4.9: 4.9 Portfolio- Filtered Memories
By the end of this section, you will be able to:
- Reflect on the composition process.
- Reflect on how the composition process affects your writing.
- Use composition for learning, critical thinking, and communicating in various rhetorical contexts.
As part of this course, you are creating a portfolio of your compositions to show your development as a writer. As you add to this portfolio, you assess your growth as a writer. The portfolio of your work allows you not only to have a record of your compositions but also to reflect on those compositions and the process involved in writing them. In this section, you will write a reflection on the process you followed in writing your turningpoint story, including revision and editing. Consider your original intention when creating your essay and how close your final composition comes to fulfilling that intention.
Reflective Task
Answer the following questions before writing several paragraphs reflecting on the personal turning point you wrote about in this chapter. Consider both the process and the outcome of the writing assignment.
- Why did you choose the turning-point moment or event that you did?
- Which part of the writing process came the most naturally? Which was the most difficult?
- Which parts of the composition do you think are the strongest? Why?
- Which parts of the composition would you still like to improve?
- How did you develop your voice as a narrator as you wrote?
- If you were to write another turning-point story, what would you do differently? (Consider topic, organization, structure, point of view, and voice.)
- How did your story change through the revision and peer-editing processes? Name at least two changes you made, and reflect on how they made the piece stronger.
- What did you learn about yourself, both as a writer and as a person, from this exercise?
Further Reading
The following titles are good examples of essays, memoirs, and poems that tell personal stories.
Akbar, Kaveh. “How I Found Poetry in Childhood Prayer: In the Presence of the Long Faultless Tongue of God.” Literary Hub , 12 Sept. 2017, https://lithub.com/kaveh-akbar-how-i...ldhood-prayer/ .
Angelou, Maya. I Know Why the Caged Bird Sings . Random House, 1969.
Anzaldúa, Gloria. Borderlands / La Frontera: The New Mestiza . 4th ed., Aunt Lute Books, 2012.
Baldwin, James. Notes of a Native Son . 1955. Beacon Press, 2012.
Beasley, Sandra. Don’t Kill the Birthday Girl: Tales from an Allergic Life . Crown, 2011.
Bryson, Bill. I’m a Stranger Here Myself: Notes on Returning to America after 20 Years Away . Broadway Books, 1999.
Lamott, Anne. Bird by Bird: Some Instructions on Writing and Life . Pantheon Books, 1994.
McCourt, Frank. Angela’s Ashes . Scribner, 1996.
O’Brien, Tim. The Things They Carried . Houghton Mifflin, 1990.
Pausch, Randy. The Last Lecture . With Jeffrey Zaslow, Hyperion, 2008.
Schutz, Samantha. I Don’t Want to Be Crazy . Scholastic, 2006.
Walls, Jeanette. The Glass Castle: A Memoir . Scribner, 2006.
Woodson, Jacqueline. Brown Girl Dreaming . Nancy Paulsen Books, 2014.
Works Cited
Abrams, Shane. EmpoWord: A Student-Centered Anthology and Handbook for College Writers . Portland State U, 2018, https://pdx.pressbooks.pub/empoword/ .
Belcher, Diane, and Alan Hirvela, editors. Linking Literacies: Perspectives on L2 Reading-Writing Connections . U of Michigan P, 2001.
Coates, Ta-Nehisi. “Ta-Nehisi Coates on the Writing Process.” Interview by Martha Teichner. YouTube , uploaded by CBS Sunday Morning, 5 Nov. 2017, http://www.youtube.com/watch?v=Z1bOLOmix1Y .
Dickens, Charles. The Pickwick Papers . Chapman & Hall, 1836. Project Gutenberg, https://www.gutenberg.org/ files/580/580-h/580-h.htm.
Ferris, Dana R. Teaching College Writing to Diverse Student Populations . U of Michigan P, 2009.
Guptill, Amy. Writing in College: From Competence to Excellence . Open SUNY, 2016, https://milneopentextbooks.org/writi...to-excellence/ .
Hawthorne, Nathaniel. “The Haunted Mind.” Twice-Told Tales , American Stationers, 1837. Project Gutenberg, https://www.gutenberg.org/files/9209...9-h/9209-h.htm .
Malcolm X. “Literacy behind Bars.” The Autobiography of Malcolm X , Ballantine, 1964. Northern Local School District, https://www.nlsd.k12.oh.us/userfiles/63/Classes/4521/ malcolm%20x%20-%20literacy%20behind%20bars.pdf.
Twain, Mark. Life on the Mississippi . James R. Osgood, 1883. Project Gutenberg, http://www.gutenberg.org/ files/245/old/orig245-h/245-h.htm.


2.4: Frameworks to Inform Your Enterpreneurial Path
By the end of this section, you will be able to:
- Identify common frameworks used to shape an entrepreneurial venture
- Compare how some frameworks better fit certain venture types
- Define an action plan and identify tools available for creating an action plan
- Describe some common types of entrepreneurs
In designing a venture that is sustainable or capable of being self-funded, it is helpful to use specific tools to manage information. One such tool is a framework —a structure or outlined process that can be used to accomplish entrepreneurial goals through problem solving, idea generation and validation, and brainstorming.
Selecting a Framework
You can choose any of several popular frameworks to help with the design and integration of your business experience and entrepreneurial thinking. The most widely used frameworks that have been developed as integrative tools to support entrepreneurial thinking include:
- Business Model Canvas (BMC) offers a simple, one-page tool used to design an innovative business model that can be presented to key stakeholders ( Figure 2.22 ). The business model canvas is discussed more fully in Business Model and Plan .
- Lean Strategy Canvas is a spinoff of the BMC that introduces a potential customer feedback loop for continuous product or idea improvement to meet the market’s needs ( Figure 2.23 ). The lean strategy canvas is discussed more fully in Launch for Growth to Success .
- Design Thinking Process supports a systematic, logical approach for addressing and solving problems with multiple solutions ( Figure 2.24 ). Design thinking was first applied in relation to STEM fields—science, technology, engineering, and mathematics. Due to the success of this process, design thinking has become popular in many other areas.
Design thinking approaches problem solving or the creation of a new venture from the perspective of the customer. For example, Amazon provides easy-to-open packages after observing the challenges customers had in opening the delivered products. Design thinking is covered in greater detail in Problem Solving and Need Recognition Techniques with applications to starting an entrepreneurial venture, product design, and improvements to existing products.
- Four Lenses Strategic Framework is used for the development of social enterprises; assesses four strategic areas (stakeholder engagement, resource mobilization, knowledge development, and culture management) to address a social problem and provide sustainable social impact ( Figure 2.25 ).
The typical application of each framework is shown in Table 2.3 . The process of selecting the most appropriate framework for your entrepreneurial interests might help you understand how to develop your idea. We recommend that you try all four frameworks before selecting one. Even though each framework is identified for a general class of venture, each one provides a different perspective for developing your venture.
| Framework | Description | Typical Use |
|---|---|---|
| Business Model Canvas | A one-page tool that maps out nine basic building blocks that are necessary for a successful business model | Helps prepare a sustainable business model |
| Lean Startup | Outlines a quick feedback loop through customer input | Used for fast-paced industries and quick idea validation |
| Design Thinking Process | Outlines a systematic, results-oriented process to address and solve problems | Used for the development of STEM fields with expansion into entrepreneurial ventures, products, and processes; applicable to all areas |
| Four Lenses Strategic Framework | A practitioner-driven model that considers four perspectives to support and develop a client-focused ecosystem | Used for the development of social ventures |
Applying the Framework through an Action Plan
At some point during your venture development process, it becomes critical to capture your thoughts and intentions in a meaningful and productive way. Creating a customized action plan —an organized, step-by-step outline or guide that pulls together the ideas, thoughts, and key steps necessary to help set the stage for entrepreneurial success—at an early stage will make the entrepreneurial process much smoother and potentially more successful in the long run. Applying an appropriate framework will provide you with a visible, tangible, strong foundation for your future venture. In completing the framework, you should identify gaps as well as ideas for further development, then add both to your action plan. Just as you can choose from several types of frameworks, you can apply any of a variety of action plans. This section introduces some widely used action-planning tools but is not exhaustive. These selected action plans are presented as a way to jumpstart your thinking for the venture creation process.
Action Plans
You may have heard stories about potential entrepreneurs who hesitated to start a venture, largely out of fear of creating a business plan. Historically, business plan creation has required significant amounts of time, resources, and research. Although business plans are still enormously valuable (and are discussed in depth in Business Model and Plan ), some useful business-plan-like tools have emerged: These are essentially variations on the development, content, and structure of a traditional business plan or one of its components. One other concern about business plans is how entrepreneurs use them once they are completed. In many cases, when the venture is launched, the entrepreneurial team discovers that the business plan does not reflect the realities that the team faces. A wide range of variables can often negate the value of the business plan. The true benefit of completing the business plan is that it forces the entrepreneurial team to think through their decisions as reflected in the plan. Even if the venture and the business plan change, the process of creating the business plan encourages critical thinking and improved decisions. In real time, you will need to make changes to your business plan and your venture. Throughout the venture’s life span, you should continue your background research and projections to adapt the business plan.
Unlike with the business plan, the purpose of an action plan is to pull together the ideas, thoughts, and actions necessary to help you set the stage for entrepreneurial success. Consider what kind of action plan you need to prepare a holiday meal. We have a vision of the end result—friends and family gathered together to share a delicious, festive meal. We will need to select the right location for the holiday meal, identify the guests to invite, and create a financial budget for the related costs of the holiday meal. Then we would need to create our action plan—similar to a business plan—to identify what actions are necessary to support the event. In our action plan, we would include inviting guests to the event, drawing up a menu and a grocery list, designing a timeline to ensure that all the dishes of the holiday meal are completed in the correct sequence: We want all the food to be ready at the right time. Our action plan would also include the clean-up process and any after-dinner activities that we want at our event. As you can see, both the business plan and the action plan are necessary for success.
 The framework offers a big picture of what you want to create and the resources required for that goal, whereas the action plan provides you with concrete actions for starting along your entrepreneurial path and, later, for supporting the business plan.
Action plans can also result from using the tools listed in Table 2.4 . These tools can help you visualize the process necessary to reach your end goal by clarifying the necessary actions. They are also tangible guidelines for innovating, exploring, and creating solutions to entrepreneurial problems or opportunities. You might also use your action plan to get “unstuck” during any challenging phase of the entrepreneurial process. One word of caution regarding these tools: You need to use them to get results. So be sure you are realistic about your interests, abilities, and availability when you create your plans. For example, wireframing is a technique for webpage design used early in the development process in which content, layout, and functionality are identified prior to the actual creation of the webpage. Coincidentally, this is another application of design thinking through the focus on the end user’s interaction with the website. As you can see from this example, searching for popular tools used within specific industries will provide you with support in building your framework. Table 2.4 provides a few examples of action planning tools that are used to delve into the specific topic.
| Tool | Description | Use |
|---|---|---|
| Vision or Dream Board | A visual tool to present the ideal situation that you are working towards achieving | Wireframing |
| Storyboard | A scene-by-scene visual of the activity process from start to finish | Downloading method (IDEO) |
| Mind map | A visual tool that assists with categorizing brainstorming types of ideas | Brainstorming |
| Hypothesis | A proposition or statement as a basis for further testing or investigating | Interviewing |
| Logic Map | Visual representation of relationships between various components or variables | Questionnaires |
The action plan support tools presented in Table 2.4 are a sample list of representative tools that are useful in motivating, inspiring, identifying, and clarifying needed actions. This list is not exhaustive; if you have something that works for you, then use it. Several apps are also available to help you capture ideas to create action plans, as shown in the Suggested Resources . The idea is to find and use a visual or tangible tool that inspires you to get focused, organized, and committed to taking the actions necessary to turn your entrepreneurial dream into a reality. Let’s say you know that you want to start a venture that helps people recover after some type of disaster. You could use one of these action plan support tools, such as a mind map, to help you brainstorm possible needs resulting from a disaster in a city ( Figure 2.26 ). You can categorize the ideas to help people and animals, or to repair the city’s infrastructure. After completing the mind map, you would then consider which areas fit your interests, passions, and skills. From this point, you could identify the type of venture you want to create and the necessary actions to move forward with your idea. Using these types of tools assists in identifying actions that need to be addressed in your action plan.
Types of Entrepreneurs
Recall from The Entrepreneurial Perspective that for some people, the entrepreneurial pathway is clear cut and logical. For example, a career in a biomedical lab may involve research and clinical trials that lead to patent applications for a product to sell in the marketplace, leading to a new venture. Others experience the entrepreneurial pathway through nontraditional methods, as when an unexpected opportunity arises. As the global marketplace continues to evolve, new entrepreneurial opportunities will open for individuals who are open to opportunities that build on creativity and innovation.
Traditional entrepreneurs were perceived as individuals who did not fit in a typical organizational structure or as people who had the brains, creativity, imagination, and money to launch out on their own. However, this perception is changing with increasing support to reduce barriers to enable access to entrepreneurship for all demographic groups. According to a 2018 Capitol Hill discussion on women, minorities, and entrepreneurship, the current entrepreneurial demographics show that only 12 percent of US innovators are women and that US-born minorities accounted for 8 percent, with African Americans making up only one-half of 1 percent of this group. 46 According to a National Academies of Sciences report, as cited in the same Capitol Hill discussion, women- and minority-owned small businesses received less than 16 percent of all Small Business Innovation Research (SBIT) program awards. Even though women account for 51 percent of the US population and own 29 percent of businesses, they received only 6 percent of SBIT awards.
Also cited at the Capitol Hill briefing, a 2015 report by the US Department of Commerce showed that women-owned small businesses have a 21 percent lower rate of winning federal contracts. One result from this Capitol Hill briefing was the passage of the Promoting Women in Entrepreneurship Act to require the National Science Foundation to encourage entrepreneurial programs to recruit and support women in commercial activities rather than purely laboratory-based activities. Some of the challenges identified in this discussion for groups other than the traditional entrepreneurs described include life choices such as childbearing, access to funding, and lack of support and follow-through to support women and minorities in their interests related to potential entrepreneurial activities.
Other findings drawn from the Census Bureau data and reported by the Kauffman Foundation found that over 70 percent of Asian, Hispanic, and African American entrepreneurs relied on personal and family savings as their main source of startup capital. Women also face challenges in funding, receiving just 2.2 percent of venture capital funding in 2018. 47 A bill called the Support Startup Businesses Act, reintroduced in the US Senate in 2019, would address these challenges by increasing overall funding to help startups, creating more flexibility in funding, and expanding services for startups. 48 The panelists ended the Capitol Hill discussion by noting that “bro culture” has proliferated at the expense of women- and minority-based entrepreneurial ideas. Cultural barriers, including historical disenfranchisement of women, minorities, and immigrants, arise from biases, with one speaker noting that investors ask more difficult and probing questions of male entrepreneurs but pose more skeptical questions to women.
Today, opportunities have expanded for businesses and organizations that respond to current challenges, which may include trying to improve a negative situation or finding a need in a positive situation, with an increasing awareness of the benefits provided through entrepreneurial activities. As more global, cultural, and economic issues and opportunities arise, more individuals will explore entrepreneurship as a response to these challenges. For example, noting the challenges that women and minorities face in starting a new venture, Alan Donegan and his team train people on how to turn their entrepreneurial visions into a reality through his PopUp Business School. The point is that opportunities should be available to everyone, as long as we keep an open mind when considering how change contributes to new venture creation.
Barriers to Funding
Given this list of cultural factors and economic factors, what can you do to assist in solving the challenges related to biases? Consider how Alan Donegan’s responded to the need to educate people on how to start their own business. He created a company to address this need. You may also consider reading statistical information such as census data and news reports to identify unique target markets and potential needs that could result in a new venture.
The Kauffman Foundation reports these issues, summarized in Table 2.5 .
| Potential Barrier | Challenges |
|---|---|
| Geographical barriers | Close to 80 percent of about $21.1 billion in venture-capital funding in the first quarter of 2018 was disbursed in five regional clusters—San Francisco (North Bay Area), Silicon Valley (South Bay Area), New England, New York City metro, and LA/ Orange County—with slightly more than 44 percent in the North and South Bay Areas. |
| Gender bias | Women are substantially less likely to start businesses than men. In 1996, the rate of new entrepreneurs for women was 260 per 100,000 people, compared to 380 per 100,000 for men. In 2017, the rate of new entrepreneurs for women was 270 per 100,000 for men. |
| Racial and ethnic bias | The landscape of entrepreneurship in the United States is marked by significant differences across racial and ethnic groups. Minority-owned firms are found to face significant barriers to capital. For example, minority-owned firms are disproportionately denied when they need and apply for additional credit. One study compared sources of finance and found that new black-owned businesses start with almost three times less in terms of overall capital than new white-owned businesses, and that this gap does not close as firms mature. |
| Lack of initial wealth | Low-income individuals without initial (pre-existing) wealth also face significant barriers to capital. Research on liquidity constraints showed that the top ninety-fifth percentile of wealthy individuals in the United States is more likely to start businesses than other income groups, and that personal and household wealth are important drivers of entry. Research at the neighborhood level found that in New York City, the richer third of neighborhoods had more than twice the rate of self-employment than the poorest third. A higher household net worth of a founder is linked to larger amounts of external funding received, even after accounting for human capital, venture characteristics, and demand for funds. |
| Shift in the banking industry | Large banks have become larger, while there are fewer small and medium-size banks. Larger banks survived the Great Recession with balance sheets restored, while small banks—the ones more likely to lend to entrepreneurs—were limited by both economic conditions and new regulatory barriers |
| Information asymmetry | The persistence of information asymmetry in capital markets between the supply of capital (investors) and the demand for capital (entrepreneurs) gives rise to barriers faced by entrepreneurs. Entrepreneurs face a larger challenge than established businesses in accessing capital because established businesses can leverage their longer track records and existing relationships. |
- What challenges do you face in these areas?
- What steps might you take to help mediate those challenges?
As the traditional view of entrepreneurship evolves, different types of entrepreneurship are emerging and are worth noting as you contemplate your entrepreneurial journey. The types presented here are among the most common today, each with its own unique opportunities and challenges.
- College Entrepreneur: As the cost of higher education continues to rise, more college students are seeking ways to reduce reliance on tuition loans by launching a venture. The college entrepreneur might launch an enterprise while attending or after graduating from college. Entrepreneurship courses might require a student to create and launch a venture as part of the curriculum, and this can turn into an actual earnings opportunity.
- Corporate Intrapreneur: If you work for a progressive company that seeks innovative solutions for growth and opportunities, you can become an intrapreneur by organizing the necessary resources to pursue a venture of organizational interest.
- Franchise Entrepreneur: Since a franchise grants a license to an entrepreneur to trade under the franchise’s name, a franchise entrepreneur gains a head start in an industry by launching the franchise.
- Immigrant Entrepreneur: With increasing global unrest, more immigrants are traveling to new countries. In the United States, ethnic communities of immigrants welcome their compatriots and assist them in becoming independent through entrepreneurship. These communities pool together the necessary resources to support the new immigrant until the business is self-sustaining.
- Internet Entrepreneur: As access to technology and its related platforms increases, so do opportunities for Internet-based businesses. Internet entrepreneurs utilize social media platforms, smartphones and tablets, applications (apps), and any other form of accessible technology as their product or venture. The critical structure for these ventures is the inclusion of an e-commerce or online payment processing capability.
- Woman or Minority Entrepreneur: Women have a unique perspective and potential to capitalize on new or already existing niches in many entrepreneurial fields. Many cultural groups, such as Haitians, Cubans, or Jamaicans, also have unique marketplace skills and demands.
- Part-time Entrepreneur: In response to economic downturns, underemployment, and unemployment, more individuals are supplementing income through part-time activities, casually referred to as “side hustles.” These individuals may launch businesses through multilevel marketing firms, such as Avon, Mary Kay, Stella & Dot, and others. This category may also include self-employed freelancers. Examples include writers, graphic designers, artists, web developers, and massage therapists.
- Social Entrepreneur: Some entrepreneurs are driven to offer innovative solutions to existing and emerging social problems, such as poverty, hunger, human trafficking, and environmental degradation. Most social enterprises are structured as nonprofit entities. However, increased interest in for-profit entities that marry business and social goals has given rise to a subcategory has emerged known as a B-corp (see Business Structures Options: Legal, Tax, and Risk Issues ), or benefits corporation. The B-corp designation is a voluntary certification that is managed by the nonprofit group B Lab to ensure that corporations adhere to specific guidelines, rules, and accountability.


7.6: Key Words and Terms
|
active transport of protons |
energy efficiency of glucose metabolism |
PEP carboxylase |
|
ATP synthase |
energy flow in glycolysis |
pH gradient |
|
bacterial flagellum |
energy flow in the Krebs Cycle |
pheophytin |
|
C4 photosynthesis |
F1 ATPase |
photosynthesis |
|
Calvin Cycle |
FAD |
Photosystems |
|
CAM photosynthesis |
FADH 2 |
proton gate |
|
carotene |
Light-dependent reactions |
proton gradient |
|
chemiosmotic mechanism |
Light-independent reactions |
proton pump |
|
Chlorophyll a |
Malic acid |
PSI electron acceptor |
|
Chlorophyll b |
mitochondria |
PSII electron acceptor |
|
molecular motor |
redox reactions |
|
|
complex I, II, III, IV |
NAD+ |
reduced electron carriers |
|
Crassulaceae |
NADH |
RUBISCO |
|
cristal membrane |
NADP+ |
RuBP |
|
Cyclic photophosphorylation |
NADPH |
Splitting water |
|
cytochromes |
outer membrane |
stoichiometry of glycolysis |
|
Dark Reactions |
oxidative phosphorylation |
stoichiometry of the Krebs Cycle |
|
electrochemical gradient |
oxidative phosphorylation |
substrate-level phosphorylation |
|
electron transport system (chain) |
P 680 |
Z-scheme |
|
endosymbiotic theory |
P 700 |


7.2.36: Memory Game (Usable with Workarounds)
Query \(\PageIndex{1}\)
Overall Usability
Usable with Workarounds:
- The activity requires significant adjustments by content authors.
Important Reminders
H5P does not provide an option to stop, pause, or hide the automatically updating timer that is part of the activity. Some users may find the timer distracting or disruptive and may require an alternative version of the activity. Provide instructions for users that describe the auto-updating content. H5P does not provide an option to include a separate question label (such as a correctly structured heading) or separate instructions for questions.
Use best practices for content authoring, including:
- If more than one question is used on a page, add unique labels for questions on the page outside of the H5P content
- Add instructions for questions on the page outside of the H5P content
- Add alt-text descriptions up to 150 characters for meaningful images
- Use good color contrast for meaningful visual information in images.
Users who require good color contrast may require an alternative version of the activity.
Provide instructions for users of screen reading software describing when cards are automatically turned over.
Visible Indication of Focus
The activity buttons have a visible indication of focus.
Color Contrast
The visible indication of focus has good contrast. The visible indication of focus is a thin black line. Users who require a visible indication of focus with more contrast may use assistive technology. The cards do not have good color contrast. The color contrast of meaningful visual elements and text with backgrounds must be ensured manually by the content author. Users who require good color contrast may require an alternative version of the activity.
Keyboard Usability
The activity is usable with the keyboard.
No Keyboard Traps
There are no keyboard traps.
Magnification
The activity supports both magnification up to 200% and reflow of content with magnification up to 400% without a loss of content or functionality and without requiring horizontal scrolling.
Usability with Screen Reading Software
The activity is usable with screen reading software. With Windows, the activity may be more usable with NVDA screen reading software than JAWS screen reading software. After selecting a card, users of JAWS screen reading software may need to read the current element to announce the contents of the card. When popup content is open, H5P does not keep the keyboard focus inside the popup content. When two cards have been selected, without finding a match, and a third card is selected, H5P has not set up the activity so that screen reading software will announce that the first two cards are automatically turned over. Provide instructions for users of screen reading software describing when cards are automatically turned over.


17.5: Conductors in Electrostatic Equilibrium via Gauss's Law
By the end of this section, you will be able to:
- Describe the electric field within a conductor at equilibrium
- Describe the electric field immediately outside the surface of a charged conductor at equilibrium
- Explain why if the field is not as described in the first two objectives, the conductor is not at equilibrium
So far, we have generally been working with charges occupying a volume within an insulator. We now study what happens when free charges are placed on a conductor. Generally, in the presence of a (generally external) electric field, the free charge in a conductor redistributes and very quickly reaches electrostatic equilibrium. The resulting charge distribution and its electric field have many interesting properties, which we can investigate with the help of Gauss’s law and the concept of electric potential.
The Electric Field inside a Conductor Vanishes
If an electric field is present inside a conductor, it exerts forces on the free electrons (also called conduction electrons), which are electrons in the material that are not bound to an atom. These free electrons then accelerate. However, moving charges by definition means nonstatic conditions, contrary to our assumption. Therefore, when electrostatic equilibrium is reached, the charge is distributed in such a way that the electric field inside the conductor vanishes.
If you place a piece of a metal near a positive charge, the free electrons in the metal are attracted to the external positive charge and migrate freely toward that region. The region the electrons move to then has an excess of electrons over the protons in the atoms and the region from where the electrons have migrated has more protons than electrons. Consequently, the metal develops a negative region near the charge and a positive region at the far end (Figure \(\PageIndex{1}\)). As we saw in the preceding chapter, this separation of equal magnitude and opposite type of electric charge is called polarization . If you remove the external charge, the electrons migrate back and neutralize the positive region.
The polarization of the metal happens only in the presence of external charges. You can think of this in terms of electric fields. The external charge creates an external electric field. When the metal is placed in the region of this electric field, the electrons and protons of the metal experience electric forces due to this external electric field, but only the conduction electrons are free to move in the metal over macroscopic distances. The movement of the conduction electrons leads to the polarization, which creates an induced electric field in addition to the external electric field (Figure \(\PageIndex{2}\)). The net electric field is a vector sum of the fields of \(+q\) and the surface charge densities \(-\sigma_A\) and \(+\sigma_B\). This means that the net field inside the conductor is different from the field outside the conductor.
The redistribution of charges is such that the sum of the three contributions at any point \(P\) inside the conductor is
\[\vec{E}_p = \vec{E}_q + \vec{E}_B + \vec{E}_A = \vec{0}. \nonumber\]
Now, thanks to Gauss’s law , we know that there is no net charge enclosed by a Gaussian surface that is solely within the volume of the conductor at equilibrium. That is, \(q_{enc} = 0\) and hence
\[\vec{E}_{net} = \vec{0} \, \mathrm{(at points inside a conductor)}.\]
Charge on a Conductor
An interesting property of a conductor in static equilibrium is that extra charges on the conductor end up on the outer surface of the conductor, regardless of where they originate. Figure \(\PageIndex{3}\) illustrates a system in which we bring an external positive charge inside the cavity of a metal and then touch it to the inside surface. Initially, the inside surface of the cavity is negatively charged and the outside surface of the conductor is positively charged. When we touch the inside surface of the cavity, the induced charge is neutralized, leaving the outside surface and the whole metal charged with a net positive charge.
To see why this happens, note that the Gaussian surface in iFigure \(\PageIndex{4}\) (the dashed line) follows the contour of the actual surface of the conductor and is located an infinitesimal distance within it. Since \(E = 0\) everywhere inside a conductor,
\[\oint \vec{E} \cdot \hat{n} dA = 0.\]
Thus, from Gauss’ law, there is no net charge inside the Gaussian surface. But the Gaussian surface lies just below the actual surface of the conductor; consequently, there is no net charge inside the conductor. Any excess charge must lie on its surface.
This particular property of conductors is the basis for an extremely accurate method developed by Plimpton and Lawton in 1936 to verify Gauss’s law and, correspondingly, Coulomb’s law. A sketch of their apparatus is shown in Figure \(\PageIndex{5}\). Two spherical shells are connected to one another through an electrometer E, a device that can detect a very slight amount of charge flowing from one shell to the other. When switch S is thrown to the left, charge is placed on the outer shell by the battery B. Will charge flow through the electrometer to the inner shell?
No. Doing so would mean a violation of Gauss’s law. Plimpton and Lawton did not detect any flow and, knowing the sensitivity of their electrometer, concluded that if the radial dependence in Coulomb’s law were \(1/r^{2+\delta}\), \(\delta\) would be less than \(2 \times 10^{-9}\) 1 . . More recent measurements place \(\delta\) at less than \(3 \times 10^{-16}\) 2 , a number so small that the validity of Coulomb’s law seems indisputable.
The Electric Field at the Surface of a Conductor
If the electric field had a component parallel to the surface of a conductor, free charges on the surface would move, a situation contrary to the assumption of electrostatic equilibrium. Therefore, the electric field is always perpendicular to the surface of a conductor.
At any point just above the surface of a conductor, the surface charge density \(\delta\) and the magnitude of the electric field \(\vec{E}\) are related by
\[E = \dfrac{\sigma}{\epsilon_0}.\]
To see this, consider an infinitesimally small Gaussian cylinder that surrounds a point on the surface of the conductor, as in Figure \(\PageIndex{6}\). The cylinder has one end face inside and one end face outside the surface. The height and cross-sectional area of the cylinder are \(\delta\) and \(\Delta A\), respectively. The cylinder’s sides are perpendicular to the surface of the conductor, and its end faces are parallel to the surface. Because the cylinder is infinitesimally small, the charge density \(\sigma\) is essentially constant over the surface enclosed, so the total charge inside the Gaussian cylinder is \(\sigma\Delta A\). Now \(\vec{E}\) is perpendicular to the surface of the conductor outside the conductor and vanishes within it, because otherwise, the charges would accelerate, and we would not be in equilibrium. Electric flux therefore crosses only the outer end face of the Gaussian surface and may be written as \(E\Delta A\) since the cylinder is assumed to be small enough that \(\vec{E}\) is approximately constant over that area. From Gauss’ law,
\[E\Delta A = \dfrac{\sigma \Delta A}{\epsilon_0}.\]
Thus
\[E = \dfrac{\sigma}{\epsilon_0}.\]
The infinite conducting plate in Figure \(\PageIndex{7}\) has a uniform surface charge density \(\sigma\). Use Gauss’ law to find the electric field outside the plate. Compare this result with that previously calculated directly.
Strategy
For this case, we use a cylindrical Gaussian surface, a side view of which is shown.
Solution
The flux calculation is similar to that for an infinite sheet of charge from the previous chapter with one major exception: The left face of the Gaussian surface is inside the conductor where \(\vec{E} = \vec{0}\), so the total flux through the Gaussian surface is \(EA\) rather than 2\(EA\). Then from Gauss’ law,
\[EA = \dfrac{\sigma A}{\epsilon_0} \nonumber\]
and the electric field outside the plate is
\[E = \dfrac{\sigma}{\epsilon_0}. \nonumber\]
Significance
This result is in agreement with the result from the previous section, and consistent with the rule stated above.
Two large conducting plates carry equal and opposite charges, with a surface charge density \(\sigma\) of magnitude \(6.81 \times 10^{-7} C/m^2\), as shown in Figure \(\PageIndex{8}\). The separation between the plates is \(l = 6.50 \, mm\). What is the electric field between the plates?
Strategy Note that the electric field at the surface of one plate only depends on the charge on that plate. Thus, apply \(E = \sigma /\epsilon_0\) with the given values.
Solution The electric field is directed from the positive to the negative plate, as shown in the figure, and its magnitude is given by
\[E = \dfrac{\sigma}{\epsilon_0} = \dfrac{6.81 \times 10^{-7} C/m^2}{8.85 \times 10^{-12} C^2/Nm^2} = 7.69 \times 10^4 N/C\]
Significance
This formula is applicable to more than just a plate. Furthermore, two-plate systems will be important later.
The isolated conducting sphere (Figure \(\PageIndex{9}\)) has a radius \(R\) and an excess charge \(q\). What is the electric field both inside and outside the sphere?
Strategy The sphere is isolated, so its surface change distribution and the electric field of that distribution are spherically symmetrical. We can therefore represent the field as \(\vec{E} = E(r) \hat{r}\). To calculate \(\vec{E}(r)\), we apply Gauss’s law over a closed spherical surface \(S\) of radius \(r\) that is concentric with the conducting sphere.
Solution
Since r is constant and \(\hat{n} = \hat{r}\) on the sphere,
\[\oint_S \vec{E} \cdot \hat{n} dA = E(r) \oint_S dA = E(r) 4\pi r^2.\]
For \(r < R\), \(S\) is within the conductor, so \(q_{enc} = 0\), and Gauss’s law gives
\[E(r) = 0,\]
as expected inside a conductor. If \(r > R\), \(S\) encloses the conductor so \(q_{enc} = q\). From Gauss’s law,
\[E(r) 4\pi r^2 = \dfrac{q}{\epsilon_0}.\]
The electric field of the sphere may therefore be written as
\[\vec{E} = \vec{0} \, (r < R),\]
\[\vec{E} = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \hat{r} \, (r \geq R).\]
Significance Notice that in the region \(r \geq R\), the electric field due to a charge \(q\) placed on an isolated conducting sphere of radius \(R\) is identical to the electric field of a point charge \(q\) located at the center of the sphere. The difference between the charged metal and a point charge occurs only at the space points inside the conductor. For a point charge placed at the center of the sphere, the electric field is not zero at points of space occupied by the sphere, but a conductor with the same amount of charge has a zero electric field at those points (Figure \(\PageIndex{10}\)). However, there is no distinction at the outside points in space where \(r > R\), and we can replace the isolated charged spherical conductor by a point charge at its center with impunity.
How will the system above change if there are charged objects external to the sphere?
- Answer
-
If there are other charged objects around, then the charges on the surface of the sphere will not necessarily be spherically symmetrical; there will be more in certain direction than in other directions.
For a conductor with a cavity, if we put a charge \(+q\) inside the cavity, then the charge separation takes place in the conductor, with \(-q\) amount of charge on the inside surface and a \(+q\) amount of charge at the outside surface (Figure \(\PageIndex{11a}\)). For the same conductor with a charge \(+q\) outside it, there is no excess charge on the inside surface; both the positive and negative induced charges reside on the outside surface (Figure \(\PageIndex{11b}\)).
If a conductor has two cavities, one of them having a charge \(+q_a\) inside it and the other a charge \(-q_b\)the polarization of the conductor results in \(-q_a\)on the inside surface of the cavity \(a\), \(+q_b\) on the inside surface of the cavity \(b\), and \(q_a - q_b\) on the outside surface (Figure \(\PageIndex{12}\)). The charges on the surfaces may not be uniformly spread out; their spread depends upon the geometry. The only rule obeyed is that when the equilibrium has been reached, the charge distribution in a conductor is such that the electric field by the charge distribution in the conductor cancels the electric field of the external charges at all space points inside the body of the conductor.


8: Introduction to Utilitarian Valuation of Biodiversity
Determining the value or worth of biodiversity is complex. Economists typically subdivide utilitarian or use values of biodiversity into direct use value for those goods that are consumed directly, such as food or timber, and indirect use value for those services that support the items that are consumed, including ecosystem functions like nutrient cycling.
There are several less tangible values that are sometimes called non-use or passive values, for things that we don't use but would consider as a loss if they were to disappear; these include existence value, the value of knowing something exists even if you will never use it or see it, and bequest value, the value of knowing something will be there for future generations (Moran and Pearce 1994). Potential or Option value refers to the use that something may have in the future; sometimes this is included as a use value, we have chosen to include it within the passive values here based on its abstract nature. The components included within the category of "utilitarian" values vary somewhat in the literature. For example, some authors classify spiritual, cultural, and aesthetic values as indirect use values, whiles others consider them to be non-use values, differentiated from indirect use values -- such as nutrient cycling -- because spiritual, cultural, and aesthetic values for biodiversity are not essential to human survival. Still others consider these values as separate categories entirely. (See also, Callicott 1997, Hunter 2002, Moran and Pearce 1994, Perlman and Adelson 1997, Primack 2002, Van Dyke 2003). In this module, we include spiritual, cultural and aesthetic values as a subset of indirect values or services, as they provide a service by enriching our lives (Table).
| Direct Use Value (Goods) | Indirect Use Value (Services) | Non-Use Values | |
|---|---|---|---|
| Food, medicine, building material, fiber, fuel | Atmospheric and climate regulation, pollination, nutrient recycling | Potential (or Option) Value | Future value either as a good or a service |
| Cultural, Spiritual, and Aesthetic | Existence Value | Value of knowing something exists | |
| Bequest Value | Value of knowing that something will be there for future generations |
Table \(\PageIndex{1}\) Categories of Values of Biodiversity
Note
Some authors choose to differentiate Cultural, Spiritual, Aesthetic, and Non-Use Values from those services that provide basic survival needs such as the air we breathe.
Glossary
- direct use value
- refers to products or goods which are consumed directly such as food or timber
- indirect use value
- refers to the services that support the products that are consumed, this includes ecosystems functions like nutrient cycling
- non use or passive value
- refers to the value for things that we don't use but would feel a loss if they were to disappear
- existence value
- the value of knowing something exists even if you will never use it or see it
- bequest value
- the value of knowing something will be there for future generations
- potential or option value
- refers to the use that something may have in the future


2: Definition of Biodiversity
Biodiversity, a contraction of the phrase "biological diversity," is a complex topic, covering many aspects of biological variation. In popular usage, the word biodiversity is often used to describe all the species living in a particular area. If we consider this area at its largest scale - the entire world - then biodiversity can be summarized as "life on earth." However, scientists use a broader definition of biodiversity, designed to include not only living organisms and their complex interactions, but also interactions with the abiotic (non-living) aspects of their environment. Definitions emphasizing one aspect or another of this biological variation can be found throughout the scientific and lay literature (see Gaston, 1996: Table 1.1). For the purposes of this module, biodiversity is defined as:
the variety of life on Earth at all its levels, from genes to ecosystems, and the ecological and evolutionary processes that sustain it.
Genetic diversity is the “fundamental currency of diversity” (Williams and Humphires, 1996) that is responsible for variation between individuals, populations and species. Therefore, it is an important aspect of any discussion of biodiversity. The interactions between the individual organisms (e.g., reproductive behavior, predation, parasitism) of a population or community, and their specializations for their environment (including ways in which they might modify the environment itself) are important functional aspects of biodiversity. These functional aspects can determine the diversity of different communities and ecosystems.
There is also an important spatial component to biodiversity. The structure of communities and ecosystems (e.g. the number of individuals and species present) can vary in different parts of the world. Similarly, the function of these communities and ecosystems (i.e. the interactions between the organisms present) can vary from one place to another. Different assemblages of ecosystems can characterize quite diverse landscapes, covering large areas. These spatial patterns of biodiversity are affected by climate, geology, and physiography (Redford and Richter, 1999).
The structural, functional, and spatial aspects of biodiversity can vary over time; therefore there is a temporal component to the analysis of biodiversity. For example, there can be daily, seasonal, or annual changes in the species and number of organisms present in an ecosystem and how they interact. Some ecosystems change in size or structure over time (e.g. forest ecosystems may change in size and structure because of the effects of natural fires, wetlands gradually silt up and decrease in size). Biodiversity also changes over a longer-term, evolutionary, time-scale. Geological processes (e.g., plate tectonics, orogenesis, erosion), changes in sea-level (marine transgressions and regressions), and changes in climate cause significant, long-term changes to the structural and spatial characteristics of global biodiversity. The processes of natural selection and species evolution, which may often be associated with the geological processes, also result in changes to local and global flora and fauna.
Many people consider humans to be a part of nature, and therefore a part of biodiversity. On the other hand, some people (e.g., Redford and Richter, 1999 ) confine biodiversity to natural variety and variability, excluding biotic patterns and ecosystems that result from human activity, even though it is difficult to assess the "naturalness" of an ecosystem because human influence is so pervasive and varied ( Hunter, 1996; Angermeier, 2000; Sanderson et al.,2002). If one takes humans as part of nature, then cultural diversity of human populations and the ways that these populations use or otherwise interact with habitats and other species on Earth are a component of biodiversity too. Other people make a compromise between totally including or excluding human activities as a part of biodiversity. These biologists do not accept all aspects of human activity and culture as part of biodiversity, but they do recognize that the ecological and evolutionary diversity of domestic species, and the species composition and ecology of agricultural ecosystems are part of biodiversity. (For further discussion see the modules on Human evolution and Cultural Diversity; in preparation.)
Glossary
- Biodiversity
- the variety of life on Earth at all its levels, from genes to ecosystems, and the ecological and evolutionary processes that sustain it
- Plate Tectonics
- the forces acting on the large, mobile pieces (or "plates") of the Earth's lithosphere (the upper part of the mantle and crust of the Earth where the rocks are rigid compared to those deeper below the Earth's surface) and the movement of those "plates".
- Orogenesis
- the process of mountain building.


7.10: Video- Las vacacionesChangetoCultura en vivo Tijuana/San Diego
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Professor López and his students are talking about their plans for their next vacation. Some plan to travel to many interesting places of different Spanish speaking countries. Pay attention to the places they mention. Have you heard of these places? Have you visited any of these places? Do you know anyone from these places? ( Video Transcript )
Now, complete the activities below. Base your answers in the video.
As you complete the activities below, copy and paste the following special characters as needed: á, é, í, ó, ú, ñ, ¿, ¡
Mark each statement as "cierto" (true) or "falso" (false).


6.2: Least Squares Regression Method
Learning Outcomes
- Use the least-squares regression method to create a regression line on a graph of cost data
This method uses all of the data available to separate the fixed and variable portions of a mixed cost. A regression line is fitted into the data using the following formula:
Y= a + bX
Y= Maintenance costs
X= Number of dogs groomed
a= the total fixed cost
b= the variable cost per unit of activity
So using a calculator available HERE .
If you use the data from the dog groomer example you should be able to calculate the following chart:
This method is more accurate, using all of the available data. The table below shows the difference in calculation using the two methods:
| Costs | High-low | Least Squares Regression Method |
|---|---|---|
| Variable cost estimate per dog groomed | $.080 | $.074 |
| Fixed cost estimate per month | $340 | $362.24 |
Practice Questions


27.18: Intentions Left Unsaid…To An Extent
Herman Melville uses irony in Benito Cereno by not fully revealing Babo’s intentions, thereby leaving the reader guessing. Douglass differs from Melville in such a way that Madison’s goals and intentions are clear throughout the story and even though the end results are both in a revolt, the characters’ futures come to an opposite conclusion.
Melville portrays Babo as a character that many readers may not come to like, because of how sneaky and mysterious he is throughout the text. There is some evidence that shows that Babo has something deeper going on when he does not let Delano speak to Benito without his (Babo) presence. Once Delano goes to talk to Cereno, “he [hears] another footstep, keeping in time with his. From the opposite door, a salver in hand, the servant was likewise advancing” (200). Before this, Babo is on top of the deck, which makes Delano believe there is no way he is able to catch up to him, but, next thing he knows, Babo is one step ahead of him. The mystery of this part of the text is that the reader is alongside Delano in an unawareness that something is going on with Babo, despite the fact this is one of the first clear signs Babo is far from Cereno’s faithful servant.
The fact that Melville doesn’t show what is really happening with Babo is ironic because it can play into what Delano and most likely the readers are feeling. In contrast, Douglass puts forth Madison’s feelings and intentions word-for-word. Indeed, Madison’s intentions are clear throughout the book, but the reader sees a sharp difference between Babo and Madison when the revolt happens; Madison states that “we do not thirst for your blood, we demand only our rightful freedom” (49). The reader knows at this point that all Madison wants for himself and others is freedom, and that even though he’ll do anything it takes, he doesn’t believe there needs to be violence. Babo orchestrates violence by having his men kill purposefully, despite not doing any of the killing himself. Many people die in Benito Cereno , including Babo, but in The Heroic Slave only a couple people die and Madison is able to leave with his freedom.
Reading these texts back-to-back allows me to see that writing can be a tremendous influence on how an audience sees a group of people or a person a certain way. Most people, including myself, initially thought that Babo was evil. In reality, he wanted the very same as Madison: to be free. If Melville wrote the exact same way Douglass did, there more than likely would not have been as much obvious violence or Babo’s intentions may have been displayed in a more clear manner as Madison’s were. Both Babo and Madison want freedom and they are both put through many awful things none of us can imagine. These two texts will continue to open my eyes as I read more about slave rebellion and black resistance, because it shifted my perspective. As Melville so ironically put at the end of his writing: “So far may even the best men err, in judging the conduct of one with the recesses of who condition he is not acquainted” (216-217). In short, don’t judge something or someone you don’t understand!


19.6: Programming Exercises
A Boolean or switching function on \(n\) variables is a map \(f : \{O, I\}^n \rightarrow \{ 0, I\}\text{.}\) A Boolean polynomial is a special type of Boolean function: it is any type of Boolean expression formed from a finite combinatio4n of variables \(x_1, \ldots, x_n\) together with \(O\) and \(I\text{,}\) using the operations \(\vee\text{,}\) \(\wedge\text{,}\) and \('\text{.}\) The values of the functions are defined in Table \(19.33\). Write a program to evaluate Boolean polynomials.
\(Table \text { } 19.33.\) Boolean polynomials
| \(x\) | \(y\) | \(x'\) | \(x \vee y\) | \(x \wedge y\) |
| \(0\) | \(0\) | \(1\) | \(0\) | \(0\) |
| \(0\) | \(1\) | \(1\) | \(1\) | \(0\) |
| \(1\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(1\) | \(1\) | \(0\) | \(1\) | \(1\) |


14.10: Personal Application Questions
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31 .
Think about someone in middle adulthood whom you admire. How does this individual demonstrate generativity in their personal or professional life? Provide examples of their contributions and reflect on what you think motivates them.
32 .
Erikson’s theory emphasizes generativity versus stagnation during middle adulthood. Reflect on your goals as you progress through adulthood. How do you hope to contribute to your family, work, or community, and what challenges might you face?
33 .
Imagine how your personality traits might evolve as you enter middle adulthood. Based on what you’ve read, which traits do you think are likely to remain stable, and which might change? Explain your reasoning with references to the concepts of stability and change in personality.
34 .
Consider Sternberg’s triangular theory of love, which identifies intimacy, passion, and commitment as key components of relationships. Reflect on a relationship you’ve observed (it could be a relative, friend, or public figure) and analyze how these three components are balanced or unbalanced in that relationship. How do you think this balance impacts their overall satisfaction?
35 .
Marriage and divorce rates have changed significantly over recent decades. Reflect on societal changes, such as increased age at first marriage, marriage equality, and higher rates of cohabitation, and how these trends have influenced your views or expectations of relationships in adulthood. What role do you think cultural norms and values play in these shifts?
36 .
Reflect on the dynamics of your current household structure. What are some of the benefits and challenges you experience within this structure? If you were to imagine a different household setup (e.g., living alone, living in a multigenerational household, or living as part of an LGBTQ+ family), how do you think your life might change? Consider the potential social, financial, and emotional impacts of these changes.
37 .
Think about a family member or close friend who is in middle adulthood and balancing multiple caregiving roles (e.g., for children, aging parents, or grandchildren). How do you see them managing these responsibilities? What factors do you think are key in helping them cope with the demands of their household?
38 .
Reflect on the concept of the “sandwich generation”—adults who are simultaneously caring for aging parents and supporting their own children. Do you know someone who is or was part of the sandwich generation? How do you think this dual responsibility might affect someone’s emotional well-being, stress levels, and personal relationships?
39 .
The process of launching young adult children involves letting go while providing support. Reflect on a real-life situation (it could be from your own family, a friend’s experience, or even a public figure) where a parent had to navigate this transition. How did the parent balance offering guidance while promoting independence? How do you think cultural expectations or family dynamics influenced their approach?
40 .
Becoming a grandparent can be both a rewarding and complex experience. Reflect on the role that grandparents play in your family or community. How do their interactions influence family dynamics, whether positively or negatively? If you were to become a grandparent, what type of role would you envision for yourself, and why?
41 .
Consider how cultural values influence your ideas of happiness and life satisfaction. In your community or family, do you observe more individualistic or collectivistic values? How do these values impact your perception of what it means to lead a satisfying life, especially as you think about your middle adult years?
42 .
Imagine yourself at the age of 55. What does a successful middle adulthood look like to you? Describe your ideal life in terms of career, relationships, health, and personal growth. What steps can you take now to align your current goals with your vision for middle adulthood?


4.8: Mechanical Vibrations
Let us look at some applications of linear second order constant coefficient equations.
Some examples
Our first example is a mass on a spring. Suppose we have a mass \(m > 0 \) (in kilograms) connected by a spring with spring constant \( k > 0 \) (in newtons per meter) to a fixed wall. There may be some external force \( F(t) \) (in newtons) acting on the mass. Finally, there is some friction measured by \( c \geq 0 \) (in newton-seconds per meter) as the mass slides along the floor (or perhaps there is a damper connected).
Let \(x\) be the displacement of the mass ( \( x = 0 \) is the rest position), with \(x\) growing to the right (away from the wall). The force exerted by the spring is proportional to the compression of the spring by Hooke’s law. Therefore, it is \(kx\) in the negative direction. Similarly the amount of force exerted by friction is proportional to the velocity of the mass. By Newton’s second law we know that force equals mass times acceleration and hence \( mx'' = F(t) - cx' - kx \) or
\[mx'' + cx' + kx = F(t) \nonumber \]
This is a linear second order constant coefficient ODE. We set up some terminology about this equation. We say the motion is
- forced , if \( F \not \equiv 0\) (if \(F\) is not identically zero),
- unforced or free , if \(F \equiv 0 \) (if \(F\) is identically zero),
- damped , if \( c > 0 \), and
- undamped , if \( c = 0 \).
This system appears in lots of applications even if it does not at first seem like it. Many real-world scenarios can be simplified to a mass on a spring. For example, a bungee jump setup is essentially a mass and spring system (you are the mass). It would be good if someone did the math before you jump off the bridge, right? Let us give two other examples.
Here is an example for electrical engineers. Consider the pictured \(RLC\) circuit. There is a resistor with a resistance of \(R\) ohms, an inductor with an inductance of \(L\) henries, and a capacitor with a capacitance of \(C\) farads. There is also an electric source (such as a battery) giving a voltage of \(E(t)\) volts at time \(t\) (measured in seconds). Let \(Q(t)\) be the charge in coulombs on the capacitor and \(I(t)\) be the current in the circuit. The relation between the two is \(Q' = I\). By elementary principles we find \(L I' + RI + \frac{Q}{C} = E\). We differentiate to get
\[L I''(t) + R I'(t) + \frac{1}{C} I(t) = E'(t) . \nonumber \]
This is a nonhomogeneous second order constant coefficient linear equation. As \(L, R\), and \(C\) are all positive, this system behaves just like the mass and spring system. Position of the mass is replaced by current. Mass is replaced by inductance, damping is replaced by resistance, and the spring constant is replaced by one over the capacitance. The change in voltage becomes the forcing function—for constant voltage this is an unforced motion.
Our next example behaves like a mass and spring system only approximately. Suppose a mass \(m\) hangs on a pendulum of length \(L\). We seek an equation for the angle \(\theta(t)\) (in radians). Let \(g\) be the force of gravity. Elementary physics mandates that the equation is \[\theta'' + \frac{g}{L} \sin \theta = 0 . \nonumber \]
Let us derive this equation using Newton's second law: force equals mass times acceleration. The acceleration is \(L \theta''\) and mass is \(m\). So \(mL\theta''\) has to be equal to the tangential component of the force given by the gravity, which is \(m g \sin \theta\) in the opposite direction. So \(mL\theta'' = -mg \sin \theta\). The \(m\) curiously cancels from the equation.
Now we make our approximation. For small \(\theta\) we have that approximately \(\sin \theta \approx \theta\). This can be seen by looking at the graph. In Figure \(\PageIndex{4}\) we can see that for approximately \(-0.5 < \theta < 0.5\) (in radians) the graphs of \(\sin \theta\) and \(\theta\) are almost the same.
Therefore, when the swings are small, \(\theta\) is small and we can model the behavior by the simpler linear equation \[\theta'' + \frac{g}{L} \theta = 0 . \nonumber \]
The errors from this approximation build up. So after a long time, the state of the real-world system might be substantially different from our solution. Also we will see that in a mass-spring system, the amplitude is independent of the period. This is not true for a pendulum. Nevertheless, for reasonably short periods of time and small swings (that is, only small angles \(\theta\)), the approximation is reasonably good.
In real-world problems it is often necessary to make these types of simplifications. We must understand both the mathematics and the physics of the situation to see if the simplification is valid in the context of the questions we are trying to answer.
Free Undamped Motion
In this section we will only consider free or unforced motion, as we cannot yet solve nonhomogeneous equations. Let us start with undamped motion where \( c = 0 \). We have the equation
\[ mx'' + kx = 0 \nonumber \]
If we divide by \( m\) and let \(w_0 = \sqrt { \dfrac {k}{m} } \), then we can write the equation as
\[ x'' + w^2_0 x = 0 \nonumber \]
The general solution to this equation is
\[ x(t) = A \cos (w_0t) + B \sin (w_0t) \nonumber \]
By a trigonometric identity, we have that for two different constants \(C\) and \( \gamma\), we have
\[ A \cos (w_0t) + B \sin (w_0t) = C \cos (w_0t - \gamma ) \nonumber \]
It is not hard to compute that \( C = \sqrt { A^2 + B^2 } \) and \( \tan \gamma = \dfrac {B}{A} \). Therefore, we let \(C\) and \( \gamma \) be our arbitrary constants and write \( x(t) = C \cos (w_0t - \gamma )\).
Below is a video on solving a differential equation that comes from a free undamped spring system.
Justify the above identity and verify the equations for \(C\) and \( \gamma\) . Hint: Start with \( \cos (\alpha - \beta ) = \cos ( \alpha) \cos (\beta) + \sin ( \alpha) \sin (\beta) \) and multiply by \( C\) . Then think what should \( \alpha \) and \( \beta \) be.
While it is generally easier to use the first form with \( A\) and \(B\) to solve for the initial conditions, the second form is much more natural. The constants \(C\) and \( \gamma\) have very nice interpretation. We look at the form of the solution
\[ x(t) = C \cos (w_0t - \gamma ) \nonumber \]
We can see that the amplitude is \( C \), \(w_0\) is the (angular) frequency, and \( \gamma \) is the so-called phase shift . The phase shift just shifts the graph left or right. We call \( w_0\) the natural (angular) frequency . This entire setup is usually called simple harmonic motion .
Let us pause to explain the word angular before the word frequency. The units of \(w_0\) are radians per unit time, not cycles per unit time as is the usual measure of frequency. Because we know one cycle is \(2 \pi\) radians, the usual frequency is given by \( \dfrac {w_0}{ 2 \pi} \). It is simply a matter of where we put the constant \( 2 \pi\), and that is a matter of taste.
The period of the motion is one over the frequency (in cycles per unit time) and hence \( \dfrac {2 \pi}{w_0} \). That is the amount of time it takes to complete one full oscillation.
Below is a video on solving a initial value problem that comes from a free undamped spring system.
Suppose that \( m = 2kg \) and \(k = 8 \dfrac {N}{m}\). The whole mass and spring setup is sitting on a truck that was traveling at \( 1 \dfrac {m}{s} \). The truck crashes and hence stops. The mass was held in place \(0.5\) meters forward from the rest position. During the crash the mass gets loose. That is, the mass is now moving forward at \( 1 \dfrac {m}{s} \), while the other end of the spring is held in place. The mass therefore starts oscillating. What is the frequency of the resulting oscillation and what is the amplitude. The units are the mks units (meters-kilograms-seconds).
The setup means that the mass was at half a meter in the positive direction during the crash and relative to the wall the spring is mounted to, the mass was moving forward (in the positive direction) at \( 1 \dfrac {m}{s} \). This gives us the initial conditions.
So the equation with initial conditions is
\[ 2x'' + 8x = 0, \quad x(0) = 0.5,\quad x'(0) = 1 \nonumber \]
We can directly compute \( w_0 = \sqrt { \dfrac {k}{m} } = \sqrt {4} = 2 \). Hence the angular frequency is 2. The usual frequency in Hertz (cycles per second) is \( \dfrac {2}{2 \pi} = \dfrac {1}{ \pi} \approx 0.318 \).
The general solution is
\[ x(t) = A \cos (2t) + B \sin (2t) \nonumber \]
Letting \( x(0) = 0.5 \) means \(A = 0.5 \). Then \(x'(t) = -2(0.5) \sin (2t) + 2B \cos (2t) \). Letting \( x'(0) = 1\) we get \(B = 0.5\). Therefore, the amplitude is \( C = \sqrt { A^2 + B^2} = \sqrt { 0.25 + 0.25 } = \sqrt {0.5} \approx 0.707\). The solution is
\[ x(t) = 0.5 \cos (2t) + 0.5 \sin (2t) \nonumber \]
A plot of \( x(t) \) is shown in Figure \(\PageIndex{5}\).
In general, for free undamped motion, a solution of the form
\[ x(t) = A \cos (w_0t) + B \sin (w_0t) \nonumber \]
corresponds to the initial conditions \( x(0) = A \) and \(x'(0) = w_0B\). Therefore, it is easy to figure out \(A\) and \(B\) from the initial conditions. The amplitude and the phase shift can then be computed from \(A\) and \(B\). In the example, we have already found the amplitude \(C\). Let us compute the phase shift. We know that \( \tan \gamma = \dfrac {B}{A} = 1\). We take the arctangent of 1 and get approximately 0.785. We still need to check if this \( \gamma\) is in the correct quadrant (and add \( \pi\) to \( \gamma\) if it is not). Since both \(A\) and \(B\) are positive, then \( \gamma\) should be in the first quadrant, and 0.785 radians really is in the first quadrant.
Many calculators and computer software do not only have the
atan
function for arctangent, but also what is sometimes called
atan2
. This function takes two arguments, \(B\) and \(A\), and returns a \( \gamma\) in the correct quadrant for you.
Below is a video on solving a differential equation that comes from an undamped spring system.
Free Damped Motion
Let us now focus on damped motion. Let us rewrite the equation
\[ mx'' + cx' + kx = 0 \nonumber \]
as
\[ x'' + 2px' + w^2_0x = 0 \nonumber \]
where
\[ w_0 = \sqrt {\dfrac {k}{m}},\quad p = \dfrac {c}{2m} \nonumber \]
The characteristic equation is
\[ r^2 + 2pr + w^2_0 = 0 \nonumber \]
Using the quadratic formula we get that the roots are
\[ r = -p \pm \sqrt { p^2 - w^2_0} \nonumber \]
The form of the solution depends on whether we get complex or real roots. We get real roots if and only if the following number is nonnegative:
\[ p^2 - w^2_0 = { ( \dfrac {c}{2m} )}^2 - \dfrac {k}{m} = \dfrac {c^2 -4km}{4m^2} \nonumber \]
The sign of \( p^2 - w^2_0 \) is the same as the sign of \(c^2 - 4km\). Thus we get real roots if and only if \( c^2 - 4km \) is nonnegative, or in other words if \( c^2 \ge 4km \).
Overdamping
When \( c^2 - 4km > 0\), we say the system is overdamped . In this case, there are two distinct real roots \(r_1\) and \(r_2\). Notice that both roots are negative. As \( \sqrt {p^2 - w^2_0} \) is always less than \(P\), then \( -P \pm \sqrt {P^2 - w^2_0}\) is negative.
The solution is \[ x(t) = C_1e^{r_1t} + C_2e^{r_2t} \nonumber \]
Since \(r_1, r_2\) are negative, \( x(t) \rightarrow 0 \) as \(t \rightarrow \infty\). Thus the mass will tend towards the rest position as time goes to infinity. For a few sample plots for different initial conditions ( Figure \(\PageIndex{6}\) ).
Do note that no oscillation happens. In fact, the graph will cross the \(x\) axis at most once. To see why, we try to solve \(0 = C_1e^{r_1t} + C_2e^{r_2t} \). Therefore, \( C_1e^{r_1t} = -C_2e^{r_2t} \) and using laws of exponents we obtain
\[ \dfrac {-C_1}{C_2} = e^{{(r_2 - r_1)}t} \nonumber \]
This equation has at most one solution \(t \ge 0\). For some initial conditions the graph will never cross the \(x\) axis, as is evident from the sample graphs.
Suppose the mass is released from rest. That is \( x(0) = x_0 \) and \(x'(0) = 0 \). Then
\[ x(t) = \dfrac {x_0}{r_1 - r_2} (r_1e^{r_2t} - r_2e^{r_1t}) \nonumber \]
It is not hard to see that this satisfies the initial conditions.
Below is a video on solving a differential equation that comes from a damped spring system.
Critical damping
When \( c^2 - 4km = 0 \), we say the system is critically damped . In this case, there is one root of multiplicity 2 and this root is \( -P\). Therefore, our solution is
\[ x(t) = C_1e^{-pt} + C_2te^{-pt} \nonumber \]
The behavior of a critically damped system is very similar to an overdamped system. After all a critically damped system is in some sense a limit of overdamped systems. Since these equations are really only an approximation to the real world, in reality we are never critically damped, it is a place we can only reach in theory. We are always a little bit underdamped or a little bit overdamped. It is better not to dwell on critical damping.
Underdamping
When \( c^2 - 4km < 0 \), we say the system is underdamped . In this case, the roots are complex.
\[\begin{align}\begin{aligned} r &= -p \pm \sqrt { p^2 - w^2_0} \\ &= -p \pm \sqrt {-1} \sqrt {w^2_0 - p^2} \\ & = -p \pm iw_1 \end{aligned}\end{align} \nonumber \]
where \( w_1 = \sqrt { w^2_0 - p^2 } \). Our solution is
\[ x(t) = e^{-pt} ( A \cos (w_1t) + B \sin (w_1t) \nonumber \]
or
\[ x(t) = Ce^{-pt} \cos ( w_1t - \gamma) \nonumber \]
An example plot is given in Figure \(\PageIndex{7}\) . Note that we still have that \( x(t) \rightarrow 0 \) as \( t \rightarrow \infty \).
In the figure we also show the envelope curves \( Ce^{-pt} \) and \( - Ce^{pt} \). The solution is the oscillating line between the two envelope curves. The envelope curves give the maximum amplitude of the oscillation at any given point in time. For example if you are bungee jumping, you are really interested in computing the envelope curve so that you do not hit the concrete with your head.
The phase shift \( \gamma\) just shifts the graph left or right but within the envelope curves (the envelope curves do not change if \( \gamma\) changes).
Finally note that the angular pseudo-frequency \(^{1}\) (we do not call it a frequency since the solution is not really a periodic function) \(w_1\) becomes smaller when the damping \(c\) (and hence \(P\)) becomes larger. This makes sense. When we change the damping just a little bit, we do not expect the behavior of the solution to change dramatically. If we keep making \(c\) larger, then at some point the solution should start looking like the solution for critical damping or overdamping, where no oscillation happens. So if \(c^2\) approaches \(4km\), we want \(w_1\) to approach 0.
On the other hand when \(c\) becomes smaller, \(w_1\) approaches \(w_0\) ( \(w_1\) is always smaller than \(w_0\) ), and the solution looks more and more like the steady periodic motion of the undamped case. The envelope curves become flatter and flatter as \(c\) (and hence \(P\) ) goes to 0.
Footnotes
[1] We do not call \(\omega_{1}\) a frequency since the solution is not really a periodic function.


Appendix B: WI DCF 251 Licensing Rules for Group Child Care Center Highlights
The complete licensing regulation can be found in the DCF 251 Licensing Rules for Group Child Care Centers and Child Care programs. Licensing Rules for Group Child Care Centers DCF 251, DCF-P-PFS205 (wisconsin.gov)
DCF 251 Administrative Code regulations that relate to chapter 2 include:
DCF 251.055 Supervision and Grouping of Children
Maximum Number of Children in a Group
|
Age Group |
Ratio |
Maximum Group Size |
|
Birth to 2 Years |
1:4 |
8 |
|
2 years to 2½ Years |
1:6 |
12 |
|
2½ Years to 3 Years |
1:8 |
16 |
|
3 Years to 4 Years |
1:10 |
20 |
|
4 Years to 5 Years |
1:13 |
26 |
|
5 Years and Over |
1:18 |
36 |
- Each child shall be supervised by a child care worker who is within the sight and sound of the children to guide the children’s behavior and activities, prevent harm, and ensure safety.
- At least one child care teacher shall supervise each group of children.
- The center shall implement and adhere to a procedure to ensure that the number, names, and whereabouts of children in care are known to assigned child care workers at all times.
- A child care worker may not provide care for children at the center for more than 12 hours in any 24-hour period
- A child may not be in care for more than 14 hours in any 24-hour period.
-
During naptime, an adjustment in group size and staff-to-child ratios in may be made as follows:
- One child care worker shall be within sight or sound of each group of sleeping children.
- Staff-to-child ratios shall be maintained in the center during naptime.
- Maximum group size requirements do not apply to napping groups.
- As children wake, sight and sound supervision and staff-to-child ratios shall be maintained for awake children.
DCF 251 Administrative Code regulations that relate to chapter 3 include:
- The indoor and outdoor premises shall be free of hazards including any recalled products.
- Materials harmful to children, including power tools, flammable or combustible materials, insecticides, matches, drugs, cleaning supplies, bleaches, and other hazardous, toxic, or poisonous articles shall be appropriately labeled and stored in areas inaccessible to children
- Stairs, walks, ramps and porches shall be maintained in a safe condition and free from the accumulation of water, ice or snow.
- Cleaning aids such as mops and brooms shall be clean. Buckets used with mops shall be emptied and stored in areas inaccessible to children
- Garbage containers in the building shall be rigid, covered, watertight, and emptied daily or more often as needed. Compactors need not be emptied daily. Garbage and refuse stored out of doors shall be kept in leak-proof containers equipped with tight-fitting covers and shall be disposed of as necessary to prevent decomposition or overflow.
- Windows and doors that are used for ventilation shall be screened.
- The space used by children shall be no less than 35 square feet of usable floor space for each child, exclusive of passageways, kitchens, bathrooms, coat storage areas, offices, storage areas, isolation quarters, staff room, furnace room, parts of rooms occupied by stationary equipment, and areas not at all times available to children, including areas used exclusively for large muscle activity, napping or eating.
- There shall be at least 75 square feet of outdoor play space for each child 2 years of age or older using the space at a given time, and at least 35 square feet of outdoor play space for each child under 2 years of age using the space at a given time.
-
Indoor furnishings and equipment shall be safe and durable. The equipment and furnishings shall be:
- Scaled to the developmental level, size and ability of the children.
- Of sturdy construction with no sharp, rough, loose, protruding, pinching or pointed edges, or areas of entrapment, in good operating condition, and anchored when necessary.
- Placed to avoid danger of injury or collision and to permit freedom of action.
- Placed over an energy-absorbing surface, when equipment is 4 feet or more in height.
- Used in accordance with all manufacturer’s instructions and any manufacturer’s recommendations that may affect the safety of children in care.
DCF 251 Administrative Code regulations that relate to chapter 4 include:
DCF 251.06 Physical Plant and Equipment
Outdoor Play Space
- Requirement for outdoor play space. A center shall have outdoor play space if children are present for more than 3 hours per day or if outdoor play is included in the center program.
- There shall be at least 75 square feet of outdoor play space for each child 2 years of age or older using the space at a given time.
- There shall be at least 35 square feet of outdoor play space for each child under 2 years of age using the space at a given time.
- An energy-absorbing surface, such as loose sand, pea gravel, or pine or bark mulch, in a depth of at least 9 inches is required under climbing equipment, swings, and slides whenever the play equipment is 4 feet or more in height and in a fall zone of 4 feet beyond the equipment. Shredded rubber and poured surfacing shall be installed to the manufacturer’s specifications based on the height of the equipment.
- he boundaries of the outdoor play space shall be defined by a permanent enclosure not less than 4 feet high to protect the children. Fencing, plants, or landscaping may be used to create a permanent enclosure. The permanent enclosure may not have any open areas that are greater than 4 inches.
DCF 251 Administrative Code regulations that relate to chapter 5 include:
DCF 251.06 Physical Plant and Equipment
-
Each center shall have a written plan for taking appropriate action in the event of an emergency, including fire, tornado, or flood; extreme heat or cold; loss of building services, including heat, water, electricity, or telephone; human- caused events, such as threats to the building or its occupants; allergic reactions; lost or missing children; vehicle accidents; or other circumstances requiring immediate attention. The plan shall include procedures for all of the following:
- Evacuation, relocation, shelter-in-place, and lock-down.
- Ensuring the needs of children under 2 years of age and children with disabilities are met.
- Communication with parents.
-
Connecting children with their parents if the center is required to evacuate the building.
- Each center shall do all of the following:
- Post the fire evacuation route and tornado shelter areas.
- Practice the fire evacuation plan monthly, and tornado drills monthly from April through October.
- Ensure that all staff members know what their duties are if there is an emergency.
- Keep a written record of dates and times all fire and tornado drills practiced.
DCF 251 Administrative Code regulations that relate to chapter 6 include:
DCF 251.05 Staff
- Documentation of a completed child care background check that indicates the person is eligible to work in a child care program is required
- child care worker, center administrator, center director, and volunteers counted in staff-to-child ratio shall be physically, mentally, and emotionally able to provide responsible care for all children, including children with disabilities.
- Each child care worker, center administrator, center director, volunteer counted in staff-to-child ratio, or substitute who provides care and supervision to children under 5 years of age shall document completion of department−approved training in shaken baby syndrome and abusive head trauma, and appropriate ways to manage crying, fussing, or distraught children prior to beginning to work with children under 5 years of age.
DCF 251.04 Operational requirements
- A licensee, employee or volunteer at a child care center who knows or has reasonable cause to suspect that a child has been abused or neglected as defined in ss. 48.02 (1) and 48.981 (1), Stats., shall immediately contact the county department of social services or human services or a local law enforcement agency, as required by s. 48.981, Stats. https://docs.legis.wisconsin.gov/sta...atutes/48/i/02
DCF 251 Administrative Code regulations that relate to chapter 7 include:
DCF 251.04 Operational Requirements
-
The licensee shall maintain a current written record obtained prior to the child’s first day of attendance or subsequent re-enrollment at the center on each child enrolled and shall make the record available to the licensing representative on request. Each record shall include
- Enrollment information - name, birthdate, address, phone number, emergency contact information, etc.
- Written consent from the parent for emergency medical care or treatment.
- uthorization from the parent outlining the plan for a child to come to the center from school, home or other activities or to go from the center to school, home or other activities unless the child is accompanied by a parent or other authorized person or transported by the center.
- Documentation of each child’s health history on a form provided by the department.
- Documentation of each child’s most recent physical examination.
DCF 251 Administrative Code regulations that relate to chapter 8 include:
251.07(6) Isolation for Illness
- A center shall have an isolation area or the care of children who appear to be ill. If the area is not a separate room, it shall be separated from space used by other children by a partition, screen, or other means to keep other children away from the ill child.
- The child shall be isolated until the child can be removed from the center.
- The child in isolation areas shall be within sight and sound supervision of a staff member.
251.06(2)(h) Smoking Prohibition
- Smoking is prohibited on the premises of a Child Care Center or in a vehicle used to transport children.
251.06 (2)(a-o) Buildings and Grounds
- The center shall be clean, safe, sanitary and in good repair at all times.
251.07(3)(h)
- There shall be sufficient storage space for the clothing and personal belongings of each child in attendance.
251.07(4) Napping Equipment
- Sleeping surface may be a bed, cot, padded mat, sleeping bag, crib or playpen.
- The sleeping surface must be clean, safe, washable and placed at least 2 feet from the nearest sleeping child.
- Bedding must be maintained and stored in a clean and sanitary manner.
- Bedding shall be washed after every 5 uses
251.09 Infant Care Personal Services
- Plan toilet training in cooperation with the parent for consistency between the center and the child’s home.
- Change wet or soiled diaper promptly
- Soiled or wet clothing provided by the infant’s representative shall be placed in a labeled plastic bag and given to the representative at the end of the day.
- Towels and washcloths used for cleaning infants shall not be shared and shall be washed after each use.
- The changing table shall be cleaned with soap & water and disinfectant solution after each use.
- 251.06(10)2.
- Whenever a potty chair is used, it shall be emptied, rinsed, and disinfected immediately after use
- 251.06(6)(i)
- Children’s hands shall be washed with soap and warm running water. Hands and face washed when soiled. Children under one year of age, hands may be washed with a wet fabric or paper washcloth that is used once and discarded.
251.07(6)(g) Infant Care General Sanitation
- Surfaces exposed to bodily secretions including walls, floors, toys, equipment and furnishings shall be washed with soap & water and disinfected.
251.09(2)
- The non-walking child who can creep or crawl shall have a safe, clean, open, warm and uncluttered area to move freely.
- Place disposable soiled diapers and gloves, if used, I a plastic-lined, hands-free, covered container immediately.
- Remove soiled diapers from containers at least daily. Containers shall be washed and disinfected daily.
- Objects that are used by infants that are mouthed shall be washed and disinfected.
251.07(6)(i)2.
- Persons shall wash their hands with soap and warm running water before handling food, before & after assisting with toileting and diapering, after wiping bodily secretions from a child with a disposable tissue, and after exposure to blood or bodily fluids.
DCF 251 Administrative Code regulations that relate to chapter 9 include:
- Observation. Each child upon arrival at a center shall be observed by a staff person for symptoms of illness and injury. For an apparently ill child, the procedure under par. (c) shall be followed.
- Isolation. A center shall have an isolation area for the care of children who appear to be ill. If the area is not a separate room, it shall be separated from space used by other children by a partition, screen, or other means to keep other children away from the ill child.
-
Ill child procedure. The following procedures shall apply when a child with an illness or condition that has the potential to affect the health of other persons, such as vomiting, diarrhea, unusual lethargy, or uncontrolled coughing, is observed in the child care center:
- The child shall be isolated until the child can be removed from the center
- The child in the isolation area shall be within sight and sound supervision of a staff member
- The child shall be provided with a bed, crib, playpen, cot, or padded mat and a sheet and blanket or a sleeping bag.
- The child’s parent or emergency contact shall be contacted as soon as possible after the illness is discovered and arrangements shall be made for the child to be removed from the center
DCF 251 Administrative Code regulations that relate to chapter 10 include:
- Center staff may give prescription or non-prescription medication, such as pain relievers, teething gels or cough syrup, to a child only under the following conditions:
-
A written authorization that includes the child’s name and birthdate, name of medication, administration instructions, medication intervals and length of the authorization dated and signed by the parent is on file. Blanket authorizations that exceed the length of time specified on the label are prohibited. In centers where the licensee chooses to handle medication:
- Medications must be kept inaccessible to children.
- Prescription medications must be administered in accordance with the label directions as prescribed by the child’s physician and there must be written approval and instructions from the child’s authorized representative before giving medication to the child.
DCF 251 Administrative Code regulations that relate to chapter 11 include:
-
Each group child care center shall have a written program of activities which are suitable for the developmental level of each child and each group of children. The program shall provide each child with experiences which will promote all of the following
- Wisconsin Early Childhood Collaborating Partners website at http://www.collaboratingpartners.com/ or through the Child Care Information Center at 1-800-362-7353.
- Self-esteem and positive self-image.
- Social interaction.
- Self-expression and communication skills.
- Creative expression.
- Large and small muscle development.
- Intellectual growth.
- Literacy
DCF 251 Administrative Code regulations that relate to chapter 12 include:
- 251.06(9)(c)3m: Meals shall be prepared on the premises, in a central kitchen operated by the child care center or in another location that has been inspected by a representative of a state agency.
Licensed child care providers must
- follow the Federal Child and Adult Care Food Program (CACFP) meal plan requirements.
- Participate in the orientation under s. DCF 251.05 (4) (a) and shall document annual training of at least 4 hours in kitchen sanitation, food handling, and nutrition.
DCF 251 Administrative Code regulations that relate to chapter 13 include:
- Food shall be provided by the center based on the amount of time children are present as specified in Table 251.07.
|
TABLE 251.07 Meal and Snack Requirements for each Child in a Group Child Care Center |
|
|
Time a Child is Present |
Number of Meals and Snacks |
|
At least 2½ but less than 4 hours |
1 snack |
|
At least 4 but less than 8 hours |
1 snack and 1 meal |
|
At least 8 but less than 10 hours |
2 snacks and 1 meal |
|
At least 10 or more hours |
2 meals and 2 or 3 snacks |
- Food shall be served at flexible intervals, but no child may go without nourishment for longer than 3 hours.
DCF 251 Administrative Code regulations that relate to chapter 14 include:
- At a minimum, children shall be provided food for each meal and snack that meets the U.S. department of agriculture child and adult care food program minimum meal requirements for amounts and types of food
- Additional portions of vegetables, fruits, bread, and milk shall be available
- Cooks, staff members, child care workers and substitutes having direct contact with the children shall be informed about food allergies and other allergies of specific children.
- Eating surfaces, including high chairs, shall be washed and sanitized before and after each use
DCF 251.09 (3) Feeding
- Feed each infant and each toddler on the child’s own feeding schedule.
- Ensure that food, breast milk, and formula brought from home are labeled with the child’s name and the date and are refrigerated, if required.
- Ensure that food, breast milk, and formula offered to infants and toddlers is consistent with the requirements of the U.S. department of agriculture child and adult care food program
DCF 251 Administrative Code regulations that relate to chapter 15 include:
- At a minimum, children shall be provided food for each meal and snack that meets the U.S. department of agriculture child and adult care food program minimum meal requirements for amounts and types of food
- Additional portions of vegetables, fruits, bread, and milk shall be available
- Cooks, staff members, child care workers and substitutes having direct contact with the children shall be informed about food allergies and other allergies of specific children.
- Eating surfaces, including high chairs, shall be washed and sanitized before and after each use
DCF 251.09 (3) Feeding
- Feed each infant and each toddler on the child’s own feeding schedule.
- Ensure that food, breast milk, and formula brought from home are labeled with the child’s name and the date and are refrigerated, if required.


3.11: Checklist for Obtaining a Health History
-
- Last updated
- Save as PDF
Use the checklist below to review the steps for completion of “Obtaining a Health History.”
Steps
Disclaimer: Always review and follow agency policy regarding this specific skill.
- Gather supplies: health history agency form.
- Knock, enter the room, greet the patient, and provide for privacy.
-
Perform safety steps:
- Perform hand hygiene.
- Check the room for transmission-based precautions.
- Introduce yourself, your role, the purpose of your visit, and an estimate of the time it will take.
- Confirm patient ID using two patient identifiers (e.g., name and date of birth).
- Explain the process to the patient and ask if they have any questions.
- Be organized and systematic.
- Use appropriate listening and questioning skills.
- Listen and attend to patient cues.
- Ensure the patient’s privacy and dignity.
- Assess ABCs.
- Address patient needs (pain, toileting, glasses/hearing aids) prior to starting. Note if the patient has signs of distress such as difficulty breathing or chest pain. If signs are present, defer the health history and obtain emergency assistance per agency policy.
-
Complete a health history interview, including the following components per your instructor’s instructions:
- Demographic and Biological Data
- Reason for Seeking Health Care
- Current and Past Medical History
- Family History
- Functional Health
- Review of Body Systems
-
Ensure five safety measures before leaving the room:
- CALL LIGHT: Within reach
- BED: Low and locked (in lowest position and brakes on)
- SIDE RAILS: Secured
- TABLE: Within reach
- ROOM: Risk-free for falls (scan room and clear any obstacles)
- Document the health history findings and report any concerns according to agency policy.