7.4: An Integrated Approach to Measurement Validation
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A complete and adequate assessment of validity must include both theoretical and empirical approaches. As shown in Figure 7.4, this is an elaborate multi-step process that must take into account the different types of scale reliability and validity.
The integrated approach starts in the theoretical realm. The first step is conceptualizing the constructs of interest. This includes defining each construct and identifying their constituent domains and/or dimensions. Next, we select (or create) items or indicators for each construct based on our conceptualization of these construct, as described in the scaling procedure in Chapter 5. A literature review may also be helpful in indicator selection. Each item is reworded in a uniform manner using simple and easy-to-understand text. Following this step, a panel of expert judges (academics experienced in research methods and/or a representative set of target respondents) can be employed to examine each indicator and conduct a Q-sort analysis. In this analysis, each judge is given a list of all constructs with their conceptual definitions and a stack of index cards listing each indicator for each of the construct measures (one indicator per index card). Judges are then asked to independently read each index card, examine the clarity, readability, and semantic meaning of that item, and sort it with the construct where it seems to make the most sense, based on the construct definitions provided. Inter-rater reliability is assessed to examine the extent to which judges agreed with their classifications. Ambiguous items that were consistently missed by many judges may be reexamined, reworded, or dropped. The best items (say 10-15) for each construct are selected for further analysis. Each of the selected items is reexamined by judges for face validity and content validity. If an adequate set of items is not achieved at this stage, new items may have to be created based on the conceptual definition of the intended construct. Two or three rounds of Q-sort may be needed to arrive at reasonable agreement between judges on a set of items that best represents the constructs of interest.
Next, the validation procedure moves to the empirical realm. A research instrument is created comprising all of the refined construct items, and is administered to a pilot test group of representative respondents from the target population. Data collected is tabulated and subjected to correlational analysis or exploratory factor analysis using a software program such as SAS or SPSS for assessment of convergent and discriminant validity. Items that do not meet the expected norms of factor loading (same-factor loadings higher than 0.60, and cross-factor loadings less than 0.30) should be dropped at this stage. The remaining scales are evaluated for reliability using a measure of internal consistency such as Cronbach alpha. Scale dimensionality may also be verified at this stage, depending on whether the targeted constructs were conceptualized as being unidimensional or multi-dimensional. Next, evaluate the predictive ability of each construct within a theoretically specified nomological network of construct using regression analysis or structural equation modeling. If the construct measures satisfy most or all of the requirements of reliability and validity described in this chapter, we can be assured that our operationalized measures are reasonably adequate and accurate.
The integrated approach to measurement validation discussed here is quite demanding of researcher time and effort. Nonetheless, this elaborate multi-stage process is needed to ensure that measurement scales used in our research meets the expected norms of scientific research. Because inferences drawn using flawed or compromised scales are meaningless, scale validation and measurement remains one of the most important and involved phase of empirical research.


19.7: Exercises
LAB 19 EXERCISES \(\PageIndex{1}\)
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Identify the following
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Match the following
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LAB 19 EXERCISES \(\PageIndex{2}\)
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Identify the following components of the upper repiratory tract
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LAB 19 EXERCISES \(\PageIndex{2}\)
Upper Respiratory Tract
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Label the following
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Label these components of the larynx
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LAB 19 EXERCISES \(\PageIndex{3}\)
Label the Cartilages and Ligaments
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LAB 19 EXERCISES \(\PageIndex{4}\)
Upper --> Lower Respiratory Tract
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Label the following
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Label the following
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LAB 19 EXERCISES \(\PageIndex{5}\)
Lower Respiratory Tract
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Label the following
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Label the following
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LAB 19 EXERCISES \(\PageIndex{6}\)
Histology
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Label the tracheal histology
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Lower Respiratory Tract
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Label the tracheal cartilage histology
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Label the following lung histology components
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19.9: Complications of the Second Stage of Labor
By the end of this section, you will be able to:
- Describe the conditions related to dystocia, the medical intervention(s) for them, and nursing care during the second stage of labor
- Describe the conditions related to FHR patterns, the medical intervention(s) for them, and nursing care during the second stage of labor
Second stage labor is also known as the pushing stage. Complications can occur during this and any stage of labor. The laboring person is working very hard to push, and the fetus is working hard to be born. Because of the stress of pushing, the laboring person can become fatigued, and the fetus can exhibit distress. The nurse will monitor the laboring person and fetus for signs of distress. At times, the second stage can take longer than expected and may be diagnosed as prolonged second stage. At other times, the fetus has difficulty in engaging in the pelvis, causing failure to descend. The nurse works with the person to support the pushing effort and provide guidance and encouragement.
Prolonged Second Stage
Prolonged second stage labor is defined as a second stage labor lasting longer than 3 hours for a nulliparous person and longer than 2 hours for a multiparous person. Risk factors for prolonged second stage are use of epidural anesthesia , persistent occiput posterior position, and head circumference or birth weight above the 90th percentile (Lipschuetz et al., 2018). Complications to the birthing person resulting from prolonged second stage include chorioamnionitis , postpartum hemorrhage , operative vaginal birth , extended perineal lacerations , and cesarean birth . Prolonged second stage is also a risk factor for shoulder dystocia in the newborn.
Failure to Descend
Descent of the presenting part begins in active labor. During second stage labor, the fetal presenting part descends past the cervix, into the vagina, and out past the perineum. Failure to descend is defined as lack of change in the station for at least 2 hours. Many times, this is caused by fetal malpositioning, especially occiput posterior position (ACOG, 2020). Rotation of the fetal head by the health-care provider manually or with forceps can allow for increase in station and vaginal birth. At other times, failure to descend results in cesarean birth.
Laboring Person Fatigue
Second stage labor takes a great deal of energy and effort. Many times, laboring persons become fatigued and need a break from pushing. If the person can no longer continue pushing, the person is said to have labor fatigue. As discussed previously, one of the Ps of dystocia is power. With labor fatigue, the power is deficient, and second stage dystocia occurs. Labor fatigue can be prevented by waiting until the peak of the contraction to encourage the birthing person to bear down during contractions, providing food and drink to replenish calories, and allowing the fetus to descend or “labor down.” The nurse can support the laboring person and encourage these interventions to help decrease fatigue.
End-Stage Fetal Bradycardia
During the second stage, the fetal head molds to fit through the pelvis. At times, the compression and molding cause FHR decelerations. These decelerations can be early (mirror the contraction), late ( uteroplacental insufficiency ), or prolonged (bradycardia lasting longer than 2 minutes). Monitoring the FHR can be difficult as the fetus descends. Therefore, the nurse must compare the laboring person’s heart rate to the FHR to ensure the monitor is tracing the FHR. In cases of prolonged decelerations during the second stage, the health-care provider might expedite birth by performing an episiotomy, using forceps or a vacuum extractor , or by performing an emergency cesarean birth.
Laboring Person Fatigue and Request for Cesarean Birth
During the second stage, the laboring person can become very fatigued and discouraged. Many times, they will request to have a cesarean birth to stop pushing. They are not always making a sound decision because of their exhaustion. The nurse and health-care provider must closely evaluate the laboring person to determine if the fatigue is leading to a prolonged second stage or if the person needs a rest. Periods of rest during the second stage allow the person to relax, decrease stress, and regain focus. Rest is an important nursing intervention during second stage of labor.


14: Higher order linear ODEs
We have already studied the basics of differential equations, including separable first-order equations. In this chapter, we go a little further and look at second-order equations, which are equations containing second derivatives of the dependent variable. The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Here we concentrate primarily on second-order equations with constant coefficients.
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- 14.2: The Method of Undetermined Coefficients I
- This section present the method of undetermined coefficients, which can be used to solve nonhomogeneous equations of the form ay''+by'+cy=F(x) where a, b, and c are constants and F(x) has a special form that is still sufficiently general to occur in many applications. This sections makes extensive use of the idea of variation of parameters introduced previously.
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- 14.6: Reduction of Order
- This section deals with reduction of order, a technique based on the idea of variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know one nontrivial (not identically zero) solution of the associated homogeneous equation.
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- 14.7: Variation of Parameters
- This section deals with the method traditionally called variation of parameters, which enables us to find the general solution of a nonhomogeneous linear second order equation provided that we know two nontrivial solutions (with nonconstant ratio) of the associated homogeneous equation.
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- 14.E: Higher order linear ODEs (Exercises)
- These are homework exercises to accompany Libl's "Differential Equations for Engineering" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Prerequisite for the course is the basic calculus sequence.


8: Periodic Functions
In this chapter, we will investigate graphs of sine, cosine, and other trigonometric functions.
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- 8.0: 8.1 Introduction to Periodic Functions
- Each day, the sun rises in an easterly direction, approaches some maximum height relative to the celestial equator, and sets in a westerly direction. The pattern of the sun’s motion throughout the course of a year is a periodic function. Creating a visual representation of a periodic function in the form of a graph can help us analyze the properties of the function.
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- 8.3: Inverse Trigonometric Functions
- In this section, we will explore the inverse trigonometric functions. Inverse trigonometric functions “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words, the domain of the inverse function is the range of the original function, and vice versa.
Contributors and Attributions
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Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus .


11.11: Internet and Computer Safety Tips
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The following article was retrieved from usa.kapersky.com (Kapersky.com, n.d.)
A 19-year-old running for public office in New Hampshire found out about the importance of following Internet safety rules the hard way. As Seacoast Online reports, his opponents found images in his social media posts that were sexually suggestive and referenced past drug use. Just like that, his political career crashed and burned upon takeoff. But, unfortunately, he isn't the only one, as careless Internet habits have left others exposed to scams, identity theft and physical harm at the hands of people they met online. With more users accessing the Internet through mobile devices, these risks are changing and growing quickly. Even though apps loom larger in most people's daily online interactions than traditional websites do, that does not mean that the basic Internet safety rules have changed. Hackers are still on the lookout for personal information they can use to access your credit card and bank information. Unsafe surfing can also lead to other threats—from embarrassing personal comments or images that, once online, are nearly impossible to erase, to getting mixed up with people you'd rather have had nothing to do with.
Here are the Top 10 Internet safety rules to help you avoid getting into trouble online (and offline).
- Keep Personal Information Professional and Limited: Potential employers or customers don't need to know your personal relationship status or your home address. They do need to know about your expertise and professional background, and how to get in touch with you. You wouldn't hand purely personal information out to strangers individually—don't hand it out to millions of people online.
- Keep Your Privacy Settings On: Marketers love to know all about you, and so do hackers. Both can learn a lot from your browsing and social media usage. But you can take charge of your information. As noted by Lifehacker, both web browsers and mobile operating systems have settings available to protect your privacy online. Major websites like Facebook also have privacyenhancing settings available. These settings are sometimes (deliberately) hard to find because companies want your personal information for its marketing value. Make sure you have enabled these privacy safeguards and keep them enabled.
- Practice Safe Browsing: You wouldn't choose to walk through a dangerous neighborhood—don't visit dangerous neighborhoods online. Cybercriminals use lurid content as bait. They know people are sometimes tempted by dubious content and may let their guard down when searching for it. The Internet's demimonde is filled with hard-to-see pitfalls, where one careless click could expose personal data or infect your device with malware. By resisting the urge, you don't even give the hackers a chance.
- Make Sure Your Internet Connection is Secure. Make sure your device is secure, and when in doubt, wait for a better time (i.e., until you're able to connect to a secure Wi-Fi network) before providing information such as your bank account number.
- Be Careful What You Download: A top goal of cybercriminals is to trick you into downloading malware—programs or apps that carry malware or try to steal information. This malware can be disguised as an app: anything from a popular game to something that checks traffic or the weather. Don’t download apps that look suspicious or come from a site you don't trust.
- Choose Strong Passwords: Passwords are one of the biggest weak spots in the whole Internet security structure, but there's currently no way around them. And the problem with passwords is that people tend to choose easy ones to remember (such as "password" and "123456"), which are also easy for cyber thieves to guess. Select strong passwords that are harder for cybercriminals to demystify. Password manager software can help you to manage multiple passwords so that you don't forget them. A strong password is one that is unique and complex—at least 15 characters long, mixing letters, numbers and special characters.
- Make Online Purchases from Secure Sites: Any time you make a purchase online, you need to provide credit card or bank account information—just what cybercriminals are most eager to get their hands on. Only supply this information to sites that provide secure, encrypted connections. As Boston University notes, you can identify secure sites by looking for an address that starts with https: (the S stands for secure) rather than simply Http: They may also be marked by a padlock icon next to the address bar.
- Be Careful What You Post: The Internet does not have a delete key, as that young candidate in New Hampshire found out. Any comment or image you post online may stay online forever because removing the original (say, from Twitter) does not remove any copies that other people made. There is no way for you to "take back" a remark you wish you hadn't made or get rid of that embarrassing selfie you took at a party.
 Indeed, they may not even be real. Be as cautious and sensible in your online social life as you are in your in-person social life.
- Keep Your Antivirus Program Up to Date: Internet security software cannot protect against every threat, but it will detect and remove most malware—though you should make sure it's to date. Be sure to stay current with your operating system's updates and updates to the applications you use. They provide a vital layer of security.
Keep these 10 basic Internet safety rules in mind and you'll avoid many of the nasty surprises that lurk online for the careless


11.4: Johann Sebastian Bach - Fugue in G minor and Sleepers, Wake
About 150 years later, another composer of church music was also guided by the needs and preferences of his faith community. The church music of Johann Sebastian Bach (1685- 1750) is quite different from that of Palestrina, however, both because musical tastes had changed in the intervening years and because Bach worked not for the Catholic but for the Lutheran Church. We will take a look at two of his most famous creations: a piece of music for the organ and a composition for choir and orchestra.
Bach’s Legacy
Today, the music of J.S. Bach is performed more frequently than that of almost any other composer from the European tradition. Ensembles all over the world are dedicated to his music, while countless books have detailed his life and works. He is esteemed by many as the greatest composer of all time (although, as will be discussed in the final chapter of this book, it is nearly impossible to define “greatness” in music).
All of this would have very much surprised the composer himself. During his lifetime, Bach was not particularly famous or respected, and he struggled constantly with difficult working conditions and low pay. He was better known as an organist than as a composer: While Bach was respected as a virtuoso performer, his compositions were considered old-fashioned and stuffy. Only a handful of his works—mostly for keyboard—were published before his death, and he had no reason to expect future generations to take any interest in his music.
Bach’s fortunes shifted in the early 19th century, when German musicians began to revive and popularize his music. The most significant such event took place in 1829 when the composer Felix Mendelssohn staged a performance of Bach’s St. Matthew Passion in Berlin. Like Sleepers, Wake (German: Wachet auf ), which we will examine, the St. Matthew Passion was a work for choir and orchestra intended for use in a Lutheran church service. Bach never imagined that any of his church music would be performed in concert halls or consumed as either art or entertainment. He sought only to support the work of the church. Since the time of Mendelssohn, however, Bach’s choral compositions have been a staple of the concert repertoire, and today’s listener can access thousands of recordings.
A brief examination of Bach’s life and career will serve to contextualize his work as a composer. Bach was born into a large family of German musicians that extended back for many generations. His father, grandfather, great-grandfather, uncles, and other male ancestors were all performers and in most cases composers, while his sons were all to become composers as well. From the time of his birth, therefore, there was no doubt about Bach’s future career. Musicians of his time and place generally found employment with either a court or a city, in which capacity they would produce new compositions, oversee performances, and participate in those performances as instrumentalists or singers.
Bach’s Career
Bach never lived outside the region of Thuringia, which today is located in central Germany, and he never travelled beyond the borders of the modern German nation. Following an education in Eisenach, he took a series of five professional posts. First he served as a church organist in the cities of Arnstadt (1703-1707) and Mühlhausen (1707-1708). His next position was at the ducal court in Weimar (1708-1717), where he played the organ and served as music director. After this he became music director at the court of Prince Leopold in Köthen (1717-1723). Finally, Bach took the position of music director at the St. Thomas Church in the city of Leipzig, where he remained until his death.
Famously, Bach was not the city council’s first choice for the job. They initially offered the post to a composer—Georg Philipp Telemann—whose music is only seldom performed today, but who at the time was considered to be more fashionable. Bach was in turn loathe to accept the job, which was less prestigious than the post he held in Köthen. He made the move to Leipzig, however, out of concern for his family. Bach, who was married twice, had a total of twenty children, ten of whom survived into adulthood. Leipzig had excellent schools, and he knew that his sons would have better prospects in that city. Bach’s second wife, Anna Magdalena, was herself a highly-skilled musician. She provided her husband with invaluable assistance, copying out parts by hand each week so that the church musicians could perform his music during the Sunday service.
In Leipzig, Bach was required to perform a variety of tasks on behalf of the municipal government. He was principally responsible for music at the St. Thomas Church, but also oversaw music at the city’s other three churches. As music director, Bach produced instrumental and vocal compositions for use in the church, hired musicians, ran rehearsals, and played the organ. He also taught music and Latin at the St. Thomas School, which was attached to the church. Finally, he was obliged to produce music for civic occasions, including commemorations of important events and celebrations of esteemed visitors.
Bach frequently complained about his immense workload and limited resources. He felt that the city did not provide adequate funds with which to hire the musicians that he needed, and he often had to limit his instrumentation due to budgetary concerns. Although today one can hear Bach’s music rendered by the best choirs and orchestras in the world, Bach himself was seldom able to arrange for high-quality performances of his music. He relied primarily on students from the St. Thomas School and from the nearby Leipzig University.
We will examine two pieces of church music from two different parts of his career. Bach’s Fugue in G minor dates from his early years as a church organist in Arnstadt. Like most of Bach’s music, it has survived only as a handwritten manuscript. The same it true of Sleepers, Wake , which was created in 1731 for use at the St. Thomas Church in Leipzig.
Fugue in G minor
Bach wrote hundreds of fugues, most of which had nothing to do with his work as a church musician. The term fugue refers to a compositional technique that can be applied across genres. Bach wrote most of his fugues for keyboard instruments such as the harpsichord, which resembles a small piano in appearance but creates sound using quite a different process: When the player depresses a key, it causes a string to be plucked with a quill or piece of hard leather. Most famously, Bach twice wrote keyboard fugues in every major and minor key. His two collections, known as The Well-Tempered Clavier Book I (1722) and Book II (1742), were intended to demonstrate a new method for tuning the harpsichord. Bach also wrote fugues for solo string instruments, orchestra, and choir. One of Bach’s final works, entitled The Art of Fugue , was left incomplete at his death but nonetheless demonstrated his ultimate mastery of the technique.
Fugal technique was widely employed in the 18th century. It is simple in principle, but very challenging to execute. Whether a fugue is written for instrumentalists or singers, we always speak of its “voices” and use the typical choral designations: soprano, alto, tenor, and bass. Most fugues have four voices, although Bach wrote for as few as two and as many as six.
Every fugue begins with a solo melody in one of the voices. This melody is called the subject . The composer will introduce the subject in each of the voices in turn until all have entered the texture. This section of the fugue is called the exposition . For the remainder of the fugue, sections that do not contain the subject, which are known as episodes , will alternate with statements of the subject, which constitute the development . The subject will appear in many different keys and sometimes in different forms (for example, the melody might be turned upside down) until finally it is heard one last time in the original key and the fugue concludes. A fugue might have many episodes or none at all, and there is no predetermined length or precise form.
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Time |
Form |
What to listen for |
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0’00” |
Exposition |
The subject is heard first in the soprano voice, then alto, then tenor, then bass |
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0’51” |
Episode |
All of the episodes consist largely of sequences |
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0’58” |
Subject |
After a false entrance in the tenor, the subject is heard in the soprano |
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1’11” |
Episode |
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1’17” |
Subject |
Heard in the alto |
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1’28” |
Episode |
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1’36” |
Subject |
Heard in the bass |
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1’47” |
Episode |
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1’59” |
Subject |
Heard in the soprano |
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2’10” |
Episode |
This is the most diverse and lengthy episode |
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2’31” |
Subject |
Heard in the bass; the tempo slows before the final cadence |
Fugue
Episode(s)
Bridging material for return to Subjects
Exposition
• Subject introduced
• Subject answered (in different keys)
• Countersubjects accompany subjects
Optional: Coda (at the end)
Development
• Subject presented in different keys
• Possible transformations of the subject (inversions, etc.)
Final
Episode(s)
Bridging material for return to Subjects
• Subject and countersubject in the original key
Fugues are difficult to write because the composer must follow complex rules concerning the relationships between the voices. These rules concern the distances between simultaneous pitches, the directions in which voices move, the special treatment of notes that do not belong to chords, and the keys in which the subject must appear. Because the subject itself can never be altered, the composer must employ all of their skill to avoid breaking rules. (These rules are not arbitrary, but rather emerged over decades of practice and guide the composer in creating a fugue that sounds good.) Bach seems to have enjoyed the challenge of fugue writing, and he often created subjects that were especially difficult to work with.
Although a fugue is not necessarily an example of church music, Bach composed
organ fugues for use in Lutheran church services. We will examine a fairly simple fugue that he composed in his capacity as church organist for the city of Arnstadt. He would have played his Fugue in G minor at the beginning or end of services, but also on organ recitals that were intended not for worship but for the enjoyment of the audience. While this piece of music sets a mood that is appropriately serious for worship, its complexities also reward careful listening.
In Arnstadt, Bach played on an organ that had just been installed by the builder Johann Friederich Wender. All organs work by blowing air through pipes, which might be made out of wood or metal, be of various shapes, or contain reeds. These variations allow organs to produce many different sounds. Most organs have multiple keyboards, each of which can be linked to one or more sets of pipes. In this way, the performer can quickly change from one sound quality to another by moving between keyboards. Sets of pipes are activated or deactivated by adjusting wooden rods known as stops. Organs also have an additional keyboard, termed the pedalboard, that is operated using the feet. The pedalboard is typically linked to pipes that sound in the lowest register.
Despite its complexity, the organ is actually one of the oldest instruments found in Europe. Organs were first developed in ancient Greece over two thousand years ago. Although early organs were small and portable instruments, they began to grow in the 14th century when the first permanent organ was installed in a church. By the 19th century organs had become enormous instruments capable of producing a great range of volumes and timbres. Bach’s organ in Arnstadt is still played today. Like other Baroque organs, it is fairly small, although still capable of producing a wide variety of sounds. It has two keyboards and a pedalboard, which together give the player access to twenty-one sets of pipes. Baroque organs are not capable of gradual changes in dynamic. The performer can suddenly alternate between loud and soft by changing keyboards, but all notes played using a given set of pipes are the same volume. Fugues, therefore, are most often performed without any dynamic changes.
Bach’s Fugue in G minor (often called the “Little” Fugue in G minor, to distinguish it from others in the same key) has a five-measure subject. The subject begins with long note values, but gradually incorporates shorter and shorter note values as it proceeds. This creates an impression of increased activity, even though the tempo does not accelerate. The fugue has four voices, which enter from highest to lowest in the order soprano-alto-tenor-bass. The texture quickly becomes dense, and—after the opening measures—at least one voice is moving rapidly at all times.  It can be challenging to pick out the subject, even once it has become familiar.
Bach was noted for the density and complexity of his music. Indeed, that was the characteristic that earned him scorn in his own time—just as it earned him respect a century after his death. He also preferred to establish and then maintain a single mood with each piece of music. As such, there is little expressive variation within the Fugue in G minor. Once the engine starts, it runs steadily and unerringly until the final cadence.
Sleepers, Wake
During the early years of his employment in Leipzig, J.S. Bach dedicated most of his energy to the creation of Lutheran church cantatas . This was a big job because the churches there required a new cantata every week. The Lutheran liturgical calendar is organized into a year-long cycle of Biblical readings, and the cantata corresponded with the topic of the readings and the sermon. For this reason, Bach needed to produce an appropriate cantata for every Sunday morning Mass and for special services, making for a total of sixty cantatas a year.
After his first year on the job, Bach could have begun to reuse old cantatas—but instead he completed a whole second cycle and most of three additional cycles. Bach had also written church cantatas at several of his previous jobs. However, none of his church cantatas were ever published. As a result, about two hundred are extant today, while hundreds more might have been lost.
A cantata is a multi-part work for voice(s) and instrumental accompaniment. Bach’s Lutheran church cantatas are multi-movement works for choir, soloists, and orchestra. Bach always used texts in German, the language of his congregation. He did not write the texts himself, however, but rather selected them from among the works of various theological poets. Each cantata is thirty to forty minutes long and usually contains four to seven movements. Some of the movements use the entire choir, while others feature solo singers, often paired with solo instruments.
Bach’s cantatas constituted the musical focus for worship at St. Thomas and other churches in Leipzig. Although forty minutes of choral music might seem excessive, the services themselves lasted four hours. The centerpiece was a one- hour sermon, which was preceded by opening prayers, hymns, readings, and the performance of the cantata. Communion followed the sermon.
The cantata had a very specific purpose: It reflected on the Biblical readings for the day, interpreted their meaning for the congregation, and prepared listeners to understand and appreciate the sermon. As stated above, Bach in no way regarded his cantatas as entertainment—or even, strictly speaking, art. He was deeply committed to his Lutheran faith and he understood his role in the service to be essentially spiritual. His cantatas shaped churchgoers’ understanding of their relationship with God.
We will see Bach’s approach to cantata composition in Sleepers, Wake . Although many of Bach’s cantatas are difficult to date, we know that this one was first performed on November 25, 1731. The occasion was the 27th Sunday after Trinity—a day that occurs only in years during which Easter falls very early, for in regular years the liturgical season of Advent will have already commenced. This explains why Bach had to write this cantata several years after having completed his annual cycles, for the 27th Sunday after Trinity had not occurred since 1704. Sleepers, Wake was performed only once in Bach’s lifetime, at Leipzig’s St. Nicholas Church. The Epistle reading for the 27th Sunday after Trinity is 1 Thessalonians 5: 1-11, while the Gospel reading is Matthew 25: 1-13. Both exhort Christians to be prepared for the return of Christ. In Paul’s letter to the Thessalonians, he warns that the Lord will come “like a thief in the night,” without any warning. Matthew records the parable of the wise and foolish virgins. All had gathered together to await the coming of the bridegroom (an allegory for Christ), but the foolish virgins had failed to bring extra oil for their lamps. While they were away buying oil, the bridegroom arrived and the wedding feast commenced. The wise virgins, who were prepared for the arrival, were welcomed into the hall, but the foolish virgins were shut out in the darkness. The message is clear: Christians must be prepared for Christ’s coming. If they are not, they will be excluded from heaven and denied eternal life.
Bach’s cantata reflects on this parable using a variety of musical and dramatic techniques. He began by selecting an appropriate Lutheran chorale on which to base his cantata. Chorales are unison hymns sung by the congregation, and they were first developed by Martin Luther himself in the early years of the church. One of Luther’s objections to Catholic worship was that the music was performed in Latin by professional choirs. He preferred participatory music in the language of the congregants. To develop a repertoire of chorales, he wrote new tunes, adapted Gregorian chants, and even borrowed popular melodies. Luther saw no problem with using secular music for worship. As he supposedly put it, “Why should the devil have all the good tunes?”
Because Sleepers, Wake is based on a Lutheran chorale, it is a special type of cantata known as a chorale cantata . Bach primarily wrote chorale cantatas while in Leipzig, and he developed a unique approach to their construction, of which Sleepers, Wake is a fine example. The cantata contains seven movements, three of which—the first, fourth, and seventh— include text and music from a 1599 chorale of the same name. Bach selected this chorale because it, too, comments on the parable of the wise and foolish virgins, thereby offering another layer of interpretation. Everyone in the congregation at St. Thomas would have known the chorale well and would have instantly recognized the words and music. Because the chorale is in A A B form, each of the movements based on it is in that same form.
I. Wake up, the voice calls us
Bach spreads his references to the chorale throughout the cantata, and he integrates it with his own music differently on each occasion. In the first movement, we hear the choir sing the first verse of the chorale text:
Wake up, the voice calls us
of the watchmen high up on the battlements, wake up, you city of Jerusalem!
This hour is called midnight;
they call us with a clear voice:
where are you, wise virgins? Get up, the bridegroom comes;
Stand up, take your lamps! Hallelujah! Alleluia!
Make yourselves ready for the wedding,
you must go to meet him!
Translation by Francis Browne
We also hear the chorale melody, but it is buried in a dense texture of newly- composed music. While Bach’s congregation would have recognized the chorale, many modern listeners have a hard time even picking the melody out.
|
Time |
Form |
What to listen for |
|---|---|---|
|
0’00” |
Ritornello |
The violins and oboes exchange melodic material in a call-and-response texture |
|
0’28” |
A “Wake up, the voice calls us...” |
The sopranos sing the chorale melody while the altos, tenors, and basses sing newly-composed material |
|
1’32” |
Ritornello A |
This ritornello is identical to that which opened the movement |
|
2’00” |
“This hour is called midnight...” |
The A music repeats with a new texts |
| 2’10” | Episode | This is the most diverse and lengthy episode |
| 2’31” | Subject Heard in the bass; the tempo slows before the final cadence | |
|
3’04” |
Ritornello |
This ritornello sounds different because it moves through several key areas |
| 3’24” | B | “Get up, the We hear exclamations of “get up” and “stand bridegroom up” from the choir comes...” |
| 3’57” | “Alleluia!” This passage is especially ornate; the chorale melody does not enter until near the end | |
| 4’42” | “Make yourselves The texture returns to normal ready...” | |
| 5’32” | Ritornello | We hear the complete ritornello one last time |
The first movement starts with orchestral music, the uneven dotted rhythms of which suggest a wedding march. Dotted rhythms were also associated with royalty in this era—another appropriate connotation for music about Christ’s coming. This opening passage is in fact a ritornello . In this movement Bach combines ritornello form, in which an orchestral melody returns throughout a piece of music, with the A A B form of the chorale. The resulting form is: rit A rit A rit B rit. The orchestra also provides short interjections between the verses within each section. Although a congregation might sing the first verse of the chorale in less than two minutes, the first movement of the cantata takes nearly four times as long to perform. This is due to the slowed-down chorale melody and frequent orchestral interruptions. As a result, however, the congregation has an opportunity to meditate on the text, the meaning of which is reinforced by Bach’s musical setting.
The first movement is texturally very dense. There are active parts for strings, winds, and voices, and it is seldom possible to identify a single, dominant melody. The opening ritornello contains three distinct layers. Underpinning everything is the basso continuo , a feature of almost all music composed in the Baroque era (1600-1750). Basso continuo is always performed by some combination of low- pitched instruments and instruments that can play chords. In this case, we hear cello, bassoon, organ, and harpsichord. While Baroque bass lines can be simple, Bach’s seldom are. This one contains a variety of interesting rhythmic and melodic elements, thereby attracting an unusual amount of the listener’s attention. Above that, the violins and oboe trade melodic motifs back and forth in a six-part texture. When the singers enter, the orchestra begins by repeating the musical material from the ritornello. The already-dense texture suddenly becomes much more complex. First we hear the sopranos with the familiar chorale melody. While the sopranos sing in slow, even note values, the altos, tenors, and basses sing quickly. These other voices occasionally integrate text painting as well, such as with their ascending cries on the text “wake up”. Also notable is the passage on the text “Alleluia,” which features excitable melismas in the altos, tenors, and basses.
|
ritornello orchestra |
A from chorale |
ritornello orchestra |
A from chorale |
ritornello orchestra |
B from chorale |
ritornello orchestra |
|
lines 1-3 of text with accompaniment |
lines 4-6 of text with accompaniment |
lines 7-12 of text with accompaniment |
IV. Zion hears the watchmen sing
The chorale disappears until the fourth movement,2 when we hear the second verse sung by the tenor section:
Zion hears the watchmen sing, her heart leaps for joy,
she awakes and gets up in haste.
Her friend comes from heaven in his splendour, strong in mercy, mighty in truth.
Her light becomes bright, her star rises.

Hosanna!
We all follow
to the hall of joy
and share in the Lord’s supper.
Translation by Francis Browne.
|
IV. “Zion hears the watchmen sing” from Sleepers, Wake 2. Composer: Johann Sebastian Bach Performance: American Bach Soloists (2007) |
This movement is much simpler than the first movement. In place of the orchestral ritornello, with its call-and-response texture, we have a unison ritornello melody from the violins and violas. The chorale melody, instead of being buried in a complex texture, is clearly presented without competition from other vocal parts. All the same, this movement is by no means simple. The two melodies—one in the strings, one in the tenors—weave around one another in unpredictable and extraordinary ways. While their phrases never start or end together, the parts always complement one another.
VII. May gloria be sung to you
We finally hear a straightforward presentation of the chorale in the seventh movement,3 in which the entire choir—and perhaps the congregation as well—sing the third verse in four-part harmony:
May gloria be sung to you
with the tongues of men and angels, with harps and with cymbals.
The gates are made of twelve pearls, in your city we are companions
of the angels on high around your throne. No eye has ever perceived,
no ear has ever heard such joy.
Therefore we are joyful, hurray, hurray!
for ever in sweet rejoicing.
Translation by Francis Browne
|
VII. “May gloria be sung to you” from Sleepers, Wake. 3. Composer: Johann Sebastian Bach Performance: American Bach Soloists (2007) |
Although Bach borrowed chorale melodies from the Lutheran tradition, he always created his own harmonizations. In practice, this means that the soprano part is borrowed, but the alto, tenor, and bass parts are original. Bach has the orchestral musicians double the vocal parts, playing the same melodies that are being sung. This makes the ensemble sound exceptionally full and rich without distracting from the chorale melody: a fitting culmination to the cantata.
II. He comes
Although there are four movements that do not contain the chorale melody or text, we will look at only two of them. The text was supplied by an unknown poet. It includes many references to the Song of Solomon—a passage of Biblical love poetry that was understood by Bach and his contemporaries to be a metaphor for the love between Jesus and the faithful soul. To set this expressive new text, Bach used musical forms from the opera stage: recitative and aria . Bach never wrote an opera and did not think highly of the form, but he often adapted operatic conventions for his own purposes.
The second movement4 of Sleepers, Wake is a recitative for solo tenor:
He comes, he comes, the bridegroom comes!
You daughters of Zion, come out,
he hastens his departure from on high to your mother’s house.
The bridegroom comes, who like a roedeer and a young stag
leaps on the hills
and brings to you the wedding feast.
Wake up, rouse yourselves to welcome the bridegroom!
There, see, he comes this way.
Translation by Francis Browne.
|
II. “He comes” from Sleepers, Wake 4. Composer: Johann Sebastian Bach Performance: American Bach Soloists (2007) |
He announces the coming of the bridegroom with a series of exuberant melodic leaps, accompanied only by basso continuo. This movement contains no repetition and has no particular form. In fact, the singing—as is always the case with recitative—is not particularly melodic at all. Instead, its purpose is to declaim the text with the utmost expressive force.
III. When are you coming, my salvation?
The recitative serves to introduce the third movement, which is a duet for soprano and bass. Just as recitative developed within the operatic tradition, this is clearly an operatic duet—specifically, the type of duet sung by two characters who are in love. The soprano and bass call back and forth to one another, expressing their mutual desire. Bach’s lovers, however, are a Soul (soprano) and Jesus (bass), and they offer a dramatic enactment of the desire that all Christians are meant to feel for their savior.
|
Time |
Form |
What to listen for |
|---|---|---|
|
0’00” |
Ritornello A |
This ritornello features a virtuosic violin obbligato |
|
0’26” |
“When are you coming. . .” |
The soprano and bass exchange lines while the solo violin line weaves about them |
|
1’46” |
Ritornello |
This ritornello is similar to that which opened the movement, but it is in a different key |
|
1’59” |
B “Open the hall. . .” |
This passage has the same texture as A, but the music and text are different |
|
2’32” |
Ritornello |
This passage is not closely related to the opening ritornello |
|
2’46” |
B’ |
This passage is similar to B, but it is in a different key |
|
3’18” |
Ritornello A’ |
This ritornello is very brief, for it is interrupted by the return of the A text |
|
3’25” |
“When are you coming. . .” |
This passage echoes A, but is not musically identical |
|
4’25” |
Ritornello |
The closing ritornello is identical to that which opened the movement |
Because the third movement is based in operatic conventions, it has the da capo form of an opera aria. “Da capo” literally means “from the head,” and it serves as an instruction to the performers to repeat the first of two parts, resulting in an A B A form. (The second A is not written out.) Bach accompanies his aria with basso continuo and an obbligato (or “obligatory”) instrumental solo. He intended the obbligato part in the fourth movement to be performed on the violino piccolo, a type of small violin that is tuned higher than a modern instrument. However, it can also be performed on a standard violin. The instrumental soloist provides ritornellos before, between, and after each of the sung sections, but also supplies a virtuosic accompaniment to the vocal soloists. The resulting music is beautiful and expressive—even if the text is a bit corny:
Soul: When are you coming, my salvation? Jesus: I come, your portion.
Soul: I wait with burning oil. Jesus: Open the hall
Soul: I open the hall
Both: to the heavenly feast. Soul: Come, Jesus!
Jesus: Come, lovely soul!
Translation by Francis Browne.


13.3: The Chemical Shift
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Objectives
After completing this section, you should be able to
- describe the delta scale used in NMR spectroscopy.
- perform calculations based on the relationship between the delta value (in ppm), the observed chemical shift (in Hz), and the operating frequency of an NMR spectrometer (in Hz).
Key Terms
Make certain that you can define, and use in context, the key terms below.
- chemical shift
- delta scale
- upfield/downfield
Study Notes
Although the calculations described in this section will help you understand the principles of NMR, it is the actual delta values, not the calculations, which are of greatest importance to the beginning organic chemist. Thus, we shall try to focus on the interpretation of NMR spectra, not the mathematical aspects of the technique.
In Section 13.9 we discuss 1 H NMR chemical shifts in more detail. Although you will eventually be expected to associate the approximate region of a 1 H NMR spectrum with a particular type of proton, you are expected to use a general table of 1 H NMR chemical shifts such as the one shown in Section 13.9.
Chemical Shifts
The NMR spectra is displayed as a plot of the applied radio frequency versus the absorption. The applied frequency increases from left to right, thus the left side of the plot is the low field, downfield or deshielded side and the right side of the plot is the high field, upfield or shielded side (see the figure below). The concept of shielding will be explained shortly.
The position on the plot at which the nuclei absorbs is called the chemical shift . Since this has an arbitrary value a standard reference point must be used. The two most common standards are TMS (tetramethylsilane, (Si(CH 3 ) 4 ) which has been assigned a chemical shift of zero, and CDCl 3 (deuterochloroform) which has a chemical shift of 7.26 for 1 H NMR and 77 for 13 C NMR. The scale is commonly expressed as parts per million (ppm) which is independent of the spectrometer frequency. The scale is the delta (δ) scale .
\[\delta=\frac{\text { frequency of signal }-\text { frequency of standard }}{\text { spectrometerfrequency }} \times 10^6 \nonumber \]
The range at which most NMR absorptions occur is quite narrow. Almost all 1 H absorptions occur downfield within 10 ppm of TMS. For 13 C NMR almost all absorptions occurs within 220 ppm downfield of the C atom in TMS.
Shielding in NMR
Structural features of the molecule will have an effect on the exact magnitude of the magnetic field experienced by a particular nucleus. This means that H atoms which have different chemical environments will have different chemical shifts. This is what makes NMR so useful for structure determination in organic chemistry. There are three main features that will affect the shielding of the nucleus, electronegativity, magnetic anisotropy of π systems and hydrogen bonding.
Electronegativity
The electrons that surround the nucleus are in motion so they created their own electromagnetic field. This field opposes the the applied magnetic field and so reduces the field experienced by the nucleus. Thus the electrons are said to shield the nucleus. Since the magnetic field experienced at the nucleus defines the energy difference between spin states it also defines what the chemical shift will be for that nucleus. Electron with-drawing groups can decrease the electron density at the nucleus, deshielding the nucleus and result in a larger chemical shift. Compare the data in the table below.
| Compound, CH 3 X | CH 3 F | CH 3 OH | CH 3 Cl | CH 3 Br | CH 3 I | CH 4 | (CH 3 ) 4 Si |
|---|---|---|---|---|---|---|---|
| Electronegativity of X | 4.0 | 3.5 | 3.1 | 2.8 | 2.5 | 2.1 | 1.8 |
| Chemical shift δ (ppm) | 4.26 | 3.4 | 3.05 | 2.68 | 2.16 | 0.23 | 0 |
As can be seen from the data, as the electronegativity of X increases the chemical shift, δ increases. This is an effect of the halide atom pulling the electron density away from the methyl group. This exposes the nuclei of both the C and H atoms, "deshielding" the nuclei and shifting the peak downfield.
The effects are cumulative so the presence of more electron withdrawing groups will produce a greater deshielding and therefore a larger chemical shift, i.e.
| Compound | CH 4 | CH 3 Cl | CH 2 Cl 2 | CHCl 3 |
|---|---|---|---|---|
| δ (ppm) | 0.23 | 3.05 | 5.30 | 7.27 |
These inductive effects are not only felt by the immediately adjacent atoms, but the deshielding can occur further down the chain, i.e.
| NMR signal | -CH 2 -CH 2 -CH 2 Br |
| δ (ppm) | 1.25 1.69 3.30 |
Magnetic Anisotropy: π Electron Effects
The π electrons in a compound, when placed in a magnetic field, will move and generate their own magnetic field. The new magnetic field will have an effect on the shielding of atoms within the field. The best example of this is benzene (see the figure below).
This effect is common for any atoms near a π bond, i.e.
| Proton Type | Effect | Chemical shift (ppm) |
|---|---|---|
| C 6 H 5 - H | highly deshielded | 6.5 - 8 |
| C=C- H | deshielded | 4.5 - 6 |
| C≡C- H | shielded * | ~2.5 |
| O=C- H | very highly deshielded | 9 - 10 |
| * the acetylene H is shielded due to its location relative to the π system |
Hydrogen Bonding
Protons that are involved in hydrogen bonding ( i.e. -OH or -NH) are usually observed over a wide range of chemical shifts. This is due to the deshielding that occurs in the hydrogen bond. Since hydrogen bonds are dynamic, constantly forming, breaking and forming again, there will be a wide range of hydrogen bonds strengths and consequently a wide range of deshielding. This as well as solvation effects, acidity, concentration and temperature make it very difficult to predict the chemical shifts for these atoms.
Experimentally -OH and -NH can be identified by carrying out a simple D 2 O exchange experiment since these protons are exchangeable.
- run the normal H-NMR experiment on your sample
- add a few drops of D 2 O
- re-run the H-NMR experiment
- compare the two spectra and look for peaks that have "disappeared"


4.3: Physical Assessment of Digestive Organs
By the end of this section, you should be able to:
- 4.2.1 Outline the assessment process of the digestive organs.
- 4.2.2 Analyze and prioritize assessment findings.
Mouth, Oropharynx, and Lips
Because digestion is responsible for fueling the body, proper assessment of the digestive system is imperative to prevent disease, maintain wellness, and promote nutritional health. Both a physical assessment and a review of the client’s health history should be completed. In reviewing the client’s health history, the nurse should inquire about dietary practices, dietary intolerances or allergies, eating and bowel patterns, changes in eating or bowel patterns, recent unintentional weight loss or gain, and medical diagnoses or surgical history. The nurse should obtain the client’s weight and height to calculate the client’s body mass index (BMI).
An assessment of the mouth and the oropharynx (the middle part of the throat) begins with the lips . The nurse should begin with the outer lips to judge for symmetry, color, texture, and contouring. The client should be asked to smile so that the nurse can judge lip symmetry. If the client is unable to lift both sides of the lips equally, they are not symmetrical, and this should be reported.
If the client is exhibiting asymmetry in their lips that is new, especially in conjunction with new symptoms such as facial drooping or asymmetry, arm weakness, numbness, paralysis, or speech difficulty such as slurring or aphasia, the nurse should immediately alert the provider, as this could indicate an acute stroke (American Stroke Association, n.d.).
Next, the inner lips and buccal mucosa (the inner lining of the cheeks) should be inspected for color, texture, moisture, and presence of abnormalities. To best visualize these structures, the nurse should direct the client to relax their mouth. The nurse should then pull the bottom lip outward and away from the client’s teeth by grasping the lip between the thumb and index finger on each side and carefully pulling outward (Villa, 2022). Drooling may be seen in young infants and toddlers up to age 2, but it should not be excessive.
Nurses should watch for numerous abnormalities, some more urgent than others. Assessment findings that the nurse should report immediately include cyanosis , a bluish discoloration of the skin that can indicate low oxygen levels, and new unilateral asymmetry with movement, as this could indicate a cerebrovascular accident.
Gums, Buccal Mucosa, Tongue, and Teeth
The inspection of the inside of the mouth will include the gums , buccal mucosa, tongue , and teeth . To begin, the client should open their mouth. The nurse should then use a penlight and a tongue depressor or blade to retract each cheek to complete a visual exam of all internal surfaces, looking for any abnormalities. To evaluate the teeth in the back of the mouth, the nurse should place both index fingers into one side of the mouth to retract the cheek. The client will then slowly open and close their mouth while the cheek is retracted so that tooth alignment can be visualized. This should be completed on both sides of the mouth.
Nurses should inspect the gums in this same manner. If the client has dentures, bridges, or other removable dental appliances, they should be removed if possible, and the nurse should repeat the evaluation for the gums separately. Check gums for retraction or recession, color, reddened inflammation, soreness, bleeding, edema, or lesions (Cleveland Clinic, 2022c).
If the epiglottis (the small cartilage that covers the windpipe) is swollen, the nurse should not examine with a tongue depressor , blade, or flexible laryngoscope . Further, the nurse should not manipulate the oral cavity because this could trigger laryngospasm and close off the client’s airway, causing a life-threatening emergency (Guerra & Waseem, 2022). Symptoms of epiglottitis include hyperthermia, sore throat, stridor (noisy breathing), dysphagia (trouble swallowing), and drooling (Mayo Clinic, 2022). Children may also act anxious and irritable and may sit up or lean forward to ease breathing, while adults may also have a muffled or hoarse voice or difficulty breathing.
To facilitate inspection of the tongue, the client should open their mouth and stick out their tongue to allow the nurse to inspect for position, color, texture, and moisture. The nurse should ask the client to curl their tongue upward and move it from side to side to assess mobility. Lastly, the nurse should instruct the client to push the tip of their tongue to the roof of their mouth and should inspect the base of the tongue , the mouth floor , and the frenulum (the small tissue beneath the tongue that provides support) for color, texture, bleeding, moisture, and lesions.
The hard and soft palates at the roof of the mouth should be inspected for color, shape, texture, and the presence of bony prominences. The nurse should instruct the client to open their mouth widely while tilting their head backward. The nurse should use a tongue depressor or blade to depress the tongue if needed and use a penlight to see the structures. The nurse should then ask the client to say “ah” and should inspect the uvula , the fleshy extension that hangs at the back of the throat and at the end of the soft palate, for position and mobility after inspecting the soft palate.
Palates, Uvula, Oropharynx, and Tonsils
The final steps of the assessment cover the palates , uvula, oropharynx, and tonsils. To begin, the nurse should use a tongue depressor or blade to push downward against the tongue halfway back on the same side of the tongue of the oropharynx that is being inspected. While holding the tongue out of the way, the nurse should use a penlight to examine the color, size, and discharge of the oropharynx and tonsils. The inspection should be repeated on the opposite side. Lastly, the nurse should elicit the gag reflex by using the tongue depressor or blade to stimulate the back of the throat (Sivakumar & Prabhu, 2023).
As with the teeth and gum assessment, abnormal findings that the nurse should report immediately include cyanosis and any suspicious lesions. Additional findings that the nurse should immediately report include a uvula that is not midline or freely movable (this could be a sign of a tumor, trauma, or nerve damage), a uvula in a newborn that is forked or bifurcated (a possible indication of an undetected cleft palate), plaques or drainage in oropharynx (this could indicate acute infection), or swelling or drainage of tonsils (possible sign of an acute infection).
Abdomen
The assessment of the abdomen includes the upper gastrointestinal system (stomach and duodenum), the accessory organs of the digestive system (pancreas, liver, and gallbladder), and most of the lower gastrointestinal system (large and small intestines). This assessment must occur in a specific order: inspection, auscultation, percussion, and then palpation. This is important because auscultating before percussing and palpating ensures the nurse is listening to undisturbed bowel sounds .
When preparing the client for abdominal assessment , position them so that their abdominal wall musculature is relaxed. To achieve this, the nurse should have the client lie supine and support the client’s head and behind the knees with pillows.
Step 1: Inspection
Inspection should be the first step in the abdominal assessment (Roscoe, 2022). The nurse should do this with the client in a completely supine position with the abdomen exposed (Mealie, Ali, & Manthey, 2022). The nurse should inspect the skin integrity, abdomen contour, symmetry, and vascular patterns and visually inspect the abdominal movement associated with respirations, aortic blood flow, and peristalsis (the muscle contractions that move food through the digestive tract).
Step 2: Auscultation
Auscultation is the process of listening to the sounds of the body using a stethoscope. The process of auscultating the abdomen is to assess the bowel and vascular sounds and peritoneal friction rubs. To auscultate the abdomen, the nurse should:
- With the diaphragm of the stethoscope, listen for bowel sounds in each of the four quadrants of the abdomen. Calculate the amount of time in between bowel sounds after listening for at least two and no more than 5 minutes in each quadrant.
- With the bell of the stethoscope, listen for the absence of bruits (the sound of blood flowing through a narrowed portion of an artery) in the aorta, renal arteries, femoral arteries, and iliac arteries in the abdomen. If the nurse suspects a delay in gastric emptying, place the stethoscope on the abdomen near the stomach, hold the client’s hips, and gently shake them from side to side while listening to their abdomen for a succussion splash , a sloshing or splashing sound caused by excessive fluid remaining in the stomach (Mealie, Ali, & Manthey, 2022). This may be uncomfortable for the client, so the nurse should inform the client what to expect.
- With the diaphragm of the stethoscope, listen for the absence of peritoneal friction rubs over the spleen and liver as the client breathes in deeply.
Step 3: Percussion
Percussion, or tapping , of the abdomen is an assessment technique to determine the presence of gas, fluid, or masses. The systematic process of percussion is to start in the lower-right quadrant of the abdomen and proceed to the upper-right quadrant, move to the upper-left quadrant, and finish in the lower-left quadrant. The nurse should percuss several areas of each quadrant to listen to the quality of the sound. Tympany , or a resonant, hollow sound, relates to gas, or flatus. Dullness, decrease, absence, or flatness of sound relates to fluid or solid masses.
Percussion is also used to best judge the size of the liver and may be helpful in identifying splenic enlargement (Mealie, Ali, & Manthey, 2022). To percuss for splenic size, the nurse should percuss the most inferior interspace on the left anterior axillary line, or the Castell’s point, as the client breathes in deeply, to listen for the start of dullness from tympany to discover the splenic edge. On the right side, the nurse should percuss down from the lung and up from the bowel to discover the change from tympany to dullness to locate the liver margins.
Step 4: Palpation
The last step in the abdominal assessment is palpation . There are three palpation stages: superficial or light palpation, deep palpation, and organ palpation. For superficial palpation , the nurse should hold their palm slightly above the client’s abdomen with their fingers parallel and then use their finger pads to depress the abdominal wall lightly at only about 1 cm (0.39 in), which is about the level of the subcutaneous tissue (Mealie, Ali, & Manthey, 2022). The nurse should move their finger pads in a circular motion over all the quadrants while noting any areas of tenderness, masses, crepitus (a grinding or crunching sound), irregularity in contour, or muscle guarding. If a known area of pain is already present on the abdomen, start farthest from the area of maximal pain with any form of palpation and end in that area last (Mealie, Ali, & Manthey, 2022).
After light palpation is complete, the nurse should move on to deep palpation . This is done slowly with the same hand position and in the same manner, except the depth of pressing is increased with firmer and more steady pressure to explore deeper. While palpating deeply in the periumbilical, inguinal, and suprapubic regions, the nurse should have the client cough while feeling for a mass to ensure there is no hidden hernia.
After palpation of the abdomen, the nurse should palpate the digestive organs. To palpate the liver , the nurse should place a hand below the lower-right rib margin and ask the client to exhale and then inhale deeply while the nurse feels for the margins and any nodularity. The nurse should then palpate the gallbladder by placing their palpating hand under the lower-right rib margin at the midclavicular line and asking the client to exhale deeply. The nurse then pushes in deeper and slowly, asking the client to inhale while watching for a sudden cessation of inhalation due to pain. The spleen is palpated by placing a hand in the lower-right quadrant and moving it toward the splenic flexure until the hand reaches the lower-left rib margin, at which time the client should be asked to exhale and then inhale deeply.
The nurse should not do deep palpation on clients with suspected abdominal aortic aneurysm, appendicitis, polycystic kidney disease, kidney transplant history, or a tender spleen because it could cause severe damage (Roscoe, 2022). For example, deep palpation may rupture an aneurysm, the appendix, or a cyst, or it may damage a newly transplanted kidney.
The nurse should report immediately abnormal findings that can indicate serious health problems within the lower gastrointestinal tract and accessory organs. Refer to Table 4.1 for a summary of possible abnormal findings and their causes.
| Assessment Finding | Abnormality | Sign/Symptom/Syndrome |
|---|---|---|
| Ecchymosis (discoloration) of flank and groin | Pancreatic hemorrhage | Grey Turner’s sign |
| Periumbilical ecchymosis | Retroperitoneal and intra-abdominal hemorrhage | Cullen’s sign |
| Purple-pink striae in the absence of pregnancy | Cushing’s syndrome | “ Stretch marks ” |
| Visible vascular pattern | Portal hypertension or vena cava obstruction | Vein dilation |
| Very marked aortic pulsations |
Aortic aneurysm
(Shaw et al., 2023) |
Pulsations |
| Surrounding erythema , induration , and tenderness |
Omphalitis
(specific to infants)
(Gantan & Wiedrich, 2022) |
Deviation in umbilical stump appearance |
| Combined with abdominal distention: jaundice and/or caput medusae (swollen veins) |
Cirrhosis
and
liver failure
(Mealie, Ali, & Manthey, 2022) |
Abdominal distention |
While physical assessment can aid in identifying potential nutritional deficiencies , it is not diagnostic, and other follow-up testing is necessary to determine the best treatment needs and options.
Rectum and Anus
Assessment of the anus and rectum is the final step of the lower gastrointestinal system assessment. To assess the anus , the nurse should visually evaluate the outer tissue for color, integrity, and absence of lesions. The nurse should then ask the client to bear down as if defecating. With gloves on and the index finger lubricated, the nurse should feel the rectal sphincter and slowly dilate it by sliding the lubricated finger into the rectum. Palpate inside the rectum for masses, fissures, or other foreign bodies. After removal of the finger, inspect the finger for any signs of bleeding or melena (dark, black, or sticky feces). If present, or if internal bleeding is suspected, the nurse should perform a guaiac test , which will confirm if blood is present.
For a newborn , the nurse should establish anal patency. This is done by two means: first, the nurse should insert a lubricated rectal thermometer 1 cm to 2 cm (0.39 in to 0.78 in) into the newborn’s rectum to record temperature, and second, the nurse should identify and document the passage of their first stool, called the meconium (Gantan & Wiedrich, 2022).


7.4: Exercises
Exercise \(\PageIndex{1}\)
For each of the following functions \(f(z)\) , find the real and imaginary component functions \(u(x,y)\) and \(v(x,y)\) , and hence verify whether they satisfy the Cauchy-Riemann equations.
-
\(f(z) = z\)
-
\(f(z) = z^2\)
-
\(f(z) = |z|\)
-
\(f(z) = |z|^2\)
-
\(f(z) = \exp(z)\)
-
\(f(z) = \cos(z)\)
-
\(f(z) = 1/z\)
Exercise \(\PageIndex{2}\)
Suppose a function \(f(z)\) is well-defined and obeys the Cauchy-Riemann equations at a point \(z\) , and the partial derivatives in the Cauchy-Riemann equations are continuous at that point. Show that the function is complex differentiable at that point. Hint: consider an arbitary displacement \(\Delta z = \Delta x + i \Delta y\) .
Exercise \(\PageIndex{3}\)
Prove that products of analytic functions are analytic: if \(f(z)\) and \(g(z)\) are analytic in \(D \subset \mathbb{C}\) , then \(f(z) g(z)\) is analytic in \(D\) .
- Answer
-
We will use the Cauchy-Riemann equations. Decompose \(z\) , \(f\) , and \(g\) into real and imaginary parts as follows: \(z = x + i y\) , \(f = u + i v\) , and \(g = p + i q\) . Since \(f(z)\) and \(g(z)\) are analytic in \(D\) , they satisfy \[\begin{align} \frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y},\;\; -\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}\\ \frac{\partial p}{\partial x} &= \frac{\partial q}{\partial y},\;\; -\frac{\partial p}{\partial y} = \frac{\partial q}{\partial x}.\end{align}\] This holds for all \(z \in D\) . Next, expand the product \(f(z)\,g(z)\) into real and imaginary parts: \[\begin{align}\begin{aligned} f(z)\,g(z) = A(x,y) + i B(x,y),\;\;\mathrm{where}\;\; \begin{cases}A = up - v q \\ B = uq + vp. \end{cases}\end{aligned}\end{align}\] Our goal is to prove that \(A\) and \(B\) satisfy the Cauchy-Riemann equations for \(x + i y \in D\) , which would then imply that \(fg\) is analytic in \(D\) .   Q.E.D.
Exercise \(\PageIndex{4}\)
Prove that compositions of analytic functions are analytic: if \(f(z)\) is analytic in \(D \subset \mathbb{C}\) and \(g(z)\) is analytic in the range of \(f\) , then \(g(f(z))\) is analytic in \(D\) .
Exercise \(\PageIndex{5}\)
Prove that reciprocals of analytic functions are analytic away from poles: if \(f(z)\) is analytic in \(D \subset \mathbb{C}\) , then \(1/f(z)\) is analytic everywhere in \(D\) except where \(f(z) = 0\) .
Exercise \(\PageIndex{6}\)
Show that if \(f(z = x + iy) = u(x,y) + i v(x,y)\) satisfies the Cauchy-Riemann equations, then the real functions \(u\) and \(v\) each obey Laplace’s equation: \[\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0.\] (Such functions are called “harmonic functions”.)
Exercise \(\PageIndex{7}\)
We can write the real and imaginary parts of a function in terms of polar coordinates: \(f(z) = u(r,\theta) + i v(r,\theta)\) , where \(z = re^{i\theta}\) . Show that the Cauchy-Riemann equations can be re-written in polar form as \[\frac{\partial u}{\partial r} = \frac{1}{r} \frac{\partial v}{\partial \theta}, \quad \frac{\partial v}{\partial r} = - \frac{1}{r}\, \frac{\partial u}{\partial \theta}.\]


6.3C: Microscopic Anatomy of Bone
- Classify woven bone and lamellar bone
Bones are composed of bone matrix, which has both organic and inorganic components. Bone matrix is laid down by osteoblasts as collagen, also known as osteoid. Osteoid is hardened with inorganic salts, such as calcium and phosphate, and by the chemicals released from the osteoblasts through a process known as mineralization.
The basic microscopic unit of bone is an osteon (or Haversian system). Osteons are roughly cylindrical structures that can measure several millimeters long and around 0.2 mm in diameter.
Each osteon consists of a lamellae of compact bone tissue that surround a central canal (Haversian canal). The Haversian canal contains the bone’s blood supplies. The boundary of an osteon is called the cement line. Osteons can be arranged into woven bone or lamellar bone.
Osteon : A photo taken through a microscope that shows the anatomy of compact bone with a detailed view of an osteon.
Woven Bone
Woven bone : Woven bone is characterized by the irregular organization of collagen fibers and is mechanically weak.
Woven bone is found on the growing ends of an immature skeleton or, in adults, at the site of a healing fracture. Woven bone is characterized by the irregular organization of collagen fibers and is mechanically weak, but forms quickly.
The criss-cross appearance of the fibrous matrix is why it is referred to as woven. It has a high proportion of osteocytes to hard inorganic salts that leads to its mechanical weakness.
Woven bone is replaced by lamellar bone during development. In contrast to woven bone, lamellar bone is highly organized in concentric sheets with a much lower proportion of osteocytes to surrounding tissue. The regular parallel alignment of collagen into sheets, or, lamellae, causes lamellar bone to be mechanically strong.
Lamellar Bone
Femur head showing trabecular bone : A cross-section of the head of the femur showing lamellar bone on the borders and trabecular bone in the center.
Lamellar bone makes up the compact or cortical bone in the skeleton, such as the long bones of the legs and arms. In a cross-section, the fibers of lamellar bone can be seen to run in opposite directions in alternating layers, much like in plywood, assisting in the bone’s ability to resist torsion forces.
When the same lamellar bone is loosely arranged, it is referred to as trabecular bone. Trabecular bone gets its name because of the spongy pattern it displays in an x-ray. The spaces within trabecular bone are filled with active bone marrow. After a fracture, woven bone forms initially, but it is gradually replaced by lamellar bone during a process known as bony substitution.
Key Points
- Woven bone is found on the growing ends of an immature skeleton or, in adults, at the site of a healing fracture.
- Woven bone is characterized by the irregular organization of collagen fibers and is mechanically weak, but forms quickly.
- Lamellar bone is much stronger than woven bone, and is highly organized in concentric sheets with a much lower proportion of osteocytes to mineralized tissue.
- When the same lamellar bone is loosely arranged, it is referred to as trabecular bone. Trabecular bone gets its name because of the spongy pattern it displays on an x-ray.
- After a fracture, woven bone forms initially and is gradually replaced by lamellar bone during a process known as bony substitution.
Key Terms
- osteoblast : A mononucleate cell from which bone develops.
- osteocytes : A star-shaped type of bone cell that is found in the cells of mature bone.
- lamellar bone : A bone with a regular, parallel alignment of collagen into sheets (lamellae) that is mechanically strong.
- woven bone : Characterized by an irregular organization of collagen fibers, this bone is mechanically weak.


2.4: Acceleration
Learning Objectives
By the end of this section, you will be able to:
- Define and distinguish between instantaneous acceleration, average acceleration, and deceleration.
- Calculate acceleration given initial time, initial velocity, final time, and final velocity.
In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration , the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions, but more inclusive.
Definition: Average Acceleration
The average acceleration is the rate at which velocity changes,
\[\bar{a}=\frac{Δv}{Δt}=\frac{v_f−v_0}{t_f−t_0}\]
where \(\bar{a}\) is average acceleration, \(v\) is velocity, and \( t\) is time. (The bar over the \(a\) means average .)
Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are \( m/s^2\), meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second.
Recall that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.
ACCELERATION AS A VECTOR
Acceleration is a vector in the same direction as the change in velocity, \(Δv\). Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction, or both.
Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion . When an object slows down, its acceleration is opposite to the direction of its motion. This is known as deceleration .
MISCONCEPTION ALERT: DECELERATION VS. NEGATIVE ACCELERATION
Deceleration always refers to acceleration in the direction opposite to the direction of the velocity. Deceleration always reduces speed. Negative acceleration, however, is acceleration in the negative direction in the chosen coordinate system. Negative acceleration may or may not be deceleration, and deceleration may or may not be considered negative acceleration. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down. For example, consider Figure \(\PageIndex{2}\).
Example \(\PageIndex{1}\): Calculating Acceleration: A Racehorse Leaves the Gate
A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration?
Strategy
First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity.
We can solve this problem by identifying \(Δv\) and \(Δt\) from the given information and then calculating the average acceleration directly from the Equation \ref{averagea}:
\[\bar{a}=\dfrac{Δv}{Δt}=\dfrac{v_f−v_0}{t_f−t_0}. \nonumber\]
Solution
- Identify the knowns. \(v_0=0 , v_f=−15.0 m/s\) (the negative sign indicates direction toward the west), \( Δt=1.80 s\).
- Find the change in velocity. Since the horse is going from zero to \( −15.0 m/s\), its change in velocity equals its final velocity: \[Δv=v_f=−15.0\, m/s .\nonumber\]
- Plug in the known values ( \(Δv\) and \(Δt\) ) and solve for the unknown \(\bar{a}\).
\(\bar{a}=\frac{Δv}{Δt}=\frac{−15.0 m/s}{1.80 s}=−8.33 m/s^2\).
Discussion
The negative sign for acceleration indicates that acceleration is toward the west. An acceleration of \(8.33\, m/s^2\) due west means that the horse increases its velocity by 8.33 m/s due west each second, that is, 8.33 meters per second per second, which we write as \( 8.33\, m/s^2\). This is truly an average acceleration, because the ride is not smooth. We shall see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to his weight.
Instantaneous Acceleration
Instantaneous acceleration \( a\), or the acceleration at a specific instant in time, is obtained by the same process as discussed for instantaneous velocity in Time, Velocity, and Speed —that is, by considering an infinitesimally small interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion. Figure \(\PageIndex{6}\) shows graphs of instantaneous acceleration versus time for two very different motions. In Figure \(\PageIndex{6a}\), the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about \(1.8 m/s^2\)). In Figure \(\PageIndex{6b}\), the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of \( +3.0\, m/s^2\) and \( –2.0 \,m/s^2\), respectively.
The next several examples consider the motion of the subway train shown in Figure \(\PageIndex{7}\). In Figure \(\PageIndex{7a}\) the shuttle moves to the right, and in Figure \(\PageIndex{7b}\) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems.
Example \(\PageIndex{2}\): Calculating Displacement - A Subway Train
What are the magnitude and sign of displacements for the motions of the subway train shown in parts (a) and (b) of Figure \(\PageIndex{7}\)?
Strategy
A drawing with a coordinate system is already provided, so we don’t need to make a sketch, but we should analyze it to make sure we understand what it is showing. Pay particular attention to the coordinate system. To find displacement, we use the equation \( Δx=x_f−x_0\). This is straightforward since the initial and final positions are given.
Solution
- Identify the knowns. In the figure we see that \(x_f=6.70\, km \) and \(x_0=4.70\, km\) for part (a), and \(x'_f=3.75\, km \) and \(x'_0=5.25\, km \) for part (b).
- Solve for displacement in part (a).\[\begin{align*} Δx &=x_f−x_0 \\[5pt] &=6.70\, km−4.70\, km \\[5pt] &=+2.00 \,km \end{align*}\]
- Solve for displacement in part (b).\[\begin{align*} Δx' &=x'_f−x'_0 \\[5pt] &=3.75\, km−5.25\, km \\[5pt] &=−1.50\, km \end{align*}\]
Discussion
The direction of the motion in (a) is to the right and therefore its displacement has a positive sign, whereas motion in (b) is to the left and thus has a negative sign.
Example \(\PageIndex{3}\): Comparing Distance Traveled with Displacement - A Subway Train
What are the distances traveled for the motions shown in parts (a) and (b) of the subway train in Figure \(\PageIndex{7}\) ?Strategy
To answer this question, think about the definitions of distance and distance traveled, and how they are related to displacement. Distance between two positions is defined to be the magnitude of displacement, which was found in Example \(\PageIndex{2}\). Distance traveled is the total length of the path traveled between the two positions (see Section on Displacement ). In the case of the subway train shown in Figure \(\PageIndex{7}\), the distance traveled is the same as the distance between the initial and final positions of the train.
Solution
1. The displacement for part (a) was +2.00 km. Therefore, the distance between the initial and final positions was 2.00 km, and the distance traveled was 2.00 km.
2. The displacement for part (b) was \( −1.5 km\). Therefore, the distance between the initial and final positions was 1.50 km, and the distance traveled was 1.50 km.
Discussion
Distance is a scalar. It has magnitude but no sign to indicate direction.
Example \(\PageIndex{4}\): Calculating Acceleration: A Subway Train Speeding Up
Suppose the train in Figure \(\PageIndex{7a}\) accelerates from rest to 30.0 km/h in the first 20.0 s of its motion. What is its average acceleration during that time interval?
Strategy
It is worth it at this point to make a simple sketch:
Solution
- Identify the knowns. \( v_0=0\) (the trains starts at rest), \( v_f=30.0 km/h \), and \(Δt=20.0 s \).
- Calculate \(Δv\). Since the train starts from rest, its change in velocity is \(Δv=+30.0 km/h \), where the plus sign means velocity to the right.
- Plug in known values and solve for the unknown, \(\bar{ a}\). \[\bar{a}=\dfrac{Δv}{Δt}=\dfrac{+30.0 km/h}{20.0 s}\nonumber\]
- Since the units are mixed (we have both hours and seconds for time), we need to convert everything into SI units of meters and seconds. (See Physical Quantities and Units for more guidance.) \[\bar{a}=\left(\dfrac{+30 km/h}{20.0 s}\right)\left(\dfrac{10^3m}{1 km}\right)\left(\dfrac{1h}{3600 s}\right)=0.417 m/s^2 \nonumber\]
Discussion
The plus sign means that acceleration is to the right. This is reasonable because the train starts from rest and ends up with a velocity to the right (also positive). So acceleration is in the same direction as the change in velocity, as is always the case.
Example \(\PageIndex{5}\): Calculate Acceleration
A Subway Train Slowing Down: Now suppose that at the end of its trip, the train in Figure \(\PageIndex{7a}\) slows to a stop from a speed of 30.0 km/h in 8.00 s. What is its average acceleration while stopping?
Strategy
In this case, the train is decelerating and its acceleration is negative because it is toward the left. As in the previous example, we must find the change in velocity and the change in time and then solve for acceleration.
Solution
- Identify the knowns. \(v_0=30.0 km/h , v_f=0 km/h\) (the train is stopped, so its velocity is 0), and \(Δt=8.00 s \).
- Solve for the change in velocity, \(Δv\). \[Δv=v_f−v_0=0−30.0 km/h=−30.0 km/h \nonumber\]
- Plug in the knowns, \( Δv\) and \(Δt\), and solve for \(\bar{a}\). \[\bar{a}=\dfrac{Δv}{Δt}=\dfrac{−30.0 km/h}{8.00 s} \nonumber\]
- Convert the units to meters and seconds. \[\bar{a}=\dfrac{Δv}{Δt}=\left(\dfrac{−30.0 km/h}{8.00 s}\right)\left(\dfrac{10^3m}{1 km}\right)\left(\dfrac{1 h}{3600 s}\right)=−1.04 m/s^2. \nonumber\]
Discussion
The minus sign indicates that acceleration is to the left. This sign is reasonable because the train initially has a positive velocity in this problem, and a negative acceleration would oppose the motion. Again, acceleration is in the same direction as the change in velocity, which is negative here. This acceleration can be called a deceleration because it has a direction opposite to the velocity.
The graphs of position, velocity, and acceleration vs. time for the trains in Example \(\PageIndex{4}\) and \(\PageIndex{5}\) are displayed in Figure \(\PageIndex{10}\). (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.)
Example \(\PageIndex{6}\): Calculating Average Velocity: The Subway Train
What is the average velocity of the train in part b of Example \(\PageIndex{2}\), and shown again below, if it takes 5.00 min to make its trip?
Strategy
Average velocity is displacement divided by time. It will be negative here, since the train moves to the left and has a negative displacement.
Solution
- Identify the knowns. \[ x'_f=3.75 km,\, x'_0=5.25 km,\, Δt=5.00 min. \nonumber\]
- Determine displacement, \(Δx'\). We found \( Δx'\) to be \( −1.5 km\) in Example \(\PageIndex{7}\).
- Solve for average velocity. \[\bar{v}=\dfrac{Δx'}{Δt}=\dfrac{−1.50 km}{5.00 min} \nonumber\]
- Convert units. \[\bar{v}=\dfrac{Δx'}{Δt}=(\dfrac{−1.50 km}{5.00 min})(\dfrac{60 min}{1 }h)=−18.0 km/h \nonumber\]
Discussion
The negative velocity indicates motion to the left.
Example \(\PageIndex{7}\): Calculating Deceleration: The Subway Train
Finally, suppose the train in Figure \(\PageIndex{7}\) slows to a stop from a velocity of 20.0 km/h in 10.0 s. What is its average acceleration?
Strategy
Once again, let’s draw a sketch:
Solution
- Identify the knowns. \(v_0=−20\, km/h\), \(v_f=0\, km/h\), \(Δt=10.0\, s\).
- Calculate \(Δv\). The change in velocity here is actually positive, since \[Δv=v_f−v_0=0−(−20\, km/h)=+20\, km/h. \nonumber\]
- Solve for \(\bar{ a}\). \[\bar{a}=\dfrac{Δv}{Δt}=\dfrac{+20.0\, km/h}{10.0\, s} \nonumber\]
- Convert units. \[\bar{a}=\left(\dfrac{+20.0 \,km/h}{10.0\, s}\right)\left(\dfrac{10^3\,m}{1\, km}\right)\left(\dfrac{1\, h}{3600 \,s}\right)=+0.556\, m/^s2 \nonumber\]
Discussion
The plus sign means that acceleration is to the right. This is reasonable because the train initially has a negative velocity (to the left) in this problem and a positive acceleration opposes the motion (and so it is to the right). Again, acceleration is in the same direction as the change in velocity, which is positive here. As in Example \(\PageIndex{5}\), this acceleration can be called a deceleration since it is in the direction opposite to the velocity.
Sign and Direction
Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in Example \(\PageIndex{5}\), where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in Figure Figure \(\PageIndex{11}\) is sped up by an acceleration to the left. In that case, both v and a are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down.
Exercise \(\PageIndex{1}\)
An airplane lands on a runway traveling east. Describe its acceleration.
- Answer
-
If we take east to be positive, then the airplane has negative acceleration, as it is accelerating toward the west. It is also decelerating: its acceleration is opposite in direction to its velocity.
PHET EXPLORATIONS: MOVING MAN SIMULATION
Learn about position, velocity, and acceleration graphs with the PhET Moving Man simulation . Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you.
Summary
- Acceleration is the rate at which velocity changes. In symbols, average acceleration \(\bar{ a}\) is \(\bar{a}=\dfrac{Δv}{Δt}=\dfrac{v_f−v_0}{t_f−t_0}\).
- The SI unit for acceleration is 2 .
- Acceleration is a vector, and thus has a both a magnitude and direction.
- Acceleration can be caused by either a change in the magnitude or the direction of the velocity.
- Instantaneous acceleration is the acceleration at a specific instant in time.
- Deceleration is an acceleration with a direction opposite to that of the velocity.
Glossary
- acceleration
- the rate of change in velocity; the change in velocity over time
- average acceleration
- the change in velocity divided by the time over which it changes
- instantaneous acceleration
- acceleration at a specific point in time
- deceleration
- acceleration in the direction opposite to velocity; acceleration that results in a decrease in velocity