Abstract.
This article answers questions about what a personal learning network is, why you might want to build one, and how to create one for yourself to experience the benefits it has to offer.
What is a Personal Learning Network?
One of the strengths of the digital environment is its highly networked nature and this is particularly relevant for personal learning networks where access to information and resources far surpasses anything that has come before both in terms of variety and volume. Further, in such an environment information changes more rapidly than ever before, and individuals need to engage in ongoing learning to keep pace, be it in a professional, personal or civic capacity. At one time it was adequate to read a daily newspaper, a magazine or two on special topics of interest and perhaps belong to an organization of interest that had conferences or meetings once or twice a year. In the past, an education typically set one up for life in the work of their choice and while there were changes in fields or topics of interest, they happened slowly, in terms of years. In the digital world of the 21
st
century, however, this is no longer the case and information can change in days, weeks or months. An excellent tool for handling quickly expanding and growing data is to create a Personal Learning Network (PLN) (Delaney & Redman, 2014; Perez, 2012; Trust, 2012).
PLN’s consist of formal and informal networks of individuals with similar goals and interests who interact using digital tools to share information, learn from each other, problem solve and collaborate (Ferguson, 2010; Nelson, 2012; Perez, 2012; Trust, 2012). They provide a vehicle for lifelong learning for both personal and professional development by enabling individuals to remain relevant in a world of rapidly changing information. While PLN’s can encompass both digital and face to face connections (Perez, 2012) the focus of this discussion will be on digital learning networks and why they are significant to learning in a digital age.
In addition, PLN’s provide far greater resources and information than one can muster alone or in a small group (Ross, Maninger, LaPrairie, & Sullivan, 2013). Ferguson (2010, p. 13) emphasizes this point when he says, “Before I built my professional learning network, I did all my learning by myself. If I needed to understand something for a new unit, I researched it on my own…. A PLN is a community of individuals around the world who are learning together.” Even though Ferguson specifically mentions a professional learning network, the mechanics of how the network functions is much the same as in a personal learning network, just with a different focus. The key point here is the idea of “learning together”. Within a networked environment there is less emphasis on singular sources of expertise and instead, a focus on dialogue and constructing knowledge as a group comes into play. This is facilitated by the speed with which conversations take place in a digital environment where people can connect across the globe in minutes and hours instead of weeks, months or years.
While the characteristics of a PLN include fast access to massive amounts of information, a successful PLN is also characterized by participants who are highly self-motivated and curious. Without people who have a passion for learning, a PLN would not be as valuable. Further, the ability to build one’s own network makes it possible to develop a highly customized approach to learning. As you will see, building a PLN gives one the freedom to develop tools, skills and knowledge that specifically suit one’s learning needs.
Tools for building a PLN.
A digital PLN can reflect a variety of tools and methods, depending on individual preferences and goals. Most people will develop a variety of tools for a well-rounded learning network that meets all of their needs. An easy way to think about building a PLN is to decide what functionality you need. For example, one requirement of a PLN is to gather the information that helps one stay current with developments in a particular field or interest area. Blogs, RSS feeds, email lists, websites, news groups and podcasts can provide a steady stream of up to date information. To take advantage of the experience of a wider variety of people and practitioners, another way to connect is through social networking tools. Some are smaller and focused on a specific subject such as Edmodo, Classroom 2.0 or the Educator’s PLN while others are much bigger and provide a single tool for tracking a variety of topics and information such as Twitter, Linked-in and Pinterest. Websites such as these provide an opportunity to both share and receive new information as well as find support, collaborators and thoughtful discussions.
While informal networks and learning opportunities are a hallmark of PLN’s it is not uncommon to include formal learning tools as well. Massive Open Online Courses (MOOCS) are offered by many colleges and universities. They are free and can offer opportunities to fold new topics and information into one’s existing expertise. ITunes-U has a large selection of lectures from multiple colleges and universities and there are also a number of websites offering online classes on a variety of topics such as Udemy and Khan Academy. Even YouTube is a good bet for finding instructional information for specific skills and topics. In short, a PLN is what each individual wants it to be and it reflects each individual’s personal passions, motivations and desire to know (Moreillon, 2016; Nelson, 2012).
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What a PLN Might Look Like
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Informal Learning
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Formal Learning
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Social Networking
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Blogs
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Pinterest
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Youtube
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RSS Feeds
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Udemy
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Podcasts
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Online Book clubs/groups
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Khan Academy
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Facebook
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Twitter
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Instagram
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Snapchat
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Figure 1. Categories of learning for a personal learning network (Green, 2017).
How to build a PLN.
Because there is such a wide variety of available tools and goals for PLN’s not everyone’s PLN will look the same. In addition, the tools for inclusion are always changing and some have different features and benefits. Given the variety of different tools available, it may be intimidating to get started so this last section offers some advice about how to start and build a successful PLN. The following can suggest a typical path towards developing a PLN.
The best way to get started is to choose a specific tool and work with it until it feels comfortable (Trust, 2012). Setting up sources for current information is a good place to start and member organizations’ websites have a lot to offer. For example, if one has an interest in world drumming and participating in drum circles, one might start the web site
http://www.worldmusicdrumming.com/
. They have a resources list, provide workshops, and they have a Facebook page. This is a great start and it connects me with an organization that caters to my personal interest and to a Facebook page where I will engage with other people who share my passion for drumming. Once I begin checking the Facebook page I find a really interesting post about an African song. A few clicks later I have landed on
http://pancocojams.blogspot.com/
where I find a lengthy article about traditional afro-Cuban music.
At this point I have an organizational website with information that is updated maybe monthly, a Facebook page that appears to be very active with posts every day and a related blog that is also updated at least every couple of days. This is a really good start to a PLN. At some point I may decide to take a drumming class, either online or in person and that organization will likely have a web presence that I can add to my PLN. I may also decide I need some help with a specific instrument and seek out some YouTube videos on how to play it and I can create a personal YouTube channel with my favorite videos. The important things to remember are “don’t try to read everything” and spend consistent time cultivating your sources (Perez, 2012).
A final step to consider in developing a PLN is to think about contributing your own information. The one thing that makes a PLN an excellent way to learn is to contribute your own information and expertise. Without people who contribute, a PLN would stagnate quickly. Nobody wants to hang out on a Facebook page where the last post was three months ago. So, when developing a social networking presence don’t lurk, actively participate and separate the professional from the private (Perez, 2012).
Now that one has a PLN the last tip is to tend to it. Without attention and care, the PLN will not be useful and interest will wane. Think of it as a garden. Sometimes planting, sometimes weeding, sometimes harvesting but always tending the garden so that it grows and is productive. Occasionally a resource may stop being useful so remove it. Other times a new, interesting resource can be added. Whatever the decision, keep in mind that this is your PLN and it is designed to serve your needs. There are no right and wrong ways to go about it. So, as one network source grows and matures expanding to another tool or community can help fill any gaps in the first one.
Conclusion.
In setting up a PLN, variety is a key factor to success. Engaging in different communities with different areas of focus will provide a rich environment for learning. This is especially easy to do in a digital environment where one literally has the entire world to draw from as a source of inspiration. There is no need to settle for sources that do not meet your needs. In the end, the important thing is that each individual can select those tools and websites that meet their learning needs. Those will not be the same for everyone and they may change over time as well.
Additional Resources.
How to build a PLN
Characteristics of successful PLN’s
References.
Delaney, S., & Redman, C. (2014). Incorporating collaborative, interactive experiences into a technology-facilitated professional learning network for pre-service science teachers.
Proceedings of the International Conference E-Learning 2014 – Part of the Multi Conference on Computer Science and Information Systems, MCCSIS 2014
, 369–373. Retrieved from
http://www.scopus.com/inward/record....nerID=tZOtx3y1
Ferguson, H. (2010). Join the flock!
Learning & Leading with Technology
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37
(8), 12–15. Retrieved from
http://www.iste.org/learn/publicatio...join-the-flock
!
Moreillon, J. (2016). Building Your PERSONAL LEARNING NETWORK (PLN): 21st-Century School Librarians Seek Self-Regulated Professional Development Online.
Knowledge Quest
,
44
(3), 64–69. Retrieved from
http://search.ebscohost.com/login.as...ite=ehost-live
Nelson, C. J. (2012). 64 Knowledge Quest | Personal Learning Networks, 70–74.
Perez, L. (2012). Innovative professional development.
Knowledge Quest
,
40
(3), 20–22. Retrieved from
http://ezproxy.georgetowncollege.edu...9h&AN=82563981
Ross, C. R., Maninger, R. M., LaPrairie, K. N., & Sullivan, S. (2013). The Use of Twitter in the Creation of Educational Professional Learning Opportunities.
Administrative Issues Journal: Connecting Education, Practice, and Research
,
5
(1), 55–76.
https://doi.org/10.5929/2<IP_ADDRESS>
Trust, T. (2012). Professional Learning Networks Designed for Teacher Learning.
Journal of Digital Learning in Teacher Education
,
28
(4), 133–138.
https://doi.org/10.1080/21532974.2012.10784693


2: The Story We Tell about Millennials — and Who We Leave Out (Allen)
By Reniqua Allen
So on the surface, Troy is the kind of millennial that think pieces are made of. He's arrogant, self-centered and convinced that he is smarter than people give him credit for. His favorite topics of conversation are girls, sneakers and cars -- not a surprise for someone who was a teenager just a few years ago. But Troy's mannerisms -- they reveal the patterns of someone who is scared, troubled and unsure of the future.
Now Troy also embodies the many positive qualities his generation is known for. An entrepreneurial spirit, an independent streak and a dedication to his parents. He believes in hard work and has tried gigs in both the licit and underground economies, but he hasn't had any luck and is just trying to find his way and still dances between both worlds.
When I met Troy a few years ago, he had been employed as a golf caddy at a local country club, carrying bags for rich men and women who often never even acknowledged his existence. Before that, he sold sneakers on Facebook. He even tried selling candy bars and water bottles, but he wasn't making enough money to help his parents out or save up for a car any time soon. So Troy saw how hard his immigrant mother from Jamaica worked and how little she got back in return, and he vowed -- Troy vowed to take a different path. So he ended up selling drugs. And then he got caught, and right now, he's trying to figure out his next steps.
In a country where money equals power, quick money, at least for a while, gives young men and women like him a sense of control over their lives, though he said he mainly did it because he wanted stability. "I wanted a good life," he told me. "I got greedy and I got caught." Yet the amazing thing about Troy is that he still believes in the American dream. He still believes that with hard work, despite being arrested, that he can move on up. Now, I don't know if Troy's dreams came true. He disappeared from the program for troubled youth that he was involved in and slipped through the cracks, but on that day that we spoke, I could tell that more than anything, Troy was happy that someone listened to his dreams and asked him about his future.
So I think about Troy and his optimism when I think of the reality that so many young, black millennials face when it comes to realizing their dreams. I think about all the challenges that so many black millennials have to endure in a world that tells them they can be anything they want to be if they work hard, but actually doesn't sit down to listen to their dreams or hear stories about their struggle. And we really need to listen to this generation if we hope to have a healthy and civil society going forward, because millennials of color, they make up a fair chunk of the US and the world population.
Now when we talk about millennials, a group that is often labeled as entitled, lazy, overeducated, noncommittal and narcissistic, the conversations often swirl around avocado toast, overpriced lattes and fancy jobs abroad -- you probably have heard all these things before. But millennials are not a monolith. Actress Lena Dunham may be the media's representation of this generation, but Troy and other voices like his are also part of the story. In fact, millennials are the largest and most diverse adult population in this country. 44 percent of all American millennials are nonwhite, but often, you wouldn't even know it at all.
Now sure, there are similarities within this population born between 1981 and 1996. Perhaps many of us do love avocado toast and lattes -- I know I do, right? But there are also extreme differences, often between millennials of color and white millennials. In fact, all too often, it seems as though we're virtually living in different worlds.
Now black millennials, a group that I have researched for a book I recently wrote, are the perfect example of the blind spot that we have when it comes to this group. For example, we have lower rates of homeownership, we have higher student debt, we get ID'd more at voter registration booths, we are incarcerated at higher rates ... we make less money, we have higher numbers of unemployment -- even when we do go to college, I should say -- and we get married at lower rates. And honestly, that's really just the beginning.
Now, none of these struggles are particularly new, right? Young black people in America have been fighting, really fighting hard to get their stories told for centuries. After the Civil War in the 1800s, Reconstruction failed to deliver the equality that the end of slavery should have heralded, so young people moved to the North and the West to escape discriminatory Jim Crow policies. Then, as segregation raged in much of the country, young black people helped spearhead civil rights campaigns in the 1950s and 1960s. After that, some people embraced black power and then became Black Panthers and then the next generation, they turned to hip-hop to make sure their voices were heard. And then Barack Obama, hopeful that he, too, may bring about change. And when that failed, when we realized we were still brutalized and battered, we had to let the world know that our lives still mattered. Now, when technology allows more video of our pain and struggle to be broadcast to the world, we wonder, like, what is next?
Our country feels more polarized than ever, yet we are still being told to pull up our pants, be respectable, be less angry, smile more and work harder. Even the attitudes of millennials themselves are overdue for an update. Research done by the Washington Post in 2015 about this supposedly "woke" group found that 31 percent of white millennials think that blacks are lazier than whites, and 23 percent say they're not as intelligent. These are, like, surprising things to me, and shocking. And these responses are not that much different than generations in the past, and it shows that unfortunately, this generation is repeating the same old stereotypes and tropes of the past. Now, a study conducted by David Binder Research and MTV in 2014 -- it found that 84 percent of young millennials were taught by their families that everyone should be equal. This is a really great thing, a really positive step. But only 37 percent in that group actually talked about race with their families.
So I could understand why things may be confusing to some. There are definitely black millennials who are succeeding. Marvel's "Black Panther," directed by black millennial Ryan Coogler and showcasing many others, broke all sorts of records. There's a crop of television shows by creatives like Donald Glover, Lena Waithe and Issa Rae. Beyoncé is, like, the queen, right? She is, like, everything. Young black authors are winning awards, Serena Williams is still dominating on the tennis court despite all her haters, and there's a crop of new politicians and activists running for office. So I don't want to, like, kill these moments of black joy that I too revel in, but I want to make it clear that these wins are too few and far between for a people that's been here for over 400 years. Like, that's insane, right?
And most people still don't really understand the full picture, right? Our stories are still misunderstood, our bodies are still taken advantage of, and our voices? Our voices are silenced in a world that still shows little concern for our everyday struggles. So our stories need to be told in a multitude of ways by a range of voices talking about diverse and nuanced topics, and they really need to be listened to.
And it is not just here in America, right? It's all around the world. Millennials make up 27 percent of the world's population. That's around two billion people. And with countries like India, China, Indonesia and Brazil, along with the United States, accounting for 50 percent of the world's millennials, it's clear that the white, often male, heterosexual narrative of the millennial is only telling half the story. Now, there's many people trying to broaden the palette. They're fighting to get their stories told and bust the millennial stereotype. Whether it's students in South Africa protesting statues of Cecil Rhodes, Michaela Coel making us laugh from the UK, or Uche Eze, who's framing views about Nigerian life, online. But I want to make it clear -- I want to make it really clear to everyone that just because things look more equal than they did in the 20th century, doesn't mean that things are equitable at all. It doesn't mean our experiences are equitable, and it certainly doesn't mean that a post-racial society, that thing that we talked about so much, ever became close to being a reality.
I think of Joelle, a middle-class 20-something who did everything the "right way," but she couldn't go to her dream school, because it was simply too expensive. Or Jalessa, who knows she can't be mediocre at her job the same way that her white peers can. Or Trina, who knows that people judge her unconventional family choices in a different way than if she were a white woman. Or actor AB, who knows that the roles he takes and gets in Hollywood are different because of his skin color. And then there's Simon. So Simon, by all means, would be an example of someone who's made it. He's a CFO at a tech company in San Francisco, he has a degree from MIT and he's worked at some of the hottest tech companies in the world.
But when I asked Simon if he had achieved the American dream, it took him a while to respond. While acknowledging that he had a really comfortable life, he admitted that under different circumstances, he might have chosen a different path. Simon really loves photography, but that was never a real option for him. "My parents weren't able to subsidize me through that sort of thing," Simon said. "Maybe that's something my children could do." So it's these kinds of stories -- the quieter, more subtle ones -- that reveal the often unique and untold stories of black millennials that show how even dreaming may differ between communities.
So we really need to listen and hear the stories of this generation, now more than ever, as the baby boomers age and millennials come to prominence. We can talk all we want to about pickling businesses in Brooklyn or avocado toast, but leaving out the stories and the voices of black millennials, large swaths of the population -- it will only increase divisions. So stories of black millennials, brown millennials and all millennials of color really need to be told, and they also need to be listened to. We'd be a far better-off country and world.
Thank you.
____________________
Reniqua Allen is a journalist who produces and writes for various outlets on issues of race, opportunity, politics and popular culture. She is the author of It Was All a Dream: A New Generation Confronts the Broken Promise to Black America . This essay is a transcript of her TED Talk .
The Story We Tell about Millennials — and Who We Leave Out by Reniqua Allen is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License .


6.1: Introduction
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Learning Outcomes
Learners. educators, and workers in all sectors are increasingly using emerging technologies such as cell phones, tablet PC, personal digital assistants (PDAs), web pads, and palmtop computers. As a result, these tools make learning and training materials accessible anywhere, anytime.
Today, the trend is towards learning and working “on the go”, rather than having to be at a specific location at a specific time. As learners become more mobile, they are demanding access to learning materials wherever they are and whenever they need them. This trend will increase because of ubiquitous computing, where computing devices, wireless connectivity, and transparent user interfaces are everywhere.
Educators must be prepared to design and deliver instruction using these emerging technologies. In addition to delivering learning materials, emerging technologies can be used to interact with learners, especially those who live in remote locations. At the same time, learners can use the technologies to connect with each other to collaborate on projects and to debate and discuss ideas.
This chapter provides a brief history of technology in education, outlines the benefits of using emerging technologies in e-learning, provides design guidelines for developing learning materials, describes the support required for these technologies, and discusses future trends in e-learning.


13.2: Physics and Length Scales- Cavity Laser, Coulomb Blockade, Nanoscale Magnets
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The special properties of nanomaterials do not derive from different laws of physics, which are the same for objects large and small. For example, Newton's second law (\(F = ma\)), Coulomb's law (\(E= \frac{q_{1}q_{2}}{4 \pi \varepsilon_{0}r}\)), and the laws of energy and momentum conservation are the same for buckyballs (C 60 ) and full-size soccer balls. Nevertheless, the physics of electrons, atoms, and photons naturally produce characteristic length scales , some of which we have already seen. For example, in Chapter 6 we discovered that the mean free path of an electron in a good metal is about 40 nm. In Chapter 10, we learned that the Bohr radius of an electron or hole in doped Si is about 4 nm, and that the coherence length of Cooper pairs in semiconductors is somewhere between a few nm and 1 µm. When objects become small relative to these characteristic lengths, their physical properties change in interesting ways. Materials that exist at the relevant length scale are called mesoscopic (meso = "between," scopic = "size") meaning that they cross over from one kind of behavior - the bulk behavior of large objects - to another. This length scale is different for different kinds of properties, but for many it happens between 1 and 100 nm. We illustrate this point with a few examples.
The cavity laser
A vertical cavity surface-emitting laser (or VCSEL) is a semiconductor-based device that emits light in the vertical direction relative to the plane of the chip. These devices are being developed and used for high power applications such as laser surgery, infrared illumination for military surveillance, and laser cutting tools. The basic design of a VCSEL is shown at the right. The VCSEL is basically a light-emitting diode, incorporating p-type and n-type regions of the III-V semiconductor (Al,Ga)As. However, it has two special features. First, the refractive index of the semiconductor is modulated above and below the junction to make Bragg mirrors. These mirrors reflect light emitted in the junction, so that the photon density becomes very high there, a necessary condition for stimulated emission and lasing. The Bragg mirror stack is asymmetric (thinner on the bottom), so that light can escape from the junction in one direction only. Second, the junction itself consists of a thin "quantum well" layer of (In,Ga)As, a III-V semiconductor with a smaller band gap than the surrounding (Al,Ga)As layers.
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A VCSEL device structure. This is a bottom-emitting multiple-quantum-well VCSEL. |
The
quantum well
structure, the band diagram of which is illustrated at the left, is the nanoscale part of the laser. The energies of the conduction and valence bands of (Al,Ga)As flank those of the thin (In,Ga)As layer. Therefore, electrons and holes injected into that layer cannot escape: electrons in (In,Ga)As do not have enough energy to climb "up" to the conduction band of (Al,Ga)As and holes cannot climb "down." Electrons confined to such a small well behave as a particle-in-a-box (as we learned in the context of electrides in Chapter 9).
The electron has a kinetic energy defined by the equation:
\[KE= \frac{h^{2}n^{2}}{8mL^{2}}\]
We can calculate the energy difference between the lowest (n=1) and next lowest (n=2) levels, which is inversely proportional to the square of the thickness (L) of the (In,Ga)As layer. In this calculation we need to use the effective mass of the electron in (In,Ga)As, which is about 7% of the electron rest mass [1] . With an 8 nm thick layer, this energy is:
\[E = \frac{(2^{2}-1^{2})(6.626 \cdot 10^{-34} Js)^{2}}{8 \cdot 0.07 \cdot (9.1 \cdot 10^{-31} kg)(8 \cdot 10^{-9} m)^{2}} = 4.0 \cdot 10^{-20} J = 0.25eV\]
The VCSEL cavity will thus have a resonant energy of 0.25 eV and emit photons at this energy in the infrared (λ ≈ 5000 nm). Note that because of the inverse square dependence of the cavity energy on layer thickness, lasers based on this design can only function at nanoscale dimensions. When the cavity is three times thicker, its resonant energy becomes comparable to the thermal energy at room temperature (kT = 0.026 eV), and the lasing effect is thermally "washed out."
Coulomb blockade
A capacitor is a (macroscopic) device that stores electrical charge. The basic structure of a capacitor is shown at the right. When a voltage is applied to such a device, it develops a charge (± Q) on the two plates that is proportional to the voltage:
\[C= \frac{Q}{V}\]
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Basic design of a parallel plate capacitor of area A and with a dielectric thickness of d |
The magnitude of the capacitance C is determined by the permittivity ε and the dimensions of the dielectric layer, A and d.
\[C= \frac{\varepsilon A}{d}\]
We can also calculate the work done in charging the capacitor up (i.e., the energy stored by charging the capacitor) by integrating the voltage times the charge:
\[E = int^{Q}_{0} V(q)dq = \int^{Q}_{0} \frac{q}{C} dq = \frac{1}{2} \frac{Q^{2}}{C} = \frac{1}{2}CV^{2} = \frac{1}{2}VQ\]
Now it is interesting to ask, what happens to a capacitor when we make it very small? This is of particular interest in a device called a single electron transistor , a schematic diagram of which is shown at the right. The metallic gate lead is separated from a "quantum dot," which can be a metal or semiconductor particle, by a thin dielectric layer. This metal-dielectric-metal sandwich acts as a capacitor, and from the equation above, the energy needed to charge it by a single electron (Q = e) is:
\[E= \frac{e^{2}}{2C}\]
where e is the charge of the electron, 1.602 x 10 -19 Coulomb. If the gate width and lateral dimensions are very small - say 2 nm as is readily achievable in self-assembled Coulomb blockade devices [2] [3] - then for a typical insulating dielectric, a voltage of about 200 mV is needed to charge the quantum dot by a single electron. Again, this effect is unique to the nanoscale, because a 10 times larger device area would make the single-electron charging voltage about 20 mV, which is smaller than the thermal energy kT (26 meV). Thus for devices larger than about 5-6 nm, individual electron charging events are washed out at room temperature by thermal fluctuations.
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Schematic of a single-electron transistor. |
How can a nanoscale capacitor like this act as a transistor , which functions as a switch in an electrical circuit? The effect comes from the mutual repulsion of electrons. An electron on the quantum dot repels any other electron that would be forced onto it by applying a small voltage between the source and the drain. Hence the conductance of the quantum dot is very low at a gate bias of zero volts, or at any gate bias (200, 400, 600 mV...) that places an integer number (1, 2, 3,...) of electrons on the dot. But halfway in between these voltages (e.g., at 100, 300, 500 mV) the energy is the same whether there are n or n+1 electrons on the dot. This means that electrons can hop on and off without changing their energy, i.e., they can tunnel through the dot from source to drain. This effect gives peaks in the conductance of the dot at regular steps in the gate voltage. In effect, the gate can act as a switch, as in a conventional field-effect transistor. Single-electron transistors are being researched as ultra-sensitive electrometers and single-molecule chemical sensors, since a tiny change in the electrostatic environment of the dot can switch the device on or off.
Nanoscale magnets
Ferro- and ferrimagnetic materials such as iron and chromium oxide are used for digitial storage of information in hard disks. The individual memory bits, which can be oriented perpendicular or parallel to the plane of the disk as shown at the right, store a logical "0" or "1" depending on the orientation of their magnetic dipole. To be useful, this information must be non-volatile, i.e., the magnetic bit must retain its polarization in the absence of an applied field from the read/write head.
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Longitudinal and perpendicular recording, two types of writing heads on a hard disk. |
The storage density of such magnetic memories is impressive. A 2.5" hard drive can now store 1 TB of information, using rod-shaped magnetic grains that are approximately 0.5 µm long. We now have good synthetic methods for making these same materials as crystals with dimensions of only a few nanometers. Why aren't those nanocrystals used to make even more dense memory disks?
The reason is that the energy needed to flip the magnetization (i.e., to turn a "0" into a "1" and vice-versa) is strongly size-dependent. For a ferro- or ferrimagnet this energy is equal to Mr 3 , where M is the magnetic energy per unit volume and r is the characteristic dimension (e.g., the length of the edge of a cube, or the diameter of a sphere) of the magnetic grain. For typical materials such as iron, this energy becomes comparable to kT when r is about 3-5 nm. Such small particles are superparamagnetic, meaning that they still have a large magnetic moment because of the ordering of their spins, but they do not retain a permanent polarization in the absence of an applied magnetic field. Superparamagnetic particles are thus not useful for magnetic memories, but they are interesting and practical in other ways, for example in ferrofluids, magnetic resonance imaging (MRI), and some emerging medical diagnostic and therapeutic applications.
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A ferrofluid containing superparamagnetic Fe 3 O 4 nanoparticles, which are coated with oleic acid and suspended in oil, in the field of a strong permanent magnet. |
In these three illustrative examples (involving light emission, electronic conduction, and magnetic behavior), the transition to new properties involves a crossover in which the characteristic energy of the system is comparable to the thermal energy kT. It just so happens that for many physical phenomena, this crossover occurs on the length scale of nanometers.


6: The Late Renaissance/Mannerism
Introduction
One act by a defiant German Augustinian monk changed the political and religious course of Europe in the 1500s. Martin Luther protested the materialism and corruption of the Roman Catholic Church. At this time, he listed 95 arguments against the use of indulgences by the Catholic Church. Indulgences were basically a get out of jail free card that stated if you paid the fee, you would be allowed into heaven. Indulgences could be purchased for yourself, for a spouse, or for your children, even if they had passed away. It was 1517 when Martin Luther nailed his manifesto to a church door in Wittenberg, Saxony. His criticism of the overly corrupt Roman Catholic Church was justified, and he wished to reform the Church through his arguments. However, by 1520, he was excommunicated from the church and a price was placed on his head by the Medici Pope Leo X.
The result of this protest by Luther was a split in the power of the Roman Catholic Church. For over 1000 years, the Roman Catholic Church had ruled supremely over large swaths of Europe. Now that power was challenged by Luther's arguments. The fact that he protested the tenants of the Roman Catholic Church led to the term Protestant being used to describe the new Christian faith for those practicing Christianity outside of the orthodoxy of the Roman Catholic Church. Except for Italy, France, and Spain, other traditionally Roman Catholic regions embraced the new religion that emphasized a personal relationship with Jesus and a personal faith developed from the studies of the Gospels, and the Old Testament. With this new religion, the priest was no longer in a position to act as an intermediary for a person's salvation. Stripping the Roman Catholic Church of this authority had profound effects on Christian imagery as well. The cult of the Saints and of the Virgin Mary, both strong within the Roman Catholic faith had less meaning in this new Protestant movement. The need for sacred images and icons of the faith remained in the areas where Roman Catholicism was strong, but we find a waning of these images in Protestant Europe.
Another interesting result of this religious upheaval involves King Henry VIII, who desired a male heir of his bloodline to the throne above all else. He requested an annulment from his marriage, but was denied by the pope, resulting in a bitter conflict. He then decided to split from the Roman Catholic Church and placed himself as the chief authority over the newly formed Church of England. This had far reached effects since England and France, a Roman Catholic holdover, were close graphically yet very different in terms of Christian beliefs. France remained Roman Catholic for the most part, and England was now split between Protestants and Roman Catholics. There was certainly the potential for violence, and religious persecution would ensue in the coming years against Catholics. Many Catholics leave England and would travel by ship to the Colonies in America. Baltimore, my hometown, would become one of the Catholic strongholds in the years to come. We also see the same violent tendencies spilling over in more recent years in Ireland between the Roman Catholics and the Protestants.
With all the turmoil in Europe at this time artists were also affected and influenced by the events that were taking place. The violence that occurred during the sack of Rome by Spanish and German forces shook the Roman Catholic Church and the Catholic faithful to their core. One of the most graphic responses in art is that of the Last Judgment painted by Michelangelo at the altar wall of the Sistine Chapel. In this dynamic and forceful painting, Michelangelo paints Jesus as the enforcer and ultimate judge for the sins of humanity. We see many damned falling towards the bottom of the fresco in horror and agony as they are doomed to eternity in hell. Jesus is represented as the massive, muscular, and strong Son of God with his right hand lifted in judgment. It is enough to strike fear in the faithful and to encourage repentance. This is a strong representation of the type of Counter-Reformation the Roman Catholic church will use to encourage the faithful Catholic church body to remain faithful to the teachings and sacraments of the church.
https://www.youtube.com/watch?v=ombPQ6egLE8
https://www.youtube.com/watch?v=4mpFcaeElCs
https://www.youtube.com/watch?v=W-mxg-LYaF4
Sculpture and Architecture
Vestibule of the Laurentian Library, Florence, Michelangelo begun 1523
Michelangelo’s design of the Laurentian Library vestibule shows a breaking away from the strict directives of the Classical Orders. Instead, Michelangelo uses classical forms as they suit his design. He mixes and matches curved and pointed pediments, even breaking into the shapes when it suits his needs. He originally wanted to include columns in the space, but realized the space is not large enough to accommodate those forms. So, his elegant solution is to recess the columns into the walls. Note his doubling up of the columns, a stylistic device he’ll use again in the dome of the New Saint Peter’s. The staircase itself is cascades down like an organic waterfall that spills out to the right and left. The color choices are similar to those used by Brunelleschi in his architectural works in Florence. Michelangelo would have been familiar with those buildings and aesthetic. What Michelangelo achieves in this space is a treatment of architectural design as sculpture, a pliable medium with creative possibilities. Architects who follow will have the freedom to explore more imaginative designs using classical forms, but not necessarily bound by strict stylistic codes.
Tomb of Giuliano de ‘Medici, New Sacristy, San Lorenzo, Florence, Michelangelo 1519-1534
https://www.youtube.com/watch?v=5gVlGU4zUeY
The funerary chapel for the Medici family was commissioned by Pope Leo X, with subsequent support by his cousin Pope Clement VII. This project was never completed and so we only have a glimpse at Michelangelo’s architectural and artistic vision for the space. This, along with The Laurentian Library, show Michelangelo using classical forms for artistic, not functional purposes. He is foremost a sculptor, and approaches these design projects through the lens of sculpture.
Pietà , Michelangelo 1546
The Pietà at the Opera del Duomo in Florence was sculpted by Michelangelo from a huge block of white Carrara marble between 1547 and about 1555, on the eve of his 80th birthday. It is one of the three Pietà sculpted by the great artist and expresses his life experiences and human suffering. Unlike the other two—his youthful work in the Vatican and the later Rondanini—the body of Christ is not only supported by Mary but also by Magdalene and the elderly Nicodemus, to whom Michelangelo gave his own face. This detail is confirmed by two biographers, the artist’s contemporaries Giorgio Vasari and Ascanio Condivi, thanks to whom we know that the sculpture was made for an altar in a church in Rome, beneath which the artist wished to be buried. The Pietà of Florence, Michelangelo’s masterpiece, “like other sculptures by Buonarroti,” said Timothy Verdon, director of the Museum, “is considered an unfinished work, though the most suitable wording for it would be that of the 16th century when it was still called an infinite work. ”
https://www.friendsofflorence.org/projects/michelangelos-pieta-opera-del-duomo/
Incomplete Work by Michelangelo
https://www.youtube.com/watch?v=h44bnIGogu8
Palazzo dei Conservatori, Campidoglio, Rome, Michelangelo designed 1545
The Campidoglio (engraving by Étienne Dupérac, 1569) Michelangelo
https://museicapitolini.org/en/infopage/palazzo-dei-conservatori
St. Peter’s Rome, Michelangelo 1546-1564; dome completed by Giacomo della Porta, 1590
With similarities to central plan churches and to the now famous dome of the Florence Cathedral, Michelangelo designs the dome for the New Saint Peter’s. Novel elements include the double faux columns used to create a visual texture.
https://smarthistory.org/st-peters-basilica/
https://www.youtube.com/watch?v=ua_lEuQguIA
Courtyard, Palazzo del Te, Mantua Giulio Romano, 1527-1534
The Palazzo del Te is a wonderfully designed space evoking the spirit of the classics. References to classical Greek temple forms are apparent throughout the design with playful and creative artistic alterations such as the lowered triglyphs in the running entablature. The faux double columns remind us of the double columns used by Michelangelo. Simple Doric columns create an elegant simplicity and textural contrast with the rectangular stonework filling the vertical spaces. Alternating arches and rectangular faux windows, doorways, and openings create a logical rhythm to the space.
Courtyard of the Palazzo Pitti, Florence, Bartolommeo Ammanati 1558-1570
Texture is the main art element standing out in the design of the courtyard of the Palazzo Pitti. As in the Ancient Roman Colosseum, the three classical orders are featured with Doric on the first floor, Ionic on the second story, and Corinthian columns featured on the third story. Arches, classical pediments are designed into the structure as ornamentation. Following the sculptural approach to architecture that Michelangelo presents in earlier designs, Ammanati exercises artistic freedom while incorporating classic themes.
http://www.museumsinflorence.com/musei/Pitti_palace.html
Façade of Uffizi, Florence, Giorgio Vasari, begun 1560
Vasari pays homage to the greats who came before him in this grand architectural design. We see the classical balance and functional columns used by Brunelleschi as well as the doubling of columns as introduced by Michelangelo. Balance, logical geometric spaces, Roman arches, and subtle contrasting colors and values come together in Vasari’s design.
https://www.uffizi.it/en/the-uffizi
https://www.florenceartmuseums.com/uffizi-gallery/
Façade of II Gesù, Rome, Giacomo della Porta, 1575-1584
Working in the style of Alberti, della Porta creates a classical façade for the Jesuit church.
https://www.youtube.com/watch?v=aj2el_owSg4
The Rape of the Sabine Women , Loggia dei Lanzi, Florence, Giovanni Bologna, 1583
After the death of Michelangelo, a possible successor is Giovanni Bologna who carves in marble a difficult piece with three intertwined figures. Although the intent was to successfully incorporate the three nudes, it was later suggested that the scene resembles the legendary Rape of the Sabine Women. The title stuck and so we now know the work by that name. It was a very difficult proposition to sculpt a work with the figures rising up in such fashion.
Painting in Italy
A style of painting emerging in the 16th century is known as Mannerism. The paintings in this style are often bizarre and oddly styled with juxtaposed figures, odd compositional devices, and multiple lighting sources. There is a distinct departure from the balanced and beautiful paintings by Renaissance artists like Rafael and the Italian painters who proceeded him. It appears that these artists knew that they could not compete with the great works of the High Renaissance artists and instead went their own creative direction. It is a distinct style that is easily recognizable because of the bright color palettes and eccentric and emotional depictions of the figures.
The Descent from the Cross , Rosso Fiorentino, 1521
The Descent from the Cross by Fiorentino was painted for followers of the cross of Jesus a specific sect of monks. In this painting, Jesus is being carried down from the cross upon which he was crucified. The monks who funded this painting are depicted in the scene. The scene itself is very chaotic and unconventional in terms of the composition and spacing. Visual forces within the painting itself direct the viewers eyes across the painting in awkward movements. There are strong diagonals. Even the figure on the lower right in agony directs us out of the painting which would be unheard of in Early and High Renaissance painting. The colors are bright and unusual. Jesus’s body has a strange off color hue in greyed green depicting the lifeless form. The figures of the monks are very animated and agitated as they argue over how to bring Jesus's body down from the cross. Although Jesus reportedly passed away around 3:00 PM in the afternoon, the scene itself has a very dark sky and is illuminated from the upper right and lower right with strong lighting. This creates strong areas of contrast and drama within the painting.
The Entombment , Santa Felicità, Florence Jacopo da Pontormo 1526-1528
The Entombment painted by Pontormo is an intriguing work of art that once again shows some of the chaos and drama in Mannerist paintings. Pontormo places himself to the left of Mary in the background in a self-portrait. The space is packed with figures, and in many ways does not make visual sense. We have no less than four light sources illuminating the scene. The cloud in the upper left of the scene is lit from above left. Yet, the figure on the upper right in the green top is illuminated from upper right. Mary is also illuminated from the upper right, yet the figure coming to support her from the lower right is illuminated from the lower right. The two figures supporting the body of Christ both look very concerned and worried and turn to engage the viewers. They are illuminated by light coming from the lower left. Jesus himself is illuminated from the right. Jesus's body is also a slightly different hue suggesting his passing on the cross.
Allegory of Venus , Angolo Bronzino, 1546
Allegory of Venus is a puzzling and intriguing painting by Angolo Bronzino in 1546 that is both erotic and incestuous.  The Cupid-like figure behind Venus is ready to throw flowers on the two of them, while a male figure, often referred to as Father Time is pulling back the cloth that was covering them. Other figures may represent Jealousy and Fraud. While the beautiful childlike face behind the Cupid-like figure has a body resembling a serpent while her hand holds out a honeycomb. The color Palette is pure Mannerist. The visual space Is packed with figures and very dynamic, although oddly composed. It is believed that this painting was commissioned by Cosimo I de Medici as a gift for King Francis I of France. Paintings like this were meant for personal delight and as conversation pieces, as well as visual puzzles to be solved. We do not know the exact meaning of the figures in the painting since there is not any related written documentation.
Portrait of Eleanora of Toledo and Her Son Giovanni de‘Medici , Angolo Bronzino, 1550
Bronzino gives us another painting commissioned by the Medici family in the Portrait of Eleonora of Toledo and her son Giovanni de’ Medici by Bronzino. This is a more traditional painting that emphasizes the family heir To Medici power. Eleanora, although the subject of the painting, is not presented to us in a flattering manner other than her elaborate dress, which is exquisitely embroidered showing the wealth and power of the Medici family. Also featured in the painting is her son Giovanni who in many ways is the most important subject in the painting, since he is the Medici heir.
Jupiter and Io , Correggio, 1532
Jupiter and Io , by Correggio, is a magnificent painting showing a moment in time when the two embraced. You can see Jupiter's face and hand in the smoky mist as he embraces Io and kisses her on the cheek. She welcomes this embrace. Io is painted with her clothing beside and under her as she leans back on the rocks. Paintings like this were often gifts given to royalty and for private consumption.
The Assumption of the Virgin , Dome of Parma Cathedral, Parma Italy, Correggio 1522-1530
Correggio gives us a fantastic Counter Reformation painting of Mary being lifted into heaven in bodily form. She is uncorrupted and ever virgin in this heavenly scene painted in fresco, with the Saints and religious dignitaries represented in the clouds as she rises up towards the light of God in what is an impressive use of three- point perspective on the Dome of the Parma Cathedral. Never had such an illusionistic fresco been created. It is really a great visual illusionistic achievement by Correggio.
Self Portrait , Parmigianino 1524
Parmigianino was a child prodigy and displays his great talent in this self-portrait painted in his early 20s. He was fascinated by seeing a convex form and decided to actually paint on a convex surface. In this painting, he gives us the illusion of this visual distortion represented in a reflected convex surface.
The Madonna with the Long Neck , Parmigianino 1534. La Belle Jardinière , Raphael 1507
Parmigianino painted The Madonna with the Long Neck in 1534 in a Manneristic style. The body proportions are quite odd, whereby the torsos are rather short, and the legs are massive and long. Mary looks down adoringly at her son Jesus, while a crowd of young girls press forward in anticipation to see the child. The upper left lighting focuses on Mary and Jesus. Her blue cape billows out on her left side, like the painting techniques used by Raphael in his paintings of Mary and Jesus. What appears to be an Old Testament prophet holding a scroll stands in the background in front of a huge column from an unfinished structure. Representations of figures in interior spaces with landscapes going off into the distance became popular after Leonardo da Vinci's painting of the Mona Lisa. We see similar paintings by artists such as Giorgione and Titian with reclining nudes in interior spaces and landscapes visible through the window openings. This is an odd painting where the child Jesus looks as if he's going to slide right off Mary's lap. This leaves the viewer feeling a little unsettled and anxious. Once again, this Mannerist painting gives us an odd juxtaposition of beauty and an unbalanced composition.
https://www.youtube.com/watch?v=suIUUGdNyWk
Portrait of a Noblewoman , Lavinia Fontana 1580
Lavinia Fontana is a female painter. Who created several works of note.
The Venetian Painters
Venus of Urbino , Titian 1538
The Venus of Urbino by Titian in 1538 gives homage to his painting mentor Giorgione and his painting of Sleeping Venus from 1509. The paintings are very similar, although in Titian's painting, Venus is gazing directly at us and is very alert and aware. The sleeping dog at the base of the bed represents fidelity, and other objects within the painting have specific meaning to Venus herself, such as the Myrtle tree and the flowers. The interior is like Venetian interiors of the day, and the two attendants appear to be gathering her garments at this early morning hour as the sun rises. It is an erotic painting showing Titian at the height of his artistic abilities
Danaë , Titian 1544-46
Titian, who painted religious scenes as well as secular themes, also delves into Pagan mythologies and legends. You can see this in several of his paintings, including Rape of Europa painted in 1559 to 1562, and in this painting of Denaë . In Denaë , we see a more relaxed female nude represented in a reclined position with, once again, the landscape background painted in beyond the interior. Drapery Is beautifully rendered just as observed in Venus of Urbino .
Pietà , Titian 1576 Pietà , Michelangelo 1546
Both Titian and Michelangelo produce work through both the High Renaissance and Mannerist Periods. Both continue to create well into their late years, and we see a distinct change in attitude and feelings in their work. Titian’s work becomes less refined and more emotional as does Michelangelo's. Both works depicting the Pietà are humble and much less confident as in their earlier works of similar subjects. Michelangelo even includes himself as the man supporting the body of Jesus. The body of Jesus in this sculptural work feels very heavy, with the limbs awkwardly placed. The body is being supported by three figures from the sides and from the back in what is a triangular compositional shape. One really feels the weight of the world bared by Jesus as he sinks towards the ground with gravity pulling on his form. It is a very emotional work, far from the sublime and beautiful Pietà he carved in his 20s. while in Rome.
Feast in the House of Levi (The Last Supper) , Paolo Veronese 1573
The Counter Reformation led to a formal body of tribunal judges referred to as the Spanish Inquisition. Artists were sometimes attacked for their creative endeavors. Such is the case with the painting The Last Supper by Veronese in 1573. Veronese was called in front of the tribunal to defend his work and a transcript of that event documents the back and forth between the Inquisitors and the artist. They felt that Veronese was making a mockery of the Last Supper and pointedly spoke to specifics of the painting with which they disagreed. In the end, Veronese was given three months to change the painting to match the newly established Counter Reformation standards and requirements of the Roman Catholic Church and the Inquisitors. Instead of making those changes to the painting, he just changed the title of the painting to avoid prosecution. He changed the name to the Feast in the House of Levi . Context is everything, so just changing the name changed the meaning of the painting and moved it out of the reach of the church’s tribunal.
The Last Supper , San Giorgio Maggiore, Venice Jacopo Tintoretto 1594
Tintoretto covers the subject of the Last Supper in vibrant and dynamic fashion. Instead of the stable and refined one-point perspective painting by da Vinci in Milan, he presents a space with diagonal recession, bustling attendants, angels, pets, and the apostles seated beside Christ. Christ offers bread and his halo radiates, as he states, “this is my body, which will be given up for you.” It is a dramatic moment representing this important and significant part of Jesus’s journey here on earth.
Architecture in Venice and Northern Italy
San Giorgio Maggiore, Venice Andrea Palladio 1565
Alberti’s innovative design, including the Colossal Order at Sant’ Andrea Mantua is reflected in the façade of San Giorgio Maggiore, in Venice. This is a façade that resembles closely a Greek or Roman Temple with large Corinthian columns and a classical pediment atop the four large columns. The result is stately, balanced, and harkens back to classical antiquity.
https://collections.vam.ac.uk/item/O617617/elevation-of-the-church-of-drawing-unknown-circle-of/
Villa La Rotonda, Vicenza Andrea Palladio 1567-1570
Palladio gives us a complete vision of a classical Ionic building in the Villa Rotunda. True columns support the portico, and the doors and windows are symmetrically placed. Sculpture strategically placed on the roof edges mimic the designs of antiquity.


1.5: Mitosis and Meiosis I
In eukaryotic cells, the time and phases from the beginning of one cell division until the beginning of the next cell division is called the cell cycle (Figure 1) . The first phase of the cell cycle is interphase. Interphase is the time during which the cell performs its normal functions and prepares for cell division. Cells spend most of their time in this phase. During interphase, chromosomes are not visible because they are decondensed (present only as a tangled mass of thin threads of DNA with associated proteins, called chromatin). The nuclear membrane is present, and visible, as is the nucleolus .
Figure 1. The cell cycle.
Interphase includes two gap phases, G1 and G2, where the cell increases in size and synthesizes new organelles, enzymes, and other proteins that are needed for cell division. In between the two gap phases, the DNA replicates in preparation for cell division. This stage is called S phase . At the beginning of S phase, chromosomes are single and unreplicated. By the end of S phase, each chromosome has made an exact copy and consists of two sister chromatids . At this point in the cell cycle the sister chromatids are held together tightly at the centromere . While the two sister chromatids are physically joined together they are still considered one replicated chromosome (Figure 2). When the sister chromatids physically separate, later during the cell cycle, they are then considered to be individual chromosomes.
In animal cells, interphase is also when the centrosome (consisting of two centrioles ) is replicated. Spindle fibers form from and radiate outward from the centrosomes to attach to and move chromosomes during cell division.
Figure 2. Chromosomes and sister chromatids.
Interphase is followed by mitosis (in the somatic cells) or meiosis (in reproductive cells), which is when replicated chromosomes and cytoplasm separate, during the process of karyokinesis and cytokinesis respectively.
Chromosomes that are the same length, have the same centromere location and the same gene sequences and positions are called homologous chromosomes. Human somatic cells contain pairs of homologous chromosomes. Each homologous pair consists of one maternal chromosome and one paternal chromosome. Cells that contain two copies of each chromosome are called diploid (2n, where n is the number of different chromosomes in a single set). Human sex cells (eggs and sperm) contain only one copy of each chromosome. Cells with only one copy of each chromosome are haploid (n). When the haploid sperm (n) and egg (n) combine during fertilization this forms a diploid zygote (2n).
MITOSIS
Mitosis is nuclear division that results in two cells containing the same number of chromosomes as the parent cell. Most human cells (skin, muscle, bone, etc.) divide by mitosis. This process is necessary for the normal growth and development of a multicellular eukaryotic organism from a zygote (fertilized egg), as well as growth and the repair and replacement of cells and tissues. At the end of mitosis, two daughter cells are formed that are identical to the original (parent) cell. Mitosis is also a form of asexual reproduction in unicellular eukaryotes.
Mitosis is a complex and highly regulated process. It occurs in the following 4 separate phases: prophase, metaphase, anaphase, and telophase. Telophase is quickly followed by cytokinesis.
Prophase: Cells prepare for division by coiling and condensing their chromatin into chromosomes. By late prophase, individual chromosomes can be seen, each consisting of two sister chromatids joined at a centromere. Spindle fibers begin to form from the centrosomes, which have begun to migrate to opposite “poles” of the cell. The nucleoli and the nuclear membrane degrade. (Figure 3)
Metaphase: Spindle fibers (called kinetochore microtubules or kinetochore spindle fibers) that emanate from the centromeres attach to the kinetochore (a proteinaceous area) on the sister chromatids. The fibers pull and otherwise manipulate the chromosomes to align them on the plane that passes through the center of the cell (metaphase plate) (Figure 4). This “plate” is not an actual structure; it merely signifies the location of replicated chromosomes prior to their impending separation.
Other non-kinetochore spindle fibers or tubules (aka polar microtubules), emanating from the two centrosomes, elongate and eventually overlap with each other near the metaphase plate.
Figure 4. Spindle fibers attaching to kinetochores in metaphase.
Anaphase: The centromeres divide, with the help of separase enzymes, and separate the sister chromatids (Figure 5). This happens simultaneously in all the chromosomes. The kinetochore spindles shorten and pull each chromatid to which they are attached toward the pole (and centrosome) from which they originate. This equally distributes exactly half the chromosomal material to each side of the cell. In late anaphase, the non-kinetochore spindles begin to elongate, lengthening the cell.
Figure 5. Cell in anaphase.
Telophase: The non-kinetochore microtubules continue to elongate, further elongating the cell in preparation for cytokinesis (splitting of the cytoplasm). The chromosomes reach their respective poles. The kinetochores disappear. The two nuclear membranes (one in each half of the cell) begin to form around the chromosomes. Nucleoli reappear and the chromosomes in each soon-to-be new cell begin to decondense back into chromatin. Mitosis is complete at the end of this stage.
Cytokinesis (splitting of the cytoplasm):
In animal cells and all other eukaryotes without a cell wall, cytokinesis is achieved by means of a constricting “belt” of protein fibers that slide past each other near the equator of the cell. As this occurs, the diameter of the belt decreases, pinching the cell to form a cleavage furrow around the cell’s circumference. As constriction proceeds, the furrow deepens until it eventually slices its way into the center of the cell. At this point, the cell is divided into two.
Plant cell walls are far too rigid to be split apart by contracting proteins. Instead, these cells assemble membrane proteins (in vesicles that bud off the Golgi apparatus) in their interior at right angles to the spindle apparatus. This expanding membrane partition, called a cell plate, continues to grow outward until it reaches the interior surface of the plasma membrane and fuses with it. This divides the cell in two.
The phases of Mitosis
Exercise 1: Modeling the Phases of Mitosis
Materials:
- Several sheets of blank paper (continuous printer paper is ideal)
-
Chromosome modeling kits
- Commercially available pop bead kits (e.g Carolina Biological Supply Company, Item #171100)
- Homemade kits may consist of pipe cleaners or yarn or socks, etc. to represent chromosomes
-
The following procedure will be described using a homemade kit consisting of pipe cleaners to represent chromosomes. The pipe cleaner chromosome kit contains:
- 10 each – short red pipe cleaner sticks, short blue pipe cleaner sticks, long pipe cleaner red stick, long blue pipe cleaner sticks (Use as 2 homologous chromosome pairs)
- 5 each – short red plastic lacing cord, short blue plastic lacing cord, long red plastic lacing cord, long blue plastic lacing cord (Use as 2 homologous chromatin pairs)
- 20 white or grey beads (Use as centromeres)
- Several red and blue beads (Use as genes for meiosis crossing-over)
Procedure:
Using models is a great way to represent natural structures and processes that are too small, or too large, or too complex to observe directly. By building chromosomes from the pipe cleaners and manipulating them to model cell division (mitosis and meiosis) you will enhance your understanding of the nature of chromosomes and the cellular structures needed to perform cell division. The pipe cleaner and plastic cord strands are intended to represent two pairs of homologous chromosomes. One pair of homologous chromosomes is longer than the other. Half of each pair is red and represents maternal DNA (genetic material contributed by a female’s egg). The other half of each pair is blue and represents paternal DNA (genetic material contributed by a father’s sperm). Thus, for each pair of homologous chromosomes, one should be red and one should be blue. The thin plastic lacing cord represents chromatin when chromosomes are in an uncoiled, decondensed state. The thicker pipe cleaner chromosomes represent the condensed chromosomes as they prepare for DNA replication and cell division. Alert your instructor if the chromosomes in your bag differ from those below.
This diploid cell with 2 homologous pairs of chromosomes will be modeled as it moves through the following phases of mitosis:
- Interphase (uncondensed DNA) Before Synthesis of DNA (G1)
- Interphase (uncondensed DNA) After Synthesis of DNA (G2)
- Prophase
- Metaphase
- Anaphase
- Telophase
- Cytokinesis
- Use the lace cording chromosomes to model the G1 phase of interphase (before synthesis of the DNA). On the paper draw the cell membrane, nucleus, nucleolus, centrioles.
- Use the lace cording chromosomes to model the G2 phase of interphase (after each chromosome was replicated during S phase). Use white beads to represent centromeres. Thread sister chromatids through a white bead to represent the duplicated chromosomes attached at the centromere. Centrioles would move toward opposite poles of the nucleus. Be sure to draw the cell membrane, nucleus, nucleolus, and centrioles on the paper.
- Use the pipe cleaner chromosomes to model the prophase stage of mitosis. The chromosomes are condensed and distributed throughout the nucleus. On your paper be sure to note that the nuclear membrane begins to break down and spindle fibers begin to form and radiate toward the chromosomes. Spindle fibers attach to kinetochore proteins at the centromeres of the chromosomes.
- Use the pipe cleaner chromosomes to model metaphase . Line up the individual chromosomes on the equator (middle) of the cell. Sister chromatids remain attached at the centromere during metaphase.
- Model anaphase by removing the white beads (centromere) from the sister chromatids to separate and move them toward opposite poles of the cell. After separation at the centromere, the chromatids are now called chromosomes.
- Anaphase ends and telophase begins when chromosomes reach opposite poles of the cell. Nuclear division happens in telophase. Nuclear envelopes and nucleoli reappear. Condensed chromosomes begin to decondense and uncoil. The formation of separate nuclear envelopes divide the nuclei and marks the end of telophase.
- Model cytokinesis by drawing the formation of a cleavage furrow to divide the cytoplasm into two and form two separate cells.
How do the daughter cells you formed compare to the original parent cell? _______________________________________________________________________
Observe the phases of Mitosis in Plant Cells
Exercise 2: Observing the Phases of Mitosis in the Onion Root Tip
Materials:
- Prepared slide of the onion root tip
- Compound light microscope
Procedure:
- Examine a slide of a longitudinal section of an onion root tip. Adjust the slide to view the region just above the root cap, where there are likely to be dividing cells.
- Focus on the dividing cells using the 4x scanning objective lens, then switch to the 10x objective and then the 40x objective.
- Survey the slide to find a cell in each phase of mitosis. Draw a cell for each phase below.
Interphase
DNA is uncondensed and in the form of chromatin. Individual chromosomes are not visible. The nuclear membrane is intact. The nucleolus is visible.
Prophase
Chromatin begins to condense into distinguishable chromosomes. These “puffy” structures are seen throughout the nucleus. Nucleoli begin to disappear. In late prophase (often called prometaphase) the nuclear membrane is no longer visible.
Metaphase
The chromosomes line up in the middle of the cell. Spindle fibers attach to kinetochores at the centromere and extend to the poles of the cell.
Anaphase
Centromeres split, separating each former chromatid into two individual chromosomes. The chromosomes move toward opposite poles.
Telophase and Cytokinesis
Chromosomes reach the poles. The nuclear envelopes begin to reform. The formation of a cell plate forms between the two cells to carry out cytokinesis.
Observe the phases of Mitosis in Animal Cells
Exercise 3: Observing the Phases of Mitosis in the Whitefish Blastula
Materials:
- Prepared slide of whitefish blastula
- Compound light microscope
Procedure:
The blastula is an early embryonic stage where many of the cells are dividing at any one time.
- Focus on the dividing cells using the 4x scanning objective lens, then switch to the 10x objective and then the 40x objective.
- Survey the slide to find a cell in each phase of mitosis. Draw a cell for each phase below.
Interphase
The DNA is uncondensed and in the form of chromatin. Individual chromosomes are not visible. The nuclear membrane is intact. The nucleolus is visible.
Prophase
Chromatin begins to condense and chromosomes are distinguishable. These “puffy” structures are seen throughout the nucleus. The nucleoli begin to disappear. In late prophase (often called prometaphase) the nuclear membrane is no longer visible.
Metaphase
The chromosomes line up in the middle of the cell. Spindle fibers attach to kinetochores at the centromere and extend to the poles of the cell.
Anaphase
Centromeres split, separating each former chromatid into two individual chromosomes. The chromosomes move toward opposite poles.
Telophase and Cytokinesis
Chromosomes reach the poles. The nuclear envelopes begin to reform. A cleavage furrow forms between the two cells to carry out cytokinesis.
How Long Does a Cell Spend in Each Phase of the Cell Cycle?
Exercise 4: Determining Time Spent in Different Phases of the Cell Cycle (Optional)
Materials:
- Laptop
- Calculator
Procedure:
- Obtain a laptop.
- Open a web browser and go to the following site: http://www.biology.arizona.edu
- This site will provide an interactive test of your ability to identify the stages of mitosis. It will also allow you to calculate the duration of the stages identified in the laboratory exercise you just completed, but the website will give standard results for the entire class.
- Go to the Cell Biology section, and find the activity “ Online Onion Root Tips ”.
- Keep clicking on “Next” at the bottom of the page until you get to the screen: Determining Time Spent in Different Phases of the Cell Cycle
- Click on “Next” at the bottom of the page.
- Identify each stage shown to you by the program. When a picture of a cell pops up in a stage of mitosis, simply click on the phase in which the cell belongs. If you make a mistake, read the explanation for why you were mistaken before making a new selection.
- When you are finished, use the formula given below and record your results in the table.
- The duration of each stage of mitosis can be determined by using the following formula. Compute the length of time for each stage and place your calculations in the table below.
(Number of cells in a stage ÷ Total number of cells) x 1440 (min in a day) = minutes a cell spends
in each stage in one day
|
Number of cells in each stage |
Interphase ______ |
Prophase ______ |
Metaphase ______ |
Anaphase _______ |
Telophase _______ |
Total 36 |
|
Percent of cells (# of cells / Total) |
100% |
|||||
|
Time (in minutes) spent in Stage – use calculation above |
References
Belwood, Jacqueline; Rogers, Brandy; and Christian, Jason, Foundations of Biology Lab Manual (Georgia Highlands College). “Lab 10: Mitosis & Meiosis,” (2019). Biological Sciences Open Textbooks . 18. CC-BY
https://oer.galileo.usg.edu/biology-textbooks/18
MEIOSIS
Introduction to Meiosis (aka “Reduction Division”)
Meiosis is a special type of cell division in which the daughter cells produced have half the number of chromosomes (n) as their parent cell. This division occurs in the reproductive organs (gonads -- testes of males or ovaries of females) of species that reproduce sexually, and results in the formation of gametes (eggs or sperm) that contain half the number of chromosomes as the original cell. Sexual reproduction involves the joining of gametes (fertilization) to form a zygote, which then has two copies of each chromosome (2n). Meiosis is a critical process, as it increases genetic diversity within a species.
Cells that divide by meiosis prepare for cellular division (during interphase) much like every other cell. Meiosis progresses through the same phases as mitosis (prophase, anaphase, metaphase, telophase, and cytokinesis). However, unlike mitosis, meiosis involves two rounds of cellular division (meiosis I and meiosis II). Meiosis involves only one round of DNA replication where each chromosome replicates to form sister chromatids. Therefore, when meiosis is completed, each daughter cell contains only half the number (n) of chromosomes as the original cell.
The stages of meiosis:
Meiosis I
Prophase l
Metaphase l
Anaphase l
Telophase l
Cytokinesis l
Meiosis II
Prophase ll
Metaphase ll
Anaphase ll
Telophase ll
Cytokinesis ll
Prophase I: During prophase of meiosis I, the chromosomes join in homologous pairs. Homologous chromosomes (aka homologs) are the same length, and carry genetic information (genes) for the same traits, but not necessarily the same versions (alleles) of the gene.
For example, human chromosome #19 contains a gene for eye color. In one person, one allele might code for blue eyes and the other allele codes for green eyes. Since every human inherits two copies of chromosome 19 (one from the mother’s egg and one from the father’s sperm) a person could have 2 blue alleles, 2 green alleles, or one of each.
Paired homologous chromosomes are called tetrads and are said to be in synapsis. During synapsis, equivalent pieces of homologous chromatids are exchanged between the chromosomes. This is called crossing-over and can occur several times along the length of the chromosomes. As occurs in the mitotic division, prophase of meiosis I also involves the degradation of the nuclear membrane and formation of spindle fibers.
Metaphase I: Metaphase of meiosis I occurs when the joined homologous chromosome pairs are moved to the center of the cell by spindle fibers (Figure 6). The fibers arrange the pairs so that homologs are on opposite sides of the metaphase plate (aka equatorial plane).
Figure 6. Homologous pairs line up at the equatorial plate in Metaphase l.
Anaphase I follows, as homologs are pulled apart, toward opposite poles of the cell (Figure 7). [*Note: this is significantly different from the separation of sister chromatids that occurs during mitosis].
Figure 7. In Anaphase l mitotic spindles pull homologs to opposite poles of the cell.
Telophase I marks the end of meiosis I, as new nuclei form and cytokinesis separates the cytoplasm forming two daughter cells.
At the end of meiosis I, the two daughter cells have half the number of chromosomes as did their parent cell. Thus, the cells have been reduced from diploid (2n) to haploid (n) (Figure 8). [n refers to the number of chromosomes in a set that are characteristic for a species. Humans have one set (n) of 23 unique chromosomes (n = 23). A diploid human cell has 2 sets (2n) of 23 unique chromosomes (2n = 46).
Figure 8. Meiosis l results in two haploid cells.
Meiosis II follows meiosis I, which proceeds very much like mitosis.
During Prophase II , chromosomes containing two sister chromatids are lined up on the equator of each daughter cell by the spindle fibers. This is completed by the end of Metaphase II (Figure 9).
Figure 9. Metaphase ll of Meiosis
The centromeres separate and sister chromatids are pulled to each pole of the cell during Anaphase ll (Figure 10) . Cytokinesis II occurs after Telophase II to complete cell division and ultimately the production of four (4) daughter cells (Figure 11). The cells produced (egg or sperm, in humans) are haploid (n rather than 2n) and will either unite (via fertilization) or die. They do not divide further on their own as meiosis is not a cycle.
Figure 10. Anaphase ll of Meiosis
Figure 11. Meiosis results in four haploid cells.
The Phases of Meiosis
Exercise 1: Modeling the Phases of Meiosis
Materials:
- Several sheets of blank paper (continuous printer paper is ideal)
-
Chromosome modeling kits
- Commercially available pop bead kits (e.g Carolina Biological Supply Company, Item #171100)
- Homemade kits may consist of pipe cleaners or yarn or socks, etc. to represent chromosomes
-
The following procedure will be described using a homemade kit consisting of pipe cleaners to represent chromosomes. The pipe cleaner chromosome kit contains:
- 10 each – short red pipe cleaner sticks, short blue pipe cleaner sticks, long pipe cleaner red stick, long blue pipe cleaner sticks (Use as 2 homologous chromosome pairs)
- 5 each – short red plastic lacing cord, short blue plastic lacing cord, long red plastic lacing cord, long blue plastic lacing cord (Use as 2 homologous chromatin pairs)
- 20 white or grey beads (Use as centromeres)
- Several red and blue beads (Use as genes for meiosis crossing-over)
Procedure:
A diploid cell with 2 homologous pairs of chromosomes (as in the previous modeling exercise) will be modeled as it moves through the meiosis. First, you will model meiosis l. Then, you will model meiosis ll as described below.
Model Meiosis l (1 diploid cell → 2 haploid cells)
-
tetrads form, crossing over occurs, homologues separate
- Interphase Before Synthesis of DNA (G1)
- Interphase After Synthesis of DNA (G2)
- Prophase l
- Metaphase l
- Anaphase l
- Telophase l
- Cytokinesis l
Model Meiosis ll (2 haploid cells → 4 haploid cells)
- sister chromatids separate
- Prophase ll
- Metaphase ll
- Anaphase ll
- Telophase ll
- Cytokinesis ll
Modeling Meiosis l
- Use the lace cording chromosomes to model the G1 phase of interphase (before synthesis of the DNA). On the paper draw the cell membrane, nucleus, nucleolus, centrioles.
- Use the lace cording chromosomes to model the G2 phase of interphase (after each chromosome was replicated during S phase). Use white beads to represent centromeres. Thread sister chromatids through a white bead to represent the duplicated chromosomes attached at the centromere. Centrioles would move toward opposite poles of the nucleus. Be sure to draw the cell membrane, nucleus, nucleolus and centrioles on the paper.
- Use the pipe cleaner chromosomes to model prophase l of meiosis. Prophase l of meiosis is notably different than prophase of mitosis. Arrange condensed chromosomes so that homologous chromosomes are paired. Put homologous chromosomes next to each other on your lab bench to simulate this process of synapsis (homologs pairing). Homologous pairs of chromosomes paired in this way are called tetrads .
When tetrads form, the inner non-sister chromatids of the tetrad pair can exchange DNA by a process known as crossing over . Pieces of equivalent segments of non-sister chromatids can be exchanged from one chromatid to the other. Crossing over can occur several times along the length of the chromosomes. Use red and blue beads to represent exchanged segments of chromatids on the inner non-sister chromatids of the tetrad pairs. Place a blue bead on an inner red (maternal) chromatid to represent DNA exchanged from the paternal chromatid. Place a red bead on an inner blue (paternal) chromatid to represent DNA from the maternal chromatid. Make a minimum of 1 crossover for each pair of homologous chromosomes.
- Use the pipe cleaner chromosomes to model metaphase l . Metaphase l of meiosis is also notably different than metaphase of mitosis. Spindle fibers move homologous pairs to line up along the equatorial plane of the cell. Line up the homologous pairs of chromosomes so that homologues are paired together but the maternal and paternal chromosomes are on opposite sides of the equatorial plate along the middle of the cell. Every pair of chromosomes is arranged independent of another. The side of the equatorial plate where each chromosome is arranged is completely random and independent of the side of the equatorial plate on which other chromosomes are located. Therefore, there are several different arrangements that can occur (Figure 12).
Figure 12. Independent Assortment in a cell with 2 homologous pairs.
- In anaphase l of meiosis, homologous chromosomes separate toward opposite poles of the cell. For each homologous pair, move the duplicated maternal chromosome and duplicated paternal chromosome to opposite poles of the cell.
- Anaphase ends and telophase l begins when chromosomes reach opposite poles of the cell. Arrange the chromosomes in groups at opposite ends of the cell. Nuclear division happens in telophase. The formation of separate nuclear envelopes divide the nuclei and mark the end of telophase.
- Model cytokinesis l by drawing the formation of a cleavage furrow to divide the cytoplasm into two and form two separate cells. These two cells will enter meiosis ll. Note that each daughter cell has half the number of chromosomes as the parental cell. Thus, the cells have been reduced from diploid (2n) to haploid (n) . [n refers to the number of pairs of chromosomes that are characteristic for a species . Humans have a “n” of 23, so a diploid human cell has 2(23), or 46 chromosomes].
Meiosis ll
- There is no DNA replication before the second cell division stage of meiosis. The stages of meiosis ll proceed very much like mitosis. The two cells created in meiosis l will enter into prophase ll . The chromosomes in each cell contain two sister chromatids, which are condensed and distributed throughout the nucleus. On your paper be sure to note that the nuclear membrane begins to break down and spindle fibers begin to form and radiate toward the chromosomes. Spindle fibers attach to kinetochore proteins at the centromeres of the chromosomes.
- Use the pipe cleaner chromosomes to model metaphase ll . Line up the individual chromosomes on the equator (middle) of the cell. Sister chromatids remain attached at the centromere during metaphase ll.
- Model anaphase ll by removing the white beads (centromere) from the sister chromatids to separate and move them toward opposite poles of the cell. After separation at the centromere, the chromatids are now called chromosomes.
- Anaphase ll ends and telophase ll begins when chromosomes reach opposite poles of the cell. Nuclear division happens in telophase. Nuclear envelopes and nucleoli reappear. Condensed chromosomes begin to decondense and uncoil. The formation of separate nuclear envelopes divide the nuclei and marks the end of telophase.
- Model cytokinesis ll by drawing the formation of a cleavage furrow to divide the cytoplasm of each cell into two. A total of four cells exist at the end of cytokinesis ll.
How many chromosomes are in the original parental cell? ___________________
How many chromosomes are in each daughter cell? _______________________
Are the chromosomes in daughter cells identical to the chromosomes in the original parental cell? ________________________________________________
Questions for Review
- What is the meaning of diploid? What abbreviation do we use to represent diploid? Name 2 diploid cells in humans.
- What is the meaning of haploid? What abbreviation do we use to represent haploid? Name 2 haploid cells in humans.
- In what stage of the cell cycle does S phase occur? Explain why the DNA must be duplicated during the S phase of the cell cycle, prior to mitosis taking place.
- What specific feature of cytokinesis in animal cells can you use to distinguish this process from cytokinesis in plant cells?
Practical Challenge Questions
- In the circle below, sketch a 2n=6 diploid cell in metaphase of mitosis. Be sure to label the centromere, centrioles, and spindle fibers.
- In the circle below, sketch a 2n=6 haploid cell in metaphase l of meiosis.
- A monogenic gene gives rise to a trait from a single set of alleles. A polygenic gene gives rise to a trait from several sets of alleles. Give an example of a monogenic and polygenic trait.
References
Belwood, Jacqueline; Rogers, Brandy; and Christian, Jason, Foundations of Biology Lab Manual (Georgia Highlands College). “Lab 10: Mitosis & Meiosis,” (2019). Biological Sciences Open Textbooks . 18. CC-BY
https://oer.galileo.usg.edu/biology-textbooks/18


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37.9: Regulation of Body Processes - Hormonal Control of Blood Calcium Levels
- Explain how blood calcium levels are regulated by parathyroid hormone
Regulation of blood calcium concentrations is important for generation of muscle contractions and nerve impulses, which are electrically stimulated. If calcium levels get too high, membrane permeability to sodium decreases and membranes become less responsive. If calcium levels get too low, membrane permeability to sodium increases and convulsions or muscle spasms may result.
Blood calcium levels are regulated by parathyroid hormone (PTH), which is produced by the parathyroid glands. PTH is released in response to low blood calcium levels. It increases calcium levels by targeting the skeleton, the kidneys, and the intestine. In the skeleton, PTH stimulates osteoclasts, which are cells that cause bone to be reabsorbed, releasing calcium from bone into the blood. PTH also inhibits osteoblasts, cells which deposit bone, reducing calcium deposition in bone. In the intestines, PTH increases dietary calcium absorption and in the kidneys, PTH stimulates re-absorption of the calcium. While PTH acts directly on the kidneys to increase calcium re-absorption, its effects on the intestine are indirect. PTH triggers the formation of calcitriol, an active form of vitamin D, which acts on the intestines to increase absorption of dietary calcium. PTH release is inhibited by rising blood calcium levels.
Hyperparathyroidism results from an overproduction of PTH, which leads to excessive amounts of calcium being removed from bones and introduced into blood circulation. This may produce structural weakness of the bones, which can lead to deformation and fractures, plus nervous system impairment due to high blood calcium levels. Hypoparathyroidism, the underproduction of PTH, results in extremely low levels of blood calcium, which causes impaired muscle function and may result in tetany (severe sustained muscle contraction).
The hormone calcitonin, which is produced by the parafollicular (or C) cells of the thyroid, has the opposite effect on blood calcium levels as PTH. Calcitonin decreases blood calcium levels by inhibiting osteoclasts, stimulating osteoblasts, and stimulating calcium excretion by the kidneys. This results in calcium being added to the bones to promote structural integrity. Calcitonin is most important in children (when it stimulates bone growth), during pregnancy (when it reduces maternal bone loss), and during prolonged starvation (because it reduces bone mass loss). In healthy, nonpregnant, unstarved adults, the role of calcitonin is unclear.
Key Points
- The parathyroid hormone (PTH), secreted by the parathyroid glands, is responsible for regulating blood calcium levels; it is released whenever blood calcium levels are low.
- PTH increases blood calcium levels by stimulating osteoclasts, which break down bone to release calcium into the blood stream.
- PTH increases blood calcium levels by increasing the amount of calcium resorbed by the kidneys before it can be excreted in the urine.
- PTH increases blood calcium levels by triggering the formation of calcitriol, which increases absorption of dietary calcium through the intestines.
- Calcitonin, a hormone produced by the thyroid, acts in opposition to PTH by inhibiting osteoclasts, stimulating osteoblasts, and increasing excretion of calcium into the urine by the kidneys.
Key Terms
- osteoblast : a mononucleate cell from which bone develops
- parathyroid hormone : a polypeptide hormone that is released by the chief cells of the parathyroid glands and is involved in raising the levels of calcium ions in the blood
- calcitonin : a hormone, secreted by parenchymal cells, that regulates calcium and phosphate metabolism
- hypoparathyroidism : deficiency of parathyroid hormone
- hyperparathyroidism : an abnormal increase in parathyroid gland activity
- calcitriol : the active metabolite 1,25-dihydroxycholecalciferol of vitamin D3 that is involved in the absorption of calcium
- osteoclast : a large multinuclear cell associated with the resorption of bone


10.4: Summary
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Key Takeaways
The momentum vector, \(\vec p\) , of a point particle of mass, \(m\) , with velocity, \(\vec v\) , is defined as:
\[\begin{aligned} \vec p = m\vec v\end{aligned}\]
We can write Newton’s Second Law for a point particle in term of its momentum:
\[\begin{aligned} \frac{d}{dt}\vec p = \sum \vec F = \vec F^{net}\end{aligned}\]
where the net force on the particle determines the rate of change of its momentum. In particular, if there is no net force on a particle, its momentum will not change.
The net impulse vector, \(\vec J^{net}\) , is defined as the net force exerted on a particle integrated from a time \(t_A\) to a time \(t_B\) :
\[\begin{aligned} \vec J^{net} = \int_{t_A}^{tB} \vec F^{net} dt\end{aligned}\]
The net impulse vector is also equal to the change in momentum of the particle in that same period of time:
\[\begin{aligned} \vec J^{net} = \Delta \vec p = \vec p_B - \vec p_A\end{aligned}\]
When we define a system of particles, we can distinguish between internal and external forces. Internal forces are those forces exerted by the particles in the system on each other. External forces are those forces on the particles in the system that are not exerted by the particles on each other. The sum over all of the forces on all of the particles in the system will be equal to the sum over the external forces, because the sum over all internal forces on a system is always zero (Newton’s Third Law).
The total momentum of a system, \(\vec P\) , is the sum of the momenta, \(\vec p_i\) , of all of the particles in the system:
\[\begin{aligned} \vec P = \sum \vec p_i\end{aligned}\]
The rate of change of the momentum of a system is equal to the sum of the external forces exerted on the system:
\[\begin{aligned} \frac{d}{dt}\vec P = \sum \vec F^{ext}\end{aligned}\]
which can be thought of as an equivalent description as Newton’s Second Law, but for the system as a whole. If the net (external) force on a system is zero, then the total momentum of the system is conserved.
Collisions are those events when the particles in a system interact (e.g. by colliding) and change their momenta. When modeling collisions, it is usually beneficial to first define a system for which the total momentum is conserved before and after the collision.
Collisions can be elastic or inelastic. Elastic collisions are those where, in addition to the total momentum, the total mechanical energy of the system is conserved. The total mechanical energy can usually be taken as the sum of the kinetic energies of the particles in the system.
Inelastic collisions are those in which the total mechanical energy of the system is not conserved. One can usually identify if mechanical energy was introduced or removed from the system and determine if the collision is elastic. It is important to identify when momentum and mechanical energy are conserved. Momentum is conserved if no net force is exerted on the system, whereas mechanical energy is conserved if no net work was done on the system by non-conservative forces (internal or external) or by external conservative forces.
We can always choose in which frame of reference to model a collision. In some cases, it is convenient to use the frame of reference of the center of mass of the system, because in that frame of reference, the total momentum of the system is zero.
If a system has a total mass \(M\) , then one can use Newton’s Second Law to describe its motion:
\[\begin{aligned} \sum \vec F^{ext} &= M \vec a_{CM}\\[4pt] \sum \vec F^{ext} &=\frac{d}{dt} \vec P\end{aligned}\]
where the sum of the forces is over all of the external forces on the system. The acceleration vector, \(\vec a_{CM}\) , describes the motion of the “center of mass” of the system. \(\vec P=M\vec v_{CM}\) is the total momentum of the system.
The center of mass of a system is a mass-weighted average of the positions of all of the particles of mass \(m_i\) and position \(\vec r_i\) that comprise the system:
\[\begin{aligned} \vec r_{CM} &=\frac{1}{M}\sum_i m_i\vec r_i\end{aligned}\]
The vector equation can be broken into components to find the \(x\) , \(y\) , and \(z\) component of the position of the center of mass. Similarly, one can also define the velocity of the center of mass of the system, in terms of the individual velocities, \(\vec v_i\) , of the particles in the system:
\[\begin{aligned} \vec v_{CM} &= \frac{1}{M}\sum_i m_i\vec v_i\end{aligned}\]
Finally, one can define the acceleration of the center of mass of the system, in terms of the individual accelerations, \(\vec a_i\) , of the particles in the system:
\[\begin{aligned} \vec a_{CM} &= \frac{1}{M}\sum_i m_i\vec a_i\end{aligned}\]
If the system is a continuous object, we can find its center of mass using a sum (integral) of infinitesimally small mass elements, \(dm\) , weighted by their position:
\[\begin{aligned} \vec r_{CM} &=\frac{1}{M}\int \vec r dm\\[4pt] \therefore x_{CM} &= \frac{1}{M}\int x dm\\[4pt] \therefore y_{CM} &= \frac{1}{M}\int y dm\\[4pt] \therefore z_{CM} &= \frac{1}{M}\int z dm\end{aligned}\]
The strategy to set up the integrals above is usually to express the mass element, \(dm\) , in terms of the position and density of the material of which the object is made. One can then integrate over position in the range defined by the dimensions of the object.
Important Equations
Momentum of a point particle:
\[\begin{aligned} \vec p = m\vec v \\[4pt] \frac{d}{dt}\vec p = \sum \vec F = \vec F^{net}\end{aligned}\]
Impulse:
\[\begin{aligned} \vec J^{net} = \int_{t_A}^{tB} \vec F^{net} dt \\[4pt] \vec J^{net} = \Delta \vec p = \vec p_B - \vec p_A\end{aligned}\]
Momentum of a system:
\[\begin{aligned} \vec P = \sum \vec p_i \\[4pt] \frac{d}{dt}\vec P = \sum \vec F^{ext}\end{aligned}\]
Newton's Second Law for a system:
\[\begin{aligned} \sum \vec F^{ext} &= M \vec a_{CM}\\[4pt] \sum \vec F^{ext} &=\frac{d}{dt} \vec P\end{aligned}\]
Position of the Center of Mass of a system:
\[\begin{aligned} \vec r_{CM} &=\frac{1}{M}\sum_i m_i\vec r_i \end{aligned}\]
Velocity of the Center of Mass of a system:
\[\begin{aligned} \vec v_{CM} &= \frac{1}{M}\sum_i m_i\vec v_i \\[4pt]\end{aligned}\]
Acceleration of the Center of Mass of a system:
\[\begin{aligned} \vec a_{CM} &= \frac{1}{M}\sum_i m_i\vec a_i \\[4pt]\end{aligned}\]
Position of the Center of Mass for a continuous object:
\[\begin{aligned} \vec r_{CM} &=\frac{1}{M}\int \vec r dm\\[4pt] \therefore x_{CM} &= \frac{1}{M}\int x dm\\[4pt] \therefore y_{CM} &= \frac{1}{M}\int y dm\\[4pt] \therefore z_{CM} &= \frac{1}{M}\int z dm\end{aligned}\]
Important Definitions
Definition
Momentum: The product of velocity and mass. SI units: \([\text{kgms}^{-1}]\). Common variable(s): \(\vec p\) .
Definition
Impulse: A property of matter which describes an object’s resistance to rotational motion. SI units: \([\text{Ns}]\). Common variable(s): \(\vec J\) .