2.4: #metoo and Twitter- The Feminist Movement on Social Media
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A Sociolinguist's Perspective
Claire-Anne Ferrière
On October 15, 2017 the actress Alyssa Milano ignited a spark with a Tweet.
With this tweet, Miliano was urging those in her publics who had ever been victims of sexual assault and/or harassment to identify as such by saying, ‘me too.’ She did not launch the hashtag as such, but it appeared very quickly – in the following minutes.
Alyssa Milano and her friend were not the only women who came up with the idea of ‘me too’ to help people express ugly circumstances and violation. Tarana Burke, a civil rights activist, had launched an offline “me too” movement back in 2006 to “give young women, particularly young women of color from low wealth communities, a sense of empowerment from the understanding that they are not alone in their circumstances” (Burke, 2013). In 2007, activist Tarana Burke began a movement for women of color which was echoed a decade later by Milano’s tweet and the responses. Milano’s access to expansive networked publics gave a similar sentiment to Burke’s the visibility and affordances of an online movement.
Burke’s and Milano’s uses of the phrase “me too” have come to be considered as two benchmarks in one movement due to the connected nature of events they describe. Through the hashtag #metoo survivors’ accounts of being harassed and violated are aggregated , or digitally pulled or presented together as related.  By aggregating survivors’ memories of these events, the Me Too movement has led to a cultural shift that centralizes survivor experiences, and shifts the blame to those who caused their pain.
The rapid spread of the Me Too movement is directly linked to the Harvey Weinstein case which started on October 5th, 2017, when The New York Times published an article entitled “Harvey Weinstein paid off sexual harassment accusers for years”, written by Megan Twohey and Jodi Kantor (Kantor & Twohey, 2017). Dozens of women have come out denouncing sexual abuse and harassment from the movie mogul, and demanding that the situation in Hollywood change. #Metoo occurs after that case, opening the demands and claims out of Hollywood. Several other events also followed from #metoo, such as the second Women’s March in January 2018 and the opposition to the nomination of Brett Kavanaugh to the Supreme Court of the United States over the summer of 2018, to mention only a few. All those events are therefore linked together and organized around common claims, which is why we can consider that all those events constitute a movement, and #metoo plays an essential role in the construction of this social movement. However, it does not play exactly the same role as social media did in the movements of performative activism, tackled in the previous chapter.
In order to get a full grasp of the “me too” movement on social media and how it is used, sociolinguistics comes in handy. Sociolinguistics studies the relationship between language on the one hand (-linguistics – the study of language) and society on the other hand (socio-), more particularly social relations; in other words, it studies how human beings use language and to what purpose. It is therefore an interesting approach to study how social media are used, and how they shape social relations. In this feminist movement, women spoke up, through language, denouncing systemic abuse and demanding that changes be made in society and in social relations between men and women. The movement was built progressively; interestingly, social media was not used the same way throughout the movement, and sociolinguistics helps us understand why it is so.
1. The Harvey Weinstein Case
In order to understand the relevance of the #metoo movement within the larger feminist movement as well as the relevance of the use of social media, we need to go back to the Weinstein case and its characteristics.
Indeed, if #metoo really turned to social media, the Weinstein case developed primarily through traditional media – i.e. in those forms of mass media that existed before the advent of the digital age (also called the new media age, as a matter of fact), for example, print media, radio broadcasting, and the television among other things. However, it is still important to keep in mind that, even if the denunciations against Harvey Weinstein were mainly done through traditional media (72 out of the 90 denunciations that were made public), all of them can be found on the internet, facilitating their diffusion. Nevertheless, it is interesting to note that those women should first choose traditional media to make their denunciations and claims public. On the contrary, very few were made through social media. One element that is essential to take into account is that the two newspapers that extensively covered the case, with in-depth investigations and analyses, were The New York Times (“Harvey Weinstein Paid Off Sexual Harassment Accusers for Decades”, by Megan Twohey and Judi Kantor) and the The New Yorker (“From Aggressive Overtures to Sexual Assault: Harvey Weinstein’s Accusers Tell their Stories,” by Ronan Farrow (Farrow, 2017)), two renowned and reliable newspapers. As a matter of fact, these two newspapers were awarded the Pulitzer prize for public service in 2018 for the reporting done by the aforementioned three journalists.
The stories these women were reporting were disruptive – Harvey Weinstein was a very prominent man in the business and even outside of it – and as such, they needed to establish their legitimacy in order to be believed. One way they had of doing so was actually by having their stories backed up by established, reliable newspapers. Not all newspapers through which the victims testified are as renown as The New Yorker or The New York Times – some denunciations were published in Variety or Deadline, for example – but the important element here is that it feels like the victims are never alone. Either the stories are reported by journalists, who lend their legitimacy to the victims, or if the victims write in their own name in opinion sections, they still have the legitimacy and weight of the newspapers backing them up . The same goes for the TV or radio shows.
The theory of denunciation put forward by Luc Boltanski et al. (Boltanski et al., 1984) supports that idea. In a nutshell, they argue that the higher the profile of the alleged aggressor and the bigger the difference between their status in society and that of the accuser is, the less “normal” the denunciation seems to be, “normal” in the sense of plausible, even possible. To put it bluntly, the more an alleged aggressor has to lose – status, some important position, wealth, etc. – the more suspicious people are because they might believe that the denunciation is not completely disinterested. In our case, even if some of the women who came out first against Harvey Weinstein were somewhat high-profile people too, Ashley Judd for example, some others were “mere” employees or former employees of one of Weinstein’s companies, and in any case, none of the accusers were actually as high-profile as Weinstein was. Having the support of newspapers such as The New York Times or The New Yorker is therefore extremely important to restore some balance between accuser and alleged aggressor.
This need to legitimize their stories is also apparent in the language they use, more particularly in how they speak about what they went through. Here is an example of a testimony delivered by one of Harvey Weinstein’s alleged victims, Louisette Geiss.
(from 5’29 to 10’47)
One or more interactive elements has been excluded from this version of the text. You can view them online here: https://opentextbooks.library.arizona.edu/hrsmwinter2022/?p=132#oembed-1
In linguistics, we often start from our own impressions, how we react to a text, and then we try and find some formal elements to back this up. We can think in terms of positive impressions, as in I felt/think that, or negative impressions, I didn’t expect that/I would have thought it would be more like that.
One of the first things that is striking in the denunciations against Harvey Weinstein is the very “stick to the facts” style. Indeed, these denunciations are very descriptive, detailed, and sometimes very graphic in what they describe. They are presented as if everything that was done and said during the assault is being reported. Moreover, the denunciations are symptomatic of what we call a paratactic style , also called the additive style: elements are presented, one simple sentence after the other, and there are few elaborate sentences, as follows:
After about 30 minutes, he asked to excuse himself and go to the bathroom.  He told me to keep talking about my film and that he was going to hop into his hot tub that was adjacent to the room, just steps away. When I finished my pitch, I was obviously nervous, and he just kept asking to watch him masturbate. I told him I was leaving. I quickly got out of the tub, he grabbed my forearm as I was trying to grab my purse, and he led me to his bathroom, pleading that I just watch him masturbate.
This gives, once again, the impression that they are just providing the plain truth and facts as they happened. On the contrary, it seems that emotions are not so present, as one could have expected of a woman recounting and reliving the assault she suffered. These denunciations are therefore more fact-oriented than emotion-oriented. This also participates in legitimising and having their denunciations accepted and believed.
At that stage – or at the very least in the first ten days of the contestation – the movement was precisely not a movement: it was only centered on one particular man, Harvey Weinstein, and when claims were made; they were also quite specific, around the issues of the relationship between men and women in Hollywood. It was therefore limited to the world of Hollywood, which is also fitting with Boltanski’s theory of denunciation that was previously mentioned: it is only logical that such a disruptive movement in society should start with women who are renown and have some status compared to “ordinary” people. The movement and its claims needed to be legitimized by well-established people before being broadened to society at large with #metoo.
What we now consider to be the next stage of the movement, #metoo, is quite different from the Weinstein case, on different accounts.
2. #metoo
Tarana Burke gave a Ted talk in November 2018 called “Me too is a movement, not a moment,” (Burke, 2018) and indeed, this stage of #me too was essential in the development of the Weinstein case as a moment into a fully-fledged feminist movement.
First, what we witness quite evidently in the #metoo movement compared to the Weinstein case is an opening up of the denunciations and demands. Indeed, #metoo is not limited to Hollywood anymore, nor is it concerned with the deeds of one particular man.
With #metoo, tens of thousands of people are saying that they too, were victims of similar kinds of assault. This was made possible through the use of social media, in particular Twitter, as a new and massive medium of protest. Twitter is accessible to anyone with an account: in the United States, in the third quarter of 2017, there were 69 million monthly active users according to the numbers released by Twitter (Clement, 2019). This represents a great change compared to the 90 women or so who reported having been assaulted by Harvey Weinstein. Moreover, more than numbers, the status of the people denouncing abuse changed. They are now “ordinary” people, like you and me, not Hollywood stars: the movement is not limited to Hollywood anymore, but broadened to society in general.
This is confirmed by the linguistic analysis of “me too” tweets. More than fifty thousand such tweets were posted in the first twenty-four hours following Alyssa Milano’s first tweet; for purpose of ease, we will only focus on the first hour of “me too”, which represents 3,847 tweets. When analyzed with specific software tools that scan the corpus (here, the set of tweets) to identify some of its linguistic characteristics, those tweets show that the most significant themes and concepts that come up are all linked to sexual assault and harassment that women suffer. Indeed, some of the most recurrent words in those tweets are “me too” of course, but also “sexually,” “harassed,” “assaulted,” and “women.” The results therefore show a direct thematic link with the Harvey Weinstein case, and outline the scope of the movement.
What these results also show is a tendency towards generalization. Indeed, there are very few details about the types and circumstances of the assault that people tweeting “me to” are reporting; contrary to the Weinstein case in which denunciations were very specific, with the “me too” movement, specificities are somewhat smoothed and all different types of assaults are gathered under the umbrella expression “sexually harassed or assaulted”, which is used 273 times in the first hour. On the contrary, any more specific kinds of assault, such as reference to rape, being groped, etc. only occur sixty times in the corpus (out of the 3,847 tweets). This tendency not to go into details with the “me too” movement as opposed to what happened in the Weinstein case can of course be imputed to the format of the tweets itself. Indeed, tweets are limited to 240 characters, which prevents victims from giving too much detail of their individual experiences of assault. It is nonetheless significant that the movement should have moved from traditional media to social media, and not any media, to Twitter in particular, with the character limit. It shows that Alyssa Milano’s intention in launching this movement on Twitter was not to provide people with a space to explain what they went through, but rather to identify as part of a community.
This community is built on what the members feel they have in common, in this case, the fact that they all suffered experiences of sexual abuse or assault. The specificities of the assault do not matter at this stage anymore. This is also quite evident in the use that is being made of the expression “me too”. What is interesting with this expression is how it is used. Indeed, the software tools can identify what we call collocations , that is to say words that often occur together in the corpus. In our case, some of the most significant collocations are “me too”, which was expected, but also “too me,” “me me,” and “too too,” which is striking. This shows that not only is the expression “me too” often used, it is often used on its own, with no details provided, creating series of tweets only composed of the expression “me too,” which the software tools then identify as collocations. Here is an excerpt taken from the corpus:
This tendency towards generalization is also evident in the reference to women that is made in the corpus. The term women is referenced 471 times in the corpus, and only 65 times in the singular form woman. We can therefore conclude that in this corpus, women are presented as a group, not as individuals, which confirms this generalization tendency: what matters is not the individual experiences of every woman who identifies as a victim, but the commonalities between all the different experiences of these women (which is actually one of the reproaches that was made to the movement).
What comes out of the analysis of those first tweets of the “me too” movement is therefore the generalization process that is put into place. The different experiences and victims are gathered together into one community, a community that is built through this network of tweets. There are several ways through which this network is constructed, and which are already evident in the first hour of the movement.
First, it is important to keep in mind that the online movement began from one tweet, posted by Alyssa Milano, encouraging people to write “me too” as a reply if they identified as victims of sexual assault too. This process of reply is essential in the building of the network: indeed, out of the 3,847 first tweets, 2,053 are direct replies to Alyssa Milano’s tweet. All the replies are therefore linked together through the original tweet posted by Alyssa Milano. Moreover, 227 tweets tag Alyssa Milano directly with the @ function of social media, and there are also 1,207 tweets using the hashtag #metoo, although it was not mentioned in Alyssa Milano’s tweet. All those means participate in building a network of tweets, with people identifying with a community of victims of sexual assault, on a general level.
The “me too” movement therefore stages a depersonalization and a generalization of the movement, from the particular Harvey Weinstein case to thousands of women saying that they too, suffered sexual assault. Specificities and individualities are left aside to focus only on commonalities. What is at stake with the “me too” movement is the move from the personal to the political, which is an essential step in transforming a moment into a movement, to use Tarana Burke’s expressions. This notion of the personal being political dates back to 1970: Carol Hanisch (Hanisch, 1970) explained in an essay bearing the same name that according to her, individual, personal experiences of oppression that women were experiencing were not isolated but on the contrary repeated, systemic at the level of society, and as such, required not individual solutions, but social, political ones. The important step is therefore to recognise the commonality of all individual experiences, to connect them:
So the reason I participate in these meetings is not to solve any personal problem. One of the first things we discover in these groups is that personal problems are political problems. There are no personal solutions at this time. There is only collective action for a collective solution. I went, and I continue to go to these meetings because I have gotten a political understanding which all my reading, all my “political discussions,” all my “political action,” all my four-odd years in the movement never gave me.  (Hanisch, 1970).
The meetings she mentions are consciousness-raising meetings at which women share their personal experiences in order for them to recognise patterns in other women’s experiences. This is what the “me too” movement is doing. It is connecting thousands of similar experiences and highlighting the systemic, social character of those experiences in order to move from the personal to the political. Thus, the depersonalisation and generalisation process that is at stake with the “me too” movement also entails a politicization of the movement, that is to say, recognising that this is a social problem that needs to be addressed at the level of society. It is therefore an essential stage in the construction of the movement, which transforms an individual and seemingly isolated case into a social and political movement.
Nevertheless, social media do not play the same part here as they did with other social movements, such as the Zapatistas, or the Arab Spring movement. In cases of performative activism, social media are an organising and communication tool supporting a movement offline. #Metoo strictly speaking has no reality, existence outside of social media. It led to other stages, such as the Women’s March of January 2018 and the opposition to the nomination of Brett Kavanaugh at the Supreme Court of the United States in the summer of 2018, which, as far as they are concerned can be considered as forms of performative activism. However, the movement #metoo itself only occurred on social media, in particular Twitter. In a sense, we can even say that it was its own performance: the sheer scale of the movement is in itself performance enough in the sense that it makes an impression and is a very clear statement for the rest of the movement.
3. Social media after #metoo
The use of social media in the feminist movement after #metoo is quite different and is on the contrary closer to creative online performative activism. Social media was for example used during the different Women Marches to organize the events, communicate, and motivate people. The different elements defining performative activism can be identified here too. For example, after Trump’s election, activists needed to organize with speed to mobilize people for the first Women’s March, which was held the day after Trump’s inauguration. If speed is less of the essence for the following Marches since organizers could coordinate the events ahead in time, speed was also important for the opposition to the nomination of Brett Kavanaugh. Social media enabled organizers to spread the word and mobilise people.
As far as visuals are concerned, the different events organized around the feminist movement also rely on symbolic visuals, such as the feminist symbol, pictures from previous marches, or, as in the case of the “National Walk Out” event organized as part of the protest against Brett Kavanaugh, pictures of women walking out of the homes or places of work as a sign of dissent. The visuals are both shared before events to mobilise people and during the performances. Hashtags are also used a lot in the different events, also creating a form of unifying visual element.
There are performances that are organised, such as marches of course, but also rallies against Brett Kavanaugh when, among other events, hundreds of protesters flooded a Senate office building, shouted slogans, and displayed banners to express their anger. Chants are also used in those events as another form of performance.
The latest feminist movement really emphasized its inclusiveness to all women – or more particularly to all people who identify as women – all races, religions, social backgrounds, and so on. Inclusiveness, or intersectionality as it is theorised in feminism (the theory according to which different forms of oppression intersect and must be taken into account – for example, a black woman suffers discrimination both from being a woman and a black person. It also aims to recognise that not all experiences of women are similar), is an essential element for the feminists involved in this movement.
Finally, organizers are not masked per se, on the contrary, they do not hesitate to publicly express their minds, unmasked, and in their own names. Nevertheless, the movement is also coordinated through different organizations, either feminist groups which existed before the movement or organizations created as part of the movement. Organizations such as the Women’s March communicate on social media under the name of the organization, through which activists therefore take a step back and therefore let the faceless organization speak.
We can therefore clearly see that the use of social media in the #metoo movement and the rest of the feminist movement is not the same. #Metoo is quite particular in this respect and developed solely online, contrary to the other events, for which social media is used as a communication and organising tool and which therefore corresponds to creative online performative activism.
Core Concepts and Questions
Core Concepts
aggregate
to pull or present content together online as related
sociolinguistics
the study of how human beings use language and to what purpose
theory of denunciation
the more a “called out” person has to lose – status, some important position, wealth, etc. – the more suspicious people are of those who call them out or denounced them, because they might believe that the denunciation is not completely disinterested
paratactic style
also called the additive style, this is a linguistic style in which elements are presented, one simple sentence after the other, and there are few elaborate sentences; often used by speakers wanting to be taken seriously
collocations
a collection of words that often occurs together
intersectionality
the theory according to which different forms of oppression intersect and can worsen oppression overall
Core Questions
A. Questions for qualitative thought
- Why does the author assert women used traditional media rather than social media to initially denounce Harvey Weinstein? What do you believe would have happened had they relied more on social media initially?
Bibliography:
- Allred, G. (2017, octobre 10). Gloria Allred Represents a New Accuser of Harvey Weinstein (Louisette Geiss). https://www.youtube.com/watch?v=O4x4vtuJKDs
- Boltanski, L., Darré, Y., & Schiltz, M.-A. (1984). La Dénonciation. Actes de la Recherche en Sciences Sociales, 51(1), 3‑40. https://doi.org/10.3406/arss.1984.2212
- Burke, T. (2013). The me too Movement. JustBeInc. https://justbeinc.wixsite.com/justbe...-movement-c7cf
- Burke, T. (2018, novembre 30). Me too is a movement, not a moment. https://www.ted.com/talks/tarana_bur...pt?language=fr
- Clement, J. (2019, août 9). Twitter Monthly Active Users in the United States 2019. Statista. https://www.statista.com/statistics/...united-states/
- Farrow, R. (2017, octobre 10). From Aggressive Overtures to Sexual Assault : Harvey Weinstein’s Accusers Tell Their Stories. The New Yorker. https://www.newyorker.com/news/news-...-their-stories
- Hanisch, C. (1970). The Personal is Political. In S. Firestone & A. Koedt (Éd.), Notes from the Second Year : Women’s Liberation.
- Kantor, J., & Twohey, M. (2017, octobre 5). Harvey Weinstein Paid Off Sexual Harassment Accusers for Decades. The New York Times. https://www.nytimes.com/2017/10/05/u...legations.html
Media Attributions
- 512px-Tarana_Burke_from_She’s_Revolutionary © Brittany "B.Monét" Fennell is licensed under a CC BY-SA (Attribution ShareAlike) license
- Screenshot_2020-08-12 #metoo and Twitter The Feminist Movement on Social Media


13.2: Union, Intersection, and Complement
Commonly sets interact.  At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.
The union of two sets contains all the elements contained in either set (or both sets).
The union is notated \(A \cup B\)
More formally, \(x \in A \cup B\) if \(x \in A\) or \(x \in B\) (or both)
The intersection of two sets contains only the elements that are in both sets.
The intersection is notated \(A \cap B\)
More formally, \(x \in A \cap B\) if \(x \in A\) and \(x \in B\)
The complement of a set A contains everything that is not in the set A .
The complement is notated \(A\), or \(A^{c}\), or sometimes \(\sim A\).
Consider the sets:
\(\quad A=\{\text { red, green, blue }\} \quad B=\{\text { red, yellow, orange }\} \quad C=\{\text { red, orange, yellow, green, blue, purple }\}\)
- Find \(A \cup B\)
- Find \(A \cap B\)
- Find \(A^{c} \cap C\)
Solution
a) The union contains all the elements in either set: \(A \cup B=\{\text { red, green, blue, yellow, orange }\}\)
Notice we only list red once.
b) The intersection contains all the elements in both sets: \(A \cap B=\{\text { red }\}\)
c) Here we're looking for all the elements that are not in set \(A\) and are also in \(C\).
\(A^{c} \cap C=\{\text { orange, yellow, purple }\}\)
Using the sets from the previous example, find \(A \cup C\) and \(B^{c} \cap A\)
- Answer
-
\(A \cup C=\{\text { red, orange, yellow, green, blue purple }\}\)
\(B^{c} \cap A=\{\text { green, blue }\}\)
Notice that in the example above, it would be hard to just ask for \(A^{c},\) since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.
A universal set is a set that contains all the elements we are interested in. This would have to be defined by the context.
A complement is relative to the universal set, so \(A^{C}\) contains all the elements in the universal set that are not in \(A\).
- If we were discussing searching for books, the universal set might be all the books in the library.
- If we were grouping your Facebook friends, the universal set would be all your Facebook friends.
- If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers
Suppose the universal set is \(U=\) all whole numbers from \(1\) to \(9 .\) If \(A=\{1,2,4\}\), then
\(A^{c}=\{3,5,6,7,8,9\}\)
As we saw earlier with the expression \(A^{c} \cap C,\) set operations can be grouped together. Grouping symbols can be used like they are with arithmetic - to force an order of operations.
Suppose
\(H=\{\text { cat, dog, rabbit, mouse }\}, F=\{\text { dog, cow, duck, pig, rabbit }\} \quad W=\{\text { duck, rabbit, deer, frog, mouse }\}\)
- Find \((H \cap P) \cup W\)
- Find \(H \cap(F \cup W)\)
- Find \((H \cap P) \cap W\)
Solution
a) We start with the intersection: \(H \cap F=\{\text { dog, rabbit }\}\)
Now we union that result with \(W:(H \cap F) \cup W=\{\text{dog, duck, rabbit, deer, frog, mouse }\}\)
b) We start with the union: \(F \cup W=\{\text{dog, cow, rabbit, duck, pig, deer, frog, mouse }\}\)
Now we intersect that result with \(H: H \cap(F \cup W)=\{\text { dog, rabbit, mouse }\}\)
c) We start with the intersection: \(H \cap F=\{\mathrm{dog}, \text { rabbit }\}\)
Now we want to find the elements of \(W\) that are not in \(\mathrm{H} \cap F\)
\((H \cap P)^{c} \cap W=\{\text { duck, deer, frog, mouse }\}\)


3.27: Introduction to Linear Regression
What you’ll learn to do: For a linear relationship, use the least squares regression line to model the pattern in the data and to make predictions.
In this section, we present steps for finding the simple linear regression formula given a set of data. This formula is derived to find the line that has the smallest total squared error from the line to the observed data. In addition, we interpret the constants in a real-world context and explore the ways in which we can use the linear regression model to form predictions or good “guesses” for new values.
Contributors and Attributions
CC licensed content, Shared previously
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


4.21: Increasing Sales with Extended Problem Solving
- Describe how a retailer can increase sales from customers engaged in extended problem solving
Consumers with an extended problem solving mindset put a great deal of effort into their purchase decision, gathering information through research and taking care to evaluate all options, before arriving at a decision. Because of the time and energy committed to the search, this diligence is more likely dedicated to the selection and purchase of high-consideration or high-value items like cars, electronics and appliances. Or, it may be focused on something that is new or infrequently purchased. Thus, the consumer feels compelled to do more research to ensure their needs will be satisfied.
While it may be tempting to assume that these shoppers are mostly concerned with quantitative assessment of the alternatives, motivations can also be qualitative, building on external influences like cultural norms and family influences. Yet, it should be noted that these customers are deliberate in their process and are unlikely to be swayed directly by advertising, merchandising and promotion. As such, salespeople can be important in helping the consumer arrive at a decision.
For these shoppers, a salesperson will need to be able to engage the consumer to understand what their specific needs and concerns are, relative to the purchase. That is, what are they specifically hoping to get by buying the product– not the item itself, but what benefits it will provide? Further, the salesperson will need to be able to speak to how well specific features will meet the consumer’s stated needs. And, they will need to be educated on the features & benefits of both the goods they’re selling and those of competitive items, as they will likely need to compare and contract specific differences.
Because these consumers with an extended problem solving mindset are deliberate in their shopping process, salespeople should expect that they will not “close the sale,” during their first interaction. Instead, they may need to nurture the relationship with the customer, helping them arrive at their purchase decision over time. Thus, effective salespeople will be those who engage in follow-up with the shopper, making themselves available to answer questions or provide perspective.
Contributors and Attributions
- Increasing Sales with Extended Problem Solving. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


2.3: Maximum Likelihood
Section 2.3a: What is a likelihood?
Since all of the approaches described in the remainer of this chapter involve calculating likelihoods, I will first briefly describe this concept. A good general review of likelihood is Edwards (1992) . Likelihood is defined as the probability, given a model and a set of parameter values, of obtaining a particular set of data. That is, given a mathematical description of the world, what is the probability that we would see the actual data that we have collected?
To calculate a likelihood, we have to consider a particular model that may have generated the data. That model will almost always have parameter values that need to be specified. We can refer to this specified model (with particular parameter values) as a hypothesis, H. The likelihood is then:
\[L(H|D)=Pr(D|H) \label{2.1}\]
Here, L and Pr stand for likelihood and probability, D for the data, and H for the hypothesis, which again includes both the model being considered and a set of parameter values. The | symbol stands for “given,” so equation 2.1 can be read as “the likelihood of the hypothesis given the data is equal to the probability of the data given the hypothesis.” In other words, the likelihood represents the probability under a given model and parameter values that we would obtain the data that we actually see.
For any given model, using different parameter values will generally change the likelihood. As you might guess, we favor parameter values that give us the highest probability of obtaining the data that we see. One way to estimate parameters from data, then, is by finding the parameter values that maximize the likelihood; that is, the parameter values that give the highest likelihood, and the highest probability of obtaining the data. These estimates are then referred to as maximum likelihood (ML) estimates. In an ML framework, we suppose that the hypothesis that has the best fit to the data is the one that has the highest probability of having generated that data.
For the example above, we need to calculate the likelihood as the probability of obtaining heads 63 out of 100 lizard flips, given some model of lizard flipping. In general, we can write the likelihood for any combination of H “successes” (flips that give heads) out of n trials. We will also have one parameter, p H , which will represent the probability of “success,” that is, the probability that any one flip comes up heads. We can calculate the likelihood of our data using the binomial theorem:
\[ L(H|D)=Pr(D|p)= {n \choose H} p_H^H (1-p_H)^{n-H} \label{2.2} \]
In the example given, n = 100 and H = 63, so:
\[ L(H|D)= {100 \choose 63} p_H^{63} (1-p_H)^{37} \label{2.3} \]
We can make a plot of the likelihood, L , as a function of p H (Figure 2.2). When we do this, we see that the maximum likelihood value of p H , which we can call $\hat{p}_H$, is at $\hat{p}_H = 0.63$. This is the “brute force” approach to finding the maximum likelihood: try many different values of the parameters and pick the one with the highest likelihood. We can do this much more efficiently using numerical methods as described in later chapters in this book.
We could also have obtained the maximum likelihood estimate for p H through differentiation. This problem is much easier if we work with the ln-likelihood rather than the likelihood itself (note that whatever value of p H that maximizes the likelihood will also maximize the ln-likelihood, because the log function is strictly increasing). So:
\[ \ln{L} = \ln{n \choose H} + H \ln{p_H}+ (n-H)\ln{(1-p_H)} \label{2.4} \]
Note that the natural log (ln) transformation changes our equation from a power function to a linear function that is easy to solve. We can differentiate:
\[ \frac{d \ln{L}}{dp_H} = \frac{H}{p_H} - \frac{(n-H)}{(1-p_H)}\label{2.5} \]
The maximum of the likelihood represents a peak, which we can find by setting the derivative $\frac{d \ln{L}}{dp_H}$ to zero. We then find the value of p H that solves that equation, which will be our estimate $\hat{p}_H$. So we have:
\[ \begin{array}{lcl} \frac{H}{\hat{p}_H} - \frac{n-H}{1-\hat{p}_H} & = & 0\\ \frac{H}{\hat{p}_H} & = & \frac{n-H}{1-\hat{p}_H}\\ H (1-\hat{p}_H) & = & \hat{p}_H (n-H)\\ H-H\hat{p}_H & = & n\hat{p}_H-H\hat{p}_H\\ H & = & n\hat{p}_H\\ \hat{p}_H &=& H / n\\ \end{array} \label{2.6}\]
Notice that, for our simple example, H / n = 63/100 = 0.63, which is exactly equal to the maximum likelihood from figure 2.2.
Maximum likelihood estimates have many desirable statistical properties. It is worth noting, however, that they will not always return accurate parameter estimates, even when the data is generated under the actual model we are considering. In fact, ML parameters can sometimes be biased. To understand what this means, we need to formally introduce two new concepts: bias and precision. Imagine that we were to simulate datasets under some model A with parameter a. For each simulation, we then used ML to estimate the parameter $\hat{a}$ for the simulated data. The precision of our ML estimate tells us how different, on average, each of our estimated parameters $\hat{a}_i$ are from one another. Precise estimates are estimated with less uncertainty. Bias, on the other hand, measures how close our estimates $\hat{a}_i$ are to the true value a . If our ML parameter estimate is biased, then the average of the $\hat{a}_i$ will differ from the true value a . It is not uncommon for ML estimates to be biased in a way that depends on sample size, so that the estimates get closer to the truth as sample size increases, but can be quite far off in a particular direction when the number of data points is small compared to the number of parameters being estimated.
In our example of lizard flipping, we estimated a parameter value of $\hat{p}_H = 0.63$. For the particular case of estimating the parameter of a binomial distribution, our ML estimate is known to be unbiased. And this estimate is different from 0.5 – which was our expectation under the null hypothesis. So is this lizard fair? Or, alternatively, can we reject the null hypothesis that p H = 0.5? To evaluate this, we need to use model selection.
Section 2.3b: The likelihood ratio test
Model selection involves comparing a set of potential models and using some criterion to select the one that provides the “best” explanation of the data. Different approaches define “best” in different ways. I will first discuss the simplest, but also the most limited, of these techniques, the likelihood ratio test. Likelihood ratio tests can only be used in one particular situation: to compare two models where one of the models is a special case of the other. This means that model A is exactly equivalent to the more complex model B with parameters restricted to certain values. We can always identify the simpler model as the model with fewer parameters. For example, perhaps model B has parameters x, y, and z that can take on any values. Model A is the same as model B but with parameter z fixed at 0. That is, A is the special case of B when parameter z = 0. This is sometimes described as model A is nested within model B, since every possible version of model A is equal to a certain case of model B, but model B also includes more possibilities.
For likelihood ratio tests, the null hypothesis is always the simpler of the two models. We compare the data to what we would expect if the simpler (null) model were correct.
For example, consider again our example of flipping a lizard. One model is that the lizard is “fair:” that is, that the probability of heads is equal to 1/2. A different model might be that the probability of heads is some other value p, which could be 1/2, 1/3, or any other value between 0 and 1. Here, the latter (complex) model has one additional parameter, p H , compared to the former (simple) model; the simple model is a special case of the complex model when p H = 1/2.
For such nested models, one can calculate the likelihood ratio test statistic as
\[ \Delta = 2 \cdot \ln{\frac{L_1}{L_2}} = 2 \cdot (\ln{L_1}-\ln{L_2}) \label{2.7}\]
Here, Δ is the likelihood ratio test statistic, L 2 the likelihood of the more complex (parameter rich) model, and L 1 the likelihood of the simpler model. Since the models are nested, the likelihood of the complex model will always be greater than or equal to the likelihood of the simple model. This is a direct consequence of the fact that the models are nested. If we find a particular likelihood for the simpler model, we can always find a likelihood equal to that for the complex model by setting the parameters so that the complex model is equivalent to the simple model. So the maximum likelihood for the complex model will either be that value, or some higher value that we can find through searching the parameter space. This means that the test statistic Δ will never be negative. In fact, if you ever obtain a negative likelihood ratio test statistic, something has gone wrong – either your calculations are wrong, or you have not actually found ML solutions, or the models are not actually nested.
To carry out a statistical test comparing the two models, we compare the test statistic Δ to its expectation under the null hypothesis. When sample sizes are large, the null distribution of the likelihood ratio test statistic follows a chi-squared ( χ 2 ) distribution with degrees of freedom equal to the difference in the number of parameters between the two models. This means that if the simpler hypothesis were true, and one carried out this test many times on large independent datasets, the test statistic would approximately follow this χ 2 distribution. To reject the simpler (null) model, then, one compares the test statistic with a critical value derived from the appropriate χ 2 distribution. If the test statistic is larger than the critical value, one rejects the null hypothesis. Otherwise, we fail to reject the null hypothesis. In this case, we only need to consider one tail of the χ 2 test, as every deviation from the null model will push us towards higher Δ values and towards the right tail of the distribution.
For the lizard flip example above, we can calculate the ln-likelihood under a hypothesis of p H = 0.5 as:
\[ \begin{array}{lcl} \ln{L_1} &=& \ln{\left(\frac{100}{63}\right)} + 63 \cdot \ln{0.5} + (100-63) \cdot \ln{(1-0.5)} \nonumber \\ \ln{L_1} &=& -5.92\nonumber\\ \end{array} \label{2.8}\]
We can compare this to the likelihood of our maximum-likelihood estimate :
\[ \begin{array}{lcl} \ln{L_2} &=& \ln{\left(\frac{100}{63}\right)} + 63 \cdot \ln{0.63} + (100-63) \cdot \ln{(1-0.63)} \nonumber \\ \ln{L_2} &=& -2.50\nonumber \end{array} \label{2.9}\]
We then calculate the likelihood ratio test statistic:
\[ \begin{array}{lcl} \Delta &=& 2 \cdot (\ln{L_2}-\ln{L_1}) \nonumber \\ \Delta &=& 2 \cdot (-2.50 - -5.92) \nonumber \\ \Delta &=& 6.84\nonumber \end{array} \label{2.10}\]
If we compare this to a χ 2 distribution with one d.f., we find that P = 0.009. Because this P-value is less than the threshold of 0.05, we reject the null hypothesis, and support the alternative. We conclude that this is not a fair lizard. As you might expect, this result is consistent with our answer from the binomial test in the previous section. However, the approaches are mathematically different, so the two P-values are not identical.
Although described above in terms of two competing hypotheses, likelihood ratio tests can be applied to more complex situations with more than two competing models. For example, if all of the models form a sequence of increasing complexity, with each model a special case of the next more complex model, one can compare each pair of hypotheses in sequence, stopping the first time the test statistic is non-significant. Alternatively, in some cases, hypotheses can be placed in a bifurcating choice tree, and one can proceed from simple to complex models down a particular path of paired comparisons of nested models. This approach is commonly used to select models of DNA sequence evolution (Posada and Crandall 1998) .
Section 2.3c: The Akaike Information Criterion (AIC)
You might have noticed that the likelihood ratio test described above has some limitations. Especially for models involving more than one parameter, approaches based on likelihood ratio tests can only do so much. For example, one can compare a series of models, some of which are nested within others, using an ordered series of likelihood ratio tests. However, results will often depend strongly on the order in which tests are carried out. Furthermore, often we want to compare models that are not nested, as required by likelihood ratio tests. For these reasons, another approach, based on the Akaike Information Criterion (AIC), can be useful.
The AIC value for a particular model is a simple function of the likelihood L and the number of parameters k:
\[AIC = 2k − 2\ln L \label{2.11}\]
This function balances the likelihood of the model and the number of parameters estimated in the process of fitting the model to the data. One can think of the AIC criterion as identifying the model that provides the most efficient way to describe patterns in the data with few parameters. However, this shorthand description of AIC does not capture the actual mathematical and philosophical justification for equation (2.11). In fact, this equation is not arbitrary; instead, its exact trade-off between parameter numbers and log-likelihood difference comes from information theory (for more information, see Burnham and Anderson 2003, Akaike (1998) ) .
The AIC equation (2.11) above is only valid for quite large sample sizes relative to the number of parameters being estimated (for n samples and k parameters, n / k > 40). Most empirical data sets include fewer than 40 independent data points per parameter, so a small sample size correction should be employed:
\[ AIC_C = AIC + \frac{2k(k+1)}{n-k-1} \label{2.12}\]
This correction penalizes models that have small sample sizes relative to the number of parameters; that is, models where there are nearly as many parameters as data points. As noted by Burnham and Anderson (2003) , this correction has little effect if sample sizes are large, and so provides a robust way to correct for possible bias in data sets of any size. I recommend always using the small sample size correction when calculating AIC values.
To select among models, one can then compare their A I C c scores, and choose the model with the smallest value. It is easier to make comparisons in A I C c scores between models by calculating the difference, Δ A I C c . For example, if you are comparing a set of models, you can calculate Δ A I C c for model i as:
\[ΔAIC_{c_i} = AIC_{c_i} − AIC_{c_{min}} \label{2.13}\]
where A I C c i is the A I C c score for model i and A I C c m i n is the minimum A I C c score across all of the models.
As a broad rule of thumb for comparing A I C values, any model with a Δ A I C c i of less than four is roughly equivalent to the model with the lowest A I C c value. Models with Δ A I C c i between 4 and 8 have little support in the data, while any model with a Δ A I C c i greater than 10 can safely be ignored.
Additionally, one can calculate the relative support for each model using Akaike weights. The weight for model i compared to a set of competing models is calculated as:
\[ w_i = \frac{e^{-\Delta AIC_{c_i}/2}}{\sum_i{e^{-\Delta AIC_{c_i}/2}}} \label{2.14} \]
The weights for all models under consideration sum to 1, so the w i for each model can be viewed as an estimate of the level of support for that model in the data compared to the other models being considered.
Returning to our example of lizard flipping, we can calculate A I C c scores for our two models as follows:
\[ \begin{array}{lcl} AIC_1 &=& 2 k_1 - 2 ln{L_1} = 2 \cdot 0 - 2 \cdot -5.92 \\\ AIC_1 &=& 11.8 \\\ AIC_2 &=& 2 k_2 - 2 ln{L_2} = 2 \cdot 1 - 2 \cdot -2.50 \\\ AIC_2 &=& 7.0 \\\ \end{array} \label{2.15} \]
Our example is a bit unusual in that model one has no estimated parameters; this happens sometimes but is not typical for biological applications. We can correct these values for our sample size, which in this case is n = 100 lizard flips:
\[ \begin{array}{lcl} AIC_{c_1} &=& AIC_1 + \frac{2 k_1 (k_1 + 1)}{n - k_1 - 1} \\\ AIC_{c_1} &=& 11.8 + \frac{2 \cdot 0 (0 + 1)}{100-0-1} \\\ AIC_{c_1} &=& 11.8 \\\ AIC_{c_2} &=& AIC_2 + \frac{2 k_2 (k_2 + 1)}{n - k_2 - 1} \\\ AIC_{c_2} &=& 7.0 + \frac{2 \cdot 1 (1 + 1)}{100-1-1} \\\ AIC_{c_2} &=& 7.0 \\\ \end{array} \label{2.16} \]
Notice that, in this particular case, the correction did not affect our A I C values, at least to one decimal place. This is because the sample size is large relative to the number of parameters. Note that model 2 has the smallest A I C c score and is thus the model that is best supported by the data. Noting this, we can now convert these A I C c scores to a relative scale:
\[ \begin{array}{lcl} \Delta AIC_{c_1} &=& AIC_{c_1}-AIC{c_{min}} \\\ &=& 11.8-7.0 \\\ &=& 4.8 \\\ \end{array} \label{2.17} \]
\[ \begin{array}{lcl} \Delta AIC_{c_2} &=& AIC_{c_2}-AIC{c_{min}} \\\ &=& 7.0-7.0 \\\ &=& 0 \\\ \end{array} \]
Note that the Δ A I C c i for model 1 is greater than four, suggesting that this model (the “fair” lizard) has little support in the data. This is again consistent with all of the results that we've obtained so far using both the binomial test and the likelihood ratio test. Finally, we can use the relative AICc scores to calculate Akaike weights:
\[ \begin{array}{lcl} \sum_i{e^{-\Delta_i/2}} &=& e^{-\Delta_1/2} + e^{-\Delta_2/2} \\\ &=& e^{-4.8/2} + e^{-0/2} \\\ &=& 0.09 + 1 \\\ &=& 1.09 \\\ \end{array} \label{2.18}\]
\[ \begin{array}{lcl} w_1 &=& \frac{e^{-\Delta AIC_{c_1}/2}}{\sum_i{e^{-\Delta AIC_{c_i}/2}}} \\\ &=& \frac{0.09}{1.09} \\\ &=& 0.08 \end{array} \]
\[ \begin{array}{lcl} w_2 &=& \frac{e^{-\Delta AIC_{c_2}/2}}{\sum_i{e^{-\Delta AIC_{c_i}/2}}} \\\ &=& \frac{1.00}{1.09} \\\ &=& 0.92 \end{array} \]
Our results are again consistent with the results of the likelihood ratio test. The relative likelihood of an unfair lizard is 0.92, and we can be quite confident that our lizard is not a fair flipper.
AIC weights are also useful for another purpose: we can use them to get model-averaged parameter estimates. These are parameter estimates that are combined across different models proportional to the support for those models. As a thought example, imagine that we are considering two models, A and B, for a particular dataset. Both model A and model B have the same parameter p , and this is the parameter we are particularly interested in. In other words, we do not know which model is the best model for our data, but what we really need is a good estimate of p . We can do that using model averaging. If model A has a high AIC weight, then the model-averaged parameter estimate for p will be very close to our estimate of p under model A; however, if both models have about equal support then the parameter estimate will be close to the average of the two different estimates. Model averaging can be very useful in cases where there is a lot of uncertainty in model choice for models that share parameters of interest. Sometimes the models themselves are not of interest, but need to be considered as possibilities; in this case, model averaging lets us estimate parameters in a way that is not as strongly dependent on our choice of models.


10.3: Rotation with Constant Angular Acceleration
- Derive the kinematic equations for rotational motion with constant angular acceleration
- Select from the kinematic equations for rotational motion with constant angular acceleration the appropriate equations to solve for unknowns in the analysis of systems undergoing fixed-axis rotation
- Use solutions found with the kinematic equations to verify the graphical analysis of fixed-axis rotation with constant angular acceleration
In the preceding section, we defined the rotational variables of angular displacement, angular velocity, and angular acceleration. In this section, we work with these definitions to derive relationships among these variables and use these relationships to analyze rotational motion for a rigid body about a fixed axis under a constant angular acceleration. This analysis forms the basis for rotational kinematics. If the angular acceleration is constant, the equations of rotational kinematics simplify, similar to the equations of linear kinematics discussed in Motion along a Straight Line and Motion in Two and Three Dimensions . We can then use this simplified set of equations to describe many applications in physics and engineering where the angular acceleration of the system is constant. Rotational kinematics is also a prerequisite to the discussion of rotational dynamics later in this chapter.
Kinematics of Rotational Motion
Using our intuition, we can begin to see how the rotational quantities \(\theta\), \(\omega\), \(\alpha\), and t are related to one another. For example, we saw in the preceding section that if a flywheel has an angular acceleration in the same direction as its angular velocity vector, its angular velocity increases with time and its angular displacement also increases. On the contrary, if the angular acceleration is opposite to the angular velocity vector, its angular velocity decreases with time. We can describe these physical situations and many others with a consistent set of rotational kinematic equations under a constant angular acceleration. The method to investigate rotational motion in this way is called kinematics of rotational motion .
To begin, we note that if the system is rotating under a constant acceleration, then the average angular velocity follows a simple relation because the angular velocity is increasing linearly with time. The average angular velocity is just half the sum of the initial and final values:
\[\bar{\omega} = \frac{\omega_{0} + \omega_{f}}{2} \ldotp \label{10.9}\]
From the definition of the average angular velocity, we can find an equation that relates the angular position, average angular velocity, and time:
\[\bar{\omega} = \frac{\Delta \theta}{\Delta t} \ldotp\]
Solving for \(\theta\), we have
\[\theta_{f} = \theta_{0} + \bar{\omega} t, \label{10.10}\]
where we have set t 0 = 0. This equation can be very useful if we know the average angular velocity of the system. Then we could find the angular displacement over a given time period. Next, we find an equation relating \(\omega\), \(\alpha\), and t. To determine this equation, we start with the definition of angular acceleration:
\[\alpha = \frac{d \omega}{dt} \ldotp\]
We rearrange this to get \(\alpha\)dt = d\(\omega\) and then we integrate both sides of this equation from initial values to final values, that is, from t 0 to t and \(\omega_{0}\) to \(\omega_{f}\). In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals:
\[\alpha \int_{t_{0}}^{t} dt' = \int_{\omega_{0}}^{\omega_{f}} d \omega \ldotp\]
Setting t 0 = 0, we have
\[\alpha t = \omega_{f} - \omega_{0} \ldotp\]
We rearrange this to obtain
\[\omega_{f} = \omega_{0} + \alpha t, \label{10.11}\]
where \(\omega_{0}\) is the initial angular velocity. Equation \ref{10.11} is the rotational counterpart to the linear kinematics equation v f = v 0 + at. With Equation \ref{10.11}, we can find the angular velocity of an object at any specified time t given the initial angular velocity and the angular acceleration.
Let’s now do a similar treatment starting with the equation \(\omega = \frac{d \theta}{dt}\). We rearrange it to obtain \(\omega\)dt = d\(\theta\) and integrate both sides from initial to final values again, noting that the angular acceleration is constant and does not have a time dependence. However, this time, the angular velocity is not constant (in general), so we substitute in what we derived above:
\[\begin{split} \int_{t_{0}}^{t_{f}} (\omega_{0} + \alpha t') dt' & = \int_{\theta_{0}}^{\theta_{f}} d \theta; \\ \int_{t_{0}}^{t} \omega_{0} dt + \int_{t_{0}}^{t} \alpha tdt & = \int_{\theta_{0}}^{\theta_{f}} d \theta = \Bigg[ \omega_{0} t' + \alpha \left(\dfrac{(t')^{2}}{2}\right)^{2} \Bigg]_{t_{0}}^{t} = \omega_{0} t + \alpha \left(\dfrac{t^{2}}{2}\right) = \theta_{f} - \theta_{0} \ldotp \end{split}\]
where we have set t 0 = 0. Now we rearrange to obtain
\[\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2} \ldotp \label{10.12}\]
Equation \ref{10.12} is the rotational counterpart to the linear kinematics equation found in Motion Along a Straight Line for position as a function of time. This equation gives us the angular position of a rotating rigid body at any time t given the initial conditions (initial angular position and initial angular velocity) and the angular acceleration.
We can find an equation that is independent of time by solving for t in Equation \ref{10.11} and substituting into Equation \ref{10.12}. Equation \ref{10.12} becomes
\[\begin{split} \theta_{f} & = \theta_{0} + \omega_{0} \left(\dfrac{\omega_{f} - \omega_{0}}{\alpha}\right) + \frac{1}{2} \alpha \left(\dfrac{\omega_{f} - \omega_{0}}{\alpha}\right)^{2} \\ & = \theta_{0} + \frac{\omega_{0} \omega_{f}}{\alpha} - \frac{\omega_{0}^{2}}{\alpha} + \frac{1}{2} \frac{\omega_{f}^{2}}{\alpha} - \frac{\omega_{0} \omega_{f}}{\alpha} + \frac{1}{2} \frac{\omega_{0}^{2}}{\alpha} \\ & = \theta_{0} + \frac{1}{2} \frac{\omega_{f}^{2}}{\alpha} - \frac{1}{2} \frac{\omega_{0}^{2}}{\alpha}, \\ \theta_{f} - \theta_{0} & = \frac{\omega_{f}^{2} - \omega_{0}^{2}}{2 \alpha} \end{split}\]
or
\[\omega_{f}^{2} = \omega_{0}^{2} + 2 \alpha (\Delta \theta) \ldotp \label{10.13}\]
Equation \ref{10.10} through Equation \ref{10.13} describe fixed-axis rotation for constant acceleration and are summarized in Table 10.1.
Table 10.1 - Kinematic Equations
| Angular displacement from average angular velocity | $$\theta_{f} = \theta_{0} + \bar{\omega} t$$ |
| Angular velocity from angular acceleration | $$\omega_{f} = \omega_{0} + \alpha t$$ |
| Angular displacement from angular velocity and angular acceleration | $$\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2}$$ |
| Angular velocity from angular displacement and angular acceleration | $$\omega_{f}^{2} = \omega_{0}^{2} + 2 \alpha (\Delta \theta)$$ |
Applying the Equations for Rotational Motion
Now we can apply the key kinematic relations for rotational motion to some simple examples to get a feel for how the equations can be applied to everyday situations.
A deep-sea fisherman hooks a big fish that swims away from the boat, pulling the fishing line from his fishing reel. The whole system is initially at rest, and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of 110 rad/s 2 for 2.00 s (Figure \(\PageIndex{1}\)).
- What is the final angular velocity of the reel after 2 s?
- How many revolutions does the reel make?
Strategy
Identify the knowns and compare with the kinematic equations for constant acceleration. Look for the appropriate equation that can be solved for the unknown, using the knowns given in the problem description.
Solution
- We are given \(\alpha\) and t and want to determine \(\omega\). The most straightforward equation to use is \(\omega_{f} = \omega_{0} + \alpha t\), since all terms are known besides the unknown variable we are looking for. We are given that \(\omega_{0}\) = 0 (it starts from rest), so $$\omega_{f} = 0 + (110\; rad/s^{2})(2.00\; s) = 220\; rad/s \ldotp$$
- We are asked to find the number of revolutions. Because 1 rev = 2\(\pi\) rad, we can find the number of revolutions by finding θ in radians. We are given \(\alpha\) and t, and we know \(\omega_{0}\) is zero, so we can obtain \(\theta\) by using $$\begin{split} \theta_{f} & = \theta_{i} + \omega_{i} t + \frac{1}{2} \alpha t^{2} \\ & = 0 + 0 + (0.500)(110\; rad/s^{2})(2.00\; s)^{2} = 220\; rad \ldotp \end{split}$$Converting radians to revolutions gives $$Number\; of\; rev = (220\; rad) \left(\dfrac{1\; rev}{2 \pi\; rad}\right) = 35.0\; rev \ldotp$$
Significance
This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.)
In the preceding example, we considered a fishing reel with a positive angular acceleration. Now let us consider what happens with a negative angular acceleration.
Now the fisherman applies a brake to the spinning reel, achieving an angular acceleration of −300 rad/s 2 . How long does it take the reel to come to a stop?
Strategy
We are asked to find the time t for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is \(\omega_{0}\) = 220 rad/s and the final angular velocity \(\omega\) is zero. The angular acceleration is given as \(\alpha\) = −300 rad/s 2 . Examining the available equations, we see all quantities but t are known in \(\omega_{f} = \omega_{0} + \alpha t\), making it easiest to use this equation.
Solution
The equation states
\[\omega_{f} = \omega_{0} + \alpha t \ldotp\]
We solve the equation algebraically for t and then substitute the known values as usual, yielding
\[t = \frac{\omega_{f} - \omega_{0}}{\alpha} = \frac{0 - 220.0\; rad/s}{-300.0\; rad/s^{2}} = 0.733\; s \ldotp\]
Significance
Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish is slower, requiring a smaller acceleration.
A centrifuge used in DNA extraction spins at a maximum rate of 7000 rpm, producing a “g-force” on the sample that is 6000 times the force of gravity. If the centrifuge takes 10 seconds to come to rest from the maximum spin rate: (a) What is the angular acceleration of the centrifuge? (b) What is the angular displacement of the centrifuge during this time?
Figure \(\PageIndex{2}\) shows a graph of the angular velocity of a propeller on an aircraft as a function of time. Its angular velocity starts at 30 rad/s and drops linearly to 0 rad/s over the course of 5 seconds. (a) Find the angular acceleration of the object and verify the result using the kinematic equations. (b) Find the angle through which the propeller rotates during these 5 seconds and verify your result using the kinematic equations.
Strategy
- Since the angular velocity varies linearly with time, we know that the angular acceleration is constant and does not depend on the time variable. The angular acceleration is the slope of the angular velocity vs. time graph, \(\alpha = \frac{d \omega}{dt}\). To calculate the slope, we read directly from Figure \(\PageIndex{2}\), and see that \(\omega_{0}\) = 30 rad/s at t = 0 s and \(\omega_{f}\) = 0 rad/s at t = 5 s. Then, we can verify the result using \(\omega = \omega_{0} + \alpha t\).
- We use the equation \(\omega = \frac{d \theta}{dt}\); since the time derivative of the angle is the angular velocity, we can find the angular displacement by integrating the angular velocity, which from the figure means taking the area under the angular velocity graph. In other words: $$\int_{\theta_{0}}^{\theta_{f}} d \theta = \theta_{f} - \theta_{0} = \int_{t_{0}}^{t_{f}} \omega (t)dt \ldotp$$Then we use the kinematic equations for constant acceleration to verify the result.
Solution
- Calculating the slope, we get $$\alpha = \frac{\omega - \omega_{0}}{t - t_{0}} = \frac{(0 - 30.0)\; rad/s}{(5.0 - 0)\; s} = -6.0\; rad/s^{2} \ldotp$$We see that this is exactly Equation \ref{10.11} with a little rearranging of terms.
- We can find the area under the curve by calculating the area of the right triangle, as shown in Figure \(\PageIndex{3}\).
\[\Delta \theta = area(triangle) = \frac{1}{2} (30\; rad/s)(5\; s) = 75\; rad \ldotp$$We verify the solution using Equation \ref{10.12}: $$\theta_{f} = \theta_{0} + \omega_{0} t + \frac{1}{2} \alpha t^{2} \ldotp$$Setting \(\theta_{0}\) = 0, we have $$\theta_{0} = (30.0\; rad/s)(5.0\; s) + \frac{1}{2} (-6.0\; rad/s^{2})(5.0\; s)^{2} = 150.0 - 75.0 = 75.0\; rad \ldotp$$This verifies the solution found from finding the area under the curve.
Significance
We see from part (b) that there are alternative approaches to analyzing fixed-axis rotation with constant acceleration. We started with a graphical approach and verified the solution using the rotational kinematic equations. Since \(\alpha = \frac{d \omega}{dt}\), we could do the same graphical analysis on an angular acceleration-vs.-time curve. The area under an \(\alpha\)-vs.-t curve gives us the change in angular velocity. Since the angular acceleration is constant in this section, this is a straightforward exercise.


5.1D: Skin Color
Skin color is determined largely by the amount of melanin pigment produced by melanocytes in the skin.
LEARNING OBJECTIVE
Explain how differing degrees of pigmentation are produced
Key Takeaways
Key Points
- Skin color is mainly determined by a pigment called melanin.
- Melanin is produced by melanocytes through a process called melanogenesis.
- The difference in skin color between lightly and darkly pigmented individuals is due to their level of melanocyte activity; it is not due to the number of melanocytes in their skin.
Key Terms
- melanin : Any of a group of naturally occurring dark pigments responsible for the color of skin.
- melanocyte : A cell in the skin that produces the pigment melanin.
- keratinocytes : Cells that take up and store melanin.
- eumelanin : The type of melanin mainly responsible for brown and black skin.
- stratum basale : The epidermal layer where melanocytes are found.
Melanin
Skin color is largely determined by a pigment called melanin but other things are involved. Your skin is made up of three main layers, and the most superficial of these is called the epidermis. The epidermis itself is made up of several different layers.
Melanocyte: Cross-section of skin showing melanin in melanocytes
The deepest of the epidermal layers is called the stratum basale or stratum germinativum. In this layer lie important cells called melanocytes. Their name is derived from two parts: melano-, which means black or darkness, and -cyte, which means cell.
Melanocytes are irregularly shaped cells that produce and store a pigment called melanin. The most abundant type of melanin is called eumelanin. This pigment is stored in organelles called melanosomes.
Eumelanin is responsible for the brown and black pigmentation of human skin or the lack thereof if little of it is produced. The production of melanin is called melanogenesis—genesis means formation or development.
How Skin Color is Determined
Regardless of background, every person has largely the same number of melanocytes, but the genetics of each person is what determines how much melanin is produced and how it is distributed throughout the skin. For example, light skinned individuals may have darker places like nipples and moles. Conversely, dark skinned individuals have a lighter tone to the palms of their hands.
Another critical factor, exposure to sunlight, triggers the production of melanin as well. This is what gives us a tan. The melanin produced in response to the sun’s rays protects our skin and the rest of the body from the harmful effects of the sun’s burn and cancer-inducing U.V. radiation.
The Role of Keratinocytes
People with darker skin have more active melanocytes compared to people with lighter skin. However, the pigment of our skin also involves the most abundant cells of our epidermis, the keratinocytes.
While melanocytes produce, store, and release melanin, keratinocytes are the largest recipients of this pigment. The transfer of melanin from melanocytes to keratinocytes occurs thanks to the long tentacles each melanocyte extends to upwards of 40 keratinocytes.
If a person is unable to produce melanin, they have a condition called albinism.
Other Skin Color Determinants
Tanned Skin: Exposure to UV radiation through tanning causes changes in the pigmentation of the skin by increasing melanin production.
Besides melanin, other factors play a role in general or local skin color. These include:
- The amount of carotene found in the stratum corneum of the epidermis and the deepest layer of the skin, the hypodermis. Carotene is a yellow-orange pigment found in carrots. Your skin may turn this color if you eat a lot of carotene-rich foods. The skin may turn yellow due to another factor, called icterus or jaundice, which occurs with serious liver disease. In this instance, bile pigments are deposited within the skin and impart a yellow color to it.
- The amount of oxygen-saturated hemoglobin found in the blood vessels of the middle layer of our skin, the dermis. Hemoglobin is the iron-containing protein pigment of our blood cells. A lack of oxygen saturation imparts a paler, grayer, or bluer color to the skin. Skin may also become paler as a result of anemia (a reduced number of hemoglobin and/or red blood cells), low blood pressure, or poor circulation of blood.
- Conversely, light-skinned individuals (compared to dark-skinned ones) may have a rosy effect to their skin thanks to the relatively more oxygen-rich hemoglobin flowing through the blood vessels of their dermis. Red-colored skin may also occur as a result of blood vessels in or near the skin dilating (expanding) due to embarrassment, fever, allergy, or inflammation.
- Finally, the skin may have red, black, blue, purple, and green bruises—all as a result of the escape of blood into surrounding tissues. As the blood (namely, the hemoglobin) disintegrates and is processed and removed by various cells, it and the bruise changes color with time.
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- integumentary system. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/integumentary_system . License : CC BY-SA: Attribution-ShareAlike
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- mechanoreceptor. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/mechanoreceptor . License : CC BY-SA: Attribution-ShareAlike
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9.1: Morley’s Miracle
Probably you have heard of the impossibility of trisecting an angle. (Hold on for a quick rant about the importance of understanding your hypotheses) What’s actually true is that you can’t trisect a generic angle if you accept the restriction of using the old-fashioned tools of Euclidean geometry: the compass and straight-edge. There are a lot of constructions that can’t be done using just a straight-edge and compass–angle trisection, duplication of a cube 1 , squaring a circle, constructing a regular heptagon, et cetera .
If you allow yourself to use a ruler – i.e. a straight-edge with marks on it (indeed you really only need two marks a unit distance apart) then angle trisection be done via what is known as a neusis construction. Nevertheless, because of the central place of Euclid’s Elements in mathematical training throughout the centuries, and thereby, a very strong predilection towards that which possible via compass and straight-edge alone, it is perhaps not surprising that a perfectly beautiful result that involved trisecting angles went undiscovered until \(1899\), when Frank Morley stated his Trisector Theorem. There is much more to this result than we will state here – so much more that the name "Morley’s Miracle" that has been given to the Trisector theorem is truly justified – but even the simple, initial part of this beautiful theory is arguably miraculous! To learn more about Morley’s theorem and its extension see [8] .
So, let’s state the theorem!
Start with an arbitrary triangle \({\triangle}ABC\) . Trisect each of its angles to obtain a diagram something like that in Figure \(9.1.1\).
The six angle trisectors that we’ve just drawn intersect one another in quite a few points.
You could literally count the number of intersection points between the angle trisectors on the diagram, but you should also be able to count them (perhaps we should say “double-count them”) combinatorially. Give it a try!
Among the points of intersection of the angle trisectors there are three that we will single out – the intersections of adjacent trisectors. In Figure \(9.1.2\) the intersection of adjacent trisectors are indicated, additionally, we have connected them together to form a small triangle in the center of our original triangle.
Are you ready for the miraculous part? Okay, here goes!
The points of intersection of the adjacent trisectors in an arbitrary triangle \(\triangle ABC\) form the vertices of an equilateral triangle.
In other words, that little blue triangle in Figure \(9.1.2\) that kind of like it might be equilateral actually does have all three sides equal to one another. Furthermore, it doesn’t matter what triangle we start with, if we do the construction above we’ll get a perfect \(60^\circ - 60^\circ - 60^\circ\) triangle in the middle!
Sources differ, but it is not clear whether Morley ever proved his theorem. The first valid proof (according to R. K. Guy in [8] was published in \(1909\) by M. Satyanarayana [15] . There are now other proofs known, for instance the cut-the-knot website ( http://www.cut-the-knot.org/ ) exposits no less than nine different proofs. The proof by Satyanarayana used trigonometry. The proof we’ll look at here is arguably the shortest ever produced and it is due to John Conway. It is definitely a "Book proof"!
Let us suppose that an arbitrary triangle \(\triangle ABC\) is given. We want to show that the triangle whose vertices are the intersections of the adjacent trisectors is equilateral – this triangle will be referred to as the Morley triangle . Let’s also denote by \(A\), \(B\) and \(C\) the measures of the angles of \(\triangle ABC\). (This is what is generally known as an “abuse of notation” – we are intentionally confounding the vertices (\(A\), \(B\) and \(C\)) of the triangle with the measure of the angles at those vertices.) It turns out that it is fairly hard to reason from our knowledge of what the angles \(A\), \(B\) and \(C\) are to deduce that the Morley triangle is equilateral. How does the following plan sound: suppose we construct a triangle, that definitely does have an equilateral Morley triangle, whose angles also happen to be \(A\), \(B\) and \(C\). Such a triangle would be similar 2 to the original triangle \(\triangle ABC\) – if we follow the similarity transform from the constructed triangle back to \(\triangle ABC\) we will see that their Morley triangles must coincide; thus if one is equilateral so is the other!
One of the features of Conway’s proof that leads to its great succinctness and beauty is his introduction of some very nice notation. Since we are dealing with angle trisectors, let \(a\), \(b\) and \(c\) be angles such that \(3a = A\), \(3b = B\) and \(3c = C\). Furthermore, let a superscript star denote the angle that is \(\dfrac{π}{3}\) (or \(60^{\circ}\) if you prefer) greater than a given angle. So, for example,
\[ a^{\star}= a + \dfrac{\pi}{3} \]
and
\[ a^{\star\star} = a + \dfrac{2}{3}. \]
Now, notice that the sum \(a+b+c\) must be \(\dfrac{\pi}{3}\) . This is an immediate consequence of \(A+B+C=\pi\) which is true for any triangle in the plane. It follows that by distributing two stars amongst the three numbers \(a\) , \(b\) and \(c\) we will come up with three quantities which sum to \(\pi\) . In other words, there are Euclidean triangles having the following triples as their vertex angles:
\((a, b, c^{\star\star}) \;\;\;\;\;\;\;\;(a, b^{\star} , c^{\star} ) \\ (a, b^{\star\star}, c) \;\;\;\;\;\;\;\;(a^{\star} , b^{\star} , c) \\ (a^{\star\star}, b, c) \;\;\;\;\;\;\;\;(a^{\star}, b, c^{\star} )\)
In a nutshell, Conway’s proof consists of starting with an equilateral triangle of unit side length, adding appropriately scaled versions of the six triangles above and ending up with a figure (having an equilateral Morley triangle) similar to \({\triangle}ABC\) . The generic picture is given in Figure \(9.1.3\). Before we can really count this argument as a proof, we need to say a bit more about what the phrase "appropriately scaled" means. In order to appropriately scale the triangles (the small acute ones) that appear green in Figure \(9.1.3\) we have a relatively easy job – just scale them so that the side opposite the trisected angle has length one; that way they will join perfectly with the central equilateral triangle.
The triangles (these are the larger obtuse ones) that appear purple in Figure \(9.1.3\) are a bit more puzzling. Ostensibly, we have two different jobs to accomplish – we must scale them so that both of the edges that they will share with green triangles have the correct lengths. How do we know that this won’t require two different scaling factors? Conway also developed an elegant argument that handles this question as well. Consider the purple triangle at the bottom of the diagram in Figure \(9.1.3\)– it has vertex angles \((a,b,c^{\star\star})\) . It is possible to construct triangles similar (via reflections) to the adjacent green triangles \((a, b^\star, c^\star)\) and \((a^\star, b, c^\star)\) of triangle \((a,b,c^{\star\star})\) . To do this just construct two lines that go through the top vertex (where the angle \(c^{\star\star}\) is) that cut the opposite edge at the angle \(c^\star\) in the two possible senses – these two lines will coincide if it should happen that \(c^\star\) is precisely \(\dfrac{\pi}{2}\) but generally there will be two and it is evident that the two line segments formed have the same length. We scale the purple triangle so that this common length will be \(1\). See Figure \(9.1.4\).
If it should happen that \(c^{\star} = \dfrac{π}{2}\), what can we say about \(C\)?
Of course, the other two obtuse triangles can be handled in a similar way.
Exercises:
What value should we get if we sum all of the angles that appear around one of the interior vertices in the finished diagram? Verify that all three have the correct sum.
In this section, we talked about similarity. Two figures in the plane are similar if it is possible to turn one into the other by a sequence of mappings: a translation, a rotation and a scaling.
Geometric similarity is an equivalence relation. To fix our notation, let \(T(x, y)\) represent a generic translation, \(R(x, y)\) a rotation and \(S(x, y)\) a scaling – thus a generic similarity is a function from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) that can be written in the form \(S(R(T(x, y)))\).
Discuss the three properties of an equivalence relation (reflexivity, symmetry and transitivity) in terms of geometric similarity.


4.4: Who Was Meant to See or Read the Texts
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With the question of who was meant to see or read a given text, we are concerned with issues such as audience and accessibility of texts. The earliest proto-cuneiform texts were not designed to be read outside the administrative sphere, and so to the historian today they can be laconic and obtuse in their content. Even their language (or languages?) is unreachable through the distribution of incised signs, some 900 of them, that early scribes marked on the soft clay tablets with their styli. Their most approachable component is the array of numerical systems employed to count and account for quantities of people, animals, and products as they made their way through the bureaucratic world (Chrisomalis 2010 provides a brilliant comparative study of the world’s systems of numerical notation, including those of ancient Mesopotamia).
How were cuneiform texts read? Charpin (2010: 20–22) proposes that in almost every case a cuneiform text would be read aloud either by a literate reader to him/herself or by a scribe to a non-literate listener, such as an official or royal recipient. Additionally, Charpin cites rare evidence that scribes might silently read texts to themselves as a means of rapidly checking their content. When we sit today in the British Museum study room and see the distinguished cuneiform scholars silently working their way through trays of broken clay tablets, perhaps we should encourage them occasionally to voice their readings aloud so that we might share in the aurality of the text and thereby gain an idea of how an ancient reader/listener might have encountered the written word (for a bold attempt at spoken Akkadian, by Irving Finkel of the British Museum, see http://www.bbc.co.uk/news/world-middle-east-13733615 ).