9.12: Electric Hazards and the Human Body
Learning Objectives
- Define thermal hazard, shock hazard, and short circuit.
- Explain what effects various levels of current have on the human body.
There are two known hazards of electricity—thermal and shock. A thermal hazard is one where excessive electric power causes undesired thermal effects, such as starting a fire in the wall of a house. A shock hazard occurs when electric current passes through a person. Shocks range in severity from painful, but otherwise harmless, to heart-stopping lethality. This section considers these hazards and the various factors affecting them in a quantitative manner.
Thermal Hazards
Electric power causes undesired heating effects whenever electric energy is converted to thermal energy at a rate faster than it can be safely dissipated. A classic example of this is the short circuit , a low-resistance path between terminals of a voltage source. An example of a short circuit is shown in Figure \(\PageIndex{1}\). Insulation on wires leading to an appliance has worn through, allowing the two wires to come into contact. Such an undesired contact with a high voltage is called a short . Since the resistance of the short, \(r\), is very small, the power dissipated in the short, \(P=V^{2} / r\), is very large. For example, if \(V\) is 120 V and \(r\) is \(0.100 ~\Omega\), then the power is 144 kW, much greater than that used by a typical household appliance. Thermal energy delivered at this rate will very quickly raise the temperature of surrounding materials, melting or perhaps igniting them.
One particularly insidious aspect of a short circuit is that its resistance may actually be decreased due to the increase in temperature. This can happen if the short creates ionization. These charged atoms and molecules are free to move and, thus, lower the resistance \(r\). Since \(P=V^{2} / r\), the power dissipated in the short rises, possibly causing more ionization, more power, and so on. High voltages, such as the 480-V AC used in some industrial applications, lend themselves to this hazard, because higher voltages create higher initial power production in a short.
Another serious, but less dramatic, thermal hazard occurs when wires supplying power to a user are overloaded with too great a current. As discussed in the previous section, the power dissipated in the supply wires is \(P=I^{2} R_{\mathrm{w}}\), where \(R_{\mathrm{w}}\) is the resistance of the wires and \(I\) the current flowing through them. If either \(I\) or \(R_{\mathrm{w}}\) is too large, the wires overheat. For example, a worn appliance cord (with some of its braided wires broken) may have \(R_{\mathrm{w}}=2.00 ~\Omega\) rather than the \(0.100 ~\Omega\) it should be. If 10.0 A of current passes through the cord, then \(P=I^{2} R_{\mathrm{w}}=200 \mathrm{~W}\) is dissipated in the cord—much more than is safe. Similarly, if a wire with a \(0.100-\Omega\) resistance is meant to carry a few amps, but is instead carrying 100 A, it will severely overheat. The power dissipated in the wire will in that case be \(P=1000 \mathrm{~W}\). Fuses and circuit breakers are used to limit excessive currents. (See Figure \(\PageIndex{2}\) and Figure \(\PageIndex{3}\).) Each device opens the circuit automatically when a sustained current exceeds safe limits.
Fuses and circuit breakers for typical household voltages and currents are relatively simple to produce, but those for large voltages and currents experience special problems. For example, when a circuit breaker tries to interrupt the flow of high-voltage electricity, a spark can jump across its points that ionizes the air in the gap and allows the current to continue flowing. Large circuit breakers found in power-distribution systems employ insulating gas and even use jets of gas to blow out such sparks. Here AC is safer than DC, since AC current goes through zero 120 times per second, giving a quick opportunity to extinguish these arcs.
Shock Hazards
Electrical currents through people produce tremendously varied effects. An electrical current can be used to block back pain. The possibility of using electrical current to stimulate muscle action in paralyzed limbs, perhaps allowing paraplegics to walk, is under study. TV dramatizations in which electrical shocks are used to bring a heart attack victim out of ventricular fibrillation (a massively irregular, often fatal, beating of the heart) are more than common. Yet most electrical shock fatalities occur because a current put the heart into fibrillation. A pacemaker uses electrical shocks to stimulate the heart to beat properly. Some fatal shocks do not produce burns, but warts can be safely burned off with electric current (though freezing using liquid nitrogen is now more common). Of course, there are consistent explanations for these disparate effects. The major factors upon which the effects of electrical shock depend are
- The amount of current \(I\)
- The path taken by the current
- The duration of the shock
- The frequency \(f\) of the current (\(f=0\) for DC)
Table \(\PageIndex{1}\) gives the effects of electrical shocks as a function of current for a typical accidental shock. The effects are for a shock that passes through the trunk of the body, has a duration of 1 s, and is caused by 60-Hz power.
| Current (mA) | Effect |
|---|---|
| 1 | Threshold of sensation |
| 5 | Maximum harmless current |
| 10–20 | Onset of sustained muscular contraction; cannot let go for duration of shock; contraction of chest muscles may stop breathing during shock |
| 50 | Onset of pain |
| 100–300+ | Ventricular fibrillation possible; often fatal |
| 300 | Onset of burns depending on concentration of current |
| 6000 (6 A) | Onset of sustained ventricular contraction and respiratory paralysis; both cease when shock ends; heartbeat may return to normal; used to defibrillate the heart |
Our bodies are relatively good conductors due to the water in our bodies. Given that larger currents will flow through sections with lower resistance (to be further discussed in the next chapter), electric currents preferentially flow through paths in the human body that have a minimum resistance in a direct path to earth. The earth is a natural electron sink. Wearing insulating shoes, a requirement in many professions, prohibits a pathway for electrons by providing a large resistance in that path. Whenever working with high-power tools (drills), or in risky situations, ensure that you do not provide a pathway for current flow (especially through the heart).
Very small currents pass harmlessly and unfelt through the body. This happens to you regularly without your knowledge. The threshold of sensation is only 1 mA and, although unpleasant, shocks are apparently harmless for currents less than 5 mA. A great number of safety rules take the 5-mA value for the maximum allowed shock. At 10 to 20 mA and above, the current can stimulate sustained muscular contractions much as regular nerve impulses do. People sometimes say they were knocked across the room by a shock, but what really happened was that certain muscles contracted, propelling them in a manner not of their own choosing. (See Figure \(\PageIndex{4}\)(a).) More frightening, and potentially more dangerous, is the “can’t let go” effect illustrated in Figure \(\PageIndex{4}\)(b). The muscles that close the fingers are stronger than those that open them, so the hand closes involuntarily on the wire shocking it. This can prolong the shock indefinitely. It can also be a danger to a person trying to rescue the victim, because the rescuer’s hand may close about the victim’s wrist. Usually the best way to help the victim is to give the fist a hard knock/blow/jar with an insulator or to throw an insulator at the fist. Modern electric fences, used in animal enclosures, are now pulsed on and off to allow people who touch them to get free, rendering them less lethal than in the past.
Greater currents may affect the heart. Its electrical patterns can be disrupted, so that it beats irregularly and ineffectively in a condition called “ventricular fibrillation.” This condition often lingers after the shock and is fatal due to a lack of blood circulation. The threshold for ventricular fibrillation is between 100 and 300 mA. At about 300 mA and above, the shock can cause burns, depending on the concentration of current—the more concentrated, the greater the likelihood of burns.
Very large currents cause the heart and diaphragm to contract for the duration of the shock. Both the heart and breathing stop. Interestingly, both often return to normal following the shock. The electrical patterns on the heart are completely erased in a manner that the heart can start afresh with normal beating, as opposed to the permanent disruption caused by smaller currents that can put the heart into ventricular fibrillation. The latter is something like scribbling on a blackboard, whereas the former completely erases it. TV dramatizations of electric shock used to bring a heart attack victim out of ventricular fibrillation also show large paddles. These are used to spread out current passed through the victim to reduce the likelihood of burns.
Current is the major factor determining shock severity (given that other conditions such as path, duration, and frequency are fixed, such as in the table and preceding discussion). A larger voltage is more hazardous, but since \(I=V / R\), the severity of the shock depends on the combination of voltage and resistance. For example, a person with dry skin has a resistance of about \(200 ~\mathrm{k} \Omega\). If he comes into contact with 120-V AC, a current \(I=(120 \mathrm{~V}) /(200 ~\mathrm{k} \Omega)=0.6 \mathrm{~mA}\) passes harmlessly through him. The same person soaking wet may have a resistance of \(10.0 ~\mathrm{k} \Omega\) and the same 120 V will produce a current of 12 mA—above the “can’t let go” threshold and potentially dangerous.
Most of the body’s resistance is in its dry skin. When wet, salts go into ion form, lowering the resistance significantly. The interior of the body has a much lower resistance than dry skin because of all the ionic solutions and fluids it contains. If skin resistance is bypassed, such as by an intravenous infusion, a catheter, or exposed pacemaker leads, a person is rendered microshock sensitive . In this condition, currents about 1/1000 those listed in Table \(\PageIndex{1}\) produce similar effects. During open-heart surgery, currents as small as \(20 ~\mu \mathrm{A}\) can be used to still the heart. Stringent electrical safety requirements in hospitals, particularly in surgery and intensive care, are related to the doubly disadvantaged microshock-sensitive patient. The break in the skin has reduced his resistance, and so the same voltage causes a greater current, and a much smaller current has a greater effect.
Factors other than current that affect the severity of a shock are its path, duration, and AC frequency. Path has obvious consequences. For example, the heart is unaffected by an electric shock through the brain, such as may be used to treat manic depression. And it is a general truth that the longer the duration of a shock, the greater its effects. Figure \(\PageIndex{5}\) presents a graph that illustrates the effects of frequency on a shock. The curves show the minimum current for two different effects, as a function of frequency. The lower the current needed, the more sensitive the body is at that frequency. Ironically, the body is most sensitive to frequencies near the 50- or 60-Hz frequencies in common use. The body is slightly less sensitive for DC (\(f=0\)), mildly confirming Edison’s claims that AC presents a greater hazard. At higher and higher frequencies, the body becomes progressively less sensitive to any effects that involve nerves. This is related to the maximum rates at which nerves can fire or be stimulated. At very high frequencies, electrical current travels only on the surface of a person. Thus a wart can be burned off with very high frequency current without causing the heart to stop. (Do not try this at home with 60-Hz AC!) Some of the spectacular demonstrations of electricity, in which high-voltage arcs are passed through the air and over people’s bodies, employ high frequencies and low currents. (See Figure \(\PageIndex{6}\).)
Section Summary
- The two types of electric hazards are thermal (excessive power) and shock (current through a person).
- Shock severity is determined by current, path, duration, and AC frequency.
- Table \(\PageIndex{1}\) lists shock hazards as a function of current.
- Figure \(\PageIndex{5}\) graphs the threshold current for two hazards as a function of frequency.
Footnotes
- 1 For an average male shocked through trunk of body for 1 s by 60-Hz AC. Values for females are 60–80% of those listed.
Glossary
- thermal hazard
- a hazard in which electric current causes undesired thermal effects
- shock hazard
- when electric current passes through a person
- short circuit
- also known as a “short,” a low-resistance path between terminals of a voltage source
- microshock sensitive
- a condition in which a person’s skin resistance is bypassed, possibly by a medical procedure, rendering the person vulnerable to electrical shock at currents about 1/1000 the normally required level


2.4: Trial and Appellate Court Functions
As demonstrated earlier in Figure 2.3.1 , the federal and state court systems include a hierarchy of courts. Now, we will explore their functions within those structures.
Trial Courts
The principal purpose of a trial court is to hear evidence and decide what happened in a case when the parties disagree on particular facts. In other words, a trial court makes findings of fact Determinations of what the facts of the case are, based on evaluating the evidence presented (e.g., determining whether someone committed a crime or violated the terms of a contract). At trial, witnesses are called, and their testimonies are recorded in a trial record for future reference. Although there are some exceptions, juries typically are responsible for making findings of fact, while judges decide questions of law Questions of law can include how to interpret the law, how to apply the law, what law is relevant to the dispute, etc. . Also, the parties can agree to have a judge serve as the finder of fact instead of a jury A trial before the judge without a jury is called a bench trial . . A losing party is entitled to appeal the case to an appellate court , discussed below.
In the federal court system , cases are filed in the US District Court, which is a federal trial court. There are 94 judicial districts in the nation (i.e., 94 federal trial courts) named for their geographic locations Eighty-nine districts are distributed among the 50 states. District courts are also located in Puerto Rico, the Virgin Islands, the District of Columbia, Guam, and the Northern Mariana Islands. . Some states with smaller populations have only one judicial district (e.g., North Dakota and Montana), while more populated states have multiple judicial districts (e.g., Wisconsin and Texas). See Figure 2.4.1 . The US Department of Justice represents the federal government in federal court in both civil and criminal cases, dividing its US Attorneys among the 94 judicial districts.
Figure \(\PageIndex{1}\) Geographic boundaries of the United States Courts of Appeals and United States District (Trial) Courts. ( Source )
Depending on the state court system, trial courts are known most commonly by names such as circuit court, superior court, or district court, but so can intermediate appellate courts . Wisconsin refers to its trial courts as circuit courts. Wisconsin has 72 counties with circuit courts, which is analogous to the 94 judicial districts in the federal system.
Figure \(\PageIndex{2}\) Geographic boundaries of the Wisconsin Courts of Appeals and Wisconsin Circuit (Trial) Courts. ( Source )
Intermediate Appellate Courts
Intermediate appellate courts —typically referred to simply as appellate courts —review the decisions of trial courts, normally to determine whether the parties received a fair trial or whether the trial judge (1) applied the appropriate law or (2) interpreted the law appropriately. Thus, these courts address questions of law Questions of law can include how to interpret the law, how to apply the law, what law is relevant to the dispute, etc. . Because the job of appellate courts is to interpret the law and determine whether the court below applied the law properly, juries do not play a role in appellate court proceedings.
A common problem with learning languages is that, frequently, different words mean or describe the same thing. For example, "bathroom," "restroom," washroom," and "lavatory" all refer to the same thing. Depending on the area of the United States, someone might refer to a carbonated beverage as "soda," "pop," or " Coke In some regions of the American South, "Coke" may refer to any brand, not just a cola produced by Coca-Cola. ." "Water fountain" commons refers to a "drinking fountain," and Wisconsin residents have a history of referring to drinking fountains as a Bubblers.)
The same nuances exist in the legal world. In the context of courts, "court of appeals" and "appellate court" both refer to an intermediate court of appeals . But despite the fact that the US Supreme Court and state courts of last resort actually are appellate courts, people do not refer to them as "appellate courts."
By contrast, it is reasonable to infer that a supreme court is at the top of the hierarchy, thus a court of last resort. Yet, New York refers to its court of last resort as its "court of appeals."
A federal trial court is called a "district court," but a trial court in Wisconsin is called a "circuit court." A federal appellate court is called a "circuit court," but an appellate court in Wisconsin is called a "district court."
While we are on the topic of trial courts, what do you think a trial court in New York is called?
- Answer
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That's right: a s upreme court . (If you already knew this, maybe you have watched Law and Order .) Now, you can amaze your friends and family with a "did you know . . .?" fun fact. (It is also possible that you will receive this reaction .)
Figures 2.4.1 (federal) and 2.4.2 (Wisconsin) demonstrate which appellate courts hear cases from which trial courts. Whether in the federal or state court system, each appellate court is responsible for hearing cases from a particular region. For example, if a party appealed the outcome of a federal case in the Western District of Wisconsin (the federal trial court located in Madison, Wisconsin), the US Court of Appeals for the Seventh Circuit (located in Chicago, Illinois) would hear the appeal. In a Wisconsin state case, an appeal from the Ozaukee County Circuit Court (a state trial court located in Port Washington, Wisconsin) would be heard by the Wisconsin Court of Appeals for the Second District (located in Waukesha, Wisconsin).
A final note about intermediate appellate courts is that they must hear any appeal made to them. Appeals to these courts are called appeals as of right An appeal that a court is legally required to hear , which is different than appeals to courts of last resort.
Courts of Last Resort
As discussed in Section 2.3, a party losing an appeal at the federal appellate court level may ask the US Supreme Court to hear its case, and this is true at the state level. However, appeals to courts of last resort are discretionary appeals An appeal that a court a court may—but it not required to—consider , so they can select which cases to hear, which usually includes cases in which the multiple federal circuit courts arrive at different conclusions about the same legal issue, cases that the Court believes have substantial national importance, and cases that present federal legal issues it has never before addressed.
The US Supreme Court agrees to hear very few cases. Each Supreme Court term, the Court receives 7,000-8,000 petitions for review, and it hears only about 80 cases (approximately one percent). State supreme courts tend to agree to hear a greater proportion. For instance, the Wisconsin Supreme Court hears approximately 10% of the appeals made.
- Supplementary Attributions
-
Source content not listed in the attribution footer below includes:
- 13.3: The Dual Court System by Glen Krutz and Sylvie, Waskiewicz is licensed by CC BY 4.0 . Original source: American Government 3e .
- 5.2: Chapter 32- Paths to the Supreme Court by David Hubert is licensed by CC BY-NC-SA . Original source: Attenuated Democracy:A Critical Introduction to U.S. Government and Politics .
- Wex Legal Dictionary and Encyclopedia by Cornell University's Legal Information Institute is licensed CC BY-NC-SA 2.5 . Original source: finding of fact ; question of law ; appeal .


1.5 Challenges
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There are, however, some barriers to the adoption of these emerging technologies. While learners may embrace them, it may take longer for institutions and corporations to adopt and implement them. Administrative policies as well as an organization’s culture can slow down or halt their adoption. Some policy makers may misunderstand the usefulness of these technologies in teaching and learning. As learners adopt new technologies, they will take more control over their own learning, which may challenge the status quo. This may gradually influence corporations and institutions to accept this new paradigm of learning. The consequences of not serving the needs of learners to keep up-to-date with these new ways of learning challenge the relevance of formal training and learning in our organizations.
Perceptions about the quality of certain technology-mediated instructional activities or environments provide additional challenges. As a prime example, the US-based College Board questions “whether Internet-based laboratories are an acceptable substitute for the hands-on culturing of gels and peering through microscopes that have long been essential ingredients of American laboratory science” (Dillon, 2006, para. 3). While emerging technologies allow us to extend nearly unlimited possibilities to those who previously did not have access to them, there may always be a group of people who feel online instruction cannot replace direct experience. Who would not want to see lions and zebras in their natural habitat in Africa instead of going to a zoo or watching a video clip online? Similarly, if it were possible to set up expensive chemistry labs in every school or college, then the virtual environments would not be necessary. They would only serve as a way to refresh knowledge, rather than to obtain it. An alternate solution may be to allow students to learn virtually, but to require them to demonstrate proficiencies in person as appropriate (e.g., before moving to a certain level of difficulty).
Intellectual property (IP) rights and digital rights management will be major challenges. Short-sighted, large corporations who expect to profit from sales (particularly in the entertainment sector) will fight widespread distribution of their product. Solutions like Creative Commons licensing will become the new way of doing business. See Chapter 15, Understanding Copyright.


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Environmental laws and regulations are implemented at three different levels: international treaties, national laws, and local laws. Each of these levels is intricately connected: international treaties influence—but also depend on—national laws to succeed, while national laws are guided by local needs and customary laws that have been in place for generations.
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International treaties and conventions provide frameworks for countries to cooperate on protecting species, ecosystems, and other levels of biodiversity. International agreements are important because: (1) many species migrate and disperse across borders, (2) ecosystems do not follow administrative boundaries, (3) pollution spreads by air and water across regions and around the globe, (4) many biological products are traded internationally, and (5) some environmental problems require global cooperation and coordination.
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National governments protect biodiversity by regulating natural resource use and preventing pollution. Subsidies and tax incentives can also be used to reward citizens and businesses that engage in environmentally-responsible behaviors.
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There is a constant need to evaluate environmental laws and regulations to ensure that they are enforced, violators are prosecuted, and new laws and amendments are passed as and when needed. Law enforcement agencies and scientists are constantly looking for new ways to address enforcement shortcomings.
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When banning environmentally detrimental activities that are not compatible with conservation goals, it is critical to help the affected people to transition toward sustainable activities. Failing that, conservation activities may unintentionally force those people to resort to illegal activities such as poaching out of desperation to obtain food and income.


13.3: More about relative sizes of sets
What is the “size” of \(\mathbb{R}\text{?}\)
We know \(\mathbb{N}\text{,}\) \(\mathbb{Z}\text{,}\) and \(\mathbb{Q}\) all have the same size (countably infinite). But \(\mathbb{R}\) is so large that it is un countable, so it seems like \(\mathbb{R}\) should be “larger” than \(\mathbb{N}\text{,}\) \(\mathbb{Z}\text{,}\) and \(\mathbb{Q}\text{.}\)
set \(B\) is larger than set \(A\) if
- \(B\) has a subset the same size as \(A\text{,}\) and
- every subset of \(B\) which is the same size as \(A\) is proper
if \(B\) is larger than \(A\text{,}\) then \(A\) is smaller than \(B\)
- Show there exists an injection \(A \hookrightarrow B\text{.}\)
- Show that every injection \(A \hookrightarrow B\) cannot also be a surjection.
However, if one or both of \(A,B\) are finite, one can instead just verify that \(\vert B \vert > \vert A \vert \text{.}\)
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Matching up the parts of the Test with the parts of the definition of
larger set
:
- The existence of an injection \(f: A \hookrightarrow B\) demonstrates that \(B\) has a subset that is the same size as \(A\text{,}\) as restricting the codomain to \(f: A \rightarrow f(A)\) creates a bijection.
- By definition, a subset \(C \subseteq B\) has the same size as \(A\) when there exists a bijection \(A \to C\text{.}\) Enlarging the codomain, such a bijection can be thought of as an injection \(A \hookrightarrow B\) whose image is \(C \subseteq B\text{.}\) If no such injection can also be surjective, then \(C \subsetneqq B\text{,}\) i.e. \(C\) is a proper subset of \(B\text{.}\)
- In the second part of the test, one can simply show that every function \(A \to B\) cannot be a surjection, in which case surely every injection \(A \hookrightarrow B\) cannot be a surjection. It may seem like it should be more difficult to prove this more general statement, but if you will find that your argument that every injection cannot be a surjection does not actually rely on the injective assumption, then there is no need for that assumption.
Set \(\mathbb{R}\) is larger than each of the sets \(\mathbb{N}\text{,}\) \(\mathbb{Z}\text{,}\) and \(\mathbb{Q}\text{.}\)
Here follows an important comparison of set sizes.
Every set is smaller than its own power set.
- Proof
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Let \(A\) represent an arbitrary set. We will apply the Larger Set Test to demonstrate that \(\scr{P}(A)\) is larger than \(A\text{.}\)
- There exists a natural injection
\begin{align*} A & \hookrightarrow \mathscr{P} (A), \\ x & \mapsto \{x\}. \end{align*}
(If \(A\) is empty, then this is just the empty function, which is always injective by Statement 1 of Proposition 12.1.1 .)- Suppose \(f: A \rightarrow \scr{P}(A)\) is an arbitrary function. Using the Surjective Function Test, we will demonstrate that it cannot be surjective by exhibiting an element \(X \in \scr{P}(A)\) for which no element \(a \in A\) satisfies \(f(a) = X\text{.}\) (We will not need to make the assumption that \(f\) is injective — see Statement 2 of Remark \(\PageIndex{1}\).)
Note that for each \(a \in A\text{,}\) the image element \(f(a) \in \scr{P} (A)\text{,}\) being a power set element, is a subset of \(A\text{.}\) So for each \(a \in A\) we can ask whether \(a\) is contained in the subset \(f(a)\) or not. Collecting together the “or not” answers, set
\begin{equation*} X = \{a \in A : a \notin f(a)\}. \end{equation*}
Note that \(X\) is a subset of \(A\text{,}\) so again this means that it is also an element of \(\scr{P}(A)\text{.}\)Could \(f(a) = X\) be possible for some \(a \in A\text{?}\) Since \(X \subseteq A\text{,}\) for each \(a \in A\) we have either \(a \in X\) or \(a \notin X\text{.}\)
Case \(a \in X\).
Then by definition of \(X\) above, we have \(a \notin f(a)\text{.}\) Since \(X\) contains element \(a\) but \(f(a)\) does not, sets \(X\) and \(f(a)\) cannot be equal.Case \(a \notin X\).
Then by definition of \(X\) above, we must have \(a \in f(a)\text{,}\) since otherwise \(a\) would satisfy the condition to be in \(X\text{.}\) But now \(f(a)\) contains element \(a\) but \(X\) does not, so again sets \(X\) and \(f(a)\) cannot be equal.Since \(f(a) = X\) is not possible in all cases, we have found an element in \(\scr{P}(A)\) that is not in the image \(f(A)\text{,}\) as required to demonstrate that \(f\) is not surjective.
Cantor's Theorem implies that there are an infinite number of “levels” of infinity! For, if \(A\) is an infinite set, then \(\scr{P}(A)\) is a larger infinite set, and \(\scr{P}(\scr{P}(A))\) is a still larger infinite set, and \(\scr{P} (\scr{P}(\scr{P}(A )))\) is a still larger infinite set, ….
The size of the set of natural numbers \(\mathbb{N}\) seems like the lowest possible level of infinity, since every subset of \(\mathbb{N}\) is either finite or has the same size as \(\mathbb{N}\text{.}\) (See Statement 1 of Proposition 13.2.1 .) The set of real numbers \(\mathbb{R}\) is larger than \(\mathbb{N}\text{,}\) since it contains \(\mathbb{N}\) as a proper subset but is not itself the same size as \(\mathbb{N}\text{.}\) So writing \(\vert \mathbb{N} \vert = \infty\) is not the same as writing \(\vert \mathbb{R} \vert = \infty\text{,}\) as they are evidently different levels of infinity. Is there any level of infinity between these two?
There does not exist a set larger than \(\mathbb{N}\) but smaller than \(\mathbb{R}\text{.}\)
It is not known whether the Continuum Hypothesis is true! In fact, it has been proved that the Continuum Hypothesis can be neither proved nor disproved in certain common axiomatic systems for set theory!
We have seen that funny things can happen with sizes of infinite sets — for example, \(\mathbb{N}\) is an proper subset of \(\mathbb{Z}\text{,}\) but the two have the same size! This is not a defect in our definitions, it just demonstrates that for infinite sets, the subset relation is not a good measure of size. But it also demonstrates that we should be vigilant about other possible unintuitive consequences of our definitions, because they might reveal defects in our definitions. For example, from our definitions of smaller and larger sets, there is no obvious reason why there could not be some weird example of a pair of sets \(A\) and \(B\) with both \(B\) larger than \(A\) and \(A\) larger than \(B\text{.}\) Luckily, that cannot happen thanks to the following theorem.
Suppose \(A\) and \(B\) are infinite sets. If there exists both an injection \(A \hookrightarrow B\) and an injection \(B \hookrightarrow A\text{,}\) then \(A\) and \(B\) have the same size.
- Proof
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Suppose \(f: A \hookrightarrow B\) and \(g: B \hookrightarrow A\) are injections. We need to exhibit a bijection from \(A\) to \(B\) (or vice versa, but we will construct one from \(A\) to \(B\)).
For every element \(a \in A\text{,}\) we can construct a chain of alternating elements from \(A\) and \(B\) as follows. Working forwards from \(a\text{,}\) the injection \(f\) maps \(a\) to some element of \(B\text{,}\) and the injection \(g\) maps that element of \(B\) to some element of \(A\text{,}\) which is mapped by \(f\) to some element of \(B\text{,}\) and so on.
\begin{align*} a_0 & = a, \\ b_0 & = f(a_0), \\ a_1 & = g(b_0), \\ b_1 & = f(a_1), \\ a_2 & = g(b_1), \\ & \vdots \end{align*}
The chain will go on infinitely because the functions \(f\) and \(g\) always provide a next element.We can also try to trace the chain backwards: starting at our original element \(a \in A\text{,}\) we can look for some element of \(B\) that \(g\) maps to \(a\text{,}\) though at first consideration it's possible that none exists. If we do find such an element of \(B\text{,}\) we can then look for some element of \(A\) that \(f\) maps to it, and so on. While the chain extends infinitely in the forward direction, we cannot be sure at this point that it will extend infinitely in the backward direction.
Now, every element of \(A\) can be placed into such a chain, and because \(f\) and \(g\) are injective the chain in which we find an element \(a\) is always the same: the next element in the chain is always \(f(a)\text{,}\) and the element before \(a\) is always the unique \(b\) in \(B\) so that \(g(b) = a\) (if such an element exists). And the elements after \(f(a)\) and before \(b\) are also uniquely determined by the injectiveness of \(f\) and \(g\text{.}\) And so on.
So we end up finding every element of \(A\) in a unique alternating chain, and each chain has one of four patterns:
- a chain with some element repeated, in which case we could force the chain to loop back on itself at the repetition to form a finite, circular chain;
- a chain with no repetition and no end or start;
- a chain with no repetition and no end, but the process to trace it backward failed at some point, and the last element in the “backward” direction (which we could view as the first element in the whole chain) is an element of \(A\text{;}\) and
- a chain with no repetition and no end, but starts with an element of \(B\text{.}\)
Now that we have cut out possible repetition by creating circular chains, every element of \(A\) appears exactly once in a unique chain, and by symmetry the same can be said about \(B\text{.}\) We can then create a bijection from \(A\) to \(B\) by mapping every element of \(A\) to the element of \(B\) that follows it in the chain it appears in. Except for elements of \(A\) that appear in a chain that has a beginning with a starting element from \(B\) — instead each of those elements of \(A\) should be mapped to the element of \(B\) that precedes it in its chain. This will create a bijective correspondence between the elements of \(A\) and \(B\text{.}\)


5.11D: Bacteriorhodopsin
- Discuss the function of bacteriorhodopsin
Bacteriorhodopsin is a protein used by Archaea, the most notable one being Halobacteria. It acts as a proton pump; that is, it captures light energy and uses it to move protons across the membrane out of the cell. The resulting proton gradient is subsequently converted into chemical energy. The resulting proton gradient is subsequently converted into chemical energy.
Bacteriorhodopsin is an integral membrane protein usually found in two-dimensional crystalline patches known as “purple membrane”, which can occupy up to nearly 50% of the surface area of the archaeal cell. The bacteriorhodopsin forms repeating elements that are arranged in chains. Each chain has seven transmembrane alpha helices and contains one molecule of retinal buried deep within, the typical structure for retinylidene proteins. It is the retinal molecule that changes its conformation when absorbing a photon, resulting in a conformational change of the surrounding protein and the proton pumping action. This releases a proton from a “holding site” into the extracellular side (EC) of the membrane. Reprotonation of the retinal molecule by restores its original isomerized form. This results in a second proton being released to the EC side. The releases the proton from its “holding site,” where a new cycle may begin.
The bacteriorhodopsin molecule is purple and is most efficient at absorbing green light (wavelength 500-650 nm, with the absorption maximum at 568 nm). Bacteriorhodopsin belongs to a family of bacterial proteins related to vertebrate rhodopsins, the pigments that sense light in the retina. Many molecules have homology to bacteriorhodopsin, including the light-driven chloride pump halorhodopsin, and some directly light-activated channels like channelrhodopsin. All other phototrophic systems in bacteria, algae, and plants use chlorophylls or bacteriochlorophylls rather than bacteriorhodopsin. These also produce a proton gradient, but in a quite different and more indirect way involving an electron transfer chain consisting of several other proteins. Furthermore, chlorophylls are aided in capturing light energy by other pigments known as “antennas”; these are not present in bacteriorhodopsin-based systems. Last, chlorophyll-based phototrophy is coupled to carbon fixation (the incorporation of carbon dioxide into larger organic molecules) and for that reason is photosynthesis, which is not true for bacteriorhodopsin-based system. Thus, it is likely that phototrophy independently evolved at least twice, once in bacteria and once in archaea.
Key Points
- Bacteriorhodopsin is a proton pump found in Archaea, it takes light energy and coverts it into chemical energy, ATP, that can be used by the cell for cellular functions.
- Bacteriorhodopsin forms chains, which contain retinal molecule within, it is the retinal molecule that absorbs a photon from light, it then changes the confirmation of the nearby Bacteriorhodopsin protein, allowing it to act as a proton pump.
- While chlorophyll based ATP generation depends on a protein gradient, like bacteriorhodopsin, but with striking differences, suggesting that phototrophy evolved in bacteria and archaea independently of each other.
Key Terms
- isomerized : converted from one isomer to another
- retinal : One of several yellow or red carotenoid pigments formed from rhodopsin by the action of light; retinene
- phototrophy : The synthesis of an organism’s food from inorganic material using light as a source of energy


44.3B: Tropical Wet Forest and Savannas
- Recognize the distinguishing characteristics of tropical wet forests and savannas
Tropical Wet Forest
Tropical wet forests, also referred to as tropical rainforests, are found in equatorial regions. The vegetation is characterized by plants with broad leaves that fall off throughout the year. Unlike the trees of deciduous forests, the trees in this biome do not have a seasonal loss of leaves associated with variations in temperature and sunlight; these forests are “evergreen” year-round.
The temperature and sunlight profiles of tropical wet forests are very stable in comparison to that of other terrestrial biomes, with the temperatures ranging from 20°C – 34°C (68°F – 93°F). Compared to other forest biomes, tropical wet forests have little variation in seasonal temperatures. This lack of seasonality leads to year-round plant growth, rather than the seasonal (spring, summer, and fall) growth seen in other biomes. In contrast to other ecosystems, tropical ecosystems do not have long days and short days during the yearly cycle. Instead, a constant daily amount of sunlight (11–12 hrs per day) provides more solar radiation and, thereby, a longer period of time for plant growth.
The annual rainfall in tropical wet forests ranges from 125-660 cm (50–200 in), with some monthly variation. While sunlight and temperature remain fairly consistent, annual rainfall is highly variable. Tropical wet forests have wet months in which there can be more than 30 cm (11–12 in) of precipitation, as well as dry months in which there are fewer than 10 cm (3.5 in) of rainfall. However, the driest month of a tropical wet forest still exceeds the annual rainfall of some other biomes, such as deserts.
Tropical wet forests have high net primary productivity because the annual temperatures and precipitation values in these areas are ideal for plant growth. Therefore, the extensive biomass present in the tropical wet forest leads to plant communities with very high species diversity. Tropical wet forests have more species of trees than any other biome. On average, between 100 and 300 species of trees are present in a single hectare (2.5 acres) of South America. One way to visualize this is to compare the distinctive horizontal layers within the tropical wet forest biome. On the forest floor is a sparse layer of plants and decaying plant matter. Above that is an understory of short shrubby foliage. A layer of trees rising above this understory is topped by a closed upper canopy: the uppermost overhead layer of branches and leaves. Some additional trees emerge through this closed upper canopy. These layers provide diverse and complex habitats for the variety of plants, fungi, animals, and other organisms within the tropical wet forests. For instance, epiphytes are plants that grow on other plants. Host plants are typically unharmed. Epiphytes are found throughout tropical wet forest biomes. Many species of animals use the variety of plants and the complex structure of the tropical wet forests for food and shelter. Some organisms live several meters above ground, having adapted to this arboreal lifestyle.
Savannas
Savannas are grasslands with scattered trees located in Africa, South America, and northern Australia. Savannas are hot, tropical areas with temperatures averaging from 24°C – 29°C (75°F – 84°F) and an annual rainfall of 10–40 cm (3.9–15.7 in). They have an extensive dry season. For this reason, forest trees do not grow as well as they do in the tropical wet forest or other forest biomes. As a result, there are relatively few trees within the grasses and forbs (herbaceous flowering plants) that dominate the savanna. Since fire is an important source of disturbance in this biome, plants have evolved well-developed root systems that allow them to quickly re-sprout after a fire.
Animals commonly found in savannas in Africa include the African elephant, lions, gazelles, giraffes, ostriches, and many other mammals, birds, plants and invertebrates. The northern Australian savannas also have many types of plants, animals, insects, and reptiles, including marsupials (kangaroos and wallabies), bats, and rodents. In addition to the native animals such as foxes and Patagonian maras (rabbit-like rodents), savannas in South America are commonly used for grazing domestic livestock, such as sheep, goats, and cattle because of their open grasslands and herbaceous layer of plants.
Key Points
- Tropical wet forests, located near the equator, have temperatures that range from 20°C – 34°C (68°F – 93°F), with little variation in seasonal temperatures.
- The lack of seasonality, constant daily sunlight, ideal temperatures, and high rate of precipitation in tropical wet forests lead to increased plant growth and high species diversity.
- The annual rainfall in tropical wet forests ranges from 125 to 660 cm (50–200 in); there is a high rate of precipitation even in the dry months.
- Savannas, grasslands with scattered trees, are located in Africa, South America, and northern Australia.
- Temperatures in savannas range from 24°C – 29°C (75°F – 84°F), with annual rainfall ranges from 10–40 cm (3.9–15.7 in).
- Because savannas are very dry, trees do not grow as well as they do in other forest biomes and diversity is minimal.
Key Terms
- deciduous : of or pertaining to trees which lose their leaves in winter or the dry season
- biome : any major regional biological community such as that of forest or desert
- understory : the layer of plants that grow in the shade of the canopy of a forest


22.1: Tracheostomy Care and Suctioning Introduction
- Safely perform nasal, oral, pharyngeal, and tracheostomy suctioning
- Provide tracheostomy care
- Explain procedure to patient
- Adapt procedure to reflect variations across the life span
- Document actions and observations
- Recognize and report significant deviations from norms
This chapter will discuss tracheostomy care and various types of suctioning (e.g., oral, nasal, pharyngeal, and tracheostomy) performed by nurses. The purpose of respiratory suctioning is to maintain a patent airway and improve oxygenation by removing mucous secretions and foreign material (e.g., vomit or gastric secretions).  Nasal and pharyngeal suctioning is performed with a sterile, soft, flexible catheter to remove accumulated saliva, pulmonary secretions, blood, vomitus, or other foreign material from nasopharyngeal areas that cannot be removed by the patient’s spontaneous cough or other less invasive procedures. [1]
Tracheostomy suctioning uses a sterile catheter that is inserted through a tracheostomy tube into a patient’s trachea. A tracheostomy tube is a tube that is inserted through a surgical opening in the neck to the trachea to create an artificial airway. Tracheostomies require routine care to prevent infection and obstruction, as well as frequent suctioning to maintain a patent airway. [2] Tracheostomy care and suctioning are performed collaboratively by nurses and respiratory therapists.
- American Association for Respiratory Care. (2004). AARC clinical practice guideline: Nasotracheal suctioning - 2004 revision & update. Respiratory Care, 49 (9), 1080-1084. https://www.aarc.org/wp-content/uploads/2014/08/09.04.1080.pdf ↵
- This work is a derivative of Clinical Procedures for Safer Patient Care by British Columbia Institute of Technology and is licensed under CC BY 4.0. ↵


2.1: Norwegian Wafer (Krumkake) Iron
Norwegian Wafer (
Krumkake
) Iron
Norwegian-American
Torgrim Fjeld/Field, c.1870
Metal (iron)
Museum acquisition
MHAHS 1991.044.0001
For many Norwegian-Americans, krumkake represents the sweet flavor of Christmas served with their favorite hot drink, coffee. After the batter is cooked in the krumkake iron, the delicately stamped cookie is rolled into the shape of a cone while still warm. Once cooled, the crisp cookie is filled with whipped cream or fruit.
Torgrim Fjeld, a Norwegian immigrant and Mount Horeb blacksmith, designed this iron. It features an ornate relief image of a sheaf of wheat encircled by the biblical prayer, “Give us this day our daily bread,” written in Norwegian. The text may reveal Fjeld’s religious response to the prominent wheat farming industry in his new home.
As the holiday season begins, many of us share warm and delightful memories of time spent with family and friends. So much of that time spent together is enriched with personal, regional, or cultural traditions. Few traditions remain as timeless as the recipes passed down from generation to generation, seasoning memories of celebration. For many Norwegian-Americans, krumkake is reminiscent of the sweet flavor of Christmas served aside their favorite hot drink, coffee. [1] This light, crisp cookie is made by setting a dollop of batter onto the iron, closing the iron, and cooking it over a stovetop or fire. Once it is cooked, the wafer is rolled into a cone and filled with whipped cream or fruit. These delicate, holiday-favored wafers annually reiterate a family’s Norwegian cultural heritage to generation long past and new ones yet to come.[2]
When Norwegian immigrants first arrived in America, they were surprised by the regions’ culinary preferences and by differences in what ingredients were available. For instance, certain Norwegian-Americans were taken aback by the preeminence of pork in the American diet and had to adapt recipes to their new home’s preferred meat source while learning to stomach a richer diet. On the other hand, they were delighted by the surplus of sugar used as a primary ingredient in traditional pastries such as the krumkake.[3]
Though referenced as an easy recipe, krumkake requires a specific cast iron mold to cook the thin wafer over an open fire. Some Norwegian-American immigrants valued their heavy krumkake irons enough to take them on the journey from Norway to America, but for those who didn’t transport the iron in their already limited luggage space, skilled immigrants like Torgrim Fjeld could make an authentic iron for them in their new hometown.[4]
Torgrim Fjeld was a Norwegian immigrant and blacksmith who owned a wagon repair shop in Mt Horeb in the nineteenth century. Born in Valdres, Norway in 1849, he moved to America in 1860 with his father Reverend John Fjeld, his mother Gunhild Torgrimsdatter Hesjebakken Fjeld and three siblings.[5] Though Fjeld primarily repaired wagons, he discovered that krumkake irons were in such high demand that he needed help delivering and selling them to neighboring communities. Marten Venden, the grandfather of Mt Horeb resident Olianna Cuneen who was locally well-known for her Norwegian arts and crafts such as rosemalling, drove a wagon to nearby towns to sell Fjeld’s krumkake irons, stamped with TJ Fjeld should anyone admire his work.
Fjeld designed this iron with an ornate relief representing his Norwegian-American identity. The iron impresses the traditional cookie with the image of a sheaf of wheat encircled by the biblical prayer, “give us this day our daily bread,” written in Norwegian.[6] The biblical text referencing wheat and bread hint at Fjeld’s personal background. As the son of a minister, he would likely be familiar with the biblical quote, but he designed the plate with something truly local to the American Midwest: a wheat sheaf. Perhaps, we can imagine the design binding together Fjeld’s religious upbringing with his new home and its prominent wheat farming industry. In turn, the plates might indicate his new community identity, Wisconsin farmers recalling their memories of home through the food they share.
References
[1] National Museum of American History, Smithsonian. Accessed March 25, 2017. http://americanhistory.si.edu/collec...t/nmah_1321759
[2] For more on food and ethnic identity, see Donna R. Gabaccia, We Are What We Eat: Ethnic Food and the Making of Americans, ( Cambridge and London: Harvard University Press, 1998).
[3] Richard J. Fapso, Norwegians in Wisconsin, ( Madison, Wisconsin: The Wisconsin Historical Society Press, 2001) 27.
[4] Wisconsin Historical Society “Ethnic Holiday” Accessed March 25, 2017. http://www.wisconsinhistory.org/muse...ic_holiday.asp
[5] “Wafer Iron.” Wisconsin Decorative Arts Database. Accessed March 23, 2017. http://content.wisconsinhistory.org/...ion/wda/id/283
[6] “Giv Os I Dag Vort Daglige Brod”


22.4: Roosevelt’s “Big Stick” Foreign Policy
While President McKinley ushered in the era of the American empire through military strength and economic coercion, his successor, Theodore Roosevelt, established a new foreign policy approach, allegedly based on a favorite African proverb, “speak softly, and carry a big stick, and you will go far” (Figure 22.4.1). At the crux of his foreign policy was a thinly veiled threat. Roosevelt believed that in light of the country’s recent military successes, it was unnecessary to use force to achieve foreign policy goals, so long as the military could threaten force. This rationale also rested on the young president’s philosophy, which he termed the “strenuous life,” and that prized challenges overseas as opportunities to instill American men with the resolve and vigor they allegedly had once acquired in the Trans-Mississippi West.
Roosevelt believed that while the coercive power wielded by the United States could be harmful in the wrong hands, the Western Hemisphere’s best interests were also the best interests of the United States. He felt, in short, that the United States had the right and the obligation to be the policeman of the hemisphere. This belief, and his strategy of “speaking softly and carrying a big stick,” shaped much of Roosevelt’s foreign policy.
THE CONSTRUCTION OF THE PANAMA CANAL
As early as the mid-sixteenth century, interest in a canal across the Central American isthmus began to take root, primarily out of trade interests. The subsequent discovery of gold in California in 1848 further spurred interest in connecting the Atlantic and Pacific Oceans, and led to the construction of the Panama Railway, which began operations in 1855. Several attempts by France to construct a canal between 1881 and 1894 failed due to a combination of financial crises and health hazards, including malaria and yellow fever, which led to the deaths of thousands of French workers.
Upon becoming president in 1901, Roosevelt was determined to succeed where others had failed. Following the advice that Mahan set forth in his book The Influence of Seapower upon History , he sought to achieve the construction of a canal across Central America, primarily for military reasons associated with empire, but also for international trade considerations. The most strategic point for the construction was across the fifty-mile isthmus of Panama, which, at the turn of the century, was part of the nation of Colombia. Roosevelt negotiated with the government of Colombia, sometimes threatening to take the project away and build through Nicaragua, until Colombia agreed to a treaty that would grant the United States a lease on the land across Panama in exchange for a payment of $10 million and an additional $250,000 annual rental fee. The matter was far from settled, however. The Colombian people were outraged over the loss of their land to the United States, and saw the payment as far too low. Influenced by the public outcry, the Colombian Senate rejected the treaty and informed Roosevelt there would be no canal.
Undaunted, Roosevelt chose to now wield the “big stick.” In comments to journalists, he made it clear that the United States would strongly support the Panamanian people should they choose to revolt against Colombia and form their own nation. In November 1903, he even sent American battleships to the coast of Colombia, ostensibly for practice maneuvers, as the Panamanian revolution unfolded. The warships effectively blocked Colombia from moving additional troops into the region to quell the growing Panamanian uprising. Within a week, Roosevelt immediately recognized the new country of Panama, welcoming them to the world community and offering them the same terms—$10 million plus the annual $250,000 rental fee—he had previously offered Colombia. Following the successful revolution, Panama became an American protectorate, and remained so until 1939.
Once the Panamanian victory was secured, with American support, construction on the canal began in May 1904. For the first year of operations, the United States worked primarily to build adequate housing, cafeterias, warehouses, machine shops, and other elements of infrastructure that previous French efforts had failed to consider. Most importantly, the introduction of fumigation systems and mosquito nets following Dr. Walter Reed’s discovery of the role of mosquitoes in the spread of malaria and yellow fever reduced the death rate and restored the fledgling morale among workers and American-born supervisors. At the same time, a new wave of American engineers planned for the construction of the canal. Even though they decided to build a lock-system rather than a sea-level canal, workers still had to excavate over 170 million cubic yards of earth with the use of over one hundred new rail-mounted steam shovels (Figure 22.4.2). Excited by the work, Roosevelt became the first sitting U.S. president to leave the country while in office. He traveled to Panama where he visited the construction site, taking a turn at the steam shovel and removing dirt. The canal opened in 1914, permanently changing world trade and military defense patterns.
This timeline of the Panama Canal illustrates the efforts involved in both the French and U.S. canal projects.
THE ROOSEVELT COROLLARY
With the construction of the canal now underway, Roosevelt next wanted to send a clear message to the rest of the world—and in particular to his European counterparts—that the colonization of the Western Hemisphere had now ended, and their interference in the countries there would no longer be tolerated. At the same time, he sent a message to his counterparts in Central and South America, should the United States see problems erupt in the region, that it would intervene in order to maintain peace and stability throughout the hemisphere.
Roosevelt articulated this seeming double standard in a 1904 address before Congress, in a speech that became known as the Roosevelt Corollary. The Roosevelt Corollary was based on the original Monroe Doctrine of the early nineteenth century, which warned European nations of the consequences of their interference in the Caribbean. In this addition, Roosevelt states that the United States would use military force “as an international police power” to correct any “chronic wrongdoing” by any Latin American nation that might threaten stability in the region. Unlike the Monroe Doctrine, which proclaimed an American policy of noninterference with its neighbors’ affairs, the Roosevelt Corollary loudly proclaimed the right and obligation of the United States to involve itself whenever necessary.
Roosevelt immediately began to put the new corollary to work. He used it to establish protectorates over Cuba and Panama, as well as to direct the United States to manage the Dominican Republic’s custom service revenues. Despite growing resentment from neighboring countries over American intervention in their internal affairs, as well as European concerns from afar, knowledge of Roosevelt’s previous actions in Colombia concerning acquisition of land upon which to build the Panama Canal left many fearful of American reprisals should they resist. Eventually, Presidents Herbert Hoover and Franklin Roosevelt softened American rhetoric regarding U.S. domination of the Western Hemisphere, with the latter proclaiming a new “Good Neighbor Policy” that renounced American intervention in other nations’ affairs. However, subsequent presidents would continue to reference aspects of the Roosevelt Corollary to justify American involvement in Haiti, Nicaragua, and other nations throughout the twentieth century. The map below (Figure 22.4.3) shows the widespread effects of Roosevelt’s policies throughout Latin America.
In 1904, Roosevelt put the United States in the role of the “police power” of the Western Hemisphere and set a course for the U.S. relationship with Central and Latin America that played out over the next several decades. He did so with the Roosevelt Corollary, in which he stated:
It is not true that the United States feels any land hunger or entertains any projects as regards the other nations of the Western Hemisphere save as such are for their welfare. All that this country desires is to see the neighboring countries stable, orderly, and prosperous. Any country whose people conduct themselves well can count upon our hearty friendship. . . . Chronic wrongdoing, or an impotence which results in a general loosening of the ties of civilized society, may in America, as elsewhere, require intervention by some civilized nation, and in the Western Hemisphere the adherence of the United States to the Monroe Doctrine may force the United States, however, reluctantly, in flagrant cases of such wrongdoing or impotence, to the exercise of an international police power.”
In the twenty years after he made this statement, the United States would use military force in Latin America over a dozen times. The Roosevelt Corollary was used as a rationale for American involvement in the Dominican Republic, Nicaragua, Haiti, and other Latin American countries, straining relations between Central America and its dominant neighbor to the north throughout the twentieth century.
AMERICAN INTERVENTION IN THE RUSSO-JAPANESE WAR
Although he supported the Open Door notes as an excellent economic policy in China, Roosevelt lamented the fact that the United States had no strong military presence in the region to enforce it. Clearly, without a military presence there, he could not as easily use his “big stick” threat credibly to achieve his foreign policy goals. As a result, when conflicts did arise on the other side of the Pacific, Roosevelt adopted a policy of maintaining a balance of power among the nations there. This was particularly evident when the Russo-Japanese War erupted in 1904.
In 1904, angered by the massing of Russian troops along the Manchurian border, and the threat it represented to the region, Japan launched a surprise naval attack upon the Russian fleet. Initially, Roosevelt supported the Japanese position. However, when the Japanese fleet quickly achieved victory after victory, Roosevelt grew concerned over the growth of Japanese influence in the region and the continued threat that it represented to China and American access to those markets (Figure 22.4.1). Wishing to maintain the aforementioned balance of power, in 1905, Roosevelt arranged for diplomats from both nations to attend a secret peace conference in Portsmouth, New Hampshire. The resultant negotiations secured peace in the region, with Japan gaining control over Korea, several former Russian bases in Manchuria, and the southern half of Sakhalin Island. These negotiations also garnered the Nobel Peace Prize for Roosevelt, the first American to receive the award.
When Japan later exercised its authority over its gains by forcing American business interests out of Manchuria in 1906–1907, Roosevelt felt he needed to invoke his “big stick” foreign policy, even though the distance was great. He did so by sending the U.S. Great White Fleet on maneuvers in the western Pacific Ocean as a show of force from December 1907 through February 1909. Publicly described as a goodwill tour, the message to the Japanese government regarding American interests was equally clear. Subsequent negotiations reinforced the Open Door policy throughout China and the rest of Asia. Roosevelt had, by both the judicious use of the “big stick” and his strategy of maintaining a balance of power, kept U.S. interests in Asia well protected.
Browse the Smithsonian National Portrait Gallery to follow Theodore Roosevelt from Rough Rider to president and beyond.
Section Summary
When Roosevelt succeeded McKinley as president, he implemented a key strategy for building an American empire: the threat, rather than the outright use, of military force. McKinley had engaged the U.S. military in several successful skirmishes and then used the country’s superior industrial power to negotiate beneficial foreign trade agreements. Roosevelt, with his “big stick” policy, was able to keep the United States out of military conflicts by employing the legitimate threat of force. Nonetheless, as negotiations with Japan illustrated, the maintenance of an empire was fraught with complexity. Changing alliances, shifting economic needs, and power politics all meant that the United States would need to tread carefully to maintain its status as a world power.
Review Questions
How did Colombia react to the United States’ proposal to construct a canal through Central America?
They preferred to build such a canal themselves.
They preferred that no canal be built at all.
They agreed to sell land to the United States to build the canal, but in a less advantageous location than the Panamanians.
They felt that Roosevelt’s deal offered too little money.
D
With the Roosevelt Corollary, Roosevelt sought to establish ________.
the consequences for any European nation that involved itself in Latin American affairs
the right of the United States to involve itself in Latin American affairs whenever necessary
the idea that Latin America was free and independent from foreign intervention
the need for further colonization efforts in the Western Hemisphere
B
Compare Roosevelt’s foreign policy in Latin America and Asia. Why did he employ these different methods?
Roosevelt’s strategy of “speaking softly and carrying a big stick” worked well in Latin America, where the United States had a strong military presence and could quickly and easily act on any threat of military action. Roosevelt’s threat of force was therefore credible in that region, and he was able to wield it effectively. In Asia, however, the United States had less of a military presence. Instead, Roosevelt sought to maintain a balance of power, wherein the various Asian countries kept each other in check and no single player grew too powerful. When the power balance tipped, Roosevelt acted to broker a peace deal between Russia and Japan as a means of restoring balance.
Glossary
- Roosevelt Corollary
- a statement by Theodore Roosevelt that the United States would use military force to act as an international police power and correct any chronic wrongdoing by any Latin American nation threatening the stability of the region


10.1.4: Catalog
The catalog page is divided into two parts, the upper part allows searching for specific books, the lower provides links to individual books. If one expands the Advanced Search box
The default search is in the centrally curated Central Bookshelves of the libraries, which presently hold 1,159 books. If one is interested in a custom book on the Campus Bookshelves and makes that selection, today there are 897 books. You can also restrict the search to a specific library, a subject in that library, an author, a campus or a publisher.
Book Front Page
After a book has been located you can click on it to bring up more information on the book
Using the pane on the right one can quickly navigate to any page in the book. The blue box has selections to read the book from the beginning on line, download the complete PDF, buy a print copy through the LibreTexts bookstore, and download the Common Cartridge for integration of chapters and pages into an LMS. One can also submit an adoption report if the book is being used for a class, or a peer review.
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6.1: Areas
Circle
Area of a circle = \(0.785 \times D^2\)
Typically, you will know the diameter of a circle, for example the diameter of a pipe or a storage tank. You may have learned the following formula (\(\pi \times r^{2}\)) for calculating the area of a circle. However, in this text we will only use the previous formula. Both formulas work, but in the water industry diameters are most commonly used.
Much of the time you will be asked to calculate the area of the opening of a pipe. However, the unit given to you will typically be given in inches. There are two ways to go about solving this type of problem.
Example \(\PageIndex{1}\)
What is the area of a 12” diameter pipe?
Solution
If you were to plug 12” into the formula (0.785 \times D^2\) you would end up with square inches and then you would most likely need to then convert to square feet.
So, the easiest way to solve all area problems where “feet” are not given is to convert to feet first and then plug the numbers into the formula.
A 12” diameter pipe equals a 1 foot diameter pipe.
\[\dfrac{12 \not{\text { inches }}}{1} \times \dfrac{1 \text { foot }}{12 \not{\text { inches }}}=1 \text { foot } \nonumber \]
By converting to feet before starting the problem you will avoid getting into units you are not familiar with.
Rectangles
Area of a rectangle = \(L \times W\)
Rectangles are common shapes of storage reservoirs, sedimentation basins, and even room in an office. Calculating the area for these types of structures is simple if you know how the length and width of the structure. Remember, the area of a square is calculated the same way. These types of dimensions are typically given in feet.
Trapezoid
Area of a trapezoid = \(\dfrac{b1+b2}{2} \times H\)
Trapezoids are not as common as circles, rectangles, and squares, but they are still found in the waterworks industry. Typically open channels and aqueducts are shaped like a trapezoid. In calculating the area of a trapezoid, you need to know the width across the aqueduct and the height of the water level. Now the width changes from the bottom of the trapezoid to the top. For example, the distance across the bottom of the trapezoid is a shorter distance then it is at the top of the trapezoid. The other point that needs to be addressed is the distance across the top of the trapezoid that is needed is the distance across the top of the water level. See the diagram below for an example of the base 1.
Once you know the distance across the trapezoid at the water level, then add it to the distance across the bottom of the trapezoid and divide that number by 2. Doing this finds the average distance across the trapezoid. Multiply this number by the height of the water level and you know have the area of the trapezoid.
Exercise 6.1
Calculate the following areas.
- What is the area of a circle with a 2 foot diameter?
- What is the area of a 36” diameter pipe?
- What is the area of a sedimentation basin that is 30 feet wide and 10 feet deep?
- What is the area of an aqueduct that is 5 feet across the bottom, 10 feet across the top and 7 feet deep?
- The roof on an above ground storage tank has a 130 foot diameter. What is the area?
- A filter is 25 feet long and 20 feet wide. What is the area?
- A water aqueduct is 10 feet wide at the bottom and 20 feet wide at the top. If it is 20 feet deep what is the area?


30.4C: Arthroplasty
Arthroplasty is a surgical procedure in which an arthritic or dysfunctional joint surface is replaced or repaired.
- Describe the purpose and process of arthroplasty
Key Points
- Arthroplasty can include removing or remodeling damaged surfaces and replacing them with silicone.
- Prosthetic replacements of hip and knee joints have become common arthroplastic surgeries.
- Wear of the bearing surfaces of components is an issue of joint replacement that can lead to damage to surrounding bone and contribute to eventual failure of the implant.
Key Terms
- arthroplasty : The surgical repair of a joint, or creation of an artificial joint, such as a hip replacement.
EXAMPLES
Hip fractures from falls are common among the elderly. Partial arthroplasty (replacing the hip socket and femur head and neck) may be done to relieve pain and improve mobility. However, as with all surgeries, complications can occur that for the elderly may prevent life-threatening challenges. Special care must be made when deciding to proceed with arthroplasty in the elderly patient.
Arthroplasty (literally “surgical repair of joint”) is an orthopedic surgical procedure in which the arthritic or dysfunctional joint surface is replaced and/or the joint is remodeled or realigned.
Planning Arthroplasty : CAD-assisted planning for a total hip arthroplasty.
Common types of arthroplasty include:
- Interpositional arthroplasty with interposition of skin, muscle, or tendon to keep inflammatory surfaces apart.
- Excisional arthroplasty in which the joint surface and bone are removed, leaving scar tissue to fill in the gap.
Other forms of arthroplasty include resectional arthroplasty, resurfacing arthroplasty, mold arthroplasty, cup arthroplasty, and silicone replacement arthroplasty, among others. Osteotomy to restore or modify joint congruity is also an arthroplasty.
For the last 45 years, the most successful and common form of arthroplasty has been the surgical replacement of degenerated joint surfaces with prostheses. For example, a hip joint that is affected by osteoarthritis may be replaced entirely (total hip arthroplasty) with a prosthetic hip. This would involve replacing both the acetabulum (hip socket) and the head and neck of the femur. The purpose of this procedure is to relieve pain, restore range of motion, and improve walking ability, thus leading to the improvement of muscle strength.
One of the main problems with joint replacements is wear of the bearing surfaces of components. This can lead to damage to surrounding bone and contribute to eventual failure of the implant. Use of alternative bearing surfaces has increased in recent years, particularly in younger patients, in an attempt to improve the wear characteristics of joint replacement components. These include ceramics and all-metal implants (as opposed to the original metal-on-plastic). The plastic (actually ultra high-molecular-weight polyethylene) can also be altered in ways that may improve durability.
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