9.11: Current Issues in Corrections
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Learning Objectives
In this final section on punishment, we review some of the contemporary special issues that are occurring in our American jails and prisons. Our decisions to incarcerate on a large scale as a solution to many issues does have significant impacts. These impacts not only affect the individuals in the justice system, but also community members as a whole. Some of these issues include overcrowding, gangs, aging prisoners, and substance abuse. This section will provide a report on some of the more pervasive issues facing corrections today. By the end of this section, students should be able to:
Critical Thinking Questions


3.6: The Future of Human Movement and Conclusion
The future dynamics of migration are very difficult to predict, but certain geographic realities provide clues for where patterns are likely to change. In those parts of the world where societies are quickly aging and fertility rates continue to decline, we can expect rates of immigration to increase. Certain countries have experienced and will continue to experience dramatic population decline (most notably Germany, Italy, Russia, and Japan), and the demand for young, workingaged immigrants will certainly continue to draw more people to those places. Japan and China represent unique cases because both are aging and yet have been resistant to allowing outsiders to become citizens or permanent residents. In spite of cultural preferences for ethnic homogeneity, it would seem likely that the culture will shift and become more accepting of outsiders, as the country needs them to take care of the elderly, pay taxes, and provide an infusion of energy into the respective countries.
Meanwhile, people living in places without sufficient opportunities will continue to move away in search of jobs and better circumstances, regardless of the attempts made by wealthy nations to keep them out. Source countries and destinations will continue to shift, as they always have. More Mexicans have returned than have left the US in recent years, and it is no longer the leading sending source country of migration to the US. Rather, more people are now coming from Central America. The demographic pressures in many African countries will absolutely drive more working-aged people out of the continent in search of better opportunities even as the journey becomes increasingly dangerous. As the Internet becomes more pervasive on that continent, more people will find the information that they need in order to plan their emigration. Finally, the highly-skilled people of the world will continue to be increasingly mobile and largely unaffected by borders or increased security. Computer programmers, nurses, doctors, engineers, and high-tech workers of all sorts will use the globe to their advantage and seek out places that best fit their desires. Television shows like “House Hunters International” demonstrate how the vast numbers of people who work online or in highly-skilled careers can virtually live anywhere.  The poor and wealthy alike will continue to move about the planet, reconstituting the human geography of our world well into the twenty-first century but for very different reasons and with very different experiences.


7.6: Probability with Permutations and Combinations
- Calculate probabilities with permutations.
- Calculate probabilities with combinations.
In our earlier discussion of theoretical probabilities, the first step we took was to write out the sample space for the experiment in question. For many experiments, that method just isn’t practical. For example, we might want to find the probability of drawing a particular 5-card poker hand. Since there are 52 cards in a deck and the order of cards doesn’t matter, the sample space for this experiment has possible 5-card hands. Even if we had the patience and space to write them all out, sorting through the results to find the outcomes that fall in our event would be just as tedious.
Luckily, the formula for theoretical probabilities doesn’t require us to know every outcome in the sample space; we just need to know how many outcomes there are. In this section, we’ll apply the techniques we learned earlier in the chapter (The Multiplication Rule for Counting, permutations, and combinations) to compute probabilities.
Using Permutations to Compute Probabilities
Recall that we can use permutations to count how many ways there are to put a number of items from a list in order. If we’re looking at an experiment whose sample space looks like an ordered list, then permutations can help us to find the right probabilities.
- In horse racing, an exacta bet is one where the player tries to predict the top two finishers in particular race in order. If there are 9 horses in a race, and a player decided to make an exacta bet at random, what is the probability that they win?
- You are in a club with 10 people, 3 of whom are close friends of yours. If the officers of this club are chosen at random, what is the probability that you are named president and one of your friends is named vice president?
- A bag contains slips of paper with letters written on them as follows: A, A, B, B, B, C, C, D, D, D, D, E. If you draw 3 slips, what is the probability that the letters will spell out (in order) the word BAD?
- Answer
-
- Since order matters for this situation, we’ll use permutations. How many different exacta bets can be made? Since there are 9 horses and we must select 2 in order, we know there are possible outcomes. That’s the size of our sample space, so it will go in the denominator of the probability. Since only one of those outcomes is a winner, the numerator of the probability is 1. So, the probability of randomly selecting the winning exacta bet is .
- There are 10 people in the club, and 2 will be chosen to be officers. Since the order matters, there are different ways to select officers. Next, we must figure out how many outcomes are in our event. We’ll use the Multiplication Rule for Counting to find that number. There is only 1 choice for president in our event, and there are 3 choices for vice president. So, there are outcomes in the event. Thus, the probability that you will serve as president with one of your friends as vice president is .
- There are 12 slips of paper in the bag, and 3 will be drawn. So, there are possible outcomes. Now, we’ll compute the number of outcomes in our event. The first letter drawn must be a B, and there are 3 of those. Next must come an A (2 of those) and then a D (4 of those). Thus, there are outcomes in our event. So, the probability that the letters drawn spell out the word BAD is .
Another bag of letters contains C, C, C, C, D, D, I, I, I, T, T, T, Y, Y, Y, Y. What is the probability that 4 letters chosen at random will spell, in order, CITY?
Combinations to Computer Probabilities
If the sample space of our experiment is one in which order doesn’t matter, then we can use combinations to find the number of outcomes in that sample space.
- Palmetto Cash 5 is a game offered by the South Carolina Education Lottery. Players choose 5 numbers from the whole numbers between 1 and 38 (inclusive); the player wins the jackpot of $100,000 if the randomizer selects those numbers in any order. If you buy one ticket for this game, what is the probability that you win the top prize by choosing all 5 winning numbers?
- There’s a second prize in the Palmetto Cash 5 game that a player wins if 4 of the player's 5 numbers are among the 5 winning numbers. What’s the probability of winning the second prize?
- Scrabble is a word-building board game. Players make hands of 7 letters by selecting tiles with single letters printed on them blindly from a bag (2 tiles have nothing printed on them; these blanks can stand for any letter). Players use the letters in their hands to spell out words on the board. Initially, there are 100 tiles in the bag. Of those, 44 are (or could be) vowels (9 As, 12 Es, 9 Is, 8 Os, 4 Us, and 2 blanks; we’ll treat Y as a consonant). What is the probability that your initial hand has no vowels?
- Answer
-
- There are 38 numbers to choose from, and the order of the 5 we pick doesn’t matter. So, there are outcomes in the sample space. Only one outcome is in our winning event, so the probability of winning is .
- As in part 1 of this example,, there are 501,492 outcomes in the sample space. The tricky part here is figuring out how many outcomes are in our event. To qualify, the outcome must contain 4 of the 5 winning numbers, plus one losing number. There are ways to choose the 4 winning numbers, and there are losing numbers. So, using the Multiplication Rule for Counting, there are outcomes in our event. Thus, the probability of winning the second prize is , which is about 0.00033.
- The number of possible starting hands is . There are consonants in the bag, so the number of all-consonant hands is . Thus, the probability of drawing all consonants is .
At a charity event with 58 people in attendance, 3 raffle winners are chosen. All receive the same prize, so order doesn’t matter. You are attending with 4 of your friends. What is the probability that at least one of you or your friends wins a raffle prize? Hint: Find the probability that none of you wins, and use the formula for complements.
If you draw a hand of 5 cards from a standard deck, what is the probability that 2 cards are \(\spadesuit\) and 3 cards are \(♡\)?
Check Your Understanding
- For the following exercises, you are drawing Scrabble tiles without replacement from a bag containing the letters A, C, E, E, I, N, N, S, S, W.
- What is the probability that you draw (in order) the letters W-I-N?
- What is the probability that you draw (in order) the letters W-I-S-E?
- What is the probability that you draw (in order) the letters S-E-E-N?
- What is the probability that you draw (in any order) the letters W-I-N?
- What is the probability that you draw (in any order) the letters W-I-S-E?
- What is the probability that you draw (in any order) the letters S-E-E-N?


8.1: Elementary and Measurable Functions
From set functions, we now return to point functions
\[
f : S \rightarrow\left(T, \rho^{\prime}\right)
\]
whose domain \(D_{f}\) consists of points of a set \(S .\) The range space \(T\) will mostly be \(E,\) i.e., \(E^{1}, E^{*}, C, E^{n},\) or another normed space. We assume \(f(x)=0\) unless defined otherwise. (In a general metric space \(T,\) we may take some fixed element \(q\) for \(0 .\) ) Thus \(D_{f}\) is all of \(S\), always.
We also adopt a convenient notation for sets:
\[
"A(P)" \text { for } "\{x \in A | P(x)\}."
\]
Thus
\[
\begin{aligned} A(f \neq a) &=\{x \in A | f(x) \neq a\}, \\ A(f=g) &=\{x \in A | f(x)=g(x)\}, \\ A(f>g) &=\{x \in A | f(x)>g(x)\}, \text { etc. } \end{aligned}
\]
A measurable space is a set \(S \neq \emptyset\) together with a set ring \(\mathcal{M}\) of subsets of \(S,\) denoted \((S, \mathcal{M})\).
Henceforth, \((S, \mathcal{M})\) is fixed.
An M-partition of a set \(A\) is a countable set family \(\mathcal{P}=\left\{A_{i}\right\}\) such that
\[
A=\bigcup_{i} A_{i}(d i s j o i n t),
\]
with \(A, A_{i} \in \mathcal{M}\).
We briefly say "the partition \(A=\bigcup A_{i} .\)"
An \(\mathcal{M}\)-partition \(\mathcal{P}^{\prime}=\left\{B_{i k}\right\}\) is a refinement of \(\mathcal{P}=\left\{A_{i}\right\}\left(\text { or } \mathcal{P}^{\prime} \text { refines }\right.\) \(\mathcal{P},\) or \(\mathcal{P}^{\prime}\) is finer than \(\mathcal{P} )\) iff
\[
(\forall i) \quad A_{i}=\bigcup_{k} B_{i k}
\]
i.e., each \(B_{i k}\) is contained in some \(A_{i}\).
The intersection \(\mathcal{P}^{\prime} \cap \mathcal{P}^{\prime \prime}\) of \(\mathcal{P}^{\prime}=\left\{A_{i}\right\}\) and \(\mathcal{P}^{\prime \prime}=\left\{B_{k}\right\}\) is understood to be the family of all sets of the form
\[
A_{i} \cap B_{k}, \quad i, k=1,2, \dots
\]
It is an \(\mathcal{M}\) -partition that refines both \(\mathcal{P}^{\prime}\) and \(\mathcal{P}^{\prime \prime}\).
A map (function) \(f : S \rightarrow T\) is elementary, or \(\mathcal{M}\)-elementary, on a set \(A \in \mathcal{M}\) iff there is an M-partition \(\mathcal{P}=\left\{A_{i}\right\}\) of \(A\) such that \(f\) is constant \(\left(f=a_{i}\right)\) on each \(A_{i} .\)
If \(\mathcal{P}=\left\{A_{1}, \ldots, A_{q}\right\}\) is finite, we say that \(f\) is simple, or \(\mathcal{M}\)-simple, on \(A .\)
If the \(A_{i}\) are intervals in \(E^{n},\) we call \(f\) a step function; it is a simple step function if \(\mathcal{P}\) is finite.
The function values \(a_{i}\) are elements of \(T\) (possibly vectors). They may be infinite if \(T=E^{*} .\) Any simple map is also elementary, of course.
A map \(f : S \rightarrow\left(T, \rho^{\prime}\right)\) is said to be measurable (or \(\mathcal{M}\) -measurable \()\) on a \(\operatorname{set} A\) in \((S, \mathcal{M})\) iff
\[
f=\lim _{m \rightarrow \infty} f_{m} \quad(\text { pointwise }) \text { on } A
\]
for some sequence of functions \(f_{m} : S \rightarrow T,\) all elementary on \(A .\) (See Chapter 4, §12 for "pointwise.")
Note 1. This implies \(A \in \mathcal{M},\) as follows from Definitions 2 and \(3 .(\mathrm{Why} \text { ? })\)
If \(f : S \rightarrow\left(T, \rho^{\prime}\right)\) is elementary on \(A,\) it is measurable on \(A .\)
- Proof
-
Set \(f_{m}=f, m=1,2, \ldots,\) in Definition \(4 .\) Then clearly \(f_{m} \rightarrow f\) on \(A\). \(square\)
If \(f\) is simple, elementary, or measurable on \(A\) in \((S, \mathcal{M}),\) it has the same property on any subset \(B \subseteq A\) with \(B \in \mathcal{M}\).
- Proof
-
Let \(f\) be simple on \(A ;\) so \(f=a_{i}\) on \(A_{i}, i=1,2, \ldots, n,\) for some finite \(\mathcal{M}\) -partition, \(A=\bigcup_{i=1}^{n} A_{i}\).
If \(A \supseteq B \in \mathcal{M},\) then
\[
\left\{B \cap A_{i}\right\}, \quad i=1,2, \ldots, n,
\]is a finite \(\mathcal{M}\) -partition of \(B(\text { why? }),\) and \(f=a_{i}\) on \(B \cap A_{i} ;\) so \(f\) is simple on \(B\).
For elementary maps, use countable partitions.
Now let \(f\) be measurable on \(A,\) i.e.,
\[
f=\lim _{m \rightarrow \infty} f_{m}
\]for some elementary maps \(f_{m}\) on \(A .\) As shown above, the \(f_{m}\) are elementary on \(B,\) too, and \(f_{m} \rightarrow f\) on \(B ;\) so \(f\) is measurable on \(B . \quad \square\)
If \(f\) is elementary or measurable on each of the (countably many \()\) sets \(A_{n}\) in \((S, \mathcal{M}),\) it has the same property on their union \(A=\bigcup_{n} A_{n}\).
- Proof
-
Let \(f\) be elementary on each \(A_{n}\) (so \(A_{n} \in \mathcal{M}\) by Note 1\()\).
By Corollary 1 of Chapter 7, §1,
\[
A=\bigcup A_{n}=\bigcup B_{n}
\]for some disjoint sets \(B_{n} \subseteq A_{n}\left(B_{n} \in \mathcal{M}\right)\).
By Corollary \(2, f\) is elementary on each \(B_{n} ;\) i.e., constant on sets of some \(\mathcal{M}\) -partition \(\left\{B_{n i}\right\}\) of \(B_{i}\).
All \(B_{n i}\) combined (for all \(n\) and all \(i )\) form an \(\mathcal{M}\)-partition of \(A\),
\[
A=\bigcup_{n} B_{n}=\bigcup_{n, i} B_{n i}.
\]As \(f\) is constant on each \(B_{n i},\) it is elementary on \(A .\)
For measurable functions \(f,\) slightly modify the method used in Corollary \(2 . \square\)
If \(f : S \rightarrow\left(T, \rho^{\prime}\right)\) is measurable on \(A\) in \((S, \mathcal{M}),\) so is the composite map \(g \circ f,\) provided \(g : T \rightarrow\left(U, \rho^{\prime \prime}\right)\) is relatively continuous on \(f[A]\).
- Proof
-
By assumption,
\[
f=\lim _{m \rightarrow \infty} f_{m} \text { (pointwise) }
\]for some elementary maps \(f_{m}\) on \(A\).
Hence by the continuity of \(g\),
\[
g\left(f_{m}(x)\right) \rightarrow g(f(x)),
\]i.e., \(g \circ f_{m} \rightarrow g \circ f\) (pointwise) on \(A\).
Moreover, all \(g \circ f_{m}\) are elementary on \(A\) (for \(g \circ f_{m}\) is constant on any partition set, if \(f_{m}\) is).
Thus \(g \circ f\) is measurable on \(A,\) as claimed. \(\square\)
If the maps \(f, g, h : S \rightarrow E^{1}(C)\) are simple, elementary, or measurable on \(A\) in \((S, \mathcal{M}),\) so are \(f \pm g, f h,|f|^{a}\) (for real \(a \neq 0 )\) and \(f / h\) (if \(h \neq 0\) on \(A ) .\)
Similarly for vector-valued \(f\) and \(g\) and scalar-valued \(h\).
- Proof
-
First, let \(f\) and \(g\) be elementary on \(A .\) Then there are two \(\mathcal{M}\)-partitions,
\[
A=\bigcup A_{i}=\bigcup B_{k},
\]such that \(f=a_{i}\) on \(A_{i}\) and \(g=b_{k}\) on \(B_{k},\) say.
The sets \(A_{i} \cap B_{k}\) (for all \(i\) and \(k )\) then form a new \(\mathcal{M}\) -partition of \(A(\text { why? })\), such that both \(f\) and \(g\) are constant on each \(A_{i} \cap B_{k}(\text { why?); hence so is } f \pm g\).
Thus \(f \pm g\) is elementary on \(A .\) Similarly for simple functions.
Next, let \(f\) and \(g\) be measurable on \(A ;\) so
\[
f=\lim f_{m} \text { and } g=\lim g_{m} \text { (pointwise) on } A
\]for some elementary maps \(f_{m}, g_{m}\).
By what was shown above, \(f_{m} \pm g_{m}\) is elementary for each \(m .\)
The rest of the theorem follows quite similarly. \(\square\)
If the range space is \(E^{n}\left(\text { or } C^{n}\right),\) then \(f\) has \(n\) real (complex) components \(f_{1}, \ldots, f_{n},\) as in Chapter 4,§3 (Part II). This yields the following theorem.
A function \(f : S \rightarrow E^{n}\left(C^{n}\right)\) is simple, elementary, or measurable on a set \(A\) in \((S, \mathcal{M})\) iff all its \(n\) component functions \(f_{1}, f_{2}, \ldots, f_{n}\) are.
- Proof
-
For simplicity, consider \(f : S \rightarrow E^{2}, f=\left(f_{1}, f_{2}\right)\).
If \(f_{1}\) and \(f_{2}\) are simple or elementary on \(A\) then (exactly as in Theorem 1\()\), one can achieve that both are constant on sets \(A_{i} \cap B_{k}\) of one and the same \(\mathcal{M}\)-partition of \(A .\) Hence \(f=\left(f_{1}, f_{2}\right),\) too, is constant on each \(A_{i} \cap B_{k},\) as required.
Conversely, let
\[
f=\overline{c}_{i}=\left(a_{i}, b_{i}\right) \text { on } C_{i}
\]for some \(\mathcal{M}\)-partition
\[
A=\bigcup C_{i}.
\]Then by definition, \(f_{1}=a_{i}\) and \(f_{2}=b_{i}\) on \(C_{i} ;\) so both are elementary (or simple) on \(A .\)
In the general case \(\left(E^{n} \text { or } C^{n}\right),\) the proof is analogous.
For measurable functions, the proof reduces to limits of elementary maps (using Theorem 2 of Chapter 3, §15). The details are left to the reader. \(\square\)
Note 2. As \(C=E^{2},\) a complex function \(f : S \rightarrow C\) is simple, elementary, or measurable on \(A\) iff its real and imaginary parts are.
By Definition \(4,\) a measurable function is a pointwise limit of elementary maps. However, if \(\mathcal{M}\) is a \(\sigma\)-ring, one can make the limit uniform. Indeed, we have the following theorem.
If \(\mathcal{M}\) is a \(\sigma\)-ring, and \(f : S \rightarrow\left(T, \rho^{\prime}\right)\) is \(\mathcal{M}\)-measurable on \(A,\) then
\[
f=\lim _{m \rightarrow \infty} g_{m} \text { (uniformly) on } A
\]
for some finite elementary maps \(g_{m}\).
- Proof
-
Thus given \(\varepsilon>0,\) there is a finite elementary map \(g\) such that \(\rho^{\prime}(f, g)<\varepsilon\)
on \(A\).
If \(\mathcal{M}\) is a \(\sigma\)-ring in \(S,\) if
\[
f_{m} \rightarrow f(\text {pointwise}) \text { on } A
\]
\(\left(f_{m} : S \rightarrow\left(T, \rho^{\prime}\right)\right),\) and if all \(f_{m}\) are \(\mathcal{M}\) -measurable on \(A,\) so also is \(f\).
Briefly: \(A\) pointwise limit of measurable maps is measurable (unlike continuous maps; cf. Chapter 4, §12).
- Proof
-
By the second clause of Theorem \(3,\) each \(f_{m}\) is uniformly approximated by some elementary map \(g_{m}\) on \(A,\) so that, taking \(\varepsilon=1 / m, m=1,2, \ldots\),
\[
\rho^{\prime}\left(f_{m}(x), g_{m}(x)\right)<\frac{1}{m} \quad \text { for all } x \in A \text { and all } m.
\]Fixing such a \(g_{m}\) for each \(m,\) we show that \(g_{m} \rightarrow f (\text { pointwise })\) on \(A,\) as required in Definition \(4 .\)
Indeed, fix any \(x \in A .\) By assumption, \(f_{m}(x) \rightarrow f(x) .\) Hence, given \(\delta>0\),
\[
(\exists k)(\forall m>k) \quad \rho^{\prime}\left(f(x), f_{m}(x)\right)<\delta.
\]Take \(k\) so large that, in addition,
\[
(\forall m>k) \quad \frac{1}{m}<\delta.
\]Then by the triangle law and by \((1),\) we obtain for \(m>k\) that
\[
\begin{aligned} \rho^{\prime}\left(f(x), g_{m}(x)\right) & \leq \rho^{\prime}\left(f(x), f_{m}(x)\right)+\rho^{\prime}\left(f_{m}(x), g_{m}(x)\right) \\ &<\delta+\frac{1}{m}<2 \delta \end{aligned}.
\]As \(\delta\) is arbitrary, this implies \(\rho^{\prime}\left(f(x), g_{m}(x)\right) \rightarrow 0,\) i.e., \(g_{m}(x) \rightarrow f(x)\) for any (fixed) \(x \in A,\) thus proving the measurability of \(f . \quad \square\)
Note 3. If
\[
\mathcal{M}=\mathcal{B} (=\text { Borel field in } S),
\]
we often say "Borel measurable" for \(\mathcal{M}\)-measurable. If
\[
\mathcal{M}=\left\{\text { Lebesgue measurable sets in } E^{n}\right\},
\]
we say "Lebesgue (L) measurable" instead. Similarly for "Lebesgue-Stieltjes (LS) measurable."


2.2: Graphs of the Secant and Cosecant Functions
Learning Objectives
- Analyze the graphs of \(y=\sec x\) and \(y=\csc x\).
- Graph variations of \(y=\sec x\) and \(y=\csc x\).
Analyzing the Graphs of \(y = \sec x\) and \(y = \csc x\)
The secant function was defined by the reciprocal identity \(sec \, x=\dfrac{1}{\cos x}\). Notice that the function is undefined when the cosine is \(0\), leading to vertical asymptotes at \(\dfrac{\pi}{2}\), \(\dfrac{3\pi}{2}\) etc. Because the cosine is never more than \(1\) in absolute value, the secant, being the reciprocal, will never be less than \(1\) in absolute value.
We can graph \(y=\sec x\) by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure \(\PageIndex{1}\). The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.
The secant graph has vertical asymptotes at each value of \(x\) where the cosine graph crosses the \(x\)-axis - this is because the inverse of 0 is undefined. We show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.
Note that, because cosine is an even function, secant is also an even function. That is, \(\sec(−x)=\sec x\).
Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant \(A\).
FEATURES OF THE GRAPH OF \(Y = A \sec(Bx)\)
- The stretching factor is \(| A |\).
- The period is \(\dfrac{2\pi}{| B |}\).
- The domain is \(x≠\dfrac{\pi}{2| B |}k\), where \(k\) is an odd integer.
- The range is \((−∞,−|A|]∪[|A|,∞)\).
- The vertical asymptotes occur at \(x=\dfrac{\pi}{2| B |}k\), where \(k\) is an odd integer.
- There is no amplitude.
- \(y=A\sec(Bx)\) is an even function because cosine is an even function.
Similar to the secant, the cosecant is defined by the reciprocal identity \(\csc x=\dfrac{1}{\sin x}\). Notice that the function is undefined when the sine is \(0\), leading to a vertical asymptote in the graph at \(0\), \(\pi\), etc. Since the sine is never more than \(1\) in absolute value, the cosecant, being the reciprocal, will never be less than \(1\) in absolute value.
We can graph \(y=\csc x\) by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure \(\PageIndex{2}\). The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.
The cosecant graph has vertical asymptotes at each value of \(x\) where the sine graph crosses the \(x\)-axis; we show these in the graph below with dashed vertical lines.
Note that, since sine is an odd function, the cosecant function is also an odd function. That is, \(\csc(−x)=−\csc x\).
The graph of cosecant, which is shown in Figure \(\PageIndex{2}\), is similar to the graph of secant.
FEATURES OF THE GRAPH OF \(Y = A \csc(Bx)\)
- The stretching factor is \(| A |\).
- The period is \(\dfrac{2\pi}{|B|}\).
- The domain is \(x≠\dfrac{\pi}{|B|}k\), where \(k\) is an integer.
- The range is \((−∞,−|A|]∪[|A|,∞)\).
- The asymptotes occur at \(x=\dfrac{\pi}{| B |}k\), where \(k\) is an integer.
- \(y=A\csc(Bx)\) is an odd function because sine is an odd function.
Graphing Variations of \(y = \sec x\) and \(y= \csc x\)
For shifted, compressed, and/or stretched versions of the secant and cosecant functions, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the cosecant function in the same way as for the secant and other functions.The equations become the following.
\[y=A\sec(Bx−C)+D \nonumber\]
\[y=A\csc(Bx−C)+D \nonumber\]
FEATURES OF THE GRAPH OF \(Y = A\sec(Bx−C)+D\)
- The stretching factor is \(|A|\).
- The period is \(\dfrac{2\pi}{|B|}\).
- The domain is \(x≠\dfrac{C}{B}+\dfrac{\pi}{2| B |}k\),where \(k\) is an odd integer.
- The range is \((−∞,−|A|]∪[|A|,∞)\).
- The vertical asymptotes occur at \(x=\dfrac{C}{B}+\dfrac{π}{2| B |}k\),where \(k\) is an odd integer.
- There is no amplitude.
- \(y=A\sec(Bx)\) is an even function because cosine is an even function.
FEATURES OF THE GRAPH OF \(Y = A\csc(Bx−C)+D\)
- The stretching factor is \(|A|\).
- The period is \(\dfrac{2\pi}{|B|}\).
- The domain is \(x≠\dfrac{C}{B}+\dfrac{\pi}{2| B |}k\),where \(k\) is an integer.
- The range is \((−∞,−|A|]∪[|A|,∞)\).
- The vertical asymptotes occur at \(x=\dfrac{C}{B}+\dfrac{\pi}{|B|}k\),where \(k\) is an integer.
- There is no amplitude.
- \(y=A\csc(Bx)\) is an odd function because sine is an odd function.
HOWTO: Given a function of the form \(y=A\sec(Bx)\), graph one period
- Sketch the function \(y=A\cos(Bx)\).
- Draw vertical asymptotes where the curve crosses the midline, which is the \(x\)-axis
- Fill in the secant curve in between the asymptotes. Where the cosine curve has a maximum, the secant curve will have an upward U. Where the cosine curve has a minimum, the secant curve will have a downward U.
Example \(\PageIndex{1}\): Graphing a Variation of the Secant Function
Graph one period of \(f(x)=2.5\sec(0.4x)\).
Solution
- Step 1. Sketch a graph of the function \(f(x)=2.5\cos(0.4x)\). \(A=2.5\) so the stretching factor is \(2.5\). \(B=0.4\) so \(P=\dfrac{2\pi}{0.4}=5\pi\). The period is \(5\pi\) units. Dividing this by 4 gives \(\dfrac{5\pi}{4}\). Each time we travel a distance of \(\dfrac{5\pi}{4}\) we will be at a maximum, on the midline, or at a minimum.
- Step 2. Draw vertical asymptotes where the curve crosses the midline, which is the \(x\)-axis
- Step 3. Fill in the secant curve in between the asymptotes. Where the cosine curve has a maximum, the secant curve will have an upward U. Where the cosine curve has a minimum, the secant curve will have a downward U. Figure \(\PageIndex{3}\) shows the graph. The green dashed curve is the cosine function, which acts at the "skeleton" for the secant curve. The blue dashed lines are the vertical asymptotes. The red curves are the acutal secant function.
Exercise \(\PageIndex{1}\)
Graph one period of \(f(x)=−2.5\sec(0.4x)\).
- Answer
-
This is a vertical reflection of the preceding graph because \(A\) is negative.
Q&A: Do the vertical shift and stretch/compression affect the secant’s range?
Yes. The range of \(f(x)=A\sec(Bx−C)+D\) is \((−∞,−|A|+D]∪[|A|+D,∞)\).
Given a function of the form \(f(x)=A\sec(Bx−C)+D\), graph one period.
- Draw the graph of \(f(x)=A\cos(Bx−C)+D\)
- Sketch the vertical asymptotes, which occur where the cosine curve passes through its midline at \(y=D\)
- Fill in the secant curve in between the asymptotes. Where the cosine curve has a maximum, the secant curve will have an upward U. Where the cosine curve has a minimum, the secant curve will have a downward U.
Example \(\PageIndex{2}\): Graphing a Variation of the Secant Function
Graph one period of \(y=\sec \left (2x−\dfrac{\pi}{2} \right )+3\).
Solution
- Step 1. Sketch the curve \(y=\cos \left (2x−\dfrac{\pi}{2} \right )+3\).The stretching/compressing factor is \(| A |=1\) so the amplitude is 1. \(B=2\) so the period is \(\dfrac{2\pi}{2}=\pi\). Dividing the period by 4 gives \(\dfrac{\pi}{4}\). Each time we travel this distance we will be at a maximum, on the midline, or a minimum. To find the phase shift, we solve the equation \(2x−\dfrac{\pi}{2}=0\), which has the solution \(x=\dfrac{\pi}{4}\). There is a phase shift \(\dfrac{\pi}{4}\) to the right. This means we can "start" the cosine curve at a maximum located at \(\left(\dfrac{\pi}{4},4\right)\). The midline is the horizontal line \(y=3\).
- Step 2. Sketch the vertical asymptotes, which occur where the cosine curve passes through its midline \(y=3\).
- Step 3. Fill in the secant curve in between the asymptotes. Where the cosine curve has a maximum, the secant curve will have an upward U. Where the cosine curve has a minimum, the secant curve will have a downward U. Figure \(\PageIndex{5}\) shows the graph. The green curve is the cosine function and the vertical blue lines are the vertical asymptotes. The curve \(y=\sec \left (2x−\dfrac{\pi}{2} \right )+3\) is in red.
Exercise \(\PageIndex{2}\)
Graph one period of \(f(x)=−6\sec(4x+2)−8\).
- Answer
Q&A: The domain of \(\csc \, x\) was given to be all \(x\) such that \(x≠k\pi\) for any integer \(k\). Would the domain of \(y=A\csc(Bx−C)+D\) be \(x≠\dfrac{C+k\pi}{B}\)?
Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function’s input.
Given a function of the form \(y=A\csc(Bx)\), graph one period.
- Sketch the function \(y=A\sin(Bx)\).
- Draw vertical asymptotes where the curve crosses the midline, which is the \(x\)-axis
- Fill in the cosecant curve in between the asymptotes. Where the sine curve has a maximum, the cosecant curve will have an upward U. Where the sine curve has a minimum, the cosecant curve will have a downward U.
Example \(\PageIndex{3}\): Graphing a Variation of the Cosecant Function
Graph one period of \(f(x)=−3\csc(4x)\).
Solution
- Step 1. Sketch a graph of the function \(f(x)=−3\sin(4x)\). \(| A |=| −3 |=3\),so the amplitude is 3. \(B=4\),so \(P=\dfrac{2\pi}{4}=\dfrac{\pi}{2}\). The period is \(\dfrac{\pi}{2}\) units. Dividing the period by 4 gives \(\dfrac{\pi}{8}\). Each time we travel a distance of \(\dfrac{\pi}{8}\) the sine curve will have a maximum, be on the midline, or have a minimum. The value of \(C\) is 0 so there is no phase shift, we can "start" the curve on the midline at \(x=0\). The value of \(D\) is 0 so there is no vertical shift - the midline is the \(x\)-axis.
- Step 2. Draw vertical asymptotes where the curve crosses the midline, which is the \(x\)-axis.
- Step 3. Fill in the cosecant curve in between the asymptotes. Where the sine curve has a maximum, the cosecant curve will have an upward U. Where the sine curve has a minimum, the cosecant curve will have a downward U. Figure \(\PageIndex{7}\) shows the graph over one period.
Exercise \(\PageIndex{3}\)
Graph \(f(x)=0.5\csc(2x)\) over at least two periods.
- Answer
Given a function of the form \(f(x)=A \csc(Bx−C)+D\), graph one period
- Draw the graph of \(f(x)=A\sin(Bx−C)+D\)
- Sketch the vertical asymptotes, which occur where the sine curve passes through its midline at \(y=D\)
- Fill in the cosecant curve in between the asymptotes. Where the sine curve has a maximum, the cosecant curve will have an upward U. Where the sine curve has a minimum, the cosecant curve will have a downward U.
Example \(\PageIndex{4}\): Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant
Sketch a graph of \(y=2\csc \left (\dfrac{\pi}{2}x \right )+1\).
Solution
- Step 1. Sketch a graph of \(y=2\sin \left (\dfrac{\pi}{2}x \right )+1\). \(| A |=2\) so the amplitude is 2. The period is \(\dfrac{2\pi}{| B |}=\dfrac{2\pi}{\dfrac{\pi}{2}}=2\pi⋅\dfrac{2}{\pi}=4\). Dividing the period by 4 gives 1. Each time we move a unit of 1 the sine curve will have a maximum, be on the midline, or have a minimum. To find the phase shift we solve the equation \(\dfrac{\pi}{2}x=0\) which has a solution \(x=0\). We can "start" the sine curve at \(x=0\) on the midline, which is at \(y=1\).
- Step 2. Sketch the vertical asymptotes, which occur where the sine curve passes through its midline at \(y=1\).
- Step 3. Fill in the cosecant curve in between the asymptotes. Where the sine curve has a maximum, the cosecant curve will have an upward U. Where the sine curve has a minimum, the cosecant curve will have a downward U.
The graph for this function is shown in Figure \(\PageIndex{9}\) in dark blue. The orange dashed line is the sine curve and the dashed vertical blue and green lines are the vertical asymptotes.
Analysis
The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of \(f(x)=2\sin \left (\frac{\pi}{2}x \right )+1\),shown as the orange dashed wave.
Exercise \(\PageIndex{4}\)
Given the graph of \(f(x)=2\cos \left (\frac{\pi}{2}x \right )+1\) shown in Figure \(\PageIndex{10}\), sketch the graph of \(g(x)=2\sec \left (\dfrac{\pi}{2}x \right )+1\) on the same axes.
- Answer
Key Equations
| Shifted, compressed, and/or stretched secant function | \(y=A \sec(Bx−C)+D\) |
| Shifted, compressed, and/or stretched cosecant function | \(y=A \csc(Bx−C)+D\) |
Key Concepts
- The secant and cosecant are both periodic functions with a period of \(2\pi\).
- \(f( x )=A\sec( Bx−C )+D\) gives a shifted, compressed, and/or stretched secant function graph. See Example \(\PageIndex{4}\) and Example \(\PageIndex{5}\).
- \(f( x )=A\csc( Bx−C )+D\) gives a shifted, compressed, and/or stretched cosecant function graph. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).
Contributors and Attributions
-
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


6.1: Arithmetic Operators
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Overview
The basic arithmetic operations are addition, subtraction, multiplication, and division. Arithmetic is performed according to an order of operations. [1]
Discussion
An operator performs an action on one or more operands. The common arithmetic operators are:
| Action | Common Symbol |
| Addition |
+
|
| Subtraction |
-
|
| Multiplication |
*
|
| Division |
/
|
| Modulus (associated with integers) |
%
|
These arithmetic operators are binary that is they have two operands. The operands may be either constants or variables.
age + 1
This expression consists of one operator (addition) which has two operands. The first is represented by a variable named age and the second is a literal constant. If age had a value of 14 then the expression would evaluate (or be equal to) 15.
These operators work as you have learned them throughout your life with the exception of division and modulus. We normally think of division as resulting in an answer that might have a fractional part (a floating-point data type). However, division, when both operands are of the integer data type, may act differently. Please refer to the next section on “Integer Division and Modulus”.
Arithmetic Assignment Operators
Many programming languages support a combination of the assignment (
=
) and arithmetic operators (
+
,
-
,
*
,
/
,
%
). Various textbooks call them “compound assignment operators” or “combined assignment operators”. Their usage can be explained in terms of the assignment operator and the arithmetic operators. In the table, we will use the variable age and you can assume that it is of integer data type.
| Arithmetic assignment examples: | Equivalent code: |
|---|---|
age += 14;
|
age = age + 14;
|
age -= 14;
|
age = age - 14;
|
age *= 14;
|
age = age * 14;
|
age /= 14;
|
age = age / 14;
|
age %= 14;
|
age = age % 14;
|
Pseudocode
Function Main ... This program demonstrates arithmetic operations. Declare Integer a Declare Integer b Assign a = 3 Assign b = 2 Output "a = " & a Output "b = " & b Output "a + b = " & a + b Output "a - b = " & a - b Output "a * b = " & a * b Output "a / b = " & a / b Output "a % b = " & a % b End
Output
a = 3 b = 2 a + b = 5 a - b = 1 a * b = 6 a / b = 1.5 a % b = 1
Flowchart
References
- cnx.org: Programming Fundamentals – A Modular Structured Approach using C++
- Flowgorithm – Flowchart Programming Language
-
Wikipedia: Arithmetic operators
↵


29.E: Special Relativity (Exercise)
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Conceptual Questions
29.1: Quantization of Energy
1. Give an example of a physical entity that is quantized. State specifically what the entity is and what the limits are on its values.
2. Give an example of a physical entity that is not quantized, in that it is continuous and may have a continuous range of values.
3. What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in its atoms and molecules?
4. If Planck’s constant were large, say \(\displaystyle 10^{34}\) times greater than it is, we would observe macroscopic entities to be quantized. Describe the motions of a child’s swing under such circumstances.
5. Why don’t we notice quantization in everyday events?
29.2: The Photoelectric Effect
6. Is visible light the only type of EM radiation that can cause the photoelectric effect?
7. Which aspects of the photoelectric effect cannot be explained without photons? Which can be explained without photons? Are the latter inconsistent with the existence of photons?
8. Is the photoelectric effect a direct consequence of the wave character of EM radiation or of the particle character of EM radiation? Explain briefly.
9. Insulators (nonmetals) have a higher BE than metals, and it is more difficult for photons to eject electrons from insulators. Discuss how this relates to the free charges in metals that make them good conductors.
10. If you pick up and shake a piece of metal that has electrons in it free to move as a current, no electrons fall out. Yet if you heat the metal, electrons can be boiled off. Explain both of these facts as they relate to the amount and distribution of energy involved with shaking the object as compared with heating it.
29.3 Photon Energies and the Electromagnetic Spectrum
11. Why are UV, x rays, and \(\displaystyle γ\) rays called ionizing radiation?
12. How can treating food with ionizing radiation help keep it from spoiling? UV is not very penetrating. What else could be used?
13. Some television tubes are CRTs. They use an approximately 30-kV accelerating potential to send electrons to the screen, where the electrons stimulate phosphors to emit the light that forms the pictures we watch. Would you expect x rays also to be created?
14. Tanning salons use “safe” UV with a longer wavelength than some of the UV in sunlight. This “safe” UV has enough photon energy to trigger the tanning mechanism. Is it likely to be able to cause cell damage and induce cancer with prolonged exposure?
15. Your pupils dilate when visible light intensity is reduced. Does wearing sunglasses that lack UV blockers increase or decrease the UV hazard to your eyes? Explain.
16. One could feel heat transfer in the form of infrared radiation from a large nuclear bomb detonated in the atmosphere 75 km from you. However, none of the profusely emitted x rays or γ rays reaches you. Explain.
17. Can a single microwave photon cause cell damage? Explain.
18. In an x-ray tube, the maximum photon energy is given by \(\displaystyle hf=qV\). Would it be technically more correct to say \(\displaystyle hf=qV+BE\), where BE is the binding energy of electrons in the target anode? Why isn’t the energy stated the latter way?
29.4: Photon Momentum
19. Which formula may be used for the momentum of all particles, with or without mass?
20. Is there any measurable difference between the momentum of a photon and the momentum of matter?
21. Why don’t we feel the momentum of sunlight when we are on the beach?
29.6: The Wave Nature of Matter
22. How does the interference of water waves differ from the interference of electrons? How are they analogous?
23. Describe one type of evidence for the wave nature of matter.
24. Describe one type of evidence for the particle nature of EM radiation.
29.7: Probability and The Heisenberg Uncertainty Principle
25. What is the Heisenberg uncertainty principle? Does it place limits on what can be known?
29.8: The Particle-Wave Duality Reviewed
26. In what ways are matter and energy related that were not known before the development of relativity and quantum mechanics?
Problems & Exercises
29.1: Quantization of Energy
27. A LiBr molecule oscillates with a frequency of \(\displaystyle 1.7×10^{13}Hz\).
(a) What is the difference in energy in eV between allowed oscillator states?
(b) What is the approximate value of \(\displaystyle n\) for a state having an energy of 1.0 eV?
Solution
(a) 0.070 eV
(b) 14
28. The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?
29. A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized.
(a) What is the difference in energy in joules between allowed oscillator states?
(b) What is the value of n for a state where the energy is 5.00 J?
(c) Can the quantization be observed?
Solution
(a) \(\displaystyle 2.21×10^{34}J\)
(b) \(\displaystyle 2.26×10^{34}\)
(c) No
29.2: The Photoelectric Effect
30. What is the longest-wavelength EM radiation that can eject a photoelectron from silver, given that the binding energy is 4.73 eV? Is this in the visible range?
Solution
263 nm
31. Find the longest-wavelength photon that can eject an electron from potassium, given that the binding energy is 2.24 eV. Is this visible EM radiation?
32. What is the binding energy in eV of electrons in magnesium, if the longest-wavelength photon that can eject electrons is 337 nm?
Solution
3.69 eV
33. Calculate the binding energy in eV of electrons in aluminum, if the longest-wavelength photon that can eject them is 304 nm.
34. What is the maximum kinetic energy in eV of electrons ejected from sodium metal by 450-nm EM radiation, given that the binding energy is 2.28 eV?
Solution
0.483 eV
35. UV radiation having a wavelength of 120 nm falls on gold metal, to which electrons are bound by 4.82 eV. What is the maximum kinetic energy of the ejected photoelectrons?
36. Violet light of wavelength 400 nm ejects electrons with a maximum kinetic energy of 0.860 eV from sodium metal. What is the binding energy of electrons to sodium metal?
Solution
2.25 eV
37. UV radiation having a 300-nm wavelength falls on uranium metal, ejecting 0.500-eV electrons. What is the binding energy of electrons to uranium metal?
38. What is the wavelength of EM radiation that ejects 2.00-eV electrons from calcium metal, given that the binding energy is 2.71 eV? What type of EM radiation is this?
Solution
(a) 264 nm
(b) Ultraviolet
39. Find the wavelength of photons that eject 0.100-eV electrons from potassium, given that the binding energy is 2.24 eV. Are these photons visible?
40. What is the maximum velocity of electrons ejected from a material by 80-nm photons, if they are bound to the material by 4.73 eV?
Solution
\(\displaystyle 1.95×10^6m/s\)
41. Photoelectrons from a material with a binding energy of 2.71 eV are ejected by 420-nm photons. Once ejected, how long does it take these electrons to travel 2.50 cm to a detection device?
42. A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal.
(a) How many electrons per second are ejected?
(b) What power is carried away by the electrons, given that the binding energy is 2.71 eV?
Solution
(a) \(\displaystyle 4.02×10^{15}/s\)
(b) 0.256 mW
43. (a) Calculate the number of photoelectrons per second ejected from a \(\displaystyle 1.00-mm^2\) area of sodium metal by 500-nm EM radiation having an intensity of \(\displaystyle 1.30 kW/m^2\) (the intensity of sunlight above the Earth’s atmosphere).
(b) Given that the binding energy is 2.28 eV, what power is carried away by the electrons?
(c) The electrons carry away less power than brought in by the photons. Where does the other power go? How can it be recovered?
44. Unreasonable Results
Red light having a wavelength of 700 nm is projected onto magnesium metal to which electrons are bound by 3.68 eV.
(a) Use \(\displaystyle KE_e=hf–BE\) to calculate the kinetic energy of the ejected electrons.
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(\displaystyle –1.90 eV\)
(b) Negative kinetic energy
(c) That the electrons would be knocked free.
45. Unreasonable Results
(a) What is the binding energy of electrons to a material from which 4.00-eV electrons are ejected by 400-nm EM radiation?
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
29.3 Photon Energies and the Electromagnetic Spectrum
46. What is the energy in joules and eV of a photon in a radio wave from an AM station that has a 1530-kHz broadcast frequency?
Solution
\(\displaystyle 6.34×10^{−9}eV, 1.01×10^{−27}J\)
47. (a) Find the energy in joules and eV of photons in radio waves from an FM station that has a 90.0-MHz broadcast frequency.
(b) What does this imply about the number of photons per second that the radio station must broadcast?
48. Calculate the frequency in hertz of a 1.00-MeV γ-ray photon.
Solution
\(\displaystyle 2.42×10^{20}Hz\)
49. (a) What is the wavelength of a 1.00-eV photon?
(b) Find its frequency in hertz.
(c) Identify the type of EM radiation.
50. Do the unit conversions necessary to show that \(\displaystyle hc=1240 eV⋅nm\), as stated in the text.
Solution
\(\displaystyle hc=(6.62607×10^{−34}J⋅s)(2.99792×10^8m/s)(\frac{10^9nm}{1 m})(\frac{1.00000 eV}{1.60218×10^{−19}J})=1239.84 eV⋅nm≈1240 eV⋅nm\)
51. Confirm the statement in the text that the range of photon energies for visible light is 1.63 to 3.26 eV, given that the range of visible wavelengths is 380 to 760 nm.
52. (a) Calculate the energy in eV of an IR photon of frequency \(\displaystyle 2.00×10^{13}Hz\).
(b) How many of these photons would need to be absorbed simultaneously by a tightly bound molecule to break it apart?
(c) What is the energy in eV of a γ ray of frequency \(\displaystyle 3.00×10^{20}Hz\)?
(d) How many tightly bound molecules could a single such γ ray break apart?
Solution
(a) 0.0829 eV
(b) 121
(c) 1.24 MeV
(d) \(\displaystyle 1.24×10^5\)
53. Prove that, to three-digit accuracy, \(\displaystyle h=4.14×10^{−15}eV⋅s\), as stated in the text.
54. (a) What is the maximum energy in eV of photons produced in a CRT using a 25.0-kV accelerating potential, such as a color TV?
(b) What is their frequency?
Solution
(a) \(\displaystyle 25.0×10^3eV\)
(b) \(\displaystyle 6.04×10^{18}Hz\)
55. What is the accelerating voltage of an x-ray tube that produces x rays with a shortest wavelength of 0.0103 nm?
56. (a) What is the ratio of power outputs by two microwave ovens having frequencies of 950 and 2560 MHz, if they emit the same number of photons per second?
(b) What is the ratio of photons per second if they have the same power output?
Solution
(a) 2.69
(b) 0.371
57. How many photons per second are emitted by the antenna of a microwave oven, if its power output is 1.00 kW at a frequency of 2560 MHz?
58. Some satellites use nuclear power.
(a) If such a satellite emits a 1.00-W flux of \(\displaystyle γ\) rays having an average energy of 0.500 MeV, how many are emitted per second?
(b) These \(\displaystyle γ\) rays affect other satellites. How far away must another satellite be to only receive one γ ray per second per square meter?
Solution
(a) \(\displaystyle 1.25×10^{13}photons/s\)
(b) 997 km
59. (a) If the power output of a 650-kHz radio station is 50.0 kW, how many photons per second are produced?
(b) If the radio waves are broadcast uniformly in all directions, find the number of photons per second per square meter at a distance of 100 km. Assume no reflection from the ground or absorption by the air.
60. How many x-ray photons per second are created by an x-ray tube that produces a flux of x rays having a power of 1.00 W? Assume the average energy per photon is 75.0 keV.
Solution
\(\displaystyle 8.33×10^{13}photons/s\)
61. (a) How far away must you be from a 650-kHz radio station with power 50.0 kW for there to be only one photon per second per square meter? Assume no reflections or absorption, as if you were in deep outer space.
(b) Discuss the implications for detecting intelligent life in other solar systems by detecting their radio broadcasts.
62. Assuming that 10.0% of a 100-W light bulb’s energy output is in the visible range (typical for incandescent bulbs) with an average wavelength of 580 nm, and that the photons spread out uniformly and are not absorbed by the atmosphere, how far away would you be if 500 photons per second enter the 3.00-mm diameter pupil of your eye? (This number easily stimulates the retina.)
Solution
181 km
63. Construct Your Own Problem
Consider a laser pen. Construct a problem in which you calculate the number of photons per second emitted by the pen. Among the things to be considered are the laser pen’s wavelength and power output. Your instructor may also wish for you to determine the minimum diffraction spreading in the beam and the number of photons per square centimeter the pen can project at some large distance. In this latter case, you will also need to consider the output size of the laser beam, the distance to the object being illuminated, and any absorption or scattering along the way.
29.4: Photon Momentum
64. (a) Find the momentum of a 4.00-cm-wavelength microwave photon.
(b) Discuss why you expect the answer to (a) to be very small.
Solution
(a) \(\displaystyle 1.66×10^{−32}kg⋅m/s\)
(b) The wavelength of microwave photons is large, so the momentum they carry is very small.
65. (a) What is the momentum of a 0.0100-nm-wavelength photon that could detect details of an atom?
(b) What is its energy in MeV?
66. (a) What is the wavelength of a photon that has a momentum of \(\displaystyle 5.00×10^{−29}kg⋅m/s\)?
(b) Find its energy in eV.
Solution
(a) 13.3 μm
(b) \(\displaystyle 9.38×10^{-2} eV\)
67. (a) A γ-ray photon has a momentum of \(\displaystyle 8.00×10^{−21}kg⋅m/s\). What is its wavelength?
(b) Calculate its energy in MeV.
68. (a) Calculate the momentum of a photon having a wavelength of \(\displaystyle 2.50 μm\).
(b) Find the velocity of an electron having the same momentum. (c) What is the kinetic energy of the electron, and how does it compare with that of the photon?
Solution
(a) \(\displaystyle 2.65×10^{−28}kg⋅m/s\)
(b) 291 m/s
(c) electron \(\displaystyle 3.86×10^{−26}J\), photon \(\displaystyle 7.96×10^{−20}J\), ratio \(\displaystyle 2.06×10^6\)
69. Repeat the previous problem for a 10.0-nm-wavelength photon.
70. (a) Calculate the wavelength of a photon that has the same momentum as a proton moving at 1.00% of the speed of light.
(b) What is the energy of the photon in MeV?
(c) What is the kinetic energy of the proton in MeV?
Solution
(a) \(\displaystyle 1.32×10^{−13}m\)
(b) 9.39 MeV
(c) \(\displaystyle 4.70×10^{−2}MeV\)
71. (a) Find the momentum of a 100-keV x-ray photon.
(b) Find the equivalent velocity of a neutron with the same momentum.
(c) What is the neutron’s kinetic energy in keV?
72. Take the ratio of relativistic rest energy, \(\displaystyle E=γmc^2\), to relativistic momentum, \(\displaystyle p=γmu\), and show that in the limit that mass approaches zero, you find \(\displaystyle E/p=c\).
Solution
\(\displaystyle E=γmc^2\) and \(\displaystyle P=γmu\), so
\(\displaystyle \frac{E}{P}=\frac{γmc^2}{γmu}=\frac{c^2}{u}\).
As the mass of particle approaches zero, its velocity u will approach c, so that the ratio of energy to momentum in this limit is
\(\displaystyle \lim_{m→0}\frac{E}{P}=\frac{c^2}{c}=c\)
which is consistent with the equation for photon energy.
73. Construct Your Own Problem
Consider a space sail such as mentioned in Example. Construct a problem in which you calculate the light pressure on the sail in \(\displaystyle N/m^2\) produced by reflecting sunlight. Also calculate the force that could be produced and how much effect that would have on a spacecraft. Among the things to be considered are the intensity of sunlight, its average wavelength, the number of photons per square meter this implies, the area of the space sail, and the mass of the system being accelerated.
74. Unreasonable Results
A car feels a small force due to the light it sends out from its headlights, equal to the momentum of the light divided by the time in which it is emitted.
(a) Calculate the power of each headlight, if they exert a total force of \(\displaystyle 2.00×10^{−2}N\) backward on the car.
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(\displaystyle 3.00×10^6\)W
(b) Headlights are way too bright.
(c) Force is too large.
29.6: The Wave Nature of Matter
75. At what velocity will an electron have a wavelength of 1.00 m?
Solution
\(\displaystyle 7.28×10^{–4}m\)
76. What is the wavelength of an electron moving at 3.00% of the speed of light?
77. At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = \(\displaystyle 10^{−15}m\).)
Solution
\(\displaystyle 6.62×10^7m/s\)
78. What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 cm (large enough for it to interfere with other billiard balls)?
79. Find the wavelength of a proton moving at 1.00% of the speed of light.
Solution
\(\displaystyle 1.32×10^{–13}m\)
80. Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s.
(a) What is the wavelength of such a neutron?
(b) What is its kinetic energy in eV?
81. (a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic.
(b) What is the neutron’s kinetic energy in MeV?
Solution
(a) \(\displaystyle 6.62×10^7m/s\)
(b) \(\displaystyle 22.9 MeV\)
82. What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?
83. What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?
Solution
15.1 keV
84. (a) Calculate the velocity of an electron that has a wavelength of \(\displaystyle 1.00 μm\).
(b) Through what voltage must the electron be accelerated to have this velocity?
85. The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light.
(a) What is the proton’s wavelength?
(b) What is its kinetic energy, assuming it is nonrelativistic?
(c) What was the equivalent voltage through which it was accelerated?
Solution
(a) 5.29 fm
(b) \(\displaystyle 4.70×10^{−12}J\)
(c) 29.4 MV
86. The kinetic energy of an electron accelerated in an x-ray tube is 100 keV. Assuming it is nonrelativistic, what is its wavelength?
87. Unreasonable Results
(a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus).
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(\displaystyle 7.28×10^{12}m/s\)
(b) This is thousands of times the speed of light (an impossibility).
(c) The assumption that the electron is non-relativistic is unreasonable at this wavelength.
29.7: Probability and The Heisenberg Uncertainty Principle
88. (a) If the position of an electron in a membrane is measured to an accuracy of \(\displaystyle 1.00 μm\), what is the electron’s minimum uncertainty in velocity?
(b) If the electron has this velocity, what is its kinetic energy in eV?
(c) What are the implications of this energy, comparing it to typical molecular binding energies?
Solution
(a) 57.9 m/s
(b) \(\displaystyle 9.55×10^{−9}eV\)
(c) From [link], we see that typical molecular binding energies range from about 1eV to 10 eV, therefore the result in part (b) is approximately 9 orders of magnitude smaller than typical molecular binding energies.
89. (a) If the position of a chlorine ion in a membrane is measured to an accuracy of \(\displaystyle 1.00 μm\), what is its minimum uncertainty in velocity, given its mass is \(\displaystyle 5.86×10^{−26}kg\)?
(b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?
90. Suppose the velocity of an electron in an atom is known to an accuracy of \(\displaystyle 2.0×10^3m/s\) (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?
Solution
29 nm,
290 times greater
91. The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?
92. A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?
Solution
\(\displaystyle 1.10×10^{−13}eV\)
93. (a) The lifetime of a highly unstable nucleus is \(\displaystyle 10^{−20}s\). What is the smallest uncertainty in its decay energy?
(b) Compare this with the rest energy of an electron.
94. The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?
Solution
\(\displaystyle 3.3×10^{−22}s\)
95. The decay energy of a short-lived nuclear excited state has an uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime it can have?
96. What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?
Solution
\(\displaystyle 2.66×10^{−46}kg\)
97. Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, \(\displaystyle ΔEΔt≈h\), using the following arguments: Since the position of a particle is uncertain by \(\displaystyle Δx≈λ\), where \(\displaystyle λ\) is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse \(\displaystyle Δx\). Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to \(\displaystyle λ\). Find \(\displaystyle Δt\) and \(\displaystyle ΔE\); then multiply them to give the approximate uncertainty principle.
29.8: The Particle-Wave Duality Reviewed
98. Integrated Concepts
The 54.0-eV electron in [link] has a 0.167-nm wavelength. If such electrons are passed through a double slit and have their first maximum at an angle of \(\displaystyle 25.0º\), what is the slit separation d?
Solution
0.395 nm
99. Integrated Concepts
An electron microscope produces electrons with a 2.00-pm wavelength. If these are passed through a 1.00-nm single slit, at what angle will the first diffraction minimum be found?
100. Integrated Concepts
A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength.
(a) What is the average photon energy in joules?
(b) How many of these photons are required to increase the temperature of a person’s shoulder by \(\displaystyle 2.0ºC\), assuming the affected mass is 4.0 kg with a specific heat of \(\displaystyle 0.83 kcal/kg⋅ºC\). Also assume no other significant heat transfer.
(c) How long does this take?
Solution
(a) \(\displaystyle 1.3×10^{−19}J\)
(b) \(\displaystyle 2.1×10^{23}\)
(c) \(\displaystyle 1.4×10^2s\)
101. Integrated Concepts
On its high power setting, a microwave oven produces 900 W of 2560 MHz microwaves.
(a) How many photons per second is this?
(b) How many photons are required to increase the temperature of a 0.500-kg mass of pasta by \(\displaystyle 45.0ºC\), assuming a specific heat of \(\displaystyle 0.900 kcal/kg⋅ºC\)? Neglect all other heat transfer.
(c) How long must the microwave operator wait for their pasta to be ready?
102. Integrated Concepts
(a) Calculate the amount of microwave energy in joules needed to raise the temperature of 1.00 kg of soup from \(\displaystyle 20.0ºC\) to \(\displaystyle 100ºC\)
.
(b) What is the total momentum of all the microwave photons it takes to do this?
(c) Calculate the velocity of a 1.00-kg mass with the same momentum. (d) What is the kinetic energy of this mass?
Solution
(a) \(\displaystyle 3.35×10^5J\)
(b) \(\displaystyle 1.12×10^{–3}kg⋅m/s\)
(c) \(\displaystyle 1.12×10^{–3}m/s\)
(d) \(\displaystyle 6.23×10^{–7}J\)
103. Integrated Concepts
(a) What is \(\displaystyle γ\) for an electron emerging from the Stanford Linear Accelerator with a total energy of 50.0 GeV?
(b) Find its momentum.
(c) What is the electron’s wavelength?
104. Integrated Concepts
(a) What is \(\displaystyle γ\) for a proton having an energy of 1.00 TeV, produced by the Fermilab accelerator?
(b) Find its momentum.
(c) What is the proton’s wavelength?
Solution
(a) \(\displaystyle 1.06×10^3\)
(b) \(\displaystyle 5.33×10^{−16}kg⋅m/s\)
(c) \(\displaystyle 1.24×10^{−18}m\)
105. Integrated Concepts
An electron microscope passes 1.00-pm-wavelength electrons through a circular aperture \(\displaystyle 2.00 μm\) in diameter. What is the angle between two just- resolvable point sources for this microscope?
106. Integrated Concepts
(a) Calculate the velocity of electrons that form the same pattern as 450-nm light when passed through a double slit.
(b) Calculate the kinetic energy of each and compare them.
(c) Would either be easier to generate than the other? Explain.
Solution
(a) \(\displaystyle 1.62×10^3m/s\)
(b) \(\displaystyle 4.42×10^{−19}J\) for photon, \(\displaystyle 1.19×10^{−24}J\) for electron, photon energy is \(\displaystyle 3.71×10^5\) times greater
(c) The light is easier to make because 450-nm light is blue light and therefore easy to make. Creating electrons with \(\displaystyle 7.43 μeV\) of energy would not be difficult, but would require a vacuum.
107. Integrated Concepts
(a) What is the separation between double slits that produces a second-order minimum at \(\displaystyle 45.0º\) for 650-nm light?
(b) What slit separation is needed to produce the same pattern for 1.00-keV protons.
Solution
(a) \(\displaystyle 2.30×10^{−6}m\)
(b) \(\displaystyle 3.20×10^{−12}m\)
108. Integrated Concepts
A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal.
(a) How many electrons per second are ejected?
(b) What power is carried away by the electrons, given that the binding energy is 2.71 eV?
(c) Calculate the current of ejected electrons.
(d) If the photoelectric material is electrically insulated and acts like a 2.00-pF capacitor, how long will current flow before the capacitor voltage stops it?
109. Integrated Concepts
One problem with x rays is that they are not sensed. Calculate the temperature increase of a researcher exposed in a few seconds to a nearly fatal accidental dose of x rays under the following conditions. The energy of the x-ray photons is 200 keV, and \(\displaystyle 4.00×10^{13}\) of them are absorbed per kilogram of tissue, the specific heat of which is \(\displaystyle 0.830 kcal/kg⋅ºC\). (Note that medical diagnostic x-ray machines cannot produce an intensity this great.)
Solution
\(\displaystyle 3.69×10^{−4}ºC\)
110. Integrated Concepts
A 1.00-fm photon has a wavelength short enough to detect some information about nuclei.
(a) What is the photon momentum?
(b) What is its energy in joules and MeV?
(c) What is the (relativistic) velocity of an electron with the same momentum?
(d) Calculate the electron’s kinetic energy.
111. Integrated Concepts
The momentum of light is exactly reversed when reflected straight back from a mirror, assuming negligible recoil of the mirror. Thus the change in momentum is twice the photon momentum. Suppose light of intensity \(\displaystyle 1.00 kW/m^2\) reflects from a mirror of area \(\displaystyle 2.00 m^2\).
(a) Calculate the energy reflected in 1.00 s.
(b) What is the momentum imparted to the mirror?
(c) Using the most general form of Newton’s second law, what is the force on the mirror?
(d) Does the assumption of no mirror recoil seem reasonable?
Solution
(a) 2.00 kJ
(b) \(\displaystyle 1.33×10^{−5}kg⋅m/s\)
(c) \(\displaystyle 1.33×10^{−5}N\)
(d) yes
112. Integrated Concepts
Sunlight above the Earth’s atmosphere has an intensity of \(\displaystyle 1.30kW/m^2\). If this is reflected straight back from a mirror that has only a small recoil, the light’s momentum is exactly reversed, giving the mirror twice the incident momentum.
(a) Calculate the force per square meter of mirror.
(b) Very low mass mirrors can be constructed in the near weightlessness of space, and attached to a spaceship to sail it. Once done, the average mass per square meter of the spaceship is 0.100 kg. Find the acceleration of the spaceship if all other forces are balanced.
(c) How fast is it moving 24 hours later?
Contributors and Attributions
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Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


19.6: Review Questions
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1.
Which teaching strategy should the nurse use to enhance learning when teaching a community CPR program?
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Use passive learning strategies
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Present materials from simple to complex
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Assume learners have the same baseline knowledge
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Choose one learning strategy to present material
2.
The community health nurse is performing a community health needs assessment prior to planning a nutrition program for older adults. Which factors will the nurse identify in this needs assessment?
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Primary language, budget, resources
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Resources, strengths, land mass
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Enthusiasm, primary language, currents needs
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Strengths, resources, current needs
3.
Which action should the nurse take when developing a community health education program based on adult learning theory?
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Assess what the learners already know about the topic
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Identify positive reinforcements to reward desired behaviors
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Utilize technology to increase connectivity
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Allow sufficient time for participant reflection
4.
Members of the health education team are meeting to review the team’s progress. Which factor may adversely affect team communication?
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Frequent brainstorming sessions
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High level of team member engagement
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Low feelings of trust
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Team member roles being clear
5.
Which activity will the nurse perform during the evaluation step of a health intervention program whose objective is to teach community members about identifying and reducing cardiac risk factors?
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Select the learning theory to be used in the program
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Identify what learners already know about cardiac risk factors
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Determine if learners can identify their cardiac risk factors
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Develop strategies for teaching about cardiac risk factors
6.
A health care team is using TeamSTEPPS® to plan and implement community health programs. Which action will the team take using this framework?
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Share leadership responsibilities
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Distribute information as needed
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Utilize an informal communication format
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Employ an interdisciplinary approach
7.
During a community assessment, the nurse identifies a need for safe after-school space for high school students to socialize, study, and play. Which basic human need does this nurse determine needs to be met?
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Security
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Physiological
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Esteem
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Self-actualization
8.
While planning a community health education program for older adults who want to exercise more frequently, the health education team decides to ask participants to sign a contract where they agree to complete a certain level of physical activity each week. Which learning theory is the team utilizing?
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Connectivism
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Constructivism
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Behaviorism
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Humanism
9.
The school health nurse is planning a community health education workshop on vaping for parents and caregivers of adolescents. To increase participation in the program, the nurse will use which strategy?
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Offer the workshop during the school day
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Require students to attend with their caregivers
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Limit the number of attendees
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Involve members of the Parent Teacher Association in the planning
10.
The nurse knows which instruction method is most effective for adolescents?
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Providing information in small segments
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Including family members in the learning
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Connecting material to past experiences
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Incorporating peer activities into the learning process


Chapter 21
(a) sodium-24; (b) aluminum-29; (c) krypton-73; (d) iridium-194
(a) (b) (c) (d)
(a) (b) (c) (d)
Nuclear reactions usually change one type of nucleus into another; chemical changes rearrange atoms. Nuclear reactions involve much larger energies than chemical reactions and have measureable mass changes.
(a), (b), (c), (d), and (e)
(a) A nucleon is any particle contained in the nucleus of the atom, so it can refer to protons and neutrons. (b) An α particle is one product of natural radioactivity and is the nucleus of a helium atom. (c) A β particle is a product of natural radioactivity and is a high-speed electron. (d) A positron is a particle with the same mass as an electron but with a positive charge. (e) Gamma rays compose electromagnetic radiation of high energy and short wavelength. (f) Nuclide is a term used when referring to a single type of nucleus. (g) The mass number is the sum of the number of protons and the number of neutrons in an element. (h) The atomic number is the number of protons in the nucleus of an element.
(a) (b) (c) (d)
(a) (b) (c) (d)
(a) 148.8 MeV per atom; (b) 7.808 MeV/nucleon
α (helium nuclei), β (electrons), β + (positrons), and η (neutrons) may be emitted from a radioactive element, all of which are particles; γ rays also may be emitted.
(a) conversion of a neutron to a proton: (b) conversion of a proton to a neutron; the positron has the same mass as an electron and the same magnitude of positive charge as the electron has negative charge; when the n:p ratio of a nucleus is too low, a proton is converted into a neutron with the emission of a positron: (c) In a proton-rich nucleus, an inner atomic electron can be absorbed. In simplest form, this changes a proton into a neutron:
The electron pulled into the nucleus was most likely found in the 1 s orbital. As an electron falls from a higher energy level to replace it, the difference in the energy of the replacement electron in its two energy levels is given off as an X-ray.
Manganese-51 is most likely to decay by positron emission. The n:p ratio for Cr-53 is = 1.21; for Mn-51, it is = 1.04; for Fe-59, it is = 1.27. Positron decay occurs when the n:p ratio is low. Mn-51 has the lowest n:p ratio and therefore is most likely to decay by positron emission. Besides, is a stable isotope, and decays by beta emission.
(a) β decay; (b) α decay; (c) positron emission; (d) β decay; (e) α decay
Half-life is the time required for half the atoms in a sample to decay. Example (answers may vary): For C-14, the half-life is 5770 years. A 10-g sample of C-14 would contain 5 g of C-14 after 5770 years; a 0.20-g sample of C-14 would contain 0.10 g after 5770 years.
or 97.3%
2 10 3 y
0.12 h –1
(a) 3.8 billion years; (b) The rock would be younger than the age calculated in part (a). If Sr was originally in the rock, the amount produced by radioactive decay would equal the present amount minus the initial amount. As this amount would be smaller than the amount used to calculate the age of the rock and the age is proportional to the amount of Sr, the rock would be younger.
c = 0; This shows that no Pu-239 could remain since the formation of the earth. Consequently, the plutonium now present could not have been formed with the uranium.
17.5 MeV
(a) (b) (c) (d)
(a) (b) (c) (d)
Two nuclei must collide for fusion to occur. High temperatures are required to give the nuclei enough kinetic energy to overcome the very strong repulsion resulting from their positive charges.
A nuclear reactor consists of the following:
- A nuclear fuel. A fissionable isotope must be present in large enough quantities to sustain a controlled chain reaction. The radioactive isotope is contained in tubes called fuel rods.
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A moderator. A moderator slows neutrons produced by nuclear reactions so that they can be absorbed by the fuel and cause additional nuclear reactions.
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A coolant. The coolant carries heat from the fission reaction to an external boiler and turbine where it is transformed into electricity.
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A control system. The control system consists of control rods placed between fuel rods to absorb neutrons and is used to adjust the number of neutrons and keep the rate of the chain reaction at a safe level.
- A shield and containment system. The function of this component is to protect workers from radiation produced by the nuclear reactions and to withstand the high pressures resulting from high-temperature reactions.
The fission of uranium generates heat, which is carried to an external steam generator (boiler). The resulting steam turns a turbine that powers an electrical generator.
Introduction of either radioactive Ag + or radioactive Cl – into the solution containing the stated reaction, with subsequent time given for equilibration, will produce a radioactive precipitate that was originally devoid of radiation.
(a) (b) 37.6 days
Alpha particles can be stopped by very thin shielding but have much stronger ionizing potential than beta particles, X-rays, and γ-rays. When inhaled, there is no protective skin covering the cells of the lungs, making it possible to damage the DNA in those cells and cause cancer.
(a) 7.64 10 9 Bq; (b) 2.06 10 −2 Ci


24.2: Transformation and Backlash
While prosperous, middle-class Americans found much to celebrate about the new era of leisure and consumption, many Americans—often those in rural areas—disagreed on the meaning of a “good life” and how to achieve it. They reacted to the rapid social changes of modern urban society with a vigorous defense of religious values and a fearful rejection of cultural diversity and equality.
NATIVISM
Beginning at the end of the nineteenth century, immigration into the United States rocketed to never-before-seen heights. Many of these new immigrants were coming from eastern and southern Europe and, for many English-speaking, native-born Americans of northern European descent, the growing diversity of new languages, customs, and religions triggered anxiety and racial animosity. In reaction, some embraced nativism, prizing white Americans with older family trees over more recent immigrants, and rejecting outside influences in favor of their own local customs. Nativists also stoked a sense of fear over the perceived foreign threat, pointing to the anarchist assassinations of the Spanish prime minister in 1897, the Italian king in 1900, and even President William McKinley in 1901 as proof. Following the Bolshevik Revolution in Russia in November 1917, the sense of an inevitable foreign or communist threat only grew among those already predisposed to distrust immigrants.
The sense of fear and anxiety over the rising tide of immigration came to a head with the trial of Nicola Sacco and Bartolomeo Vanzetti (Figure 24.2.1). Sacco and Vanzetti were Italian immigrants who were accused of being part of a robbery and murder in Braintree, Massachusetts, in 1920. There was no direct evidence linking them to the crime, but (in addition to being immigrants) both men were anarchists who favored the destruction of the American market-based, capitalistic society through violence. At their trial, the district attorney emphasized Sacco and Vanzetti’s radical views, and the jury found them guilty on July 14, 1921. Despite subsequent motions and appeals based on ballistics testing, recanted testimony, and an ex-convict’s confession, both men were executed on August 23, 1927.
Opinions on the trial and judgment tended to divide along nativist-immigrant lines, with immigrants supporting the innocence of the condemned pair. The verdict sparked protests from Italian and other immigrant groups, as well as from noted intellectuals such as writer John Dos Passos, satirist Dorothy Parker, and famed physicist Albert Einstein. Muckraker Upton Sinclair based his indictment of the American justice system, the “documentary novel” Boston , on Sacco and Vanzetti’s trial, which he considered a gross miscarriage of justice. As the execution neared, the radical labor union Industrial Workers of the World called for a three-day nationwide walkout, leading to the Great Colorado Coal Strike of 1927. Protests occurred worldwide from Tokyo to Buenos Aires to London (Figure 24.2.1).
One of the most articulate critics of the trial was then-Harvard Law School professor Felix Frankfurter, who would go on to be appointed to the U.S. Supreme Court by Franklin D. Roosevelt in 1939. In 1927, six years after the trial, he wrote in The Atlantic , “By systematic exploitation of the defendants’ alien blood, their imperfect knowledge of English, their unpopular social views, and their opposition to the war, the District Attorney invoked against them a riot of political passion and patriotic sentiment; and the trial judge connived at—one had almost written, cooperated in—the process.”
To “preserve the ideal of American homogeneity,” the Emergency Immigration Act of 1921 introduced numerical limits on European immigration for the first time in U.S. history. These limits were based on a quota system that restricted annual immigration from any given country to 3 percent of the residents from that same country as counted in the 1910 census. The National Origins Act of 1924 went even further, lowering the level to 2 percent of the 1890 census, significantly reducing the share of eligible southern and eastern Europeans, since they had only begun to arrive in the United States in large numbers in the 1890s. Although New York congressmen Fiorello LaGuardia and Emanuel Celler spoke out against the act, there was minimal opposition in Congress, and both labor unions and the Ku Klux Klan supported the bill. When President Coolidge signed it into law, he declared, “America must be kept American.”
The Library of Congress’s immigration collectioncontains information on different immigrant groups, the timelines of their immigration, maps of their settlement routes, and the reasons they came. Click the images on the left navigation bar to learn about each group.
THE KU KLUX KLAN
The concern that a white, Protestant, Anglo-Saxon United States was under siege by throngs of undesirables was not exclusively directed at foreigners. The sense that the country was also facing a threat from within its borders and its own citizenry was also prevalent. This sense was clearly reflected in the popularity of the 1915 motion picture, D. W. Griffith’s The Birth of a Nation (Figure 24.2.2). Based on The Clansman , a 1915 novel by Thomas Dixon, the film offers a racist, white-centric view of the Reconstruction Era.  The heroes of the film were the Ku Klux Klan, who saved the whites, the South, and the nation. While the film was reviled by many African Americans and the NAACP for its historical inaccuracies and its maligning of freed slaves, it was celebrated by many whites who accepted the historical revisionism as an accurate portrayal of Reconstruction Era oppression. After viewing the film, President Wilson reportedly remarked, “It is like writing history with lightning, and my only regret is that it is all so terribly true.”
In a letter dated April 17, 1915, Mary Childs Nerney, a secretary of the NAACP, wrote to a local censor to request that certain scenes be cut from The Birth of a Nation .
My dear Mr. Packard:
I am utterly disgusted with the situation in regard to “The Birth of a Nation.” As you will read in the next number of the Crisis, we have fought it at every possible point. In spite of the promise of the Mayor [of Chicago] to cut out the two objectionable scenes in the second part, which show a white girl committing suicide to escape from a Negro pursuer, and a mulatto politician trying to force marriage upon the daughter of his white benefactor, these two scenes still form the motif of the really unimportant incidents, of which I enclose a list. I have seen the thing four times and am positive that nothing more will be done about it. Jane Addams saw it when it was in its worst form in New York. I know of no one else from Chicago who saw it. I enclose Miss Addam’s opinion.
When we took the thing before the Police Magistrate he told us that he could do nothing about it unless it [led] to a breach of the peace. Some kind of demonstration began in the Liberty Theatre Wednesday night but the colored people took absolutely no part in it, and the only man arrested was a white man. This, of course, is exactly what Littleton, counsel for the producer, Griffith, held in the Magistrates’ Court when we have our hearing and claimed that it might lead to a breach of the peace.
Frankly, I do not think you can do one single thing. It has been to me a most liberal education and I purposely am through. The harm it is doing the colored people cannot be estimated. I hear echoes of it wherever I go and have no doubt that this was in the mind of the people who are producing it. Their profits here are something like $14,000 a day and their expenses about $400. I have ceased to worry about it, and if I seem disinterested, kindly remember that we have put six weeks of constant effort of this thing and have gotten nowhere.
Sincerely yours,
—Mary Childs Nerney, Secretary, NAACP
On what grounds does Nerney request censorship? What efforts to get the movie shut down did she describe?
The Ku Klux Klan, which had been dormant since the end of Reconstruction in 1877, experienced a resurgence of attention following the popularity of the film. Just months after the film’s release, a second incarnation of the Klan was established at Stone Mountain, Georgia, under the leadership of William Simmons. This new Klan now publicly eschewed violence and received mainstream support. Its embrace of Protestantism, anti-Catholicism, and anti-Semitism, and its appeals for stricter immigration policies, gained the group a level of acceptance by nativists with similar prejudices. The group was not merely a male organization: The ranks of the Klan also included many women, with chapters of its women’s auxiliary in locations across the country. These women’s groups were active in a number of reform-minded activities, such as advocating for prohibition and the distribution of Bibles at public schools. But they also participated in more expressly Klan activities like burning crosses and the public denunciation of Catholics and Jews (Figure 24.2.3). By 1924, this Second Ku Klux Klan had six million members in the South, West, and, particularly, the Midwest—more Americans than there were in the nation’s labor unions at the time. While the organization publicly abstained from violence, its member continued to employ intimidation, violence, and terrorism against its victims, particularly in the South.
The Klan’s newfound popularity proved to be fairly short-lived. Several states effectively combatted the power and influence of the Klan through anti-masking legislation, that is, laws that barred the wearing of masks publicly. As the organization faced a series of public scandals, such as when the Grand Dragon of Indiana was convicted of murdering a white schoolteacher, prominent citizens became less likely to openly express their support for the group without a shield of anonymity. More importantly, influential people and citizen groups explicitly condemned the Klan. Reinhold Niebuhr, a popular Protestant minister and conservative intellectual in Detroit, admonished the group for its ostensibly Protestant zealotry and anti-Catholicism. Jewish organizations, especially the Anti-Defamation League, which had been founded just a couple of years before the reemergence of the Klan, amplified Jewish discontent at being the focus of Klan attention. And the NAACP, which had actively sought to ban the film The Birth of a Nation , worked to lobby congress and educate the public on lynchings. Ultimately, however, it was the Great Depression that put an end to the Klan. As dues-paying members dwindled, the Klan lost its organizational power and sunk into irrelevance until the 1950s.
FAITH, FUNDAMENTALISM, AND SCIENCE
The sense of degeneration that the Klan and anxiety over mass immigration prompted in the minds of many Americans was in part a response to the process of postwar urbanization. Cities were swiftly becoming centers of opportunity, but the growth of cities, especially the growth of immigrant populations in those cities, sharpened rural discontent over the perception of rapid cultural change. As more of the population flocked to cities for jobs and quality of life, many left behind in rural areas felt that their way of life was being threatened. To rural Americans, the ways of the city seemed sinful and profligate. Urbanites, for their part, viewed rural Americans as hayseeds who were hopelessly behind the times.
In this urban/rural conflict, Tennessee lawmakers drew a battle line over the issue of evolution and its contradiction of the accepted, biblical explanation of history. Charles Darwin had first published his theory of natural selection in 1859, and by the 1920s, many standard textbooks contained information about Darwin’s theory of evolution. Fundamentalist Protestants targeted evolution as representative of all that was wrong with urban society. Tennessee’s Butler Act made it illegal “to teach any theory that denies the story of the Divine Creation of man as taught in the Bible, and to teach instead that man has descended from a lower order of animals.”
The American Civil Liberties Union (ACLU) hoped to challenge the Butler Act as an infringement of the freedom of speech. As a defendant, the ACLU enlisted teacher and coach John Scopes, who suggested that he may have taught evolution while substituting for an ill biology teacher. Town leaders in Dayton, Tennessee, for their part, sensed an opportunity to promote their town, which had lost more than one-third of its population, and welcomed the ACLU to stage a test case against the Butler Act. The ACLU and the town got their wish as the Scopes Monkey Trial, as the newspapers publicized it, quickly turned into a carnival that captured the attention of the country and epitomized the nation’s urban/rural divide (Figure 24.2.4).
Fundamentalist champion William Jennings Bryan argued the case for the prosecution. Bryan was a three-time presidential candidate and Woodrow Wilson’s Secretary of State until 1915, at which point he began preaching across the country about the spread of secularism and the declining role of religion in education. He was known for offering $100 to anyone who would admit to being descended from an ape. Clarence Darrow, a prominent lawyer and outspoken agnostic, led the defense team. His statement that, “Scopes isn’t on trial, civilization is on trial. No man’s belief will be safe if they win,” struck a chord in society.
The outcome of the trial, in which Scopes was found guilty and fined $100, was never really in question, as Scopes himself had confessed to violating the law. Nevertheless, the trial itself proved to be high drama. The drama only escalated when Darrow made the unusual choice of calling Bryan as an expert witness on the Bible. Knowing of Bryan’s convictions of a literal interpretation of the Bible, Darrow peppered him with a series of questions designed to ridicule such a belief. The result was that those who approved of the teaching of evolution saw Bryan as foolish, whereas many rural Americans considered the cross-examination an attack on the Bible and their faith.
H. L. Mencken covered the trial for Baltimore’s The Evening Sun. One of most popular writers of social satire of his age, Mencken was very critical of the South, the trial, and especially Bryan. He coined the terms “monkey trial “and “Bible belt.” In the excerpt below, Mencken reflects on the trial’s outcome and its overall importance for the United States.
The Scopes trial, from the start, has been carried on in a manner exactly fitted to the anti- evolution law and the simian imbecility under it. There hasn’t been the slightest pretense to decorum. The rustic judge, a candidate for re-election, has postured the yokels like a clown in a ten-cent side show, and almost every word he has uttered has been an undisguised appeal to their prejudices and superstitions. The chief prosecuting attorney, beginning like a competent lawyer and a man of self-respect, ended like a convert at a Billy Sunday revival. It fell to him, finally, to make a clear and astounding statement of theory of justice prevailing under fundamentalism. What he said, in brief, was that a man accused of infidelity had no rights whatever under Tennessee law. . . .
Darrow has lost this case. It was lost long before he came to Dayton. But it seems to me that he has nevertheless performed a great public service by fighting it to a finish and in a perfectly serious way. Let no one mistake it for comedy, farcical though it may be in all its details. It serves notice on the country that Neanderthal man is organizing in these forlorn backwaters of the land, led by a fanatic, rid of sense and devoid of conscience. Tennessee, challenging him too timorously and too late, now sees its courts converted into camp meetings and its Bill of Rights made a mock of by its sworn officers of the law. There are other States that had better look to their arsenals before the Hun is at their gates.
—H. L. Mencken, The Evening Sun , July 18, 1925
How does Mencken characterize Judge Raulston? About what threat is Mencken warning America?
Indicative of the revival of Protestant fundamentalism and the rejection of evolution among rural and white Americans was the rise of Billy Sunday. As a young man, Sunday had gained fame as a baseball player with exceptional skill and speed. Later, he found even more celebrity as the nation’s most revered evangelist, drawing huge crowds at camp meetings around the country. He was one of the most influential evangelists of the time and had access to some of the wealthiest and most powerful families in the country (Figure 24.2.5). Sunday rallied many Americans around “old-time” fundamentalist religion and garnered support for prohibition. Recognizing Sunday’s popular appeal, Bryan attempted to bring him to Dayton for the Scopes trial, although Sunday politely refused.
Even more spectacular than the rise of Billy Sunday was the popularity of Aimee Semple McPherson, a Canadian Pentecostal preacher whose Foursquare Church in Los Angeles catered to the large community of midwestern transplants and newcomers to California (Figure 24.2.5). Although her message promoted the fundamental truths of the Bible, her style was anything but old fashioned. Dressed in tight-fitting clothes and wearing makeup, she held radio-broadcast services in large venues that resembled concert halls and staged spectacular faith-healing performances. Blending Hollywood style and modern technology with a message of fundamentalist Christianity, McPherson exemplified the contradictions of the decade well before public revelations about her scandalous love affair cost her much of her status and following.
Section Summary
The old and the new came into sharp conflict in the 1920s. In many cases, this divide was geographic as well as philosophical; city dwellers tended to embrace the cultural changes of the era, whereas those who lived in rural towns clung to traditional norms. The Sacco and Vanzetti trial in Massachusetts, as well as the Scopes trial in Tennessee, revealed many Americans’ fears and suspicions about immigrants, radical politics, and the ways in which new scientific theories might challenge traditional Christian beliefs. Some reacted more zealously than others, leading to the inception of nativist and fundamentalist philosophies, and the rise of terror groups such as the Second Ku Klux Klan.
Review Questions
The Scopes Monkey Trial revolved around a law that banned teaching about ________ in public schools.
- the Bible
- Darwinism
- primates
- Protestantism
B
Which man was both a professional baseball player and an influential evangelist during the 1920s?
- Babe Ruth
- H. L. Mencken
- Jim Thorpe
- Billy Sunday
D
What was the platform of the Second Ku Klux Klan, and in what activities did they engage to promote it?
The reincarnated Ku Klux Klan championed an anti-black, anti-immigrant, anti-Catholic, and anti-Jewish philosophy, and promoted the spread of Protestant beliefs. The Klan publicly denounced the groups they despised and continued to engage in activities such as cross-burning, violence, and intimidation, despite their public commitment to nonviolent tactics. Women’s groups within the Klan also participated in various types of reform, such as advocating the prohibition of alcohol and distributing Bibles in public schools.
Glossary
- nativism
- the rejection of outside influences in favor of local or native customs
- Scopes Monkey Trial
- the 1925 trial of John Scopes for teaching evolution in a public school; the trial highlighted the conflict between rural traditionalists and modern urbanites
- Second Ku Klux Klan
- unlike the secret terror group of the Reconstruction Era, the Second Ku Klux Klan was a nationwide movement that expressed racism, nativism, anti-Semitism, and anti-Catholicism