning of the last scene? Has this woman liberated herself from patriarchy as an individual through her madness or has she experienced the ultimate victimization?
5. Gilman was very much an activist writer and authored texts in many different voices (the social scientist, the journalist, the utopian writer, the Gothic writer). What is she advocating in this story? What are the risks of this narrative strategy for a politically-oriented feminist writer?
6. What contemporary issues does "The Yellow Wallpaper" evoke? Why do you think it has become one of the five most popular readings in college courses?
Contributors and Attributions
Section on questions adapted from Introduction to Women and Gender Studies from MIT Open Courseware, license CC-BY-NC-SA


5.2.1: Internet, Social Media, and the Age of Algorithms
As we might already assume, most people get health information via internet sources in the 21st Century. When faced with a new symptom or health decision, the first action is often a simple internet search. Which websites top the results list may be determined by advertising dollars, or by search engine algorithms that tailor results to the person’s past search history. Further direct-to-consumer advertising by medical and pharmaceutical companies can be a result of a search, viewing a website or post on social media. These algorithms and targeted ads may affect our health decisions far more than we realize (Stark & Fins, 2013).
Algorithms also affect our social media experience. Although a user may initially choose whom to follow, social media platforms will typically recommend similar pages based on that individual’s use data. User-generated content can be quickly shared with a broad audience, often without the identification of the original source or any verification of the information. And social media platforms are often used to spread an ideology or sell products. Take, for example, the anti-vaccine sentiment after the COVID-19 vaccine rollout. It was discovered that a majority of the anti-vaccine misinformation posts on social media sites Facebook and Twitter were created by the same 12 individuals (many of whom had been sharing health disinformation for years already) (Bardosh et al., 2022). When these posts garner equal - if not more - attention in the media than posts from reputable sources, misinformation spreads farther and faster than correct information. Similar to the analogy of attempting to put out a million small fires at once, it can be difficult to combat this type of misinformation with well-researched rebuttals in a timely fashion. As a result, it is still left up to the viewer to do the due diligence of validating the information presented to them, which is often a more lengthy task than most people are willing to take on.
As humans, we already struggle with a variety of cognitive biases that cloud our decision-making. Availability bias (more of “it” must make it true), confirmation bias (searching for evidence to support what we already believe), and overconfidence bias (feeling we know more than we do), are all exacerbated by the social media and search engine algorithms (Stark & Fins, 2013). Our “engagement” with certain posts is tracked, and we are subsequently fed similar posts - thus cementing our own personal “reality bubble”. With the advent of readily available artificial intelligence the access to information is potentially limitless, but the quality of that information remains malleable by both commercial interests and machine learning algorithms.


The media categories you can include in an online course are:
-
text
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Text is typically presented on computer screens but the resources you provide can also include print-based materials.
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audio
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Audio can be heard from DVD-ROM/CD-ROM disks, computer hard drives, an intranet, and the Internet. However, an online course can also include resources like tapes (audiocassettes), radio, television, and live commentary.
-
visuals
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Visuals can be stored on DVD-ROM/CD-ROM disks, computer hard drives, an intranet, and the Internet. Other resources can include slides, photographs, overhead transparencies, and paper-based material.
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video
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Video can be retrieved from DVD-ROM/CD-ROM disks, computer hard drives, an intranet, and the Internet. Other sources can include miniDV tapes, film, and VHS tapes.
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animations
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Animations can be stored on DVD-ROM/CD-ROM disks, computer hard drives, an intranet, and the Internet. Film, VHS tapes and other sources can also contain animation resources.
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real objects
-
Real objects include actual equipment and models.
Note
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Video typically includes natural images recorded with video equipment, whereas animations are usually created artificially with computers and/or other tools.
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Video materials often include an audio component.
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There are major differences between video and film. This chapter uses the terms film and video synonymously.
Media and Learning
The media you select do not determine whether learning will occur. The media simply carry your message to the learner. However, the media you use can influence the amount of learning that occurs. If you combine the media’s strengths with instructional methods that take advantage of these strengths, you can positively influence learning.
Complete instructional packages can, but should not necessarily, include all of the different media. Note that:
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Learning from course content that includes more than one medium is usually more effective than content using only one medium. This is partly because different parts of the brain process different information. For example, some parts of the brain process text, while others process visuals. When instructional materials activate more regions of the brain, there are increases in learning and retention compared to materials that require fewer parts of the brain to process information.
Note
Teaching with more than one medium is usually more effective than teaching with only one medium.
-
In many situations, you can and should use more than one medium to teach the skill. You will need to determine the media that will complement the intended instructional strategy.
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If you use too many media at one time, you can impede learning.
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Although multi-sensory learning experiences tend to be effective, learners can only process a limited amount of information at one time. Imagine trying to read text while a supporting animation is being shown on the screen.
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Media should support and enhance each other.
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Base your media mix decision on what is being taught, how it is being taught, how it will be tested, and the characteristics of your target audience.
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Different media may be needed for different learning outcomes. For example, video may be appropriate for the attitude component but may not provide the corrective feedback necessary for the intellectual skills component.
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Do not select media simply to dazzle or for convenience.


14.2: Fluids, Density, and Pressure (Part 1)
- State the different phases of matter
- Describe the characteristics of the phases of matter at the molecular or atomic level
- Distinguish between compressible and incompressible materials
- Define density and its related SI units
- Compare and contrast the densities of various substances
- Define pressure and its related SI units
- Explain the relationship between pressure and force
- Calculate force given pressure and area
Matter most commonly exists as a solid, liquid, or gas; these states are known as the three common phases of matter. We will look at each of these phases in detail in this section.
Characteristics of Solids
Solids are rigid and have specific shapes and definite volumes. The atoms or molecules in a solid are in close proximity to each other, and there is a significant force between these molecules. Solids will take a form determined by the nature of these forces between the molecules. Although true solids are not incompressible, it nevertheless requires a large force to change the shape of a solid. In some cases, the force between molecules can cause the molecules to organize into a lattice as shown in Figure \(\PageIndex{1}\). The structure of this three-dimensional lattice is represented as molecules connected by rigid bonds (modeled as stiff springs), which allow limited freedom for movement. Even a large force produces only small displacements in the atoms or molecules of the lattice, and the solid maintains its shape. Solids also resist shearing forces. (Shearing forces are forces applied tangentially to a surface, as described in Static Equilibrium and Elasticity .)
Characteristics of Fluids
Liquids and gases are considered to be fluids because they yield to shearing forces, whereas solids resist them. Like solids, the molecules in a liquid are bonded to neighboring molecules, but possess many fewer of these bonds. The molecules in a liquid are not locked in place and can move with respect to each other. The distance between molecules is similar to the distances in a solid, and so liquids have definite volumes, but the shape of a liquid changes, depending on the shape of its container. Gases are not bonded to neighboring atoms and can have large separations between molecules. Gases have neither specific shapes nor definite volumes, since their molecules move to fill the container in which they are held (Figure \(\PageIndex{1}\)).
Liquids deform easily when stressed and do not spring back to their original shape once a force is removed. This occurs because the atoms or molecules in a liquid are free to slide about and change neighbors. That is, liquids flow (so they are a type of fluid), with the molecules held together by mutual attraction. When a liquid is placed in a container with no lid, it remains in the container. Because the atoms are closely packed, liquids, like solids, resist compression; an extremely large force is necessary to change the volume of a liquid.
In contrast, atoms in gases are separated by large distances, and the forces between atoms in a gas are therefore very weak, except when the atoms collide with one another. This makes gases relatively easy to compress and allows them to flow (which makes them fluids). When placed in an open container, gases, unlike liquids, will escape.
In this chapter, we generally refer to both gases and liquids simply as fluids, making a distinction between them only when they behave differently. There exists one other phase of matter, plasma, which exists at very high temperatures. At high temperatures, molecules may disassociate into atoms, and atoms disassociate into electrons (with negative charges) and protons (with positive charges), forming a plasma. Plasma will not be discussed in depth in this chapter because plasma has very different properties from the three other common phases of matter, discussed in this chapter, due to the strong electrical forces between the charges.
Density
Suppose a block of brass and a block of wood have exactly the same mass. If both blocks are dropped in a tank of water, why does the wood float and the brass sink (Figure \(\PageIndex{2}\))? This occurs because the brass has a greater density than water, whereas the wood has a lower density than water.
Density is an important characteristic of substances. It is crucial, for example, in determining whether an object sinks or floats in a fluid.
The average density of a substance or object is defined as its mass per unit volume,
\[\rho = \frac{m}{V} \label{14.1}\]
where the Greek letter \(\rho\) (rho) is the symbol for density, m is the mass, and V is the volume.
The SI unit of density is kg/m 3 . Table 14.1 lists some representative values. The cgs unit of density is the gram per cubic centimeter, g/cm 3 , where
\[1\; g/cm^{3} = 1000\; kg/m^{3} \ldotp\]
The metric system was originally devised so that water would have a density of 1 g/cm 3 , equivalent to 10 3 kg/m 3 . Thus, the basic mass unit, the kilogram, was first devised to be the mass of 1000 mL of water, which has a volume of 1000 cm 3 .
| Solids | (0.0 °C) | Liquids | (0.0 °C) | Gases | (0.0 °C, 101.3 kPa) |
|---|---|---|---|---|---|
| Substance | \(\rho\)(kg/m 3 ) | Substance | \(\rho\)(kg/m 3 ) | Substance | \(\rho\)(kg/m 3 ) |
| Aluminum | 2.70 x 10 3 | Benzene | 8.79 x 10 2 | Air | 1.29 x 10 0 |
| Bone | 1.90 x 10 3 | Blood | 1.05 x 10 3 | Carbon dioxide | .1.98 x 10 0 |
| Brass | 8.44 x 10 3 | Ethyl alcohol | 8.06 x 10 2 | Carbon monoxide | 1.25 x 10 0 |
| Concrete | 2.40 x 10 3 | Gasoline | 6.80 x 10 2 | Helium | 1.80 x 10 -1 |
| Copper | 8.92 x 10 3 | Glycerin | 1.26 x 10 3 | Hydrogen | 9.00 x 10 -2 |
| Cork | 2.40 x 10 2 | Mercury | 1.36 x 10 4 | Methane | 7.20 x 10 -2 |
| Earth's crust | 3.30 x 10 3 | Olive oil | 9.20 x 10 2 | Nitrogen | 1.25 x 10 0 |
| Glass | 2.60 x 10 3 | Nitrous oxide | 1.98 x 10 0 | ||
| Granite | 2.70 x 10 3 | Oxygen | 1.43 x 10 0 | ||
| Iron | 7.86 x 10 3 | ||||
| Lead | 1.13 x 10 4 | ||||
| Oak | 7.10 x 10 2 | ||||
| Pine | 3.73 x 10 2 | ||||
| Platinum | 2.14 x 10 4 | ||||
| Polystyrene | 1.00 x 10 2 | ||||
| Tungsten | 1.93 x 10 4 | ||||
| Uranium | 1.87 x 10 3 |
As you can see by examining Table 14.1, the density of an object may help identify its composition. The density of gold, for example, is about 2.5 times the density of iron, which is about 2.5 times the density of aluminum. Density also reveals something about the phase of the matter and its substructure. Notice that the densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. The densities of gases are much less than those of liquids and solids, because the atoms in gases are separated by large amounts of empty space. The gases are displayed for a standard temperature of 0.0 °C and a standard pressure of 101.3 kPa, and there is a strong dependence of the densities on temperature and pressure. The densities of the solids and liquids displayed are given for the standard temperature of 0.0 °C and the densities of solids and liquids depend on the temperature. The density of solids and liquids normally increase with decreasing temperature.
Table 14.2 shows the density of water in various phases and temperature. The density of water increases with decreasing temperature, reaching a maximum at 4.0 °C, and then decreases as the temperature falls below 4.0 °C. This behavior of the density of water explains why ice forms at the top of a body of water.
| Substance | \(\rho\)(kg/m 3 ) |
|---|---|
| Ice (0 °C) | 9.17 x 10 2 |
| Water (0 °C) | 9.998 x 10 2 |
| Water (4 °C) | 1.000 x 10 3 |
| Water (20 °C) | 9.982 x 10 2 |
| Water (100 °C) | 9.584 x 10 2 |
| Steam (100 °C, 101.3 kPa) | 1.670 x 10 2 |
| Sea water (0°C) | 1.030 x 10 3 |
The density of a substance is not necessarily constant throughout the volume of a substance. If the density is constant throughout a substance, the substance is said to be a homogeneous substance. A solid iron bar is an example of a homogeneous substance. The density is constant throughout, and the density of any sample of the substance is the same as its average density. If the density of a substance were not constant, the substance is said to be a heterogeneous substance. A chunk of Swiss cheese is an example of a heterogeneous material containing both the solid cheese and gas-filled voids. The density at a specific location within a heterogeneous material is called local density, and is given as a function of location, \(\rho\) = \(\rho\)(x, y, z) (Figure \(\PageIndex{3}\)).
Local density can be obtained by a limiting process, based on the average density in a small volume around the point in question, taking the limit where the size of the volume approaches zero,
\[\rho = \lim_{\Delta V \rightarrow 0} \frac{\Delta m}{\Delta V} \label{14.2}\]
where \(\rho\) is the density, m is the mass, and V is the volume.
Since gases are free to expand and contract, the densities of the gases vary considerably with temperature, whereas the densities of liquids vary little with temperature. Therefore, the densities of liquids are often treated as constant, with the density equal to the average density.
Density is a dimensional property; therefore, when comparing the densities of two substances, the units must be taken into consideration. For this reason, a more convenient, dimensionless quantity called the specific gravity is often used to compare densities. Specific gravity is defined as the ratio of the density of the material to the density of water at 4.0 °C and one atmosphere of pressure, which is 1000 kg/m 3 :
\[Specific\; gravity = \frac{Density\; of\; material}{Density\; of\; water} \ldotp\]
The comparison uses water because the density of water is 1 g/cm 3 , which was originally used to define the kilogram. Specific gravity, being dimensionless, provides a ready comparison among materials without having to worry about the unit of density. For instance, the density of aluminum is 2.7 in g/cm 3 (2700 in kg/m 3 ), but its specific gravity is 2.7, regardless of the unit of density. Specific gravity is a particularly useful quantity with regard to buoyancy, which we will discuss later in this chapter.
Pressure
You have no doubt heard the word ‘pressure’ used in relation to blood (high or low blood pressure) and in relation to weather (high- and low-pressure weather systems). These are only two of many examples of pressure in fluids. (Recall that we introduced the idea of pressure in Static Equilibrium and Elasticity , in the context of bulk stress and strain.)
Pressure (p) is defined as the normal force F per unit area A over which the force is applied, or
\[p = \frac{F}{A} \ldotp \label{14.3}\]
To define the pressure at a specific point, the pressure is defined as the force dF exerted by a fluid over an infinitesimal element of area dA containing the point, resulting in p = \(\frac{dF}{dA}\).
A given force can have a significantly different effect, depending on the area over which the force is exerted. For instance, a force applied to an area of 1 mm 2 has a pressure that is 100 times as great as the same force applied to an area of 1 cm 2 . That is why a sharp needle is able to poke through skin when a small force is exerted, but applying the same force with a finger does not puncture the skin (Figure \(\PageIndex{4}\)).
Note that although force is a vector, pressure is a scalar. Pressure is a scalar quantity because it is defined to be proportional to the magnitude of the force acting perpendicular to the surface area. The SI unit for pressure is the pascal (Pa), named after the French mathematician and physicist Blaise Pascal (1623–1662), where
\[1\; Pa = 1\; N/m^{2} \ldotp\]
Several other units are used for pressure, which we discuss later in the chapter.


1.14.61: Solubility Products
A given closed system at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) (which is close to ambient) contains an aqueous solution of a sparingly soluble salt \(\mathrm{MX}\); e.g. \(\mathrm{AgCl}\). The system also contains solid salt \(\mathrm{MX}\). When a soluble salt (e.g. \(\mathrm{KNO}_{3}\)) is added the solubility of salt \(\mathrm{MX}\) increases. This remarkable observation is readily accounted for. The equilibrium involving the sparingly soluble salt is represented as follows.
| \(\mathrm{MX}(\mathrm{s})\) | \(\Leftrightarrow\) | \(\mathrm{M}^{+}\mathrm{~X}^{-}(\mathrm{aq})\) |
| solid | solution |
We represent the salt \(\mathrm{MX}\) by the symbol \(j\). At equilibrium,
\[\mu_{\mathrm{j}}^{\prime \prime}(\mathrm{s})=\mu_{\mathrm{j}}^{\mathrm{eq}}(\mathrm{aq}) \nonumber \]
In terms of the solubility \(\mathrm{S}_{j}\) of the salt \(\mathrm{MX}\), a 1:1 salt,
\[\mu_{\mathrm{j}}^{*}(\mathrm{~s})=\mu_{\mathrm{j}}^{0}(\mathrm{aq})+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{\mathrm{o}}\right) \nonumber \]
By definition
\[\Delta_{\text {sol }} G^{0}=-R \, T \, \ln K_{s}=\mu_{j}^{0}(a q)-\mu_{j}^{*}(s) \nonumber \]
\(\mathrm{K}_{\mathrm{S}}\) is the solubility product, a characteristic property of salt \(\mathrm{MX}\) (at defined \(\mathrm{T}\) and \(\mathrm{p}\)).
\[\mathrm{K}_{\mathrm{S}}=\left[\mathrm{S}_{\mathrm{j}} \, \gamma_{\pm} / \mathrm{m}^{0}\right]^{2} \nonumber \]
Or,
\[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)-\ln \left(\gamma_{\pm}\right) \nonumber \]
According to the DHLL,
\[\ln \left(\gamma_{\pm}\right)=-S_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
\(\mathrm{I}\) is the ionic strength of the solution which can be changed by adding a soluble salt. From equations (e) and (f),
\[\ln \left(\mathrm{S}_{\mathrm{j}} / \mathrm{m}^{0}\right)=(1 / 2) \, \ln \left(\mathrm{K}_{\mathrm{s}}\right)+\mathrm{S}_{\gamma} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
The key point to note is the positive sign in equation (g) showing that the theory accounts for the observed salting–in of the sparingly soluble salt. Further a plot of \(\ln \left(S_{j} / m^{0}\right)\) against \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) is linear yielding an estimate for \(\mathrm{K}_{\mathrm{S}}\) from the intercept.


14.9: Viscosity and Turbulence
- Explain what viscosity is
- Calculate flow and resistance with Poiseuille's law
- Explain how pressure drops due to resistance
- Calculate the Reynolds number for an object moving through a fluid
- Use the Reynolds number for a system to determine whether it is laminar or turbulent
- Describe the conditions under which an object has a terminal speed
In Applications of Newton’s Laws , which introduced the concept of friction, we saw that an object sliding across the floor with an initial velocity and no applied force comes to rest due to the force of friction. Friction depends on the types of materials in contact and is proportional to the normal force. We also discussed drag and air resistance in that same chapter. We explained that at low speeds, the drag is proportional to the velocity, whereas at high speeds, drag is proportional to the velocity squared. In this section, we introduce the forces of friction that act on fluids in motion. For example, a fluid flowing through a pipe is subject to resistance, a type of friction, between the fluid and the walls. Friction also occurs between the different layers of fluid. These resistive forces affect the way the fluid flows through the pipe.
Viscosity and Laminar Flow
When you pour yourself a glass of juice, the liquid flows freely and quickly. But if you pour maple syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids viscosity. Juice has low viscosity, whereas syrup has high viscosity.
The precise definition of viscosity is based on laminar, or nonturbulent, flow. Figure \(\PageIndex{1}\) shows schematically how laminar and turbulent flow differ. When flow is laminar, layers flow without mixing. When flow is turbulent, the layers mix, and significant velocities occur in directions other than the overall direction of flow.
Turbulence is a fluid flow in which layers mix together via eddies and swirls. It has two main causes. First, any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Second, high speeds cause turbulence. The drag between adjacent layers of fluid and between the fluid and its surroundings can form swirls and eddies if the speed is great enough. In Figure \(\PageIndex{2}\), the speed of the accelerating smoke reaches the point that it begins to swirl due to the drag between the smoke and the surrounding air.
Figure \(\PageIndex{3}\) shows how viscosity is measured for a fluid. The fluid to be measured is placed between two parallel plates. The bottom plate is held fixed, while the top plate is moved to the right, dragging fluid with it. The layer (or lamina) of fluid in contact with either plate does not move relative to the plate, so the top layer moves at speed v while the bottom layer remains at rest. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from v to 0 as shown. Care is taken to ensure that the flow is laminar, that is, the layers do not mix. The motion in the figure is like a continuous shearing motion. Fluids have zero shear strength, but the rate at which they are sheared is related to the same geometrical factors A and L as is shear deformation for solids. In the diagram, the fluid is initially at rest. The layer of fluid in contact with the moving plate is accelerated and starts to move due to the internal friction between moving plate and the fluid. The next layer is in contact with the moving layer; since there is internal friction between the two layers, it also accelerates, and so on through the depth of the fluid. There is also internal friction between the stationary plate and the lowest layer of fluid, next to the station plate. The force is required to keep the plate moving at a constant velocity due to the internal friction.
A force F is required to keep the top plate in Figure \(\PageIndex{3}\) moving at a constant velocity v, and experiments have shown that this force depends on four factors. First, F is directly proportional to v (until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on v). Second, F is proportional to the area A of the plate. This relationship seems reasonable, since A is directly proportional to the amount of fluid being moved. Third, F is inversely proportional to the distance between the plates L. This relationship is also reasonable; L is like a lever arm, and the greater the lever arm, the less the force that is needed. Fourth, F is directly proportional to the coefficient of viscosity, \(\eta\) The greater the viscosity, the greater the force required. These dependencies are combined into the equation
\[F = \eta \frac{vA}{L} \ldotp\]
This equation gives us a working definition of fluid viscosity \(\eta\). Solving for \(\eta\) gives
\[\eta = \frac{FL}{vA} \label{14.17}\]
which defines viscosity in terms of how it is measured. The SI unit of viscosity is \(\frac{N\; \cdotp m}{[(m/s)m^{2}]}\) = (N/m 2 )s or Pa • s. Table \(\PageIndex{1}\) lists the coefficients of viscosity for various fluids. Viscosity varies from one fluid to another by several orders of magnitude. As you might expect, the viscosities of gases are much less than those of liquids, and these viscosities often depend on temperature.
| Fluid | Temperature (°C) | Viscosity \(\eta \times 10^{3}\) |
|---|---|---|
| Air | 0 | 0.0171 |
| 20 | 0.0181 | |
| 40 | 0.0190 | |
| 100 | 0.0218 | |
| Ammonia | 20 | 0.00974 |
| Carbon dioxide | 20 | 0.0147 |
| Helium | 20 | 0.0196 |
| Hydrogen | 0 | 0.0090 |
| Mercury | 20 | 0.0450 |
| Oxygen | 20 | 0.0203 |
| Steam | 100 | 0.0130 |
| Liquid water | 0 | 1.792 |
| 20 | 1.002 | |
| 37 | 0.6947 | |
| 40 | 0.653 | |
| 100 | 0.282 | |
| Whole blood | 20 | 3.015 |
| 37 | 2.084 | |
| Blood plasma | 20 | 1.810 |
| 37 | 1.257 | |
| Ethyl alcohol | 20 | 1.20 |
| Methanol | 20 | 0.584 |
| Oil (heavy machine) | 20 | 660 |
| Oil (motor, SAE 10) | 30 | 200 |
| Oil (olive) | 20 | 138 |
| Glycerin | 20 | 1500 |
| Honey | 20 | 2000-10000 |
| Maple syrup | 20 | 2000-3000 |
| Milk | 20 | 3.0 |
| Oil (corn) | 20 | 65 |
Laminar Flow Confined to Tubes: Poiseuille’s Law
What causes flow? The answer, not surprisingly, is a pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate \(Q\) is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as
\[Q = \frac{p_{2} - p_{1}}{R}\]
where \(p_1\) and \(p_2\) are the pressures at two points, such as at either end of a tube, and \(R\) is the resistance to flow. The resistance \(R\) includes everything, except pressure, that affects flow rate. For example, \(R\) is greater for a long tube than for a short one. The greater the viscosity of a fluid, the greater the value of \(R\). Turbulence greatly increases R, whereas increasing the diameter of a tube decreases \(R\).
If viscosity is zero, the fluid is frictionless and the resistance to flow is also zero. Comparing frictionless flow in a tube to viscous flow, as in Figure \(\PageIndex{4}\), we see that for a viscous fluid, speed is greatest at midstream because of drag at the boundaries. We can see the effect of viscosity in a Bunsen burner flame [part (c)], even though the viscosity of natural gas is small.
The resistance R to laminar flow of an incompressible fluid with viscosity \(\eta\) through a horizontal tube of uniform radius r and length l, is given by
\[R = \frac{8 \eta l}{\pi r^{4}} \ldotp \label{14.18}\]
This equation is called Poiseuille’s law for resistance , named after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood through the body.
Let us examine Poiseuille’s expression for R to see if it makes good intuitive sense. We see that resistance is directly proportional to both fluid viscosity \(\eta\) and the length l of a tube. After all, both of these directly affect the amount of friction encountered—the greater either is, the greater the resistance and the smaller the flow. The radius r of a tube affects the resistance, which again makes sense, because the greater the radius, the greater the flow (all other factors remaining the same). But it is surprising that r is raised to the fourth power in Poiseuille’s law. This exponent means that any change in the radius of a tube has a very large effect on resistance. For example, doubling the radius of a tube decreases resistance by a factor of 2 4 = 16.
Taken together \(Q = \frac{p_{2} - p_{1}}{R}\) and \(R = \frac{8 \eta l}{\pi r^{4}}\) give the following expression for flow rate:
\[Q = \frac{(p_{2} - p_{1}) \pi r^{4}}{8 \eta l} \ldotp \label{14.19}\]
This equation describes laminar flow through a tube. It is sometimes called Poiseuille’s law for laminar flow, or simply Poiseuille’s law (Figure \(\PageIndex{5}\)).
An air conditioning system is being designed to supply air at a gauge pressure of 0.054 Pa at a temperature of 20 °C. The air is sent through an insulated, round conduit with a diameter of 18.00 cm. The conduit is 20-meters long and is open to a room at atmospheric pressure 101.30 kPa. The room has a length of 12 meters, a width of 6 meters, and a height of 3 meters. (a) What is the volume flow rate through the pipe, assuming laminar flow? (b) Estimate the length of time to completely replace the air in the room. (c) The builders decide to save money by using a conduit with a diameter of 9.00 cm. What is the new flow rate?
Strategy
Assuming laminar flow, Poiseuille’s law states that
\[Q = \frac{(p_{2} - p_{1}) \pi r^{4}}{8 \eta l} = \frac{dV}{dt} \ldotp \nonumber\]
We need to compare the artery radius before and after the flow rate reduction. Note that we are given the diameter of the conduit, so we must divide by two to get the radius.
Solution
- Assuming a constant pressure difference and using the viscosity \(\eta = 0.0181\; mPa\; \cdotp s\), $$Q = \frac{(0.054\; Pa)(3.14)(0.09\; m)^{4}}{8(0.0181 \times 10^{-3}\; Pa\; \cdotp s)(20\; m)} = 3.84 \times 10^{-3}\; m^{3}/s \ldotp$$
- Assuming constant flow \(Q = \frac{dV}{dt} \approx \frac{\Delta V}{\Delta t}\) $$\Delta t = \frac{\Delta V}{Q} = \frac{(12\; m)(6\; m)(3\; m)}{3.84 \times 10^{-3}\; m^{3}/s} = 5.63 \times 10^{4}\; s = 15.63\; hr \ldotp \nonumber$$
- Using laminar flow, Poiseuille’s law yields $$Q = \frac{(0.054\; Pa)(3.14)(0.045\; m){4}}{8(0.0181 \times 10^{-3}\; Pa\; \cdotp s)(20\; m)} = 22.40 \times 10^{-4}\; m^{3}/s \ldotp$$Thus, the radius of the conduit decreases by half reduces the flow rate to 6.25% of the original value.
Significance
In general, assuming laminar flow, decreasing the radius has a more dramatic effect than changing the length. If the length is increased and all other variables remain constant, the flow rate is decreased:
\[\begin{split} \frac{Q_{A}}{Q_{B}} & = \frac{\frac{(p_{2} - p_{1}) \pi r_{A}^{4}}{8 \eta l_{A}}}{\frac{(p_{2} - p_{1}) \pi r_{B}^{4}}{8 \eta l_{B}}} = \frac{l_{B}}{l_{A}} \\ Q_{B} & = \frac{l_{A}}{l_{B}} Q_{A} \ldotp \end{split} \nonumber\]
Doubling the length cuts the flow rate to one-half the original flow rate.
If the radius is decreased and all other variables remain constant, the volume flow rate decreases by a much larger factor.
\[\begin{split} \frac{Q_{A}}{Q_{B}} & = \frac{\frac{(p_{2} - p_{1}) \pi r_{A}^{4}}{8 \eta l_{A}}}{\frac{(p_{2} - p_{1}) \pi r_{B}^{4}}{8 \eta l_{B}}} = \left(\dfrac{r_{A}}{r_{B}}\right)^{4} \\ Q_{B} & = \left(\dfrac{r_{B}}{r_{A}}\right)^{4} Q_{A} \end{split}\]
Cutting the radius in half decreases the flow rate to one-sixteenth the original flow rate.
Flow and Resistance as Causes of Pressure Drops
Water pressure in homes is sometimes lower than normal during times of heavy use, such as hot summer days. The drop in pressure occurs in the water main before it reaches individual homes. Let us consider flow through the water main as illustrated in Figure \(\PageIndex{6}\). We can understand why the pressure p 1 to the home drops during times of heavy use by rearranging the equation for flow rate:
\[\begin{align} Q & = \frac{p_{2} - p_{1}}{R} \\[4pt] p_{2} - p_{1} & = RQ . \label{EQ5} \end{align}\]
In this case, \(p_2\) is the pressure at the water works and \(R\) is the resistance of the water main. During times of heavy use, the flow rate \(Q\) is large. This means that \(p_2 − p_1\) must also be large. Thus \(p_1\) must decrease. It is correct to think of flow and resistance as causing the pressure to drop from p 2 to p 1 . The equation p 2 − p 1 = RQ is valid for both laminar and turbulent flows.
We can also use Equation \ref{EQ5} to analyze pressure drops occurring in more complex systems in which the tube radius is not the same everywhere. Resistance is much greater in narrow places, such as in an obstructed coronary artery. For a given flow rate Q, the pressure drop is greatest where the tube is most narrow. This is how water faucets control flow. Additionally, R is greatly increased by turbulence, and a constriction that creates turbulence greatly reduces the pressure downstream. Plaque in an artery reduces pressure and hence flow, both by its resistance and by the turbulence it creates.
Measuring Turbulence
An indicator called the Reynolds number \(N_R\) can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as
\[N_{R} = \frac{2 \rho vr}{\eta}\; (flow\; in\; tube) \label{14.20}\]
where \(\rho\) is the fluid density, v its speed, \(\eta\) its viscosity, and \(r\) the tube radius. The Reynolds number is a dimensionless quantity. Experiments have revealed that \(N_R\) is related to the onset of turbulence. For N R below about 2000, flow is laminar. For \(N_R\) above about 3000, flow is turbulent.
For values of \(N_R\) between about 2000 and 3000, flow is unstable—that is, it can be laminar, but small obstructions and surface roughness can make it turbulent, and it may oscillate randomly between being laminar and turbulent. In fact, the flow of a fluid with a Reynolds number between 2000 and 3000 is a good example of chaotic behavior. A system is defined to be chaotic when its behavior is so sensitive to some factor that it is extremely difficult to predict. It is difficult, but not impossible, to predict whether flow is turbulent or not when a fluid’s Reynold’s number falls in this range due to extremely sensitive dependence on factors like roughness and obstructions on the nature of the flow. A tiny variation in one factor has an exaggerated (or nonlinear) effect on the flow.
In Example 14.8, we found the volume flow rate of an air conditioning system to be Q = 3.84 x 10 −3 m 3 /s. This calculation assumed laminar flow.
- Was this a good assumption?
- At what velocity would the flow become turbulent?
Strategy
To determine if the flow of air through the air conditioning system is laminar, we first need to find the velocity, which can be found by
\[Q = Av = \pi r^{2} v \ldotp \nonumber\]
Then we can calculate the Reynold’s number, using the equation below, and determine if it falls in the range for laminar flow
\[R = \frac{2 \rho vr}{\eta} \ldotp \nonumber \]
Solution
- Using the values given: $$\begin{split} v & = \frac{Q}{\pi r^{2}} = \frac{3.84 \times 10^{-3}\; m^{3}/s}{3.14 (0.09\; m)^{2}} = 0.15\; m/s \\ R & = \frac{2 \rho vr}{\eta} = \frac{2 (1.23\; kg/m^{3})(0.15\; m/s)(0.09\; m)}{0.0181 \times 10^{-3}\; Pa\; \cdotp s} = 1835 \ldotp \end{split}$$Since the Reynolds number is 1835 < 2000, the flow is laminar and not turbulent. The assumption that the flow was laminar is valid.
- To find the maximum speed of the air to keep the flow laminar, consider the Reynold’s number. $$\begin{split} R & = \frac{2 \rho vr}{\eta} \leq 2000 \\ v & = \frac{2000(0.0181 \times 10^{-3}\; Pa\; \cdotp s)}{2(1.23\; kg/m^{3})(0.09\; m)} = 0.16\; m/s \ldotp \end{split}$$
Significance
When transferring a fluid from one point to another, it desirable to limit turbulence. Turbulence results in wasted energy, as some of the energy intended to move the fluid is dissipated when eddies are formed. In this case, the air conditioning system will become less efficient once the velocity exceeds 0.16 m/s, since this is the point at which turbulence will begin to occur.


15.9: Chapter 10
Your Turn
10.1
1. \(\overrightarrow {BD}\) is the ray that starts at point \(B\) and extends infinitely in the direction of point \(D\).
2. \(\overline {AB}\) represents the line segment that starts at point \(A\) and ends at point \(B\).
3. \(\overleftarrow {BA}\) represents the ray that starts at point \(A\) and extends infinitely in the direction of point \(B\).
4. \(\overleftrightarrow {AD}\) represents a line that contains the points \(A\) and \(D\). Notice the arrowheads on both ends of the line above \(\mathit{AD}\), which means that the line continues infinitely in both directions.
10.2
1. Answers will vary. One way \({\overline {AE}} ,\,{\overline {EI}} ,\,{\overline {IE}} ,\,{\overline {EB}} ,\,{\overline {BC}} ,\,{\overline {CD}} ,\,{\overline {DG}} \,\); Second way \({\overline {AH}} ,\,{\overline {HI}},\,{\overline {IE}} ,\,{\overline {EB}} ,\,{\overline {BC}} ,{\overline {CF}} ,\,{\overline {FG}} \).
10.3
1. \(\overleftrightarrow {RS}\parallel \overleftrightarrow {UV}\)
\(\overleftrightarrow {RS} \bot \overleftrightarrow {XY}\)
\(\overleftrightarrow {UV} \bot \overleftrightarrow {XY}\)
\(\overleftrightarrow {RS} \bot \overleftrightarrow {XY}\)
\(\overleftrightarrow {UV} \bot \overleftrightarrow {XY}\)
10.4
1. \({\overline {AB}} \cap {\overline {BC}} = B\)
2. \(\overrightarrow {BC} \cup \overleftarrow {CA} = \overleftrightarrow {AC}\)
3. \({\overline {BC}} \cap {\overline {AC}} = {\overline {BC}} \)
10.5
1. Point \(A\) is located at \(( - 5, - 2)\); Point \(B\) is located at \((- 3,4)\); Point \(C\) is located at \((3, - 6)\); Point \(D\) is located at \((5, - 2)\); Point \(E\) is located at \((3,2)\); Point \(F\) is located at \((5,5)\).
2. Points \(A\) and \(B\) are on the straight line \(\overleftrightarrow {AB}\).
3. The line that begins at point \(E\) in the direction of point \(F\) is a ray, \(\overrightarrow {EF}\).
4. The line from point \(C\) to point \(D\) is a line segment \(\overline {CD} \).
5. Yes, this represents a plane. One reason is that the figure contains four points that are not on the line \(\overleftrightarrow {AB}\).
10.6
1. Plane \(MNOP\) intersects with plane \(OPQR\), and plane \(OPQR\) intersects with plane \(QRST\).
10.7
1.
| Acute Angles | Obtuse Angles | Right Angles | Straight Angles |
|---|---|---|---|
|
\(\angle AOB\)
\(\angle AOC\) \(\angle BOC\) \(\angle BOD\) \(\angle COD\) \(\angle DOE\) \(\angle FOE\) |
\(\angle AOE\)
\(\angle BOF\) \(\angle COF \) |
\(\angle AOD\)
\(\angle BOE\) \(\angle DOF \) |
\(\angle AOF\) |
10.8
1. \(\begin{array}{rcl}{5x}&{ = }&
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\\ {2x + 5} & { = } &
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\end{array}\)
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10.9
1. \(m\measuredangle (6x) = {24^ \circ}\), \(m\measuredangle (9x) = {36^ \circ}\), \(m\measuredangle (7x + 2) = {30^ \circ}\)
10.10
1. \(m\measuredangle 4 = {67^ \circ }\), \(m\measuredangle 1 = {113^ \circ }\), \(m\measuredangle 3 = {113^ \circ }\)
10.11
1. \({130^ \circ }\)
10.12
1. \(m\measuredangle 5 = {120^ \circ}\), \(m\measuredangle 4 = {120^ \circ}\), \(m\measuredangle 8 = {120^ \circ}\)
10.13
1. \(m\measuredangle 1 = {132^ \circ}\), \(m\measuredangle 5 = {132^ \circ}\)
10.14
1. \(m\measuredangle 5 = {56^ \circ }\)
\(m\measuredangle 6 = {62^ \circ }\)
\(m\measuredangle 7 = {118^ \circ }\)
\(m\measuredangle 8 = {62^ \circ }\)
\(m\measuredangle 9 = {118^ \circ }\)
\(m\measuredangle 6 = {62^ \circ }\)
\(m\measuredangle 7 = {118^ \circ }\)
\(m\measuredangle 8 = {62^ \circ }\)
\(m\measuredangle 9 = {118^ \circ }\)
10.15
1. \(m\measuredangle x = {66^ \circ }\), \(m\measuredangle (x + 1) = {67^ \circ }\), \(m\measuredangle (x - 19) = {47^ \circ }\)
10.16
1. \(m\measuredangle 1 = {35^ \circ }\); \(m\measuredangle 2 = 2 = {85^ \circ }\); \(m\measuredangle 3 = {60^ \circ }\)
10.17
1. Triangle \(ABC\) is congruent to triangle \(DEF\).
10.18
1. SAS
10.19
1. ASA
10.20
1. The triangles are similar.
10.21
1. \(x = 6\), \(y = 24\)
10.22
1. The tree is 86 feet high.
10.23
1. 60 ft
10.24
1. rectangle
2. pentagon
3. heptagon
4. parallelogram
10.25
1. Shapes 1, 2, 4, and 6 are triangles; shape 3 is a pentagon; shape 5 is a parallelogram; and shape 7 is a rectangle.
10.26
1. 120 in
10.27
1. 22.4 in
10.28
1. \(a = {108^ \circ }\)
The sum of the interior angles is \({540^ \circ }\).
The sum of the interior angles is \({540^ \circ }\).
10.29
1. We have the sum of interior angles is \({360^ \circ }\). Then, \(x = - 9.\) The other angles measure \({54^ \circ }\), \({111^ \circ }\), \({44^ \circ }\).
10.30
1. \(7(51.43) = {360^ \circ }\)
10.31
1. \(C = 2(2.25)\pi = 14.14\text{ cm}\)
10.32
1. \(r = 2.5\,\text{cm}\)
10.33
1. You need to buy 15.74 feet of trim.
10.34
1. The translated hexagon has labels \(A''\), \(B''\), \(C''\), \(D''\), \(E''\), \(F''\).
10.35
1. Rotate the triangle about the rotation point \({90^ \circ }\) to the right three times.
10.36
1. This tessellation could be produced with a reflection of the triangle vertically, then each triangle is rotated \({180^ \circ }\) and translated to the right.
10.37
1. From the first square on the left, rotate the square \({30^ \circ }\) to the right, or \(A1 \to A2\). Then, reflect the square over the horizontal, or \(A2 \to A3\). Next, reflect all three squares over the vertical line. The lavender triangles comprise another pattern that tessellates, fits in with the squares, and fills the gaps.
10.38
1. No
10.39
1. Not by themselves, but by adding an equilateral triangle, the two regular polygons do tessellate the plane without gaps.
10.40
1. We made a tessellation with a regular decagon (10 sides) and an irregular hexagon. We see that the regular decagon will not fill the plane by itself. The gap is filled, however, with an irregular hexagon. These two shapes together will fill the plane.
10.41
1. \(8\,\text{cm}^2\)
10.42
1. 108 ft
2
10.43
1. 45 boxes at a cost of $2,025.00
10.44
1. \(270\,{\text{i}}{{\text{n}}^2}\)
10.45
1. 13,671 square feet; cost is $19,872.95.
10.46
1. \(132\,{{\text{in}}^2}\)
10.47
1. \({d_2} =10\)
10.48
1. 40 in
10.49
1. \(A = 96.25\,{\text{c}}{{\text{m}}^2}\)
10.50
1. \(30\,{\text{y}}{{\text{d}}^2}\)
10.51
1. \(28.3\,{\text{c}}{{\text{m}}^2}\)
10.52
1. the 15-inch pizza
10.53
1. \(A = 21.875 + 4.8 = 26.7\,{\text{f}}{{\text{t}}^2}\)
10.54
1. \(21.46\,{\text{c}}{{\text{m}}^2}\)
10.55
1. $300
10.56
1. \(\begin{array}{rcl}{SA}&{ = }&{432\text{ cm}{^2}}\\{V}&{ = }&{540\text{ cm}{^3}}\\ \end{array}\)
10.57
1. \(100\,\text{cm}{^2}\)
10.58
1.
/**/\begin{array}{rcl} {SA}&{ = }&{336\text{ cm}{^2}}\\ {V}&{ = }&{480\text{ cm}{^3}}\\ \end{array}/**/
10.59
1. \(1{,}192\text{ ft}{^2}\)
10.60
1.
/**/\begin{array}{rcl} {SA}&{ = }&{527.78\text{ cm}{^2}}\\ {V}&{ = }&{769.69\text{ cm}{^3}}\\ \end{array}/**/
10.61
1. approximately 8 1/3 cans of soup
10.62
1. 6.25 ft by 6.25 ft
10.63
1. \(1.26\,{\text{ft}}\,{\text{wide}}\, \times \,1.26\,{\text{ft}}\,{\text{long}}\, \times \,1.26\,{\text{ft}}\,{\text{high}},\) $95.26
10.64
1. \(a = 5\)
10.65
1. 1,140 ft
10.66
1. The slanted distance will be 120.4 inches.
10.67
1. The side lengths are \(15,15\sqrt 3,30.\)
10.68
1. The ladder reaches 12 feet up the wall and sits \(12\sqrt {3\,}{\text{ feet}}\) from the wall.
10.69
1. Each side \(x\) equals \(x = 4\sqrt 2 = 5.66.\)
10.70
1.
/**/\begin{array}{*{20}{l}} {r = 6.53} \\ {y = 4.2} \end{array}/**/
10.71
1. \(\alpha = {46.3^ \circ },\,\beta = {43.7^ \circ }\), \(x = 5.73\)
10.72
1. 2,241 ft
10.73
1. \(c = 8.06\), one angle is \({60^ \circ }\), and the other angle is \({30^ \circ }\).
10.74
1. \({11.9^ \circ }\)
10.75
1. 46 ft
Check Your Understanding
1. The line containing point \(D\) and point \(A\) is a line segment from point \(D\) to point \(A\), \(\overline{DA} \), or from point \(A\) to point \(D\), \(\overline {AD} \).
2. The line containing points \(C\) and \(B\) is a straight line that extends infinitely in both directions and contains points \(C\) and \(B\).
3. This is a ray that begins at point \(E\), although it does not contain point \(E\), and extends in the direction of point \(F\).
4. \({\overline {AB}} \cup {\overline {BD}} = \overline {AD} \). The union of line segment \(\overline {AB} \) and the line segment \({\overline {BD}}\) contains all points in each line segment combined.
5. \(\overrightarrow {BD} \cap {\overline {BC}} = {\overline {BC}} \). The intersection of the ray \(\overrightarrow {BD} \) and the line segment \({\overline {BC}} \) contains only the points common to each set, \({\overline {BC}} \).
6. \(\overrightarrow {BA} \cup \overrightarrow {BD} = \overleftrightarrow {AD}\). The union of the ray starting at point \(B\) and extending infinitely in the direction of \(A\) and the ray starting at point \(B\) and extending infinitely in the direction of \(D\) is the straight line extending infinitely in both directions containing points \(A\), \(B\), \(C\), and \(D\).
7. Two lines are parallel if the distance between the lines is constant implying that the lines cannot intersect.
8. Perpendicular lines intersect forming a \({90^ \circ }\) angle between them.
9. Yes, because it contains a line and a point not on the line.
10. straight
11. obtuse
12. right
13. acute
14. \(m\measuredangle 1 = {149^ \circ }\) by supplementary angles with \(\measuredangle {31^ \circ }.\)
15. \(m\measuredangle 3 = {31^ \circ }\) by vertical angles with the angle measuring \({31^ \circ }.\)
16. \(m\measuredangle 5 = {149^ \circ }\) by corresponding angles with \(\measuredangle 1.\)
17. \(x = {89^ \circ }\)
18. \(x = {67^ \circ }\)
19. \(x = {77^ \circ }\) and \(y = {26^ \circ }\)
20. These are similar triangles, so we can solve using proportions.
Then, \(x = \frac14{3}\) and \(y = \frac16{3}\).
| \(\begin{array}{*{20}{rcl}}{\frac{3}{2}}&{ = }&{\frac7{x}} \\ {3x}&{ = }&{14} \\ {x}&{ = }&{\frac14{3}} \\ \end{array}\) |
\(\begin{array}{*{20}{rcl}}{\frac{4}{2}}&{ = }&{\frac
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} \\{\frac14{3}(4)}&{ = }&{2(4 + y)} \\{\frac56{3}}&{ = }&{8 + 2y} \\{56}&{ = }&{3(8 + 2y)} \\{56}&{ = }&{24 + 6y} \\{32}&{ = }&{6y} \\{\frac16{3}}&{ = }&{y}\end{array}\)
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|
21. Set up the proportions.
\(\begin{array}{*{20}{rcl}}{\frac{6}{a}}&{ = }&{\frac1214} \\{6(14)}&{ = }&{12a} \\{84}&{ = }&{12a} \\{7}&{ = }&{a}\end{array}\)
Thus, \(t = 20\) and \(a = 7.\)
\(\begin{array}{*{20}{rcl}}{\frac{6}{a}}&{ = }&{\frac1214} \\{6(14)}&{ = }&{12a} \\{84}&{ = }&{12a} \\{7}&{ = }&{a}\end{array}\)
Thus, \(t = 20\) and \(a = 7.\)
22. pentagon
23. octagon
24. heptagon
25. \(6(5) = 30\text{ cm}\)
26. \(S = \left( {6 - 2} \right){180^ \circ } = {720^ \circ }\)
27. \({120^ \circ }\)
28. \(\begin{array}{*{20}{rcl}}{360}&{=}&{152 + 9x + (5x + 1) + (x + 12)}\\{}&{=}&{15x + 165}\\{195}&{=}&{15x}\\{13}&{=}&{x}\\{x + 12}&{=}&{25^ \circ },\,9x = {117^ \circ },\,5x + 1 = {66^ \circ }\end{array}\)
29. \(C = 2\pi (3) = 6\pi = 18.85\text{ cm}\)
30. The patterns are repeated shapes that can be transformed in such a way as to fill the plane with no gaps or overlaps.
31. Starting with the triangle with the point labeled \(A\), the triangle is translated point by point 3 units to the right and 3 units up to point \(A'\). Then, the triangle labeled \(A'\) is translated 3 units to the right and 3 units up to point \(A''\).
32. The triangle is rotated about the rotation point \(180^\circ \) to vertex \(B'\).
33. The dark triangle is reflected about the vertical line showing the light back, and then reflected about the horizontal line. The pattern is repeated leaving a white diamond between the shapes.
34. <IP_ADDRESS>.3.3
35. 7.5 cm
2
36. 25 ft
37. \(168\,{\text{c}}{\text{m}^2}\)
38. \(64.5\,{\text{c}}{{\text{m}}^2}\)
39. 201.1 in
2
40. \(116.38\,{\text{i}}{{\text{n}}^2}\)
41. \(706.86\,{\text{i}}{{\text{n}}^2}\)
42. \(386.6\,\text{cm}{^2}\)
43. \(416\,\text{in}{^2}\)
44. \(640\,\text{in}{^3}\)
45. \(942.48\,\text{cm}{^2}\)
46. \(1,570\,\text{cm}{^3}\)
47. \(511.35\,\text{in}{^2}\)
48.
/**/\begin{array}{*{20}{l}} {a = 2.5} \\ {b = 2.5\sqrt 3 } \end{array}/**/
49.
/**/\begin{array}{*{20}{l}} {b = 10} \\ {c = 10\sqrt 2 } \end{array}/**/
50. \(b = 6\)
51. 24.2 ft
52. \(c = 11.7{\text{ cm}}\)
53. \(x = 14.3{\text{ m}}\)


8.11: Why It Matters- Positioning
Why create a product or service positioning statement that aligns with a value proposition and a target segment?
Ever since you were small, you have loved special occasions and gatherings. It started with sprawling extended-family picnics when you were a child. From there, you graduated to bigger and more elaborate events: attending fairs and grand openings, crashing weddings and voter conventions, wandering through business expositions at the local convention center. In fact, one of your favorite memories is of the time you sneaked into the legendary Consumer Electronics Show when you were visiting your cousin in Las Vegas. As far as you’re concerned, the more people, the bigger the party and the better.
Now, as an adult, you’ve landed a job with Shindiggity, an events management firm that specializes in putting on conferences, trade shows, and events for industry associations and companies. You love getting to meet new people, travel, and work on making fun, memorable events.
Here’s the challenge: Shindiggity is relatively small and new on the events management scene. Although events management is a growing area for business services, it’s also a fairly crowded field. When you google “events management company,” you get 432 million results. Shindiggity is easily lost in the crowd.
At Shindiggity’s annual retreat, employees from across the company are divided into several groups, and each group is given the same task: How can we get Shindiggity to stand out from the competition?
You realize that the VP from your division has been assigned to your group and that this is a great chance for you to impress her with your creativity and initiative. You also want to help your company be successful. Right off, you jot down some questions to focus your thinking:
- What makes Shindiggity better than competitors? Is it the quality of events? The creative ideas? The amazing people? The way it uses technology to make things smoother and more efficient? Something else?
- Who do we need to talk to about Shindiggity, and what will make them decide to give us a try?
- How can we make sure they remember Shindiggity and what we stand for?
These are critical questions you need to answer in order to help your company (and, you think, impress the VP). At the core, they are questions about positioning and differentiation: What position do you hold in customers’ minds? What position do you want to hold? How are you different in positive ways that make you stand out from the pack?
If Shindiggity can’t figure out how to make a lasting impact in customers’ minds, it won’t win enough clients to stay in business. And, if it goes out of business, you’ll be job hunting instead of continuing to do the work you love. On the other hand, if you come up with great ideas, you may help Shindiggity position itself effectively against competitors, which could lead to new business and—who knows?—to more job responsibility or a promotion for you. It’s pretty obvious which scenario you decide to pursue.
Learning Outcomes
- Define positioning and differentiation, and why they are important to marketing a product or service
- Explain the process of selecting a positioning and differentiation strategy
- Develop and evaluate positioning statements based on defined criteria
- Explain repositioning and the associated risks and complexities of repositioning a product or service
- Describe the process of implementing a positioning strategy
Contributors and Attributions
- Why It Matters: Positioning and Differentiation. Provided by : Lumen Learning. License : CC BY: Attribution
- San Diego Comic-Con 2011u2014Kermit and Gonza balloon costumes. Authored by : Conmunityu2014Pop Culture Geek. Located at : https://www.flickr.com/photos/popculturegeek/5991540917/ . License : CC BY-NC: Attribution-NonCommercial


3.5: Nutrition
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- Last updated
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Breast milk is considered the ideal diet for newborns. It has the right amount of calories, fat, and protein to support overall physical and neurological development, it provides a source of iron more easily absorbed in the body than the iron found in dietary supplements, it provides a resistance against many diseases, it is more easily digested by infants than is formula, and it helps babies make a transition to solid foods more easily than if bottle fed. For all of these reasons, it is recommended that mothers breast feed their infants until at least 6 months of age and that breast milk be used in the diet throughout the first year (U.S. Department of Health and Human Services, 2004a in Berk, 2007).
However, most mothers who breastfeed in the United States stop breast feeding at about 6-8 weeks, often in order to return to work outside the home. Mothers can certainly continue to provide breast milk to their babies by expressing and freezing the milk to be bottle fed at a later time or by being available to their infants at feeding time, but some mothers find that after the initial encouragement they receive in the hospital to breast feed, the outside world is less supportive of such efforts. Some workplaces support breastfeeding mothers by providing flexible schedules and welcoming infants, but many do not. And the public support of breastfeeding is sometimes lacking. Women in Canada are more likely to breastfeed than are those in the United States and the Canadian health recommendation is for breastfeeding to continue until 2 years of age. Facilities in public places in Canada such as malls, ferries, and workplaces provide more support and comfort for the breastfeeding mother and child than found in the United States.
One early argument given to promote the practice of breastfeeding was that it promoted bonding and healthy emotional development for infants. However, this does not seem to be the case. Breastfed and bottle-fed infants adjust equally well emotionally (Ferguson and Woodward, 1999). This is good news for mothers who may be unable to breastfeed for a variety of reasons and for fathers who might feel left out as a result.
In addition to the nutritional benefits of breastfeeding, breast milk is free! Anyone who has priced formula recently can appreciate this added incentive to breastfeeding. Prices for a month’s worth of formula can easily range from $130-200. Breastfeeding also stimulates contractions in the uterus to help it regain its normal size. And women who breastfeed are more likely to space their pregnancies further apart.
A Historic Look at Breastfeeding
The use of wet nurses, or lactating women hired to nurse others’ infants, during the middle ages eventually declined and mothers increasingly breastfed their own infants in the late 1800s. In the early part of the 20th century, breastfeeding began to go through another decline and by the 1950s, it was practiced less frequently by middle class, more affluent mothers as formula began to be viewed as superior to breast milk. In the late 1960s and 1970s, greater emphasis began to be placed on natural childbirth and breastfeeding and the benefits of breastfeeding were more widely publicized. Gradually rates of breastfeeding began to climb, particularly among middle-class educated mothers who received the strongest messages to breastfeed. Today, women receive consultation from lactation specialists before being discharged from the hospital to ensure that they are informed of the benefits of breastfeeding and given support and encouragement to get their infants to get used to taking the breast. This does not always happen immediately and first time mothers, especially, can become upset or discouraged. In this case, lactation specialists and nursing staff can encourage the mother to keep trying until baby and mother are comfortable with the feeding.
Global Considerations and Malnutrition
In the 1960s, formula companies led campaigns in developing countries to encourage mothers to feed their babies on infant formula. Many mothers felt that formula would be superior to breast milk and began using formula. The use of formula can certainly be healthy under conditions in which there is adequate, clean water with which to mix the formula and adequate means to sanitize bottles and nipples. However, in many of these countries such conditions were not available and babies often were given diluted, contaminated formula which made them become sick with diarrhea and become dehydrated. Rates of breast feeding declined in Peru from 90 percent to 10 percent in just 8 years time (Berger, 2001). These conditions continue today and now many hospitals prohibit the distribution of formula samples to new mothers in efforts to get them to rely on breast feeding. Many of these mothers do not understand the benefits of breast feeding and have to be encouraged and supported in order to promote this practice. Breast feeding could save the lives of millions of infants each year, according to the World Health Organization, yet fewer than 40 percent of infants are breastfed exclusively for the first 6 months of life. Find out more at the WHO’s 10 facts on breastfeeding. Most women can breastfeed unless they are receiving chemotherapy or radiation therapy, have HIV, are dependent on illicit drugs, or have active, untreated tuberculosis.
Children in developing countries and countries experiencing the harsh conditions of war are at risk for two major types of malnutrition. Infantile marasmus refers to starvation due to a lack of calories and protein. Children who do not receive adequate nutrition lose fat and muscle until their bodies can no longer function. Babies who are breast fed are much less at risk of malnutrition than those who are bottle fed. After weaning, children who have diets deficient in protein may experience kwashiorkor or the “disease of the displaced child” often occurring after another child has been born and taken over breastfeeding. This results in a loss of appetite and swelling of the abdomen as the body begins to break down the vital organs as a source of protein.
Kwashiorkor (Photo Courtesy Centers for Disease Control and Prevention)
The Breast Milk Industry
The benefits of breast milk are well-known and publicized. The collection and distribution of breast milk has become a million dollar industry supplying hospitals and others in need of the ideal diet. For more information, go to www.prolacta.com to see a current development in the story of breast milk.
Milk Anemia in the United States
About 9 million children in the United States are malnourished (Children’s Welfare, 1998). More still suffer from milk anemia, a condition in which milk consumption leads to a lack of iron in the diet. This can be due to the practice of giving toddlers milk as a pacifier-when resting, when riding, when waking, and so on. Appetite declines somewhat during toddlerhood and a small amount of milk (especially with added chocolate syrup) can easily satisfy a child’s appetite for many hours. The calcium in milk interferes with the absorption of iron in the diet as well. Many preschools and daycare centers give toddlers a drink after they have finished their meal in order to prevent spoiling their appetites.


15.2: Video- Visualizing Data
Below you’ll see a link to a TED Talk by David McCandless titled “The Beauty of Data Visualization.” TED is a wonderful resource for intelligent, entertaining talks by experts on all kinds of subjects. You can probably find a video relevant to your own research project, if you look: http://www.ted.com/
Please watch this 18-minute video, and when finished, respond to this question:
McCandless thinks that information design is really important because “we’re all visualizers now; we’re all demanding a visual aspect to our information.” If we assume it’s true, then how has this changed our educational system? Are there particularly good examples of how teachers and schools have responded? Extreme deficits?
Responses should be 1-2 paragraphs long, and reference your own experience with visual learning, either from your personal education or what you’ve witnessed from a friend or family member’s experience.