11.1: A First Taste of Ramsey Theory
Bob likes to think of himself as a wild and crazy guy, totally unpredictable. Most guys do. But Alice says that Bob can't change his basic nature, which is excruciatingly boring. Carlos remarks that perhaps we shouldn't be so hard on Bob, because under certain circumstances, we can all be forced to be dull and repetitive.
Recall that when \(n\) is a positive integer, we let \([n]=\{1,2,…,n\}\). In this chapter, when \(X\) is a set and \(k\) is a non-negative integer with \(k \leq |X|\), we borrow from our in-line notation for binomial coefficients and let \(C(X,k)\) denote the family of all \(k\)-element subsets of \(X\). So \(|C([n],k)|=C(n,k)\) whenever \(0 \leq k \leq n\).
Recall that the Pigeon Hole Principle asserts that if \(n+1\) pigeons are placed in \(n\) holes, then there must be some hole into which two or more pigeons have been placed. More formally, if \(n\) and \(k\) are positive integers, \(t>n(k−1)\) and \(f:[t] \rightarrow [n]\) is any function, then there is a \(k\)-element subset \(H⊆[t]\) and an element \(j \in [n]\) so that \(f(i)=j\) for every \(i \in H\).
We now embark on a study of an elegant extension of this basic result, one that continues to fascinate and challenge.
Returning to the discussion at the start of this section, you might say that an induced subgraph \(H\) of a graph \(G\) is “boring” if it is either a complete subgraph or an independent set. In either case, exactly every pair of vertices in \(H\) behaves in exactly the same boring way. So is boredom inevitable? The answer is yes—at least in a relative sense. As a starter, let's show that any graph on six (or more) vertices has a boring subgraph of size three.
Let \(G\) be any graph with six of more vertices. Then either \(G\) contains a complete subgraph of size 3 or an independent set of size 3 .
- Proof
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Let \(x\) be any vertex in \(G\). Then split the remaining vertices into two sets \(S_1\) and \(S_2\) with \(S_1\) being the neighbors of \(x\) and \(S_2\) the non-neighbors. Since \(G\) has at least six vertices, we know that either \(|S_1| \geq 3\) or \(|S_2| \geq 3\). Suppose first that \(|S_1| \geq 3\) and let \(y_1, y_2\) and \(y_3\) be distinct vertices from \(S_1\). If \(y_iy_j\) is an edge in \(G\) for some distinct pair \(i,j \in \{1,23\}\), then \(\{x,y_i,y_j\}\) is a complete subgraph of size 3 in \(G\). On the other hand, if there are no edges among the vertices in \(\{y_1,y_2,y_3\}\), then we have an independent set of size .3.
The argument when \(|S_2| \geq 3\) is dual.
We note that the bound of six in the preceding lemma is sharp, as a cycle on five vertices does not contain either a complete set of size 3 nor an independent set of size 3.
Next, here is the statement that generalizes this result.
If \(m\) and \(n\) are positive integers, then there exists a least positive integer \(R(m,n)\) so that if \(G\) is a graph and \(G\) has at least \(R(m,n)\) vertices, then either \(G\) contains a complete subgraph on \(m\) vertices, or \(G\) contains an independent set of size \(n\) .
- Proof
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We show that \(R(m,n)\) exists and is at most \(\binom{m+n-2}{m-1}\). This claim is trivial when either \(m \leq 2\) or \(n \leq 2\), so we may assume that \(m,n \geq 3\). From this point, we proceed by induction on \(t=m+n\) assuming that the result holds when \(t \leq 5\).
Now let \(x\) be any vertex in \(G\). Then there are at least \(\binom{m+n-2}{m-1}-1\) other vertices, which we partition as \(S_1 \cup S_2\), where \(S_1\) are those vertices adjacent to \(x\) in \(G\) and \(S_2\) are those vertices which are not adjacent to \(s\).
We recall that the binomial coefficients satisfy
\(\dbinom{m+n-2}{m-1} = \dbinom{m+n-3}{m-2} + \dbinom{m+n-3}{m-1} = \dbinom{m+n-3}{m-2} + \dbinom{m+n-3}{n-2}\)
So either \(|S_1| \geq \binom{m+n-3}{m-2}\) or \(|S_1| \geq \binom{m+n-3}{m-1}\). If the first option holds, and \(S_1\) does not have an independent set of size \(n\), then it contains a complete subgraph of size \(m−1\). It follows that we may add \(x\) to this set to obtain a complete subgraph of size \(m\) in \(G\).
Similarly, if the second option holds, and \(S_2\) does not contain a complete subgraph of size \(m\), then \(S_2\) contains an independent set of size \(n−1\), and we may add \(x\) to this set to obtain an independent set of size \(n\) in \(G\).


11.1: Studying metaphor in corpora
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Metaphor is traditionally defined as the transfer of a word from one referent (variously called vehicle, figure or source) to another (the tenor, ground or target) (cf. e.g. Aristotle, Poetics , XXI). If metaphor were indeed located at the word level, it should be straightforwardly amenable to corpus-linguistic analysis. Unfortunately, things are slightly more complicated. First, the transfer does not typically concern individual words but entire semantic fields (or even conceptual domains, according to some theories). Second, as discussed in some detail in Chapter 4, there is nothing in the word itself that distinguishes its literal and metaphorical uses. One way around this problem is manual annotation, and there are very detailed and sophisticated proposals for annotation procedures (most notably the Pragglejaz Metaphor Identification Procedure, cf., for example, Steen et al. 2010).
However, as stressed in various places throughout this book, the manual annotation of corpora severely limits the amount of data that can be included in a research design; this does not invalidate manual annotation, but it makes alternatives highly desirable. Two broad alternatives have been proposed in corpus linguistics. Since these were discussed in some detail in Chapter 4, we will only repeat them briefly here before illustrating them in more detail in the case studies.


3.1: Health History Introduction
Learning Objectives
- Establish a therapeutic nurse-patient relationship
- Use effective verbal and nonverbal communication techniques
- Collect health history data
- Modify assessment techniques to reflect variations across the life span and cultural variations
- Document actions and observations
- Recognize and report significant deviations from norms
“‘Sickness’ is what is happening to the patient. Listen to them.” [1]
The profession of nursing is defined by the American Nurses Association as “the protection, promotion, and optimization of health and abilities, prevention of illness and injury, facilitation of healing, alleviation of suffering through the diagnosis and treatment of human response, and advocacy in the care of individuals, families, groups, communities, and populations.” [2] Simply put, nurses treat human responses to health problems and/or life processes. Nurses look at each person holistically, including emotional, spiritual, psychosocial, and physical health needs. They also consider problems and issues that the person experiences as a part of a family and a community. To collect detailed information about a patient’s human response to illness and life processes, nurses perform a health history. A health history is part of the Assessment phase of the nursing process. It consists of using directed, focused interview questions and open-ended questions to obtain symptoms and perceptions from the patient about their illnesses, functioning, and life processes. While obtaining a health history, the nurse is also simultaneously performing a general survey. Visit the “General Survey Assessment” chapter more information.
- Weed, L. L. (1975). Your health care and how to manage it . University of Vermont. ↵
- American Nurses Association. (2015). Nursing: Scope and standards of practice (3rd ed.). American Nurses Association. ↵


4.6: Using Computation to Simulate Motion in Multiple Dimensions
Let's continue where we left off with computationally simulating one-dimensional motion. We saw that we can write vectors in Visual Python using the syntax.
r = vec(x, y, z)
where x, y , and z are the components of the position vector r . Of course, we can write a vector of any quantity. For example, position, velocity, or acceleration. Continuing the example of a ball moving from before, try adding a velocity component of \(v_y = 0.5\) m/s by editing ball.vel . This will create motion in the y-direction that is the same as the x-direction. You may start from the code that ended the previous computational simulation. We will make a small change first. In the definition of the ball, we are going to create a trail of dots that shows the ball's past positions, and we will remove the velocity arrow for the this example.
ball = sphere(pos=vec(0,0,0), color=color.green, radius=0.1, make_trail=True, trail_type="points") ball.vel = vec(0.5, 0, 0) r_0 = ball.pos t = 0 dt = 1 while t<10: rate(1) ball.pos = r_0 + ball.vel * t t = t + dt
Next, let's look at an example from an earlier section that includes acceleration. The example is shown again below.
An object starts at the origin of a coordinate system at time \(t=0s\), with an initial velocity vector \(v_0=(10m/s)\hat{x}+(15m/s)\hat{y}. The acceleration in the \(x\) direction is \(0m/s^2\) and the acceleration in the \(y\) direction is \(−10 m/s^2\).
- Write an equation for the position vector as a function of time.
- Determine the position of the object at \(t=10s\).
- Plot the trajectory of the object for the first \(5s\) of motion.
- Answer
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scene2 = canvas(title='Examples of 2D Accelerated Motion', width=400, height=400, center=vector(5,0,0), background=color.black, range=300) ball = sphere(pos=vec(0,0,0), color=color.green, radius=10, make_trail=True, trail_type="points") ball.vel = vec(10, 15, 0) ball.acc = vec(0, -10, 0) r_0 = ball.pos t = 0 dt = 1 while t<=10: rate(1) ball.pos = r_0 + ball.vel * t + 0.5 * ball.acc * t**2 t = t + dt print(ball.pos)
To solve this we need to do several things to our constant velocity code above. First, we need to edit the initial velocity, ball.vel , using what is given in the example. Then, we should define ball.acc = vec(0, -10, 0) . Third, we should make the while loop go until less than or equal to 10s by writing while t <= 10: . Fourth, inside the loop, we need to edit the ball.pos to include the acceleration. In Python, raising to a power is done by two asterisks, e.g., t**2 is the square of t . The time-dependent position is
\[\vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a}t^2\]
will be written in Visual Python as
ball.pos = r_0 + ball.vel * t + 0.5 * ball.acc * t**2
Finally, we can print the ball's position after the loop finishes with the command print(ball.pos) . This print statement should be unindented after the while loop to let Python know that it occurs after the looping is complete. Of course, you are welcome to put the print statement in the loop (indented under t = t + dt) if you want to see what difference this makes. Making these changes we can show the position vs. time of the object and print the position after 10 s have passed. To summarize, the steps are
- Edit ball.vel.
- Create an acceleration of the ball, ball.acc.
- Make the loop end at 10 s.
- Edit ball.pos in the loop to include ball.acc .
- Add a print statement to see the ball's position at \(t=10 s\).
Below is a trinket that will keep the scene fixed. Copy the constant velocity code above and make these five edits. The ball radius may need to be increased if it is difficult to see. If you get stuck look at the answer to the example.
It's not necessary to simulate the motion if we only want to calculate or graph. We can use Visual Python to calculate the position of an object as we did above. To do this, we simply define the initial position, initial velocity, and acceleration. Then, we can use the kinematic equation to calculate the position at a later time. Try putting this into the trinket above.
r0 = vec(0,0,0) v0 = vec(10, 15, 0) a = vec(0, -10, 0) t = 10 rt = r0 + v0*t + 0.5*a*t**2 print(rt)
Similarly, we can use Python (not Visual Python) to plot. You'll notice the trinket below says Python3. This distinguishes it from the other trinkets above. It also does not have the first line that imports Visual Python ( Web VPython 3.2 ).
To make lists (arrays) of numbers, it is best to use a library called numpy . To make graphs also requires importing a library matplotlib for making graphs. Using the command import a as b makes library a available by using b to call it.
import numpy as np import matplotlib.pyplot as plt
We won't use vectors, but we keep our initial values for \(x_0, y_0, v_{x0}, v_{y0}, a_x\), and \(a_y\). Then, we need to make lists of \(x(t), y(t)\), and \(t\) that correspond to the points we want to graph. These are calculated using \(x_0, y_0, v_{x0}, v_{y0}, a_x, a_y\), and the list of times \(t\). Therefore, we must define \(t\) before attempting to calculate \(x(t)\), and \(y(t)\)
x_0 = 0 y_0 = 0 v_x0= 10 v_y0 = 15 a_x = 0 a_y = -10 t = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0]) xt = x_0 + v_x0 * t + 0.5 * a_x * t**2 yt = y_0 + v_y0 * t + 0.5 * a_y * t**2
That will create lists of \(x(t)\) and \(y(t)\) because Python recognizes that \(t\) is a list and calculates for each value.
Finally, we insert the code to make a graph.
plt.plot(xt, yt, 'or') plt.xlabel('x (m)') plt.ylabel('y (m)') plt.show()
The first line plots xt on the \(x\) axis and yt on the \(y\) axis. The 'or' makes circles for the points that are red. The second line labels the \(x\) axis. The third line labels the \(y\) axis. The last line puts all the plot code into an image. Try combining these three blocks of code in the trinket below to see that they plot the 2D motion of the object.


9.6: Chapter Summary
In the final chapter of this book, you have learned about academic integrity and plagiarism. This chapter provided you with skills related to style rules for in-text citations when paraphrasing and quoting. Additionally, you learned about style rules related to the creation of title pages and reference lists.
Your Writing Journey
Part of your journey is to become familiar with the style rules of academic writing in the nursing field. APA might seem like a nuisance at first, but this style is important to ensure consistency and clarity across writing. Your instructors expect you to know it and adhere to it. Familiarize yourself with academic integrity as these standards are the parameters for plagiarism and self-plagiarism. Developing academic integrity is necessary and you want to acknowledge others’ ideas and voices while finding and articulating your own.


10.2: Winds and the Coriolis Effect
Differential heating of the Earth’s surface results in equatorial regions receiving more heat than the poles. As air is warmed at the equator it becomes less dense and rises, while at the poles the cold air is denser and sinks. If the Earth was non-rotating, the warm air rising at the equator would reach the upper atmosphere and begin moving horizontally towards the poles. As the air reached the poles it would cool and sink, and would move over the surface of Earth back towards the equator. This would result in one large atmospheric convection cell in each hemisphere (Figure \(\PageIndex{1}\)), with air rising at the equator and sinking at the poles, and the movement of air over the Earth’s surface creating the winds. On this non-rotating Earth, the prevailing winds would thus blow from the poles towards the equator in both hemispheres (Figure \(\PageIndex{1}\)).
The non-rotating situation in Figure \(\PageIndex{1}\) is of course only hypothetical, and in reality the Earth’s rotation makes this atmospheric circulation a bit more complex. The paths of the winds on a rotating Earth are deflected by the Coriolis Effect . The Coriolis Effect is a result of the fact that different latitudes on Earth rotate at different speeds. This is because every point on Earth must make a complete rotation in 24 hours, but some points must travel farther, and therefore faster, to complete the rotation in the same amount of time. In 24 hours a point on the equator must complete a rotation distance equal to the circumference of the Earth, which is about 40,000 km. A point right on the poles covers no distance in that time; it just turns in a circle. So the speed of rotation at the equator is about 1600 km/hr, while at the poles the speed is 0 km/hr. Latitudes in between rotate at intermediate speeds; approximately 1400 km/hr at 30 o and 800 km/hr at 60 o . As objects move over the surface of the Earth they encounter regions of varying speed, which causes their path to be deflected by the Coriolis Effect.
To explain the Coriolis Effect, imagine a cannon positioned at the equator and facing north. Even though the cannon appears stationary to someone on Earth, it is in fact moving east at about 1600 km/hr due to Earth’s rotation. When the cannon fires the projectile travels north towards its target; but it also continues to move to the east at 1600 km/hr, the speed it had while it was still in the cannon. As the shell moves over higher latitudes, its momentum carries it eastward faster than the speed at which the ground beneath it is rotating. For example, by 30 o latitude the shell is moving east at 1600 km/hr while the ground is moving east at only 1400 km/hr. Therefore, the shell gets “ahead” of its target, and will land to the east of its intended destination. From the point of view of the cannon, the path of the projectile appears to have been deflected to the right (red arrow, Figure \(\PageIndex{2}\)). Similarly, a cannon located at 60 o and facing the equator will be moving east at 800 km/hr. When its shell is fired towards the equator, the shell will be moving east at 800 km/hr, but as it approaches the equator it will be moving over land that is traveling east faster than the projectile. So the projectile gets “behind” its target, and will land to the west of its destination. But from the point of view of the cannon facing the equator, the path of the shell still appears to have been deflected to the right (green arrow, Figure \(\PageIndex{2}\)). Therefore, in the Northern Hemisphere, the apparent Coriolis deflection will always be to the right .
In the Southern Hemisphere the situation is reversed (Figure \(\PageIndex{2}\)). Objects moving towards the equator from the south pole are moving from low speed to high speed, so are left behind and their path is deflected to the left. Movement from the equator towards the south pole also leads to deflection to the left. In the Southern Hemisphere, the Coriolis deflection is always to the left from the point of origin.
The magnitude of the Coriolis deflection is related to the difference in rotation speed between the start and end points. Between the poles and 60 o latitude, the difference in rotation speed is 800 km/hr. Between the equator and 30 o latitude, the difference is only 200 km/hr (Figure \(\PageIndex{2}\)). Therefore the strength of the Coriolis Effect is stronger near the poles, and weaker at the equator.
Because of the rotation of the Earth and the Coriolis Effect, rather than a single atmospheric convection cell in each hemisphere, there are three major cells per hemisphere. Warm air rising at the equator cools as it moves through the upper atmosphere, and it descends at around 30 o latitude. The convection cells created by rising air at the equator and sinking air at 30 o are referred to as Hadley Cells , of which there is one in each hemisphere. The cold air that descends at the poles moves over the Earth’s surface towards the equator, and by about 60 o latitude it begins to rise, creating a Polar Cell between 60 o and 90 o . Between 30 o and 60 o lie the Ferrel Cells , composed of sinking air at 30 o and rising air at 60 o (Figure \(\PageIndex{3}\)). With three convection cells in each hemisphere that rotate in alternate directions, the surface winds no longer always blow from the poles towards the equator as in the non-rotating Earth in Figure \(\PageIndex{1}\). Instead, surface winds in both hemispheres blow towards the equator between 90 o and 60 o latitude, and between 0 o and 30 o latitude. Between 30 o and 60 o latitude, the surface winds blow towards the poles (Figure \(\PageIndex{3}\)).
The surface winds created by the atmospheric convection cells are also influenced by the Coriolis Effect as they change latitudes. The Coriolis Effect deflects the path of the winds to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. Adding this deflection leads to the pattern of prevailing winds illustrated in Figure \(\PageIndex{4}\). Between the equator and 30 o latitude are the trade winds ; the northeast trade winds in the Northern Hemisphere and the southeast trade winds in the Southern Hemisphere (note that winds are named based on the direction from which they originate, not where they are going). The westerlies are the dominant winds between 30 o and 60 o in both hemispheres, and the polar easterlies are found between 60 o and the poles.
In between these wind bands lie regions of high and low pressure. High pressure zones occur where air is descending, while low pressure zones indicate rising air. Along the equator the rising air creates a low pressure region called the doldrums , or the Intertropical Convergence Zone (ITCZ)(convergence zone because this is where the trade winds converge). At 30 o latitude there are high pressure zones of descending air known as the horse latitudes , or the subtropical highs. Finally, at 60 o lies another low pressure region called the polar front . It should be noted that these high and low pressure zones are not fixed in place; their latitude fluctuates depending on the season, and these fluctuations have important implications for regional climates.
Doldrums? Horse latitudes? Trade winds?
These may seem like some odd names for these atmospheric phenomena, but many of them can be traced back to maritime traditions and lore.
The doldrums refer to regions of low pressure around the equator. In these areas, air is rising rather than moving horizontally, so these regions commonly encounter very light winds. The lack of wind could leave sailing ships becalmed for days or weeks at a time, which was not good for the morale of the ship’s crew.
Like the doldrums the horse latitudes are also areas with light winds, this time due to descending air, which could leave ships becalmed. One explanation for the term “horse latitudes” is that when these ships became stranded they ran the risk of running out of food or water. To conserve these resources, sailors would throw their dead or dying horses overboard, hence the “horse latitudes.” Another explanation is that many sailors received part of their pay before a voyage, and often spent it before departing. This meant that they would spend the first part of the voyage working without pay and in debt, a period called the “dead horse” time, which might last for a few months. When they started earning their pay once again, they had a “dead horse” ceremony and threw a pretend horse overboard. The timing of this ceremony often coincided with reaching the horse latitudes, leading to the association of the ceremony with the location. A third explanation is that a ship was referred to as “horsed” when winds were weak and the ship instead had to rely on ocean currents to move them. This could be a common occurrence in the high pressure zones around 30 o latitude, so they were referred to as the horse latitudes.
The term trade winds may have originally derived from the terms for “track” or “path”, but the term may have become more common during European exploration and commercialization of the New World. Mariners sailing from Europe to the New World could sail south until they reached the trade winds, which would then propel their ships across the Atlantic to the Caribbean. To return to Europe, ships could sail to the northeast until they entered the westerlies, which would then steer them back to Europe.


16.E: Oscillatory Motion and Waves (Exercises)
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Conceptual Questions
16.1: Hooke’s Law: Stress and Strain Revisited
1. Describe a system in which elastic potential energy is stored.
16.3: Simple Harmonic Motion: A Special Periodic Motion
2. What conditions must be met to produce simple harmonic motion?
3. (a) If frequency is not constant for some oscillation, can the oscillation be simple harmonic motion?
(b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?
4 . Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
5. Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.
6. As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.
7. Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.
16.4: The Simple Pendulum
8. Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.
16.5: Energy and the Simple Harmonic Oscillator
9. Explain in terms of energy how dissipative forces such as friction reduce the amplitude of a harmonic oscillator. Also explain how a driving mechanism can compensate. (A pendulum clock is such a system.)
16.7: Damped Harmonic Motion
10. Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)
11. How would a car bounce after a bump under each of these conditions?
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overdamping
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underdamping
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critical damping
12. Most harmonic oscillators are damped and, if undriven, eventually come to a stop. How is this observation related to the second law of thermodynamics?
16.8: Forced Oscillations and Resonance
13. Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?
16.9: Waves
14. Give one example of a transverse wave and another of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.
15. What is the difference between propagation speed and the frequency of a wave? Does one or both affect wavelength? If so, how?
16.10: Superposition and Interference
16. Speakers in stereo systems have two color-coded terminals to indicate how to hook up the wires. If the wires are reversed, the speaker moves in a direction opposite that of a properly connected speaker. Explain why it is important to have both speakers connected the same way.
16.11: Energy in Waves: Intensity
17. Two identical waves undergo pure constructive interference. Is the resultant intensity twice that of the individual waves? Explain your answer.
18. Circular water waves decrease in amplitude as they move away from where a rock is dropped. Explain why.
Problems & Exercises
16.1: Hooke’s Law: Stress and Strain Revisited
19. Fish are hung on a spring scale to determine their mass (most fishermen feel no obligation to truthfully report the mass).
(a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load?
(b) What is the mass of a fish that stretches the spring 5.50 cm?
(c) How far apart are the half-kilogram marks on the scale?
Solution
(a) \(\displaystyle 1.23×10^3N/m\)
(b) \(\displaystyle 6.88 kg\)
(c) \(\displaystyle 4.00 mm\)
20. It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg.
(a) What is the spring’s effective spring constant?
(b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85 kg team?
21. One type of BB gun uses a spring-driven plunger to blow the BB from its barrel.
(a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s.
(b) What force must be exerted to compress the spring?
Solution
(a) 889 N/m
(b) 133 N
22. (a) The springs of a pickup truck act like a single spring with a force constant of \(\displaystyle 1.30×10^5N/m\). By how much will the truck be depressed by its maximum load of 1000 kg?
(b) If the pickup truck has four identical springs, what is the force constant of each?
23. When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m.
(a) What is the force constant of the spring?
(b) Will the spring be compressed more when he hops down the road?
Solution
(a) \(\displaystyle 6.53×10^3N/m\)
(b) Yes
24. A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it.
(a) What is the force constant of the spring?
(b) What is the unloaded length of the spring?
16.2: Period and Frequency in Oscillations
25. What is the period of \(\displaystyle 60.0Hz\) electrical power?
Solution
16.7 ms
26. If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?
Solution
0.400 s/beats
27. Find the frequency of a tuning fork that takes \(\displaystyle 2.50×10^{−3}s\) to complete one oscillation.
Solution
400 Hz
28. A stroboscope is set to flash every \(\displaystyle 8.00×10^{−5}s\). What is the frequency of the flashes?
Solution
12,500 Hz
29. A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?
Solution
1.50 kHz
30 . Engineering Application
Each piston of an engine makes a sharp sound every other revolution of the engine.
(a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer?
(b) At how many revolutions per minute is the engine rotating?
Solution
(a) 93.8 m/s
(b) \(\displaystyle 11.3×10^3\) rev/min
16.3: Simple Harmonic Motion: A Special Periodic Motion
31. A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?
Solution
\(\displaystyle 2.37N/m\)
32. If the spring constant of a simple harmonic oscillator is doubled, by what factor will the mass of the system need to change in order for the frequency of the motion to remain the same?
33. A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?
Solution
0.389 kg
34. By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?
35. Suppose you attach the object with mass \(\displaystyle m\) to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length.
(a) Show that the spring exerts an upward force of \(\displaystyle 2.00mg\) on the object at its lowest point.
(b) If the spring has a force constant of \(\displaystyle 10.0N/m\) and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations.
(c) Find the maximum velocity.
36. A diver on a diving board is undergoing simple harmonic motion. Her mass is 55.0 kg and the period of her motion is 0.800 s.  What is his mass if the mass of the board is negligible?
Solution
94.7 kg
37. Suppose a diving board with no one on it bounces up and down in a simple harmonic motion with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the simple harmonic motion of a 75.0-kg diver on the board?
38. The device pictured in Figure entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring constant.
(a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its spring constant?
(b) What is the time for one complete bounce of this child? (c) What is the child’s maximum velocity if the amplitude of her bounce is 0.200 m?
This child’s toy relies on springs to keep infants entertained. (credit: By Humboldthead, Flickr)
39. A 90.0-kg skydiver hanging from a parachute bounces up and down with a period of 1.50 s. What is the new period of oscillation when a second skydiver, whose mass is 60.0 kg, hangs from the legs of the first, as seen in Figure.
The oscillations of one skydiver are about to be affected by a second skydiver. (credit: U.S. Army, www.army.mil)
Solution
1.94 s
16.4: The Simple Pendulum
As usual, the acceleration due to gravity in these problems is taken to be g=9.80m/s2, unless otherwise specified.
40. What is the length of a pendulum that has a period of 0.500 s?
Solution
6.21 cm
41. Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?
42. What is the period of a 1.00-m-long pendulum?
Solution
2.01 s
43. How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?
44. The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?
Solution
2.23 Hz
45. Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?
46. (a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is \(\displaystyle 9.79m/s^2\) is moved to a location where it the acceleration due to gravity is \(\displaystyle 9.82m/s^2\). What is its new period?
(b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.
Solution
(a) 2.99541 s
(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by \(\displaystyle (0.01)^2=0.01%\) so it is necessary to have at least 4 digits after the decimal to see the changes.
47. A pendulum with a period of 2.00000 s in one location \(\displaystyle (g=9.80m/s^2)\) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?
48. (a) What is the effect on the period of a pendulum if you double its length?
(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?
Solution
(a) Period increases by a factor of 1.41 (\(\displaystyle \sqrt{2}\))
(b) Period decreases to 97.5% of old period
49. Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is \(\displaystyle 1.63m/s^2\).
50. At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is \(\displaystyle 1.63m/s^2\), if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.
Solution
Slow by a factor of 2.45
51. Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.
52. If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?
Solution
length must increase by 0.0116%.
16.5: Energy and the Simple Harmonic Oscillator
53. The length of nylon rope from which a mountain climber is suspended has a force constant of \(\displaystyle 1.40×10^4N/m\).
(a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg?
(b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? Hint: Use conservation of energy. Ignore the energy the climber gains as the rope stretches.
Solution
(a) \(\displaystyle 1.99 Hz\)
(b) 50.2 cm
54. Engineering Application
Near the top of the Citigroup Center building in New York City, there is an object with mass of \(\displaystyle 4.00×10^5\)kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building.
(a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s?
(b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?
Solution
(a) \(\displaystyle 3.95×10^6N/m\)
(b) \(\displaystyle 7.90×10^6J\)
16.6: Uniform Circular Motion and Simple Harmonic Motion
55. (a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of \(\displaystyle 1.50×10^6N/m\), if the amplitude of the bounce is 0.200 cm?
(b)What is the maximum energy stored in the spring?
Solution
a). 0.266 m/s
b). 3.00 J
56. A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position?
(b) How many joules of kinetic energy does the object have at its maximum velocity?
57. At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of \(\displaystyle x/X\) give \(\displaystyle v=±v_{max}/2\), where \(\displaystyle X\) is the amplitude of the motion?
Solution
\(\displaystyle ±\frac{\sqrt{3}}{2}\)
58. A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?
16.7: Damped Harmonic Motion
59. The amplitude of a lightly damped oscillator decreases by \(\displaystyle 3.0\%\) during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?
16.8: Forced Oscillations and Resonance
60. How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.
Solution
384 J
61. If a car has a suspension system with a force constant of \(\displaystyle 5.00×10^4N/m\), how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?
62. (a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring?
(b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance.
(c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.
Solution
(a). 0.123 m
(b). −0.600 J
(c). 0.300 J. The rest of the energy may go into heat caused by friction and other damping forces.
63. Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction \(\displaystyle μ_s=0.100\).
(a) How far can the spring be stretched without moving the mass?
(b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is \(\displaystyle μ_k=0.0850\), what total distance does it travel before stopping? Assume it starts at the maximum amplitude.
64. Engineering Application: A suspension bridge oscillates with an effective force constant of \(\displaystyle 1.00×10^8N/m\).
(a) How much energy is needed to make it oscillate with an amplitude of 0.100 m?
(b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart \(\displaystyle 1.00×10^4J\) of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude?
Solution
(a) \(\displaystyle 5.00×10^5J\)
(b) \(\displaystyle 1.20×10^3s\)
16.9: Waves
65. Storms in the South Pacific can create waves that travel all the way to the California coast, which are 12,000 km away. How long does it take them if they travel at 15.0 m/s?
Solution
t=9.26 d
66. Waves on a swimming pool propagate at 0.750 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.0 s. How far away is the other end of the pool?
67. Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?
Solution
f=40.0 Hz
68. How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?
69. Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake it the bridge twice per second, what is the propagation speed of the waves?
Solution
\(\displaystyle v_w=16.0 m/s\)
70. What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at 0.800 m/s?
71. What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?
Solution
λ=700 m
72. Radio waves transmitted through space at \(\displaystyle 3.00×10^8m/s\) by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?
73. Your ear is capable of differentiating sounds that arrive at the ear just 1.00 ms apart. What is the minimum distance between two speakers that produce sounds that arrive at noticeably different times on a day when the speed of sound is 340 m/s?
Solution
d=34.0 cm
74. (a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, they compare the arrival times of S- and P-waves, which travel at different speeds. Figure) If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined?
(b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)
A seismograph as described in above problem.(credit: Oleg Alexandrov)
16.10: Superposition and Interference
75. A car has two horns, one emitting a frequency of 199 Hz and the other emitting a frequency of 203 Hz. What beat frequency do they produce?
Solution
\(\displaystyle f=4 Hz\)
76. The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?
77. Two tuning forks having frequencies of 460 and 464 Hz are struck simultaneously. What average frequency will you hear, and what will the beat frequency be?
Solution
462 Hz,
4 Hz
78. Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?
79. A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again.
(a) What is the speed of the wave?
(b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?
Solution
(a) 3.33 m/s
(b) 1.25 Hz
80. Three adjacent keys on a piano (F, F-sharp, and G) are struck simultaneously, producing frequencies of 349, 370, and 392 Hz. What beat frequencies are produced by this discordant combination?
16.11: Energy in Waves: Intensity
81. Medical Application
Ultrasound of intensity \(\displaystyle 1.50×10^2W/m^2\) is produced by the rectangular head of a medical imaging device measuring 3.00 by 5.00 cm. What is its power output?
Solution
0.225 W
82. The low-frequency speaker of a stereo set has a surface area of \(\displaystyle 0.05m^2\) and produces 1W of acoustical power. What is the intensity at the speaker? If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity \(\displaystyle 0.1W/m^2\)?
83. To increase intensity of a wave by a factor of 50, by what factor should the amplitude be increased?
Solution
7.07
84. Engineering Application
A device called an insolation meter is used to measure the intensity of sunlight has an area of \(\displaystyle 100 cm^2\) and registers 6.50 W. What is the intensity in \(\displaystyle W/m^2\)?
85. Astronomy Application
Energy from the Sun arrives at the top of the Earth’s atmosphere with an intensity of \(\displaystyle 1.30kW/m^2\). How long does it take for \(\displaystyle 1.8×10^9J\) to arrive on an area of \(\displaystyle 1.00m^2\)?
Solution
16.0 d
86. Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?
Solution
2.50 kW
87. Engineering Application
(a) A photovoltaic array of (solar cells) is 10.0% efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is \(\displaystyle 700W/m^2\), what area should your array have to gather energy at the rate of 100 W?
(b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 ¢ per kilowatt-hour.
88. A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally \(\displaystyle 2.00×10^{–5}W/m^2\), but is turned up until the amplitude increases by 30.0%, what is the new intensity?
Solution
\(\displaystyle 3.38×10^{–5}W/m^2\)
89. Medical Application
(a) What is the intensity in \(\displaystyle W/m^2\) of a laser beam used to burn away cancerous tissue that, when 90.0% absorbed, puts 500 J of energy into a circular spot 2.00 mm in diameter in 4.00 s?
(b) Discuss how this intensity compares to the average intensity of sunlight (about \(\displaystyle 700W/m^2\)) and the implications that would have if the laser beam entered your eye. Note how your answer depends on the time duration of the exposure.
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) .


14.9: Junction Field-effect Transistors
A field effect transistor (FET) is a unipolar device, conducting a current using only one kind of charge carrier. If based on an N-type slab of semiconductor, the carriers are electrons. Conversely, a P-type based device uses only holes.
At the circuit level, field effect transistor operation is simple. A voltage applied to the gate , input element, controls the resistance of the channel , the unipolar region between the gate regions. (Figure below) In an N-channel device, this is a lightly doped N-type slab of silicon with terminals at the ends. The source and drain terminals are analogous to the emitter and collector, respectively, of a BJT. In an N-channel device, a heavy P-type region on both sides of the center of the slab serves as a control electrode, the gate. The gate is analogous to the base of a BJT.
“Cleanliness is next to godliness” applies to the manufacture of field effect transistors. Though it is possible to make bipolar transistors outside of a clean room , it is a necessity for field effect transistors. Even in such an environment, manufacture is tricky because of contamination control issues. The unipolar field effect transistor is conceptually simple, but difficult to manufacture. Most transistors today are a metal oxide semiconductor variety (later section) of the field effect transistor contained within integrated circuits. However, discrete JFET devices are available.
Junction field effect transistor cross-section.
A properly biased N-channel junction field effect transistor (JFET) is shown in Figure above. The gate constitutes a diode junction to the source to drain semiconductor slab. The gate is reverse biased. If a voltage (or an ohmmeter) were applied between the source and drain, the N-type bar would conduct in either direction because of the doping. Neither gate nor gate bias is required for conduction. If a gate junction is formed as shown, conduction can be controlled by the degree of reverse bias.
Figure below(a) shows the depletion region at the gate junction. This is due to diffusion of holes from the P-type gate region into the N-type channel, giving the charge separation about the junction, with a non-conductive depletion region at the junction. The depletion region extends more deeply into the channel side due to the heavy gate doping and light channel doping.
N-channel JFET: (a) Depletion at gate diode. (b) Reverse biased gate diode increases depletion region. (c) Increasing reverse bias enlarges depletion region. (d) Increasing reverse bias pinches-off the S-D channel.
The thickness of the depletion region can be increased Figure above(b) by applying moderate reverse bias. This increases the resistance of the source to drain channel by narrowing the channel. Increasing the reverse bias at (c) increases the depletion region, decreases the channel width, and increases the channel resistance. Increasing the reverse bias V GS at (d) will pinch-off the channel current. The channel resistance will be very high. This V GS at which pinch-off occurs is V P , the pinch-off voltage. It is typically a few volts. In summation, the channel resistance can be controlled by the degree of reverse biasing on the gate.
The source and drain are interchangeable, and the source to drain current may flow in either direction for low level drain battery voltage (< 0.6 V). That is, the drain battery may be replaced by a low voltage AC source. For a high drain power supply voltage, to 10’s of volts for small signal devices, the polarity must be as indicated in Figure below(a). This drain power supply, not shown in previous figures, distorts the depletion region, enlarging it on the drain side of the gate. This is a more correct representation for common DC drain supply voltages, from a few to tens of volts. As drain voltage V DS increased,the gate depletion region expands toward the drain. This increases the length of the narrow channel, increasing its resistance a little. We say “a little” because large resistance changes are due to changing gate bias. Figure below(b) shows the schematic symbol for an N-channel field effect transistor compared to the silicon cross-section at (a). The gate arrow points in the same direction as a junction diode. The “pointing” arrow and “non-pointing” bar correspond to P and N-type semiconductors, respectively.
N-channel JFET electron current flow from source to drain in (a) cross-section, (b) schematic symbol.
Figure above shows a large electron current flow from (-) battery terminal, to FET source, out the drain, returning to the (+) battery terminal. This current flow may be controlled by varying the gate voltage. A load in series with the battery sees an amplified version of the changing gate voltage.
P-channel field effect transistors are also available. The channel is made of P-type material. The gate is a heavily dopped N-type region. All the voltage sources are reversed in the P-channel circuit (Figure below) as compared with the more popular N-channel device. Also note, the arrow points out of the gate of the schematic symbol (b) of the P-channel field effect transistor.
P-channel JFET: (a) N-type gate, P-type channel, reversed voltage sources compared with N-channel device. (b) Note reversed gate arrow and voltage sources on schematic.
As the positive gate bias voltage is increased, the resistance of the P-channel increases, decreasing the current flow in the drain circuit.
Discrete devices are manufactured with the cross-section shown in Figure below. The cross-section, oriented so that it corresponds to the schematic symbol, is upside down with respect to a semiconductor wafer. That is, the gate connections are on the top of the wafer. The gate is heavily doped, P + , to diffuse holes well into the channel for a large depletion region. The source and drain connections in this N-channel device are heavily doped, N + to lower connection resistance. However, the channel surrounding the gate is lightly doped to allow holes from the gate to diffuse deeply into the channel. That is the N - region.
Junction field effect transistor: (a) Discrete device cross-section, (b) schematic symbol, (c) integrated circuit device cross-section.
All three FET terminals are available on the top of the die for the integrated circuit version so that a metalization layer (not shown) can interconnect multiple components. (Figure above(c) ) Integrated circuit FET’s are used in analog circuits for the high gate input resistance.. The N-channel region under the gate must be very thin so that the intrinsic region about the gate can control and pinch-off the channel. Thus, gate regions on both sides of the channel are not necessary.
Junction field effect transistor (static induction type): (a) Cross-section, (b) schematic symbol.
The static induction field effect transistor (SIT) is a short channel device with a buried gate. (Figure above) It is a power device, as opposed to a small signal device. The low gate resistance and low gate to source capacitance make for a fast switching device. The SIT is capable of hundreds of amps and thousands of volts. And, is said to be capable of an incredible frequency of 10 gHz.
Metal semiconductor field effect transistor (MESFET): (a) schematic symbol, (b) cross-section.The Metal semiconductor field effect transistor (MESFET) is similar to a JFET except the gate is a schottky diode instead of a junction diode. A schottky diode is a metal rectifying contact to a semiconductor compared with a more common ohmic contact. In Figure above the source and drain are heavily doped (N + ). The channel is lightly doped (N - ). MESFET’s are higher speed than JFET’s. The MESET is a depletion mode device, normally on, like a JFET. They are used as microwave power amplifiers to 30 gHz. MESFET’s can be fabricated from silicon, gallium arsenide, indium phosphide, silicon carbide, and the diamond allotrope of carbon.
Review
- The unipolar junction field effect transistor (FET or JFET) is so called because conduction in the channel is due to one type of carrier
- The JFET source, gate, and drain correspond to the BJT’s emitter, base, and collector, respectively.
- Application of reverse bias to the gate varies the channel resistance by expanding the gate diode depletion region.


1.6: Measurement Uncertainty, Accuracy, and Precision
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Learning Objectives
By the end of this section, you will be able to:
- Define accuracy and precision
- Distinguish exact and uncertain numbers
- Correctly represent uncertainty in quantities using significant figures
- Apply proper rounding rules to computed quantities
Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number . By counting the eggs in a carton, one can determine exactly how many eggs the carton contains. The numbers of defined quantities are also exact. By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.
Significant Figures in Measurement
The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.
Refer to the illustration in Figure \(\PageIndex{1}\). The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.
This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram. If the coin is weighed on a more sensitive balance, the mass might be 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some uncertainty , which depends on the device used (and the user’s ability). All of the digits in a measurement, including the uncertain last digit, are called significant figures or significant digits . Note that zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows “120,” then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.
A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. But what if you were analyzing a reported value and trying to determine what is significant and what is not? Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought. We will use the terms “leading,” “trailing,” and “captive” for the zeros and will consider how to deal with them.
Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point.
Captive zeros result from measurement and are therefore always significant. Leading zeros, however, are never significant—they merely tell us where the decimal point is located.
The leading zeros in this example are not significant. We could use exponential notation (as described in Appendix B) and express the number as \(8.32407 \times
10^{−3}\); then the number 8.32407 contains all of the significant figures, and \(10^{−3}\) locates the decimal point.
The number of significant figures is uncertain in a number that ends with a zero to the left of the decimal point location. The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located. The ambiguity can be resolved with the use of exponential notation: \(1.3\times 10^3\) (two significant figures), \(1.30 \times 10^3\) (three significant figures, if the tens place was measured), or \(1.300 \times 10^3\) (four significant figures, if the ones place was also measured). In cases where only the decimal-formatted number is available, it is prudent to assume that all trailing zeros are not significant.
When determining significant figures, be sure to pay attention to reported values and think about the measurement and significant figures in terms of what is reasonable or likely when evaluating whether the value makes sense. For example, the official January 2014 census reported the resident population of the US as 317,297,725. Do you think the US population was correctly determined to the reported nine significant figures, that is, to the exact number of people? People are constantly being born, dying, or moving into or out of the country, and assumptions are made to account for the large number of people who are not actually counted. Because of these uncertainties, it might be more reasonable to expect that we know the population to within perhaps a million or so, in which case the population should be reported as 3.17 10 8 people.
Significant Figures in Calculations
A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself. Take the uncertainty in measurements into account to avoid misrepresenting the uncertainty in calculated results. One way to do this is to report the result of a calculation with the correct number of significant figures, which is determined by the following three rules for rounding numbers:
- When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places (the least certain value in terms of addition and subtraction).
- When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures (the least certain value in terms of multiplication and division).
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, “round down” and leave the retained digit unchanged; if it is more than 5, “round up” and increase the retained digit by 1. If the dropped digit is 5, and it’s either the last digit in the number or it’s followed only by zeros, round up or down, whichever yields an even value for the retained digit. If any nonzero digits follow the dropped 5, round up. (The last part of this rule may strike you as a bit odd, but it’s based on reliable statistics and is aimed at avoiding any bias when dropping the digit “5,” since it is equally close to both possible values of the retained digit.)
The following examples illustrate the application of this rule in rounding a few different numbers to three significant figures:
- 0.028675 rounds “up” to 0.0287 (the dropped digit, 7, is greater than 5)
- 18.3384 rounds “down” to 18.3 (the dropped digit, 3, is less than 5)
- 6.8752 rounds “up” to 6.88 (the dropped digit is 5, and a nonzero digit follows it)
- 92.85 rounds “down” to 92.8 (the dropped digit is 5, and the retained digit is even)
Let’s work through these rules with a few examples.
Example \(\PageIndex{1}\): Rounding Numbers
Round the following to the indicated number of significant figures:
- 31.57 (to two significant figures)
- 8.1649 (to three significant figures)
- 0.051065 (to four significant figures)
- 0.90275 (to four significant figures)
Solution
- 31.57 rounds “up” to 32 (the dropped digit is 5, and the retained digit is even)
- 8.1649 rounds “down” to 8.16 (the dropped digit, 4, is less than 5)
- 0.051065 rounds “down” to 0.05106 (the dropped digit is 5, and the retained digit is even)
- 0.90275 rounds “up” to 0.9028 (the dropped digit is 5, and the retained digit is even)
Exercise \(\PageIndex{1}\)
Round the following to the indicated number of significant figures:
- 0.424 (to two significant figures)
- 0.0038661 (to three significant figures)
- 421.25 (to four significant figures)
- 28,683.5 (to five significant figures)
- Answer
-
- 0.42
- 0.00387
- 421.2
- 28,684
Example \(\PageIndex{2}\): Addition and Subtraction with Significant Figures
Rule: When adding or subtracting numbers, round the result to the same number of decimal places as the number with the fewest decimal places (i.e., the least certain value in terms of addition and subtraction).
- Add 1.0023 g and 4.383 g.
- Subtract 421.23 g from 486 g.
Solution
(a)
\[\begin{align*}
1&.0023 \, \text{g} \\
+\quad 4&.383 \, \text{g}\\
\hline 5&.3853\, \text{g}
\end{align*} \]
Answer is 5.385 g (round to the thousandths place; three decimal places)
(b)
\[\begin{align*}
486& \, \text{g} \\
-\quad 421&.23 \, \text{g}\\
\hline 64&.77\, \text{g}
\end{align*} \]
Answer is 65 g (round to the ones place; no decimal places)
Exercise \(\PageIndex{2}\)
- Add 2.334 mL and 0.31 mL.
- Subtract 55.8752 m from 56.533 m.
- Answer
-
- 2.64 mL
- 0.658 m
Example \(\PageIndex{3}\): Multiplication and Division with Significant Figures
Rule: When multiplying or dividing numbers, round the result to the same number of digits as the number with the fewest significant figures (the least certain value in terms of multiplication and division).
- Multiply 0.6238 cm by 6.6 cm.
- Divide 421.23 g by 486 mL.
Solution
(a)
\[\underbrace{0.6238\,\text{cm}}_{\text{four significant figures}} \times \underbrace{6.6\, \text{cm}}_{\text{two significant figures}} =4.11708\, \text{cm}^2 \rightarrow \underbrace{4.1\, \text{cm}^2}_{\text{round to two significant figures}} \nonumber \]
(b)
\[\underbrace{421.23\,\text{g}}_{\text{five significant figures}} ÷ \underbrace{486\, \text{mL}}_{\text{three significant figures}} = 0.866728... \, \text{g}/\text{mL} \rightarrow \underbrace{0.867\, \text{g}/\text{mL} }_{\text{round to three significant figures}} \nonumber \]
Exercise \(\PageIndex{3}\)
- Multiply 2.334 cm and 0.320 cm.
- Divide 55.8752 m by 56.53 s.
- Answer
-
- 0.747 cm 2
- 0.9884 m/s
In the midst of all these technicalities, it is important to keep in mind the reason for these rules about significant figures and rounding—to correctly represent the certainty of the values reported and to ensure that a calculated result is not represented as being more certain than the least certain value used in the calculation.
Example \(\PageIndex{4}\): Calculation with Significant Figures
One common bathtub is 13.44 dm long, 5.920 dm wide, and 2.54 dm deep. Assume that the tub is rectangular and calculate its approximate volume in liters.
Solution
\[\begin{aligned}
V & =l \times w \times d \\
& =13.44 \, \text{dm} \times 5.920 \, \text{dm} \times 2.54 \, \text{dm} \\
& =202.09459 \ldots \, \text{dm}^3 \text { (value from calculator) } \\
& =202 \, \text{dm}^3 \text { or } 202\, \text{L} \text { (answer rounded to three significant figures) }
\end{aligned} \nonumber \]
Exercise \(\PageIndex{4}\)
What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm 3 ?
- Answer
-
1.034 g/mL
Example \(\PageIndex{5}\): Experimental Determination of Density Using Water Displacement
A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.
- Use these values to determine the density of this piece of rebar.
- Rebar is mostly iron. Does your result in (a) support this statement? How?
Solution
The volume of the piece of rebar is equal to the volume of the water displaced:
\[\text { volume }=22.4\, \text{mL} - 13.5\, \text{mL} =8.9\, \text{mL} = 8.9\, \text{cm}^3 \nonumber \]
(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)
The density is the mass-to-volume ratio:
\[\text { density }=\frac{\text { mass }}{\text { volume }}=\frac{69.658\, \text{g} }{8.9\, \text{cm}^3}=7.8\, \text{g} / \text{cm}^3 \nonumber \]
(rounded to two significant figures, per the rule for multiplication and division)
The density of iron is 7.9 g/cm 3 , very close to that of rebar, which lends some support to the fact that rebar is mostly iron.
Exercise \(\PageIndex{5}\)
An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder, with results as shown.
- Use these values to determine the density of this material.
- Do you have any reasonable guesses as to the identity of this material? Explain your reasoning.
- Answer
-
19 g/cm 3
It is likely gold; the right appearance for gold and very close to the known density of gold at \(19.3 \, \text{g}/\text{cm}^3\)
Accuracy and Precision
Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results. Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value. These characterizations can be extended to other contexts, such as the results of an archery competition (Figure \(\PageIndex{2}\)).
Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines that are meant to dispense 10 ounces (296 mL) of cough syrup into storage bottles. She proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, obtaining the results tabulated in Table \(\PageIndex{1}\).
| Dispenser #1 | Dispenser #2 | Dispenser #3 |
|---|---|---|
| 283.3 | 298.3 | 296.1 |
| 284.1 | 294.2 | 295.9 |
| 283.9 | 296.0 | 296.1 |
| 284.0 | 297.8 | 296.0 |
| 284.1 | 293.9 | 296.1 |
Considering these results, she will report that dispenser #1 is precise (values all close to one another, within a few tenths of a milliliter) but not accurate (none of the values are close to the target value of 296 mL, each being more than 10 mL too low). Results for dispenser #2 represent improved accuracy (each volume is less than 3 mL away from 296 mL) but worse precision (volumes vary by more than 4 mL). Finally, she can report that dispenser #3 is working well, dispensing cough syrup both accurately (all volumes within 0.1 mL of the target volume) and precisely (volumes differing from each other by no more than 0.2 mL).


9: World War II
American soldiers recover the dead on Omaha Beach in 1944. Library of Congress .
-
- 9.4: The United States and the European War
- While Hitler marched across Europe, the Japanese continued their war in the Pacific. In 1939 the United States dissolved its trade treaties with Japan and the following year cut off supplies of war materials by embargoing oil, steel, rubber, and other vital goods. It was hoped that economic pressure would shut down the Japanese war machine. Instead, Japan’s resource-starved military launched invasions across the Pacific to sustain its war effort.
-
- 9.5: The United States and the Japanese War
- As Americans celebrated V-E (Victory in Europe) Day, they redirected their full attention to the still-raging Pacific War. As in Europe, the war in the Pacific started slowly. After Pearl Harbor, the American-controlled Philippine archipelago fell to Japan. After running out of ammunition and supplies, the garrison of American and Filipino soldiers surrendered. The prisoners were marched eighty miles to their prisoner-of-war camp without food, water, or rest. Ten thousand died.
-
- 9.6: Soldiers' Experiences
- Almost eighteen million men served in World War II. Volunteers rushed to join the military after Pearl Harbor, but the majority—over ten million—were drafted into service. Volunteers could express their preference for assignment, and many preempted the draft by volunteering. Regardless, recruits judged I-A, “fit for service,” were moved into basic training, where soldiers were developed physically and trained in the basic use of weapons and military equipment.
-
- 9.7: The Wartime Economy
- Economies win wars no less than militaries. The war converted American factories to wartime production, reawakened Americans’ economic might, armed Allied belligerents and the American armed forces, effectively pulled America out of the Great Depression, and ushered in an era of unparalleled economic prosperity.
-
- 9.8: Women and World War II
- President Franklin D. Roosevelt and his administration had encouraged all able-bodied American women to help the war effort. He considered the role of women in the war critical for American victory, and the public expected women to assume various functions to free men for active military service. While most women opted to remain at home or volunteer with charitable organizations, many went to work or donned a military uniform.
-
- 9.10: Toward a Postwar World
- Americans celebrated the end of the war. At home and abroad, the United States looked to create a postwar order that would guarantee global peace and domestic prosperity. Although the alliance of convenience with Stalin’s Soviet Union would collapse, Americans nevertheless looked for the means to ensure postwar stability and economic security for returning veterans.
Thumbnail: Raising the Flag on Iwo Jima. (Public Domain; Joe Rosenthal @ the Associated Press).


1.5: Common Suffixes
The suffix is at the end of the word and adds meaning. When building a definition of a medical term from its components, start with the meaning of the suffix. For example, hepat/ itis is defined as “inflammation of the liver.”
Suffixes often indicate:
- Procedures: For example, -scopy means “visual examination”
- Conditions: For example, -itis means “inflammation”
- Diseases: For example, -oma means “tumor”
Common Suffixes
Here is a list of commonly used suffixes in medical terms. It is helpful to memorize these common suffixes as you build your knowledge of medical terminology.
- -ac : Pertaining to
- -ad : Toward
- -al : Pertaining to
- -algia, -dynia : Pain
- -ar : Pertaining to
- -ary : Pertaining to
- -cele : Protrusion
- -centesis : Surgical withdrawal of fluid
- -cide, -cidal : Kill, destroy
- -coccus : Berry shaped
- -cyte : Cell
- -eal : Pertaining to
- -ectomy : Surgical removal
- -emia : Pertaining to blood
- -esis : Condition
- -gen : Substance that produces/causes
- -genic : Produced by
- -gram : Record of
- -graph : Instrument used to record
- -graphy : A process of recording
- -ia : Condition of, diseased state, abnormal state
- -ic, -ior : Pertaining to
- -ism : State of, condition, theory
- -itis : Inflammation
- -logist : Specialist who studies and treats disorders
- -logy : Study of
- -lysis : Destruction, breakdown
- -megaly : Enlargement
- -meter : Instrument used to measure
- -metry : Process of measuring
- -oid : Resembling
- -oma : Tumor
- -opia : Vision condition
- -opsy : To view
- -osis : Abnormal condition
- -ous : Pertaining to
- -pathy : Disease, abnormality
- -penia : Lack of, deficiency
- -pexy : Surgical fixation, suspension
- -phagia : Eating
- -phobia : Intense fear of
- -plasia : Formation, development
- -plasm : Growth, substance, formation
- -plasty : Surgical reconstruction
- -plegia : Paralysis
- -pnea : Breathing
- -poiesis, -poietic : Production, manufacture of
- -ptosis : Drooping
- -rrhage : Excessive flow
- – rrhagia : Excessive flow
- -rrhaphy : Suturing in place, fixation
- -rrhea : Flow, discharge
- -rrhexis : Rupture or breaking away
- -sclerosis : Hardening
- -scope : Instrument used to visually examine
- -scopic : Visual examination
- -scopy : Process of visual examination
- -spasm : Sudden, involuntary
- -stasis : To stop
- -stomy : Artificial surgical opening
- -tome : Instrument used to cut
- -tomy : Process of incision, cutting into
- -tripsy : Crushing
- -trophic, -trophy : Growth, development
Note there are several suffixes that mean “pertaining to,” such as -ac, -al, -ar, -ary, -eal, -ic, -ior, and -ous.
Examples of Common Suffixes in Medical Terms
- Arachno phobia : Fear of spiders
- Amnio centesis : Procedure to withdraw fluid from the amniotic sac
- An emia : Low levels of hemoglobin in the blood
- Cardio megaly : Enlargement of the heart
- Colono scopy : Procedure that visually examines the colon
- Dys pnea : Difficulty breathing
- Hemi plegia : Paralysis of one side of the body
- Mammo graphy : Radiological process of examining the breast tissue
- My algia : Muscle pain
Additional suffixes will be discussed in each chapter related to specific body systems.


13.4: Gross Anatomy of the Kidney
The kidneys lie on either side of the spine in the retroperitoneal space between the parietal peritoneum and the posterior abdominal wall, well protected by muscle, fat, and ribs. They are roughly the size of your fist, and the male kidney is typically a bit larger than the female kidney. The kidneys are well vascularized, receiving about 25 percent of the cardiac output at rest.
External Anatomy
The left kidney is located at about the T12 to L3 vertebrae, whereas the right is lower due to slight displacement by the liver. Upper portions of the kidneys are somewhat protected by the eleventh and twelfth ribs. Each kidney weighs about 125–175 g in males and 115–155 g in females. They are about 11–14 cm in length, 6 cm wide, and 4 cm thick, and are directly covered by a fibrous capsule composed of dense, irregular connective tissue that helps to hold their shape and protect them. This capsule is covered by a shock-absorbing layer of adipose tissue called the renal fat pad, which in turn is encompassed by a tough renal fascia. The fascia and, to a lesser extent, the overlying peritoneum serve to firmly anchor the kidneys to the posterior abdominal wall in a retroperitoneal position.
On the superior aspect of each kidney is the adrenal gland. The adrenal cortex directly influences renal function through the production of the hormone aldosterone to stimulate sodium reabsorption.
Internal Anatomy
A frontal section through the kidney reveals an outer region called the renal cortex and an inner region called the medulla. The renal columns are connective tissue extensions that radiate downward from the cortex through the medulla to separate the most characteristic features of the medulla, the renal pyramids and renal papillae. The papillae are bundles of collecting ducts that transport urine made by nephrons to the calyces of the kidney for excretion. The renal columns also serve to divide the kidney into 6–8 lobes and provide a supportive framework for vessels that enter and exit the cortex. The pyramids and renal columns taken together constitute the kidney lobes.
Renal Hilum
The renal hilum is the entry and exit site for structures servicing the kidneys: vessels, nerves, lymphatics, and ureters. The medial-facing hila are tucked into the sweeping convex outline of the cortex. Emerging from the hilum is the renal pelvis, which is formed from the major and minor calyxes in the kidney. The smooth muscle in the renal pelvis funnels urine via peristalsis into the ureter. The renal arteries form directly from the descending aorta, whereas the renal veins return cleansed blood directly to the inferior vena cava. The artery, vein, and renal pelvis are arranged in an anterior-to-posterior order.
Nephrons and Vessels
The renal artery first divides into segmental arteries, followed by further branching to form interlobar arteries that pass through the renal columns to reach the cortex. The interlobar arteries, in turn, branch into arcuate arteries, cortical radiate arteries, and then into afferent arterioles. The afferent arterioles service about 1.3 million nephrons in each kidney.
Nephrons are the “functional units” of the kidney; they cleanse the blood and balance the constituents of the circulation. The afferent arterioles form a tuft of high-pressure capillaries about 200 µm in diameter, the glomerulus. The rest of the nephron consists of a continuous sophisticated tubule whose proximal end surrounds the glomerulus in an intimate embrace—this is Bowman’s capsule. The glomerulus and Bowman’s capsule together form the renal corpuscle. As mentioned earlier, these glomerular capillaries filter the blood based on particle size. After passing through the renal corpuscle, the capillaries form a second arteriole, the efferent arteriole. These will next form a capillary network around the more distal portions of the nephron tubule, the peritubular capillaries and vasa recta, before returning to the venous system. As the glomerular filtrate progresses through the nephron, these capillary networks recover most of the solutes and water, and return them to the circulation. Since a capillary bed (the glomerulus) drains into a vessel that in turn forms a second capillary bed, the definition of a portal system is met. This is the only portal system in which an arteriole is found between the first and second capillary beds. (Portal systems also link the hypothalamus to the anterior pituitary, and the blood vessels of the digestive viscera to the liver.)
Cortex
In a dissected kidney, it is easy to identify the cortex; it appears lighter in color compared to the rest of the kidney. All of the renal corpuscles as well as both the proximal convoluted tubules (PCTs) and distal convoluted tubules are found here. Some nephrons have a short loop of Henle that does not dip beyond the cortex. These nephrons are called cortical nephrons. About 15 percent of nephrons have long loops of Henle that extend deep into the medulla and are called juxtamedullary nephrons.
Contributors and Attributions
- Printing a human kidney - Anthony Atala. Provided by : TED-Ed. Located at : https://youtu.be/bX3C201O4MA . License : All Rights Reserved . License Terms : Standard YouTube License


2.3: Economic Systems
What you’ll learn to do: describe and differentiate between major different economic systems
In this section, you’ll learn about the basic organizing principles of different types of economies. Understanding the characteristics of a competitive market, in particular, is an important foundation for understanding the mechanisms of supply and demand, which you’ll learn about later in this module.
Learning Objectives
- Distinguish between market, planned, and mixed economies
Economic Systems
Consider how complex a modern economy is. It includes all production of goods and services, all buying and selling, all employment. The economic life of every individual is interrelated, at least to a small extent, with the economic lives of thousands or even millions of other individuals. Who organizes and coordinates this system? Who insures that, for example, the number of televisions a society produces is the same as the amount it needs and wants? Who insures that the right number of employees works in the electronics industry? Who makes sure that televisions are produced in the best way possible? How does it all get done? The answer to these important questions depends on the kind of economic system a society uses.
On the right : Figure \(\PageIndex{1}\) Legoland, Billund, Denmark: Picture of a Planned Economy?
In the modern world today, there is a range of economic systems, from market economies to planned (or command) economies.
Market Economies
A market is any situation that brings together buyers and sellers of goods or services. Buyers and sellers can be either individuals or businesses. In a market economy, economic decision-making happens through markets. Market economies are based on private enterprise: the means of production (resources and businesses) are owned and operated by private individuals or groups of private individuals. Businesses supply goods and services based on demand. Which goods and services are supplied depends on what products businesses think will bring them the most profit. The more a product is demanded by consumers or other businesses, the higher the price businesses can charge, and so the more of the product will be supplied. Consumer demand depends on peoples’ incomes. A person’s income is based on his or her ownership of resources (especially labor). The more society values the person’s output, the higher the income they will earn (think Lady Gaga or LeBron James).
Examples of free-market economies include Hong Kong, Singapore, and to a large extent, New Zealand, and the United States.
Free Markets
In a market economy , decisions about what products are available and at what prices are determined through the interaction of supply and demand. A competitive market is one in which there is a large number of buyers and sellers, so that no one can control the market price. A free market is one in which the government does not intervene in any way. A free and competitive market economy is the ideal type of market economy, because what is supplied is exactly what consumers demand.
Price controls are an example of a market that is not free. When government intervenes, the market outcomes will be different from those that would occur in a free and competitive market model. When markets are less than perfectly competitive (e.g., monopolistic), the market outcomes will also differ.
practice question \(\PageIndex{1}\)
Which of the following is a defining feature of free markets?
- There is a large number of buyers and sellers.
- The government does not interfere in any way with the economic decisions of buyers and sellers
- The means of production are owned by government, not private individuals.
- Answer
-
b is correct - This is the key defining feature of free markets. The government allows the forces of supply and demand to determine quantity and price.
a is incorrect - This is not a necessary feature for a free market. A market in which a limited number of buyers or sellers control the market would still be a free market just not a competitive market.
c is incorrect - Private ownership of the means of production (labor, factories, businesses,etc.) is a defining feature of market economies and therefore free markets.
practice question \(\PageIndex{2}\)
The economy of Atlantia is defined by the private ownership of the means of production by Atlantia’s citizens. While owners are usually free to utilize their property as they see fit, the government does intervene to ban certain harmful products (i.e. addictive drugs, lead paint, etc) from entering the market. Atlantia’s parliament has recently passed a minimum wage law setting minimum hourly compensation at $5.00 per hour. Atlantia is an example of a ________ economy.
- mixed market
- free market
- command
- Answer
-
a is correct - Private ownership of the means of production along with limited government intervention are characteristics of a market economy.
Planned (or Command) Economies
Command economies operate very differently. In a command economy , economic effort is devoted to goals passed down from a ruler or ruling class. Ancient Egypt was a good example: A large part of economic life was devoted to building pyramids (like the one at the left), for the pharaohs. Medieval manor life is another example: The lord provided the land for growing crops and protection in the event of war. In return, vassals provided labor and soldiers to do the lord’s bidding. In the last century, communism emphasized command economies.
In a command economy, resources and businesses are owned by the government. The government decides what goods and services will be produced and what prices will be charged for them. The government decides what methods of production will be used and how much workers will be paid. Some necessities like health care and education are provided for free, as long as the state determines that you need them. Currently, North Korea and Cuba have command economies.
The primary distinction between a free and command economy is the degree to which the government determines what can be produced and what prices will be charged. In a free market, these determinations are made by the collective decisions of the market itself (which is comprised of producers and consumers). Producers and consumers make rational decisions about what will satisfy their self-interest and maximize profits, and the market responds accordingly. In a planned economy, the government makes most decisions about what will be produced and what the prices will be, and the market must follow that plan.
Most economies in the real world are mixed; they combine elements of command and market systems. The U.S. economy is positioned toward the market-oriented end of the spectrum. Many countries in Europe and Latin America, while primarily market-oriented, have a greater degree of government involvement in economic decisions than in the U.S. economy. China and Russia, while they are closer now to having a market-oriented system than several decades ago, remain closer to the command-economy end of the spectrum.
practice question \(\PageIndex{3}\)
Jane Doe is a shop owner in the fictional country of Xanadu. Every month the government’s planning ministry sends her a large booklet (which resembles a phone book) regulating the price of essential commodities including milk, flour, and eggs. In response to these regulations Jane Doe has reduced her inventory of essential goods and switched to selling luxuries such as cakes, pies, and soft drinks. Luxury goods are not price controlled. Does Jane Doe reside in a command economy? Why or why not?
- Jane resides in a command economy, as the government regulates some prices
- Jane does not reside in a command economy.
- Jane does not reside in a command economy. She owns her own business and is free to make some production and price decisions
- Answer
-
c is correct - The government's extensive intervention in the economy combined with private ownership and limited independent market activity allows us to best characterize Xanadu's economy as 'mixed'.
a is incorrect - While Xanadu's government intervenes extensively in the economy, Jane's business is still privately owned and free to make some independent decisions. These are not features of a total command economy.
b is incorrect - While Jane does not reside in a command economy, lack of money is not a defining feature of command economies.
practice question \(\PageIndex{4}\)
You are the leader of a revolutionary movement which has recently overthrown the government of the tiny country of Tropico. During the 3 rd Party Congress, which was held shortly before your glorious victory, the party committee decreed that “to free Tropico’s struggling masses from the jackboots of capitalism and reactionary imperialism, it is necessary to immediately enforce a command economy upon realization of our final victory.” Which of the following actions would betray your mandate to create a command economy in Tropico?
- Create a Ministry of Planning which will set prices, wages, and production targets.
- Immediately expropriating all private property, which will now be state-owned
- Maintain private ownership, but legalize labor unions which will now be free to bargain with private businesses for improved wages and benefits.
- Answer
-
c is correct - This form of government intervention is limited and allows for the continued independence of privately owned businesses and resources. Their continued existence is a betrayal of the decrees issued during the 3 r d Party Congress.
The following Crash Course video provides additional information about the broad economic choices that countries make when they decide between planned and market economies. The narrators talk fast, so you’ll need to listen closely and possibly watch the video a second time!
To recap, economic systems determine the following:
- What to produce?
- How to produce it?
- Who gets it?
In a planned economy, government controls the factors of production:
- In a true communist economy, there is no private property—everyone owns the factors of production. This type of planned economy is called a command economy.
- In a socialist economy, there is some private property and some private control of industry.
In a free-market (capitalist) economy, individuals own the factors of production:
- Privately owned businesses produce products.
- Consumers choose the products they prefer causing the companies that product them to make more profit.
Even in free markets, governments will
- Maintain the rule of law
- Create public goods and services such as roads and education
- Step in when the market gets things wrong (e.g., setting minimum wage, establishing environmental standards)
In reality, economies are neither completely free-market nor completely planned. Neither exists in “pure” form, since all societies and governments regulate their economies to varying degrees. Throughout this course we will consider a number of ways in which the U.S. government influences and controls the economy.