Separation Anxiety And Its Effects On Children - 1410 Words

Separation anxiety is very common as children grow and develop. They may fear those few moments during daycare drop-off or that initial first meeting with a new babysitter. These children will often times grow out of this anxiety after they become familiar with their surroundings and caregivers. But, if they continue having these separation anxiety induced outbursts, even after they have been with a trusted caregiver for a time, parents and primary caregivers often worry about the well-being of their child and their child’s development. They may believe that their child has acquired a Separation Anxiety Disorder (SAD). I selected this topic because I work with multiple toddlers in a center and their parents have expressed some minor worry towards whether or not their child suffers with SAD or if this anxiety is causing them delays. I would love to be able to know some minor details about this broad topic and refer them to the website and articles that I found helpful when writ ing this report. This topic is important because it is has many symptoms that are similar to regular and developmentally appropriate separation anxiety. When a child suffers from SAD, it is something that needs attention by a pediatric professional in order for it not to affect the child later in life. The first article that I read and reviewed was called â€Å"Adult Agoraphobia and Childhood Separation Anxiety: Using Children s Literature to Understand the Link†. It was a fairly theoretical and made theShow MoreRelatedSeparation Anxiety And Its Effects On Children1560 Words   |  7 Pagesexample of separation anxiety, a developmental phase that most children go through during their early years. Separation anxiety is most likely to occur in child with a family history of anxiety/depression, child who are shy, children who have a lack of appropriate parental interaction, and overprotective parents. Separation anxiety is one of the most common anxiety disorders in children affecting one out of twenty. It is a developmental stage during which the child experiences anxiety when separatedRead MoreLong Term Effects of Childhood Separation Anxiety1230 Words   |  5 Pagesï » ¿Long-Term Effects of Childhood Separation Anxiety Abstract This report delves into the connection between childhood separation anxiety disorder and the long-term implications that it may have. To understand the connections I preformed secondary research through â€Å"Academic Search Complete†. I found that childhood separation anxiety disorder is connected with serious mental disorders, such as anorexia, bulimia, social phobias, depression, and behavior disorders. Many studies have shown that childhoodRead MoreSeparation Anxiety : A Type Of An Attachment Disorder1232 Words   |  5 Pagessome type of disorders. Separation anxiety is one of them. According to the www.attachment.org website, â€Å"this disorder is a type of an attachment disorder that is usually observed by young children, who feel they are getting lack of affection and attention from parents or their caregivers due to separation.† I believe that many people feel unsafe being alone and they are afraid to be alone. However, due to inescapable situation or techno logy or laziness they face separation, loneliness, and fear.Read MoreEssay about Separation Anxiety Disorder Among Children and Adolescents 1320 Words   |  6 PagesThis paper examines the various symptoms of Separation Anxiety Disorder among children and adolescents, the refusal of children diagnosed with Separation Anxiety Disorder to go to school, and the treatments that are used to treat this disorder. Studies have shown that Separation Anxiety Disorder is the third most common anxiety disorder among children. Symptoms are fairly easy to recognize, but must be addressed quickly. Refusal to go to school is one of the most significant consequences of thisRead MoreInterventions Of Separation Anxiety Disorder1046 Words   |  5 PagesInterventions of Separation Anxiety Separation anxiety disorder (SAD) is one of the most common anxiety disorders in preadolescent children and can cause serious distress in a child’s life. Having separation anxiety disorder in childhood can significantly increase the chances of internalizing problems and other anxiety disorders. Unfortunately, because separation anxiety is common in children it is hard to determine if the behaviors the children are displaying are atypical. Although there are variousRead MoreWhere Does Bad Behavior Do Children Come From?1515 Words   |  7 PagesWhere Does Bad Behavior in Children Come From? Many bystanders perceive that young children with bad behavior want attention from whoever is around. In most cases this is true, but sometimes there is an underlying root to this behavior. In order to handle an outraged child, one must consider why he/she is like this. Three of the reasons why a child could be acting out are separation anxiety, the size of the child’s family and birth order, and disorders. Separation anxiety is defined as the troubledRead MoreTreatment Of Anxiety Among Children And Adolescents1220 Words   |  5 PagesTreatment of Anxiety in Children and Adolescents Mental health is becoming more prevalent in todays society as many social groups are working to raise awareness for it. However, while this is the case, sometimes children and adolescents that face the same challenges are forgotten about. I chose to research the topic of anxiety because it is personally something I have suffered from since childhood. When we read the chapter for class on anxiety, it was very hard for me. No one recognized the anxiety I hadRead MoreSeparation And Divorce : Common Phenomena1429 Words   |  6 PagesSeparation and divorce are common phenomena in the community today, but still represent one of the major life stressor for most individuals involved, with a strong negative consequences for the physical and mental health of all members within the family. When parents separate from their children, the children experience the loss or reduction of their family unit and the security that comes with it. Separation can destabilize the inclination that the globe or the world is safe and predictable. ForRead MoreThe Quality Of The Two Programs For An Evaluation938 Words   |  4 Pagesresearched in this assignment are Children’s Anxiety Treatment with cognitive-behavioral therapy (individual and family modality), and Adult Opiate Abuse Treatment with buprenorphine and naltrexone. These studies have shown the effects of different treatments which involves buprenorphine and naltrexone formulations for relapse prevention and detoxified opioid addicts. Research on cognitive behavioural therapy for children and adolescents with anxiety is suggested to be based on sound theoreticalRead MoreTemperament721 Words   |  3 Pageseasier over time ***other infants don’t fit into these categories*** * Temperament is moderately stable through infancy, childhood, and adolescence. Stability of Temperament * Fearful preschoolers tend to be more inhibited as older children and adolescents * Inhibited more likely to be introverted adults Temperament and other aspects of development Various aspects of temperament related to: * School success * Peer interactions * Compliance with parents * Depression

Essay about Crime in Latin America - 1197 Words

Prisons for a long time have been a gateway to try to save society, when the only thing that it’s doing is hurting the social order because it’s creating more problems that are not being treated from the beginning. Crime has become a big problem during these hard times with the poor economy, but it has especially affected Latin America because of all the problems that overcrowded prisons have brought forward. In Latin America Brazil and Mexico are the two largest countries that have been affected with having the highest percentage of crimes, inmates in prisons, and concerns with overcrowded prisons. And these increase with the high crime rates in Latin America that are rising due to drug trafficking wars in Brazil and Mexico. The†¦show more content†¦But this tends to not be possible because Brazil and Mexico are having troubles trying to sustain their facilities. For example, in Brazil President Fernando Henrique Cardoso secured that he would increase federa l funding from $4.5 million to $33 million by 2001 so that Brazilian prisons can use it to build more facilities. No matter what was promised many still doubt that this money for prisons will solve them 2 main concerns that Brazil faces which is overcrowded prisons and corruption. As in now, there are about 200,000 inmates in a system built to hold fewer than half that number. This is a clear example of that issues that they have to work with in the country (BBC News). Moreover, there are not details about a budget or the funding they receive in the Mexican prisons. The only information available is that they spend about 130 pesos per day to care for 1 of the 210,000 inmates. This converts to 27 million pesos spent daily and about 9.93 billion pesos per year (Suverza 1). Plus, it’s about 6.56 million pesos per day just to make sure they get fed and clothed. And the documents say this misuse of funds directly affects by authorities to fight crime. â€Å"The lack of security is due to a lack of resources for the anticipation and prevention of acts by criminal organizations, and poor funding for investigation and efficiently processing the most serious of crimes† (SuverzaShow MoreRelatedTrafficking And Organized Crime During Latin America1859 Words   |  8 PagesIn the last decades, narco trafficking and organized crime had emerged as the main security threats in Latin America. Deviant globalization increase their power by establishing connections with similar organizations in other parts of the world, like cocaine distribution networks, for example. This perceived danger increases the pressure on the governments for effective solutions, and some of them consider that these problems had overwhelmed the capabilities of the local police. Consequently, theRead MoreAmerica s Trade Area Of The Americas1494 Words   |  6 PagesHistory Latin America is composed of seventeen countries which was colonized by Spain’s and Portugal. They are large in diverse population with four hundred and ninety million people in total. The percentage of the Indian and African that lives in Latin America is basically seventy-five percent just in the cities. The industrial and development grew since the 1960’s; also the free Trade Area of the Americas (FTAA) proposes to integrate economies of Latin America, North America and the Caribbean (exceptRead MoreThe Economic And Social Standing Of Their Country On The Neocolonial Practise863 Words   |  4 Pagesunethical. The military states of Latin America used the resources provided by operation Condor to coordinate information and resources to aid each other in quelling insurgencies in their respected countries. An instance of this was in 1978 when Uruguayan forces crossed into Brazil and captured two activist believed to support the opposition in Uruguay. Many have argued, such as Luis Roniger, that the U.S. attempted to promote western democratic values in Latin America, though their means were contradictoryRead MoreArgumentative Essay - Education in Latin America848 Words   |  4 PagesFUTURE, A LATIN-AMERICAN PERSPECTIVE by Ana-Maria Gonzalez â€Å"Education leads to a brighter future.† Quite a clichà ©d phrase, actually. So popular, that people tend to forget the true significance of it. It is known that Latin America faces numerous problems that makes looking into the future a discouraging view; but we fail to realize the lack of education may be the root of these issues, including violence, unemployment and poverty. Even though education is widely available in Latin America, peopleRead MoreThe Features Of Corruption Of Latin America876 Words   |  4 Pages Corruption exists to some extent in all countries. But in Latin America, corruption seems an inextricable part of life. Some schools of thought, particularly those in the political science circle, view corruption as functional to the maintenance of a political system. Meaning, corruption may not be ideal, but it represents a way for people to access resources that would otherwise be unavailable to them. In sharp contrast, economists point to the many downf alls of corruption: a climate of fear andRead MoreThe Impact Of Drug Trafficking And Organized Crime1099 Words   |  5 PagesUnited States has a vast illegal drug market as well as high numbers of people indulging in organized crime. Drug law enforcement personnel face problems when protecting the United States borders to avoid any drug trafficking instances. Drug trafficking involves smuggling of illegal drugs producing states such as Mexico to the consumer markets in other regions within the United States Organized Crime, on the other hand, is the practice of the offense through threats or violence and aims to collect legalRead MoreOrganization Structure of the Gang Latin Kings1488 Words   |  6 Pages The Latin Kings organization was founded in Chicago, Illinois, in 1940 after a number of Puerto Rican men and later Mexican men organized themselves into a group that was meant to protect their communities. However, this group, that was meant to protect Latino immigrants against racism and oppression, grew over the years and turned into one of the largest criminal gangs in America. From 1970 the Latin Kings started to commit a large number of crimes: murder, drug trafficking, robberies etc. SinceRead MoreFighting Drug Cartels On The Americas712 Words   |  3 Pages Committee: UNODC Country: Nigeria Topic: Fighting Drug Cartels in the Americas Delegate: Angel Rivera Fighting Drug Cartels in the Americas Background Drug cartels have arisen as a major crisis for the future in the Americas. Individuals indulge themselves on drugs for many reasons such as tradition, attempting to escape poverty, and generating revenue for rebellious activities. Drug trafficking has proven to be ludacris, with the increasing involvement of corrupt government officials in theirRead MoreThe Major Economic Problems That Have Plagued Latin America1497 Words   |  6 Pages1. What are the major economic problems that have plagued Latin America in the 20th century? During the 20th century Latin America went through a change after the U.S made the clam to directly defend Latin America. This caused a sudden trade switch from the Europe nation to the U.S. With this trade switch we start to see a big gap between the lower and upper class. With this gap the poor gets poorer and the rich become Carlos Slim. Carlos Slim was the world’s richest person form 2010-2013 and isRead MoreThe Causes Of Drug Trafficking991 Words   |  4 PagesThe crime I chose is drug trafficking. Drug Trafficking has affected nearly every inch of the world. Drug trafficking has brought untold riches to criminals and has left almost everything else in society tainted. Drug Trafficking has affected both the minority and majority members around the globe. Every country in the world criminal’s groups seem to have their hand in the drug trade. Even though almost every country criminal’s organization has their hand in the drug trade it is the unstable countrie s

Literature Review Sample - 10727 Words

siht taht noitartsnomed dna ,sdohtem fo noitceles eht ,cipot eht ot hcaorppa ralucitrap eht yfitsuj ot erutaretil eht ni saedi eht fo esu eht ,sdrow rehto ni ;sisehtnys dna sisylana evitceffe dna ,ytiverb dna ytiralc ,ycnetsisnoc dna ruogir ,htped dna htdaerb etairporppa snaem ytilauQ .seihpargoilbib detat -onna desiugsid ylniht ylno era ,tcaf ni ,sweiver ynaM .ylbaredisnoc seirav eseht fo ytilauq eht ,erutaretil eht fo sweiver dellac era tahw ecudorp od stneduts hcraeser hguohtla ,ecitcarp nI .enod ylisae ksat a sa dna suoivbo gnihtemos sa nees netfo si erutaretil fo ydob a fo weiver a gnikatrednU .ecalp tsr ® eht ni enod eb ot sdeen ti yhw ro ,hcraeser eht ni desu eb nac ti woh ,enod eb nac erutaretil detaler fo weiver a woh fo†¦show more content†¦sisylana ycilop seiduts l anoitazina gr o erut aretil yhpargoeg namuh seiduts redneg seiduts latnemnorivne seiduts lanoitacude scimonoce yrotsih laicos dna cimonoce seiduts larutluc sei d uts yt i nummoc seiduts aidem dna noit acinummoc seiduts ssenisub gninnalp nwot dna tnemnorivne tliub .tsil siht ni dedulcni neeb evah thgim ,ecnatsni rof ,ygoloeahcra ;evitsuahxe ton si tsil sihT .woleb detsil senilpicsid eht sedulcni hcihw ,secneics laicos eht nihtiw gnikrow elpoep ta demia si koob sihT .hcraeser eht fo ytilauq eht evorpmi pleh thgim taht syaw gniticxe dna wen ni saedi ezisehtnys ot dna yllacitylana daer ot desu eb nac taht seuqinhcet cireneg fo egnar a ot noitcudortni na edivorp ot edam neeb sah tpmetta na ,hcus sA .erutaretil c ®iceps-cipot a gniweiver fo yhw dna woh eht dnatsrednu ot redro ni detaicerppa eb ot deen taht ssecorp hcraeser eht fo stnemele esoht ot noitcudortni na eb ot dednetni si tI .hcraeser ni gniniart dna noitacude edivorp ohw esoht ot esu fo eb osla yam ti hguohtla ,srehcraeser tneduts rof yliramirp nettirw neeb sah koob sihT .hcraeser ni gniniart dna noitacude eht edivorp ohw esoht fo tluaf eht eb netfo nac sweiver erutaretil roop :ytiliba rieht ni gniliaf a ro tluaf rieht ylirassecen ton si tI .rehcraeser tneduts eht no demalb eb syawla to nnac erutaretil cipot a fo sweiver rooP .wen gnihtemos setubirtnoc hcraeser weiver erutaretil a gnioD 2 .elbissecca ti ekam ot redro ni ,sexedni dna stcartsba sa hcus ,sloot fo esu eht hguorhtShow MoreRelatedSample Literature Review2561 Words   |  11 Pagesexamination of the role of attachment that eating disorders play in adolescents is explored. It is hypothesized that lower quality of attachment to parents in adolescents is associated with higher level of eating disorders. The following six literature reviews attempt to display and support this hypothesis. In a research article by Bachar, Canetti, Hochfdorf, Latzer ( 2002), two questions were addressed to conduct the study. First, is the role of the family environment related to anorexic and bulimicRead MoreChapter 2 Review of Related Literature Sample1295 Words   |  6 PagesCHAPTER 2 REVIEW OF RELATED LITERATURE AND STUDIES FOREIGN STUDIES In its broadest sense, impeachment is the process by which public officials may be removed from office on the basis of their conduct. Strictly speaking, it is the decision by a legislature to accuse an official of one or more offenses that warrant removal according to constitutional standards. A vote to impeach then triggers a trial based on those charges. The most famous impeachment proceedings have involved presidents, butRead MoreThe Development Of App Preceptors Essay977 Words   |  4 PagesThe focus of the evidence in this literature review is to explore the need for and the development of APP preceptors to aid in new hire transition to practice, with a focus on CRNA preceptors. The review will begin broadly by presenting the evidence that supports the need for preceptor program development due to the common themes identified by novice APPs. Then, the need for preceptor development will be discussed. Finally, the review will conclude with current evidence that preceptor deve lopmentRead MoreApplication Of App Preceptor Programs Essay964 Words   |  4 PagesThe focus of the evidence of this literature review is to explore the need for of APP preceptor programs to aid in new hire transition to practice. The review will begin broadly by presenting the evidence that supports the need for preceptor program implementation due to common themes identified by novice APPs. Then, the need for preceptor development will be discussed. Finally, the review will conclude with current evidence that preceptor development is not only needed, but is effective in bridgingRead MoreLiterature Review On Childhood Trauma Essay1113 Words   |  5 Pages Literature Review Article Critique Jocelyn Claudio Widener University September 25, 2016 Overview The authors of this literature review evaluated studies completed on adults who were 50 years or older and also experienced trauma as a child. They reviewed the impact childhood trauma has on their mental and physical health as older adults. They collected findings from 23 studies that were published between 1996 through 2011 and concluded that childhood trauma did in fact have negativeRead MoreAnalysis Of Article Writing Style1068 Words   |  5 Pagesorganization was not clearly delineated. There was an ineffective use of headings. The research question and population sampling were embedded within the text which required the reader to search for these items. There was no Literature Review heading. The review of the literature was discussed within the Introduction section. References were listed throughout the research article. Terms or â€Å"jargon† used was defined and related to public internet social network programs. Definitions were also includedRead MoreComparision Matrix1517 Words   |  7 Pagesread empirical studies can help doctoral learners manage time more effectively (GCU Lecture 2, 2012 p. 1). In this paper, one will find a complete comparison of three articles that will review the different forms of research questions posed for the studies, sample populations used, the limitations, literature review, study conclusions, and recommendations for further research. Article one analyzed Transformational Leadership in the Public Sector: Does Structure Matter, written by Bradley WrightRead MoreA Qualitative Research Methods For Public Service Course847 Words   |  4 Pagesthat is flexible in order to study the group within its own environment (2014). The last of the three designs is case study research design. Case study involves collecting data from a sample of a larger group of similar background within a designated time frame (Baskarada, 2014). The results obtained from the sample group, theoretically represents a larger group of similar background (2014). Case study research design can be used for both qualitative and quantitative research. Quantitative researchRead MoreResearch Study On The Field Of Specialisation1473 Words   |  6 Pagesbecause it is recent, relevant to the research question and conducted in United Kingdom. Aveyard (2014) states that the critique and detailed analysis of an article for review is an important step to addressing the author’s objective for the review. Study Purpose The study’s aim was to assess understanding of UK weaning guidelines in a sample of UK mothers and to ascertain the various sources of weaning advice accessed by first time mothers in the UK (Moore et al, 2014). Parahoo (2014) states that theRead MoreThe Content Management System ( Cms ) Usage1153 Words   |  5 Pagesthe journal titled Information Technology and Libraries (ITAL). ITAL is the official scholarly peer reviewed journal of the Library and Information Technology Association (LITA), a division of the American Library Association (ALA). This journal reviews and publishes articles in the following areas related to library automation, the Internet and other aspects of information technology (American Library Association, 2015). The purpose of this study was to â€Å"examine Content Management System (CMS)

Was William Shakespeare an Author Essay - 1855 Words

Was William Shakespeare an Author? That every word doth almost tell my name / Showing their birth, and where they did proceed... Some might say that this quote from Sonnet 76 eloquently expresses the narrators desire to be heard. This is a normal enough emotion to have. In todays society, people will fight behemently for that right. In Elizabethan times, however, to be heard was not a right at all, but a privilege. The queen, Elizabeth I, had the power to silence any opposition. One could easily see how a verse like the above example could find its inspiration. Some would argue that, with the necessary information, one could just as easily see a darker purpose uncovered, William Shakespeare: Did he exist? There is no doubt†¦show more content†¦In case the traditional view is not familiar, Ill review it briefly here. It states that William Shakespeare was born in 1654 in Stratford-upon-Avon, a small town near London. He married at age 18, had three children, and died in 1616. In that time, he penned at least 1 54 sonnets and 37 plays. He lived a commoners life and was an actor with the Globe Theater in London. His death sparked no comment anywhere. This much is generally agreed upon. To our detriment, very few documents exist pertaining to him. There is no record of his ever being schooled anywhere, no birth records, and no documents written by him. The only examples of his handwriting we have are six signatures, three from his will. Interestingly enough, they are all spelled differently. I will spell it Shakespeare here to avoid confusion, but ... variants such as Shake-speare, Shakspeare, Shaksper, and even Shak--- exist. These inconsistencies were not common, but did exist in 16th century Elizabethan society (Stevens, 1992). No one ever thought to disbelieve his authenticity. His name is indeed recorded as an actor with His Majestys Servants, a popular theater troupe of the time. He did exist. That, however, is not the question at hand. For Mark Twain, the great American writer, the seeds of doubt were planted in his head while traveling to England on a riverboat. He heard the shipmaster talking in a manner entirely unfamiliar to him. TheShow MoreRelatedBiography of William Shakespeare1709 Words   |  7 PagesWilliam Shakespeare: Real or Fake? Introduction Who is the real Shakespeare? There are those who insist that William Shakespeare is the author of the many works attributed to him and reports state that there are those who believe some type of conspiracy exists to protect the real name of the author of those works. Claims state that there is no evidence to document William Shakespeare of Stratford as the author and that he did not have the aristocratic background, education, or knowledge to haveRead MoreThe Greatest Pieces Of English Literature972 Words   |  4 PagesWilliam Shakespeare has written some of the greatest pieces of English literature but some doubt he is the one responsible for it. There are some scholars who believe the conspiracy that Shakespeare did not write the work attributed to him and the true author is someone else, due to the anti-stratfordians who propose theories of who could be the real author, this once small talk has turn into a well- known conspiracy theory, although little to none evidence is all we have on a man named William ShakespeareRead MoreWilliam Shakespeare s A Good And Lasting Legacy822 Words   |  4 PagesWilliam Shakespeare William Shakespeare left a good, lasting legacy that we see almost every day and we don t even know it s there. A legacy is something that you leave behind weather good or bad, Shakespeare s left a legacy of books and being thought as one of the greatest writers in the world. In William’s life he created many pieces of literature and words that people use everywhere today. We may not know a whole lot about William Shakespeare’s life but we can see that he was a veryRead MoreShakespeares Theory Of Sir Francis Bacon Vs. William Shakespeare1437 Words   |  6 PagesBacon was, in fact, William Shakespeare. Supporters of these claims refer to Shakespeare’s writing style, lack of education, and reportedly hidden messages within his texts as some of the evidence that indicates that Bacon was the real author of the many plays and poems attributed to Shakespeare. These assertions do not conclusively substantiate the Bacon theory and only attempt to discredit the real William Shakespeare. Therefore, Sir Francis Bacon did not write as William Shakespeare. The firstRead MoreWilliam Shakespeare s Influence On Modern Culture1090 Words   |  5 PagesPeriod 1 19 November 2015 Shakespeare s Influence on Modern Culture William Shakespeare is one of the world s most influential people to ever live. â€Å"BBC audience survey names Shakespeare as Britain s Man of the Millennium.† (Andrews 2) Shakespeare’s works continue to be evident globally in modern society. Hundreds of years after William Shakespeare’s death, his influence continues to make an effect in the modern day English language, modern movies and film, and authors or artists today. Read MoreWilliam Shakespeare s Romeo And Juliet1733 Words   |  7 PagesShakespeare is a figure shrouded in mystery. In this paper, the title of â€Å"Shakespeare† will refer to the author of the works currently credited to William Shakespeare of Stratford-upon-Avon. Shakespeare is undoubtedly one of the most famous writers of all time. He created masterpieces like Hamlet, Macbeth, Romeo and Juliet, and several more. For a long period of time, William Shakespeare from Stratford-upon-Avon was considered the author of all the works credited to Shakespeare. For over a centuryRead MoreWilliam Shakespeare: A Brief Biography651 Words   |  3 Pages William Shakespeare is arguably the most well known and successful author is the history of literature. Little is known about Shakespeare’s childhood and is what questions he’s existence. Besides the lack of knowledge of his childhood, Shakespeare lived a successful adult life. His plays changed the english language lan guage forever. In all of his success, people still doubt he ever existed. William Shakespeare’s birth is unknown but church records show that he was baptised on April 26, 1564Read MoreWilliam Shakespeare s Life And Work Transcends Time1092 Words   |  5 PagesMany authors have had a lasting impact because of their literary work. During the Elizabethan time period, William Shakespeare began his remarkable career as a playwright changing/impacting blank. William Shakespeare’s life and work transcends time because of their relative relatability. William Shakespeare’s childhood was privileged in some ways. In â€Å"William Shakespeare; Life of Drama,† a documentary produced by by Rod Caird, Shakespeare was christened on April 26th in the year 1564. AccordingRead MoreEssay on Shakespeare Authorship Controversy1504 Words   |  7 PagesShakespeare, the man who wrote 37 plays and more than a hundred sonnets, is known throughout the world. Many people consider him one of the best English playwrights of our time, others say that he was a genius. William Shakspere was born in Stratford-upon Avon in 1564 and died in 1616 at the age of 52. In the mid-19th century, questions had arisen about the Shakespeare authorship controversy, and many scholars wondered whether Shakspere, the man from Stratford, wrote the plays. Ralph W. EmersonRead MoreEssay about The Shakespeare Authorship Debate1632 Words   |  7 PagesThe Shakespeare Authorship Debate Although William Shakespeare is considered to be one of the most revered and well-renowned authors of all time, controversy surrounds the belief that he actually produced his own literary works. Some rumors even go so far as to question the reality of such a one, William Shakespeare, brought on by paralleling the quality of his pieces with his personal background and education. With such farfetched allegations, it persuaded others to peek into the person we all

Physics Notes Free Essays

string(244) " lie in the equatorial plane of the earth because it must accelerate in a plane where the centre of Earth lies since the net orce exerted on the satellite is the Earth’s gravitational force, which is directed towards the centre of Earth\." Gravitation Gravitational field strength at a point is defined as the gravitational force per unit mass at that point. Newton’s law of gravitation: The (mutual) gravitational force F between two point masses M and m separated by a distance r is given by F =| GMm| (where G: Universal gravitational constant)| | r2| | or, the gravitational force of between two point masses is proportional to the product of their masses ; inversely proportional to the square of their separation. Gravitational field strength at a point is the gravitational force per unit mass at that point. We will write a custom essay sample on Physics Notes or any similar topic only for you Order Now It is a vector and its S. I. unit is N kg-1. By definition, g = F / m By Newton Law of Gravitation, F = GMm / r2 Combining, magnitude of g = GM / r2 Therefore g = GM / r2, M = Mass of object â€Å"creating† the field Example 1: Assuming that the Earth is a uniform sphere of radius 6. 4 x 106 m and mass 6. 0 x 1024 kg, find the gravitational field strength g at a point: (a) on the surface, g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / (6. 4 x 106)2 = 9. 77ms-2 (b) at height 0. 50 times the radius of above the Earth’s surface. g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / ( (1. 5 ? 6. 4 x 106)2 = 4. 34ms-2 Example 2: The acceleration due to gravity at the Earth’s surface is 9. 0ms-2. Calculate the acceleration due to gravity on a planet which has the same density but twice the radius of Earth. g = GM / r2 gP / gE = MPrE2 / MErP2 = (4/3) ? rP3rE2? P / (4/3) ? rE3rP2? E = rP / rE = 2 Hence gP = 2 x 9. 81 = 19. 6ms-2 Assuming that Earth is a uniform sphere of mass M. The magnitude of the gravitational force from E arth on a particle of mass m, located outside Earth a distance r from the centre of the Earth is F = GMm / r2. When a particle is released, it will fall towards the centre of the Earth, as a result of the gravitational force with an acceleration ag. FG = mag ag = GM / r2 Hence ag = g Thus gravitational field strength g is also numerically equal to the acceleration of free fall. Example 1: A ship is at rest on the Earth’s equator. Assuming the earth to be a perfect sphere of radius R and the acceleration due to gravity at the poles is go, express its apparent weight, N, of a body of mass m in terms of m, go, R and T (the period of the earth’s rotation about its axis, which is one day). At the North Pole, the gravitational attraction is F = GMEm / R2 = mgo At the equator, Normal Reaction Force on ship by Earth = Gravitational attraction – centripetal force N = mgo – mR? = mgo – mR (2? / T)2 Gravitational potential at a point is defined as the work done (by an external agent) in bringing a unit mass from infinity to that point (without changing its kinetic energy). ? = W / m = -GM / r Why gravitational potential values are always negative? As the gravitational force on the mass is attractive, the work done by an ext ag ent in bringing unit mass from infinity to any point in the field will be negative work {as the force exerted by the ext agent is opposite in direction to the displacement to ensure that ? KE = 0} Hence by the definition of negative work, all values of ? re negative. g = -| d? | = – gradient of ? -r graph {Analogy: E = -dV/dx}| | dr| | Gravitational potential energy U of a mass m at a point in the gravitational field of another mass M, is the work done in bringing that mass m {NOT: unit mass, or a mass} from infinity to that point. ; U = m ? = -GMm / r Change in GPE, ? U = mgh only if g is constant over the distance h; {; h;; radius of planet} otherwise, must use: ? U = m? f-m? i | Aspects| Electric Field| Gravitational Field| 1. | Quantity interacting with or producing the field| Charge Q| Mass M| 2. Definition of Field Strength| Force per unit positive charge E = F / q| Force per unit mass g = F / M| 3. | Force between two Point Charges or Masses| Coulomb’s Law: Fe = Q1Q2 / 4 or2| Newton’s Law of Gravitation: Fg = G (GMm / r2)| 4. | Field Strength of isolated Point Charge or Mass| E = Q / 4 or2| g = G (GM / r2)| 5. | Definition of Potential| Work done in bringing a unit positive charge from infinity to the point; V = W /Q| Work done in bringing a unit mass from infinity to the point; ? = W / M| 6. | Potential of isolated Point Charge or Mass| V = Q / 4 or| ? -G (M / r)| 7. | Change in Potential Energy| ? U = q ? V| ? U = m | Total Energy of a Satellite = GPE + KE = (-GMm / r) + ? (GMm / r) Escape Speed of a Satellite By Conservation of Energy, Initial KE| +| Initial GPE| =| Final KE| +| Final GPE| (? mvE2)| +| (-GMm / r)| =| (0)| +| (0)| Thus escape speed, vE = v(2GM / R) Note : Escape speed of an object is independent of its mass For a satellite in circular orbit, â€Å"the centripetal force is provided by the gravitational force† {Must always state what force is providing the centripetal force before following eqn is used! Henc e GMm / r2 = mv2 / r = mr? 2 = mr (2? / T)2 A satellite does not move in the direction of the gravitational force {ie it stays in its circular orbit} because: the gravitational force exerted by the Earth on the satellite is just sufficient to cause the centripetal acceleration but not enough to also pull it down towards the Earth. {This explains also why the Moon does not fall towards the Earth} Geostationary satellite is one which is always above a certain point on the Earth (as the Earth rotates about its axis. For a geostationary orbit: T = 24 hrs, orbital radius (; height) are fixed values from the centre of the Earth, ang velocity w is also a fixed value; rotates fr west to east. However, the mass of the satellite is NOT a particular value ; hence the ke, gpe, ; the centripetal force are also not fixed values {ie their values depend on the mass of the geostationary satellite. } A geostationary orbit must lie in the equatorial plane of the earth because it must accelerate in a p lane where the centre of Earth lies since the net orce exerted on the satellite is the Earth’s gravitational force, which is directed towards the centre of Earth. You read "Physics Notes" in category "Papers" {Alternatively, may explain by showing why it’s impossible for a satellite in a non-equatorial plane to be geostationary. } Thermal Physics Internal Energy: is the sum of the kinetic energy of the molecules due to its random motion ; the potential energy of the molecules due to the intermolecular forces. Internal energy is determined by the values of the current state and is independent of how the state is arrived at. You can read also Thin Film Solar Cell Thus if a system undergoes a series of changes from one state A to another state B, its change in internal energy is the same, regardless of which path {the changes in the p ; V} it has taken to get from A to B. Since Kinetic Energy proportional to temp, and internal energy of the system = sum of its Kinetic Energy and Potential Energy, a rise in temperature will cause a rise in Kinetic Energy and thus an increase in internal energy. If two bodies are in thermal equilibrium, there is no net flow of heat energy between them and they have the same temperature. NB: this does not imply they must have the same internal energy as internal energy depends also on the number of molecules in the 2 bodies, which is unknown here} Thermodynamic (Kelvin) scale of temperature: theoretical scale that is independent of the properties of any particular substance. An absolute scale of temp is a temp scale which does not depend on the property of any particular substance (ie the thermodynamic scale) Abs olute zero: Temperature at which all substances have a minimum internal energy {NOT: zero internal energy. } T/K = T/ °C + 273. 15, by definition of the Celsius scale. Specific heat capacity is defined as the amount of heat energy needed to produce unit temperature change {NOT: by 1 K} for unit mass {NOT: 1 kg} of a substance, without causing a change in state. c = Q / m? T Specific latent heat of vaporisation is defined as the amount of heat energy needed to change unit mass of a substance from liquid phase to gaseous phase without a change of temperature. Specific latent heat of fusion is defined as the amount of heat energy needed to change unit mass of a substance from solid phase to liquid phase without a change of temperature L = Q / m {for both cases of vaporisation ; melting} The specific latent heat of vaporisation is greater than the specific latent heat of fusion for a given substance because * During vaporisation, there is a greater increase in volume than in fusion, * Thus more work is done against atmospheric pressure during vaporisation, * The increase in vol also means the INCREASE IN THE (MOLECULAR) POTENTIAL ENERGY, ; hence, internal energy, during vaporisation more than that during melting, * Hence by 1st Law of Thermodynamics, heat supplied during vaporisation more than that during melting; hence lv ; lf {since Q = ml = ? U – W}. Note: 1. the use of comparative terms: greater, more, and; 2. the increase in internal energy is due to an increase in the PE, NOT KE of molecules 3. the system here is NOT to be considered as an ideal gas system Similarly, you need to explain why, when a liq is boiling, thermal energy is being supplied, and yet, the temp of the liq does not change. | Melting| Boiling| Evaporation| Occurrence| Throughout the substance, at fixed temperature and pressure| On the surface, at all temperatures| Spacing(vol) ; PE of molecules| Increase slightly| Increase significantly| | Temperature ; hence KE of molecules| Remains constant during process| Decrease for remaining liquid| First Law of Thermodynamics: The increase in internal energy of a system is equal to the sum of the heat supplied to the system and the work done on the system. ?U = W + Q| ? U: Increase in internal energy of the system Q: Heat supplied to the system W: work done on the system| {Need to recall the sign convention for all 3 terms} Work is done by a gas when it expands; work is done on a gas when it is ompressed. W = area under pressure – volume graph. For constant pressure {isobaric process}, Work done = pressure x ? Volume Isothermal process: a process where T = const {? U = 0 for ideal gas} ? U for a cycle = 0 {since U ? T, ; ? T = 0 for a cycle } Equation of state for an ideal gas: p V = n R T, where T is in Kelvin {NOT:  °C}, n: no. of moles. p V = N k T, where N: no. of molecules, k:Boltzmann con st Ideal Gas: a gas which obeys the ideal gas equation pV = nRT FOR ALL VALUES OF P, V ; T Avogadro constant: defined as the number of atoms in 12g of carbon-12. It is thus the number of particles (atoms or molecules) in one mole of substance. For an ideal gas, internal energy U = Sum of the KE of the molecules only {since PE = 0 for ideal gas} U = N x? m ;c2; = N x (3/2)kT {for monatomic gas} * U depends on T and number of molecules N * U ? T for a given number of molecules Ave KE of a molecule, ? m ;c2; ? T {T in K: not  °C} Dynamics Newton’s laws of motion: Newton’s First Law Every body continues in a state of rest or uniform motion in a straight line unless a net (external) force acts on it. Newton’s Second Law The rate of change of momentum of a body is directly proportional to the net force acting on the body, and the momentum change takes place in the direction of the net force. Newton’s Third Law When object X exerts a force on object Y, object Y exerts a force of the same type that is equal in magnitude and opposite in direction on object X. The two forces ALWAYS act on different objects and they form an action-reaction pair. Linear momentum and its conservation: Mass: is a measure of the amount of matter in a body, ; is the property of a body which resists change in motion. Weight: is the force of gravitational attraction (exerted by the Earth) on a body. Linear momentum: of a body is defined as the product of its mass and velocity ie p = m v Impulse of a force (I): is defined as the product of the force and the time ? t during which it acts ie I = F x ? t {for force which is const over the duration ? t} For a variable force, the impulse I = Area under the F-t graph { ? Fdt; may need to â€Å"count squares†} Impulse is equal in magnitude to the change in momentum of the body acted on by the force. Hence the change in momentum of the body is equal in mag to the area under a (net) force-time graph. {Incorrect to define impulse as change in momentum} Force: is defined as the rate of change of momentum, ie F = [ m (v – u) ] / t = ma or F = v dm / dt The {one} Newton: is defined as the force needed to accelerate a mass of 1 kg by 1 m s-2. Principle of Conservation of Linear Momentum: When objects of a system interact, their total momentum before and after interaction are equal if no net (external) force acts on the system. * The total momentum of an isolated system is constant m1 u1 + m2 u2 = m1 v1 + m2 v2 if net F = 0 {for all collisions } NB: Total momentum DURING the interaction/collision is also conserved. (Perfectly) elastic collision: Both momentum ; kinetic energy of the system are conserved. Inelastic collision: Only momentum is conserved, total kinetic energy is not conserved. Perfectly inelastic collision: Only momentum is conserved, and the particles stick togethe r after collision. (i. e. move with the same velocity. ) For all elastic collisions, u1 – u2 = v2 – v1 ie. relative speed of approach = relative speed of separation or, ? m1u12 + ? m2u22 = ? m1v12 + ? 2v22 In inelastic collisions, total energy is conserved but Kinetic Energy may be converted into other forms of energy such as sound and heat energy. Current of Electricity Electric current is the rate of flow of charge. {NOT: charged particles} Electric charge Q passing a point is defined as the product of the (steady) current at that point and the time for which the current flows, Q = I t One coulomb is defined as the charge flowing per second pass a point at which the current is one ampere. Example 1: An ion beam of singly-charged Na+ and K+ ions is passing through vacuum. If the beam current is 20 ? A, calculate the total number of ions passing any fixed point in the beam per second. (The charge on each ion is 1. 6 x 10-19 C. ) Current, I = Q / t = Ne / t where N is the no. of ions and e is the charge on one ion. No. of ions per second = N / t = I / e = (20 x 10-6) / (1. 6 x 10-19) = 1. 25 x 10-14 Potential difference is defined as the energy transferred from electrical energy to other forms of energy when unit charge passes through an electrical device, V = W / Q P. D. = Energy Transferred / Charge = Power / Current or, is the ratio of the power supplied to the device to the current flowing, V = P / I The volt: is defined as the potential difference between 2 pts in a circuit in which one joule of energy is converted from electrical to non-electrical energy when one coulomb passes from 1 pt to the other, ie 1 volt = One joule per coulomb Difference between Potential and Potential Difference (PD): The potential at a point of the circuit is due to the amount of charge present along with the energy of the charges. Thus, the potential along circuit drops from the positive terminal to negative terminal, and potential differs from points to points. Potential Difference refers to the difference in potential between any given two points. For example, if the potential of point A is 1 V and the potential at point B is 5 V, the PD across AB, or VAB , is 4 V. In addition, when there is no energy loss between two points of the circuit, the potential of these points is same and thus the PD across is 0 V. Example 2: A current of 5 mA passes through a bulb for 1 minute. The potential difference across the bulb is 4 V. Calculate: (a) The amount of charge passing through the bulb in 1 minute. Charge Q = I t = 5 x 10-3 x 60 = 0. 3 C (b) The work done to operate the bulb for 1 minute. Potential difference across the bulb = W / Q 4 = W / 0. Work done to operate the bulb for 1 minute = 0. 3 x 4 = 1. 2 J Electrical Power, P = V I = I2 / R = V2 / R {Brightness of a lamp is determined by the power dissipated, NOT: by V, or I or R alone} Example 3: A high-voltage transmission line with a resistance of 0. 4 ? km-1 carries a current of 500 A. The line is at a potential of 1200 kV at the power station and carries the current to a city lo cated 160 km from the power station. Calculate (a) the power loss in the line. The power loss in the line P = I2 R = 5002 x 0. 4 x 160 = 16 MW (b) the fraction of the transmitted power that is lost. The total power transmitted = I V = 500 x 1200 x 103 = 600 MW The fraction of power loss = 16 / 600 = 0. 267 Resistance is defined as the ratio of the potential difference across a component to the current flowing through it , R = VI {It is NOT defined as the gradient of a V-I graph; however for an ohmic conductor, its resistance equals the gradient of its V-I graph as this graph is a straight line which passes through the origin} The Ohm: is the resistance of a resistor if there is a current of 1 A flowing through it when the pd across it is 1 V, ie, 1 ? = One volt per ampere Example 4: In the circuit below, the voltmeter reading is 8. 00 V and the ammeter reading is 2. 00 A. Calculate the resistance of R. Resistance of R = V / I = 8 / 2 = 4. 0 ? | | Temperature characteristics of thermistors: The resistance (i. e. the ratio V / I) is constant because metallic conductors at constant temperature obey Ohm’s Law. | As V increases, the temperature increases, resulting in an increase in the amplitude of vibration of ions and the collision frequency of electrons with the lattice ions. Hence the resistance of the filament increases with V. | A thermistor is made from semi-conductors. As V increases, temperature increases. This releases more charge carriers (electrons and holes) from the lattice, thus reducing the resistance of the thermistor. Hence, resistance decreases as temperature increases. | In forward bias, a diode has low resistance. In reverse bias, the diode has high resistance until the breakdown voltage is reached. | Ohm’s law: The current in a component is proportional to the potential difference across it provided physical conditions (eg temp) stay constant. R = ? L / A {for a conductor of length l, uniform x-sect area A and resistivity ? Resistivity is defined as the resistance of a material of unit cross-sectional area and unit length. {From R = ? l / A , ? = RA / L} Example 5: Calculate the resistance of a nichrome wire of length 500 mm and diameter 1. 0 mm, given that the resistivity of nichrome is 1. 1 x 10-6 ? m. Resistance, R = ? l / A = [(1. 1 x 10-6)(500 x 10-3)] / ? (1 x 10-3 / 2)2 = 0. 70 ? Electromotive force (Emf) is defined as t he energy transferred / converted from non-electrical forms of energy into electrical energy when unit charge is moved round a complete circuit. ie EMF = Energy Transferred per unit charge E = WQ EMF refers to the electrical energy generated from non-electrical energy forms, whereas PD refers to electrical energy being changed into non-electrical energy. For example, EMF Sources| Energy Change| PD across| Energy Change| Chemical Cell| Chem ; Elec| Bulb| Elec ; Light| Generator| Mech ; Elec| Fan| Elec ; Mech| Thermocouple| Thermal ; Elec| Door Bell| Elec ; Sound| Solar Cell| Solar ; Elec| Heating element| Elec ; Thermal| Effects of the internal resistance of a source of EMF: Internal resistance is the resistance to current flow within the power source. It reduces the potential difference (not EMF) across the terminal of the power supply when it is delivering a current. Consider the circuit below: The voltage across the resistor, V = IR, The voltage lost to internal resistance = Ir Thus, the EMF of the cell, E = IR + Ir = V + Ir Therefore If I = 0A or if r = 0? , V = E Motion in a Circle Kinematics of uniform circular motion Radian (rad) is the S. I. unit for angle, ? and it can be related to degrees in the following way. In one complete revolution, an object rotates through 360 ° , or 2? rad. As the object moves through an angle ? , with respect to the centre of rotation, this angle ? s known as the angular displacement. Angular velocity (? ) of the object is the rate of change of angular displacement with respect to time. ? = ? / t = 2? / T (for one complete revolution) Linear velocity, v, of an object is its instantaneous velocity at any point in its circular path. v = arc length / time taken = r? / t = r? * The direction of th e linear velocity is at a tangent to the circle described at that point. Hence it is sometimes referred to as the tangential velocity * ? is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis. A body moving in a circle at a constant speed changes velocity {since its direction changes}. Thus, it always experiences an acceleration, a force and a change in momentum. Centripetal acceleration a = r? 2 = v2 / r {in magnitude} Centripetal force Centripetal force is the resultant of all the forces that act on a system in circular motion. {It is not a particular force; â€Å"centripetal† means â€Å"centre-seeking†. Also, when asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as â€Å"centripetal force†. } Centripetal force, F = m r ? 2 = mv2 / r {in magnitude} A person in a satellite orbiting the Earth experiences â€Å"weightlessness† although the gravi field strength at that height is not zero because the person and the satellite would both have the same acceleration; hence the contact force between man ; satellite / normal reaction on the person is zero {Not because the field strength is negligible}. D. C. Circuits Circuit Symbols: Open Switch| Closed Switch| Lamp| Cell| Battery| Voltmeter| Resistor| Fuse| Ammeter| Variable resistor| Thermistor| Light dependent resistor (LDR)| Resistors in Series: R = R1 + R2 + †¦ Resistors in Parallel: 1/R = 1/R1 + 1/R2 + †¦ Example 1: Three resistors of resistance 2 ? , 3 ? and 4 ? respectively are used to make the combinations X, Y and Z shown in the diagrams. List the combinations in order of increasing resistance. Resistance for X = [1/2 + 1/(4+3)]-1 = 1. 56 ? Resistance for Y = 2 + (1/4 + 1/3)-1 = 3. 71 ? Resistance for Z = (1/3 + 1/2 + 1/4)-1 = 0. 923 ? Therefore, the combination of resistors in order of increasing resistance is Z X Y. Example: Referring to the circuit drawn, determine the value of I1, I and R, the combined resistance in the circuit. E = I1 (160) = I2 (4000) = I3 (32000) I1 = 2 / 160 = 0. 0125 A I2 = 2 / 4000 = 5 x 10-4 A I3 = 2 / 32000 = 6. 25 x 10-5 ASince I = I1 + I2 + I3, I = 13. 1 mAApplying Ohm’s Law, R = 213. 1 x 10-3 = 153 ? | | Example: A battery with an EMF of 20 V and an internal resistance of 2. 0 ? is connected to resistors R1 and R2 as shown in the diagram. A total current of 4. 0 A is supplied by the battery and R2 has a resistance of 12 ?. Calculate the resistance of R1 and the power supplied to each circuit component. E – I r = I2 R2 20 – 4 (2) = I2 (12) I2 = 1A Therefore, I1 = 4 – 1 = 3 AE – I r = I1 R1 12 = 3 R1 Therefore, R1 = 4Power supplied to R1 = (I1)2 R1 = 36 W Power supplied to R2 = (I2)2 R2 = 12 W| | For potential divider with 2 resistors in series, Potential drop across R1, V1 = R1 / (R1 + R2) x PD across R1 ; R2 Potential drop across R2, V1 = R2 / (R1 + R2) x PD across R1 ; R2 Example: Two resistors, of resistance 300 k? and 500 k? respectively, form a potential divider with outer junctions maintained at potentials of +3 V and -15 V. Determine the potential at the junction X between the resistors. The potential difference across the 300 k? resistor = 300 / (300 + 500) [3 – (-15)] = 6. 75 V The potential at X = 3 – 6. 75 = -3. 75 V A thermistor is a resistor whose resistance varies greatly with temperature. Its resistance decreases with increasing temperature. It can be used in potential divider circuits to monitor and control temperatures. Example: In the figure on the right, the thermistor has a resistance of 800 ? when hot, and a resistance of 5000 ? when cold. Determine the potential at W when the temperature is hot. When thermistor is hot, potential difference across it = [800 / (800 + 1700)] x (7 – 2) = 1. 6 VThe potential at W = 2 + 1. 6 V = 3. 6 V| | A Light dependent resistor (LDR) is a resistor whose resistance varies with the intensity of light falling on it. Its resistance decreases with increasing light intensity. It can be used in a potential divider circuit to monitor light intensity. Example: In the figure below, the resistance of the LDR is 6. 0 M in the dark but then drops to 2. 0 k in the light Determine the potential at point P when the LDR is in the light. In the light the potential difference across the LDR= [2k / (3k + 2k)] x (18 – 3) = 6 VThe potential at P = 18 – 6= 12 V| | The potential difference along the wire is proportional to the length of the wire. The sliding contact will move along wire AB until it finds a point along the wire such that the galvanometer shows a zero reading. When the galvanometer shows a zero reading, the current through the galvanometer (and the device that is being tested) is zero and the potentiometer is said to be â€Å"balanced†. If the cell has negligible internal resistance, and if the potentiometer is balanced, EMF / PD of the unknown source, V = [L1 / (L1 + L2)] x E Example: In the circuit shown, the potentiometer wire has a resistance of 60 ?. Determine the EMF of the unknown cell if the balanced point is at B. Resistance of wire AB= [0. 65 / (0. 65 + 0. 35)] x 60 = 39 ? EMF of the test cell= [39 / (60 + 20)] x 12| Work, Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ? Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by â€Å"counting the squares†. } By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction} Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation: v2 = u2 +2as, as = 12 (v2 – u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 – u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them. Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = – dU / dx. Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h| Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| Useful Output Energy| x100% =| Useful Output Power| x100%| | Input Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total Time| | t| Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object moving at const speed: F = Total resistive force {equilibrium condition} * for object beginning to accelerate: F = Total resistive force + ma Forces Hooke’s Law: Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain energy = Area under the F-x graph {May need to â€Å"count the squares†} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? x2 Forces on Masses in Gravitational Fields: A region of space in which a mass experiences an (attractive) force due to the presence of another mass. Forces on Charge in Electric Fields: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the upper and lower surfaces of the object. Archimedes’ Principle: Upthrust = weight of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: * The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a maximum value {called the static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * Only exists when there is (relative) motion Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act. A couple is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple: The produce of one of the force s of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of Moments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Measurement Base quantities and their units; mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol): Base Quantities| SI Units| | Name| Symbol| Length| metre| m| Mass| kilogram| kg| Time| second| s| Amount of substance| mole| mol| Temperature| Kelvin| K| Current| ampere| A| Luminous intensity| candela| cd| Derived units as products or quotients of the base units: Derived| Quantities Equation| Derived Units| Area (A)| A = L2| m2| Volume (V)| V = L3| m3| Density (? )| ? = m / V| kg m-3| Velocity (v)| v = L / t| ms-1| Acceleration (a)| a = ? v / t| ms-1 / s = ms-2| Momentum (p)| p = m x v| (kg)(ms-1) = kg m s-1| Derived Quantities| Equation| Derived Unit| Derived Units| | | Special Name| Symbol| | Force (F)| F = ? p / t| Newton| N| [(kg m s-1) / s = kg m s-2| Pressure (p)| p = F / A| Pascal| Pa| (kg m s-2) / m2 = kg m-1 s-2| Energy (E)| E = F x d| joule| J| (kg m s-2)(m) = kg m2 s-2| Power (P)| P = E / t| watt| W| (kg m2 s-2) / s = kg m2 s-3| Frequency (f)| f = 1 / t| hertz| Hz| 1 / s = s-1| Charge (Q)| Q = I x t| coulomb| C| A s| Potential Difference (V)| V = E / Q| volt| V| (kg m2 s-2) / A s = kg m2 s-3 A-1| Resistance (R)| R = V / I| ohm| ? (kg m2 s-3 A-1) / A = kg m2 s-3 A-2| Prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: Multiplying Factor| Prefix| Symbol| 10-12| pico| p| 10-9| nano| n| 10-6| micro| ? | 10-3| milli| m| 10-2| centi| c| 10-1| decid| d| 103| kilo| k| 106| mega| M| 109| giga| G| 1012| tera| T| Estimates of physical quantities: When making an estimate, it is only reason able to give the figure to 1 or at most 2 significant figures since an estimate is not very precise. Physical Quantity| Reasonable Estimate| Mass of 3 cans (330 ml) of Coke| 1 kg| Mass of a medium-sized car| 1000 kg| Length of a football field| 100 m| Reaction time of a young man| 0. 2 s| * Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used. (eg. Topic 3 (Dynamics), N94P2Q1c) * Often, when making an estimate, a formula and a simple calculation may be involved. EXAMPLE 1: Estimate the average running speed of a typical 17-year-old? s 2. 4-km run. velocity = distance / time = 2400 / (12. 5 x 60) = 3. 2 ? 3 ms-1 EXAMPLE 2: Which estimate is realistic? | Option| Explanation| A| The kinetic energy of a bus travelling on an expressway is 30000J| A bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13. 8 to 22. 2 ms-1. Thus, its KE will be approximately ? m(182) = 162m. Thus, for its KE to be 30000J: 162m = 30000. Thus, m = 185kg, which is an absurd weight for a bus; ie. This is not a realistic estimate. | B| The power of a domestic light is 300W. | A single light bulb in the house usually runs at about 20W to 60W. Thus, a domestic light is unlikely to run at more than 200W; this estimate is rather high. | C| The temperature of a hot oven is 300 K. 300K = 27 0C. Not very hot. | D| The volume of air in a car tyre is 0. 03 m3. | | Estimating the width of a tyre, t, is 15 cm or 0. 15 m, and estimating R to be 40 cm and r to be 30 cm,volume of air in a car tyre is = ? (R2 – r2)t = ? (0. 42 – 0. 32)(0. 15) = 0. 033 m3 ? 0. 03 m3 (to one sig. fig. )| Distinction between systematic errors (including zero errors) an d random errors and between precision and accuracy: Random error: is the type of error which causes readings to scatter about the true value. Systematic error: is the type of error which causes readings to deviate in one direction from the true value. Precision: refers to the degree of agreement (scatter, spread) of repeated measurements of the same quantity. {NB: regardless of whether or not they are correct. } Accuracy: refers to the degree of agreement between the result of a measurement and the true value of the quantity. | ; ; R Error Higher ; ; ; ; ; ; Less Precise ; ; ;| v v vS Error HigherLess Accuratev v v| | | | | | Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required). For a quantity x = (2. 0  ± 0. 1) mm, Actual/ Absolute uncertainty, ? x =  ± 0. 1 mm Fractional uncertainty, ? xx = 0. 05 Percentage uncertainty, ? xx 100% = 5 % If p = (2x + y) / 3 or p = (2x – y) / 3, ? p = (2? x + ? y) / 3 If r = 2xy3 or r = 2x / y3, ? r / r = ? x / x + 3? y / y Actual error must be recorded to only 1 significant figure, ; The number of decimal places a calculated quantity should have is determined by its actual error. For eg, suppose g has been initially calculated to be 9. 80645 ms-2 ; ? g has been initially calculated to be 0. 04848 ms-2. The final value of ? g must be recorded as 0. 5 ms-2 {1 sf }, and the appropriate recording of g is (9. 81  ± 0. 05) ms-2. Distinction between scalar and vector quantities: | Scalar| Vector| Definition| A scalar quantity has a magnitude only. It is completely described by a certain number and a unit. | A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-hea d represents the direction of the vector. | Examples| Distance, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc. Common Error: Students tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar. | Displacement, velocity, moments (or torque), momentum, force, electric field etc. | Representation of vector as two perpendicular components: In the diagram below, XY represents a flat kite of weight 4. 0 N. At a certain instant, XY is inclined at 30 ° to the horizontal and the wind exerts a steady force of 6. 0 N at right angles to XY so that the kite flies freely. By accurate scale drawing| By calculations using sine and cosine rules, or Pythagoras? theorem| Draw a scale diagram to find the magnitude and direction of the resultant force acting on the kite. R = 3. 2 N (? 3. 2 cm) at ? = 112 ° to the 4 N vector. | Using cosine rule, a2 = b2 + c2 – 2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30 °) R = 3. 23 NUsing sine rule: a / sin A = b / sin B 6 / sin ? = 3. 23 / sin 30 ° ? = 68 ° or 112 ° = 112 ° to the 4 N vector| Summing Vector Components| | Fx = – 6 sin 30 ° = – 3 NFy = 6 cos 30 ° – 4 = 1. 2 NR = v(-32 + 1. 22) = 3. 23 Ntan ? = 1. 2 / 3 = 22 °R is at an angle 112 ° to the 4 N vector. (90 ° + 22 °)| Kinematics Displacement, speed, velocity and acceleration: Distance: Total length covered irrespective of the direction of motion. Displacement: Distance moved in a certain direction. Speed: Distance travelled per unit time. Velocity: is defined as the rate of change of displacement, or, displacement per unit time {NOT: displacement over time, nor, displacement per second, nor, rate of change of displacement per unit time} Acceleration: is defined as the rate of change of velocity. Using graphs to find displacement, velocity and acceleration: * The area under a velocity-time graph is the change in displacement. The gradient of a displacement-time graph is the {instantaneous} velocity. * The gradient of a velocity-time graph is the acceleration. The ‘SUVAT’ Equations of Motion The most important word for this chapter is SUVAT, which stands for: * S (displacement), * U (initial velocity), * V (final velocity), * A (acceleration) and * T (time) of a particle that is in moti on. Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly. 1. v = u +at| derived from definition of acceleration: a = (v – u) / t| 2. | s = ? (u + v) t| derived from the area under the v-t graph| 3. | v2 = u2 + 2as| derived from equations (1) and (2)| 4. | s = ut + ? at2| derived from equations (1) and (2)| These equations apply only if the motion takes place along a straight line and the acceleration is constant; {hence, for eg. , air resistance must be negligible. } Motion of bodies falling in a uniform gravitational field with air resistance: Consider a body moving in a uniform gravitational field under 2 different conditions: Without Air Resistance: Assuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free fall. With Air Resistance: If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate greater than . 81 ms-2 . This causes the time taken to reach its maximum height reached to be lower than in the case with no air resistance. The max height reached is also reduced. At the highest point, the body is momentarily at rest; air resistance becomes zero and hence the only force acting on it is the weight. The acceleration is thus 9. 81 ms-2 at this point. As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9. 81 ms-2. As air resistance increases with speed, it eventually equals its weight (but in opposite direction). From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity. Equations for the horizontal and vertical motion: | x direction (horizontal – axis)| y direction (vertical – axis)| s (displacement)| sx = ux t sx = ux t + ? ax t2| sy = uy t + ? ay t2 (Note: If projectile ends at same level as the start, then sy = 0)| u (initial velocity)| ux| uy| v (final velocity)| vx = ux + axt (Note: At max height, vx = 0)| vy = uy + at vy2 = uy2 + 2asy| a (acceleration)| ax (Note: Exists when a force in x direction present)| ay (Note: If object is falling, then ay = -g)| (time)| t| t| Parabolic Motion: tan ? = vy / vx ?: direction of tangential velocity {NOT: tan ? = sy / sx } Forces Hooke’s Law: Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain ener gy = Area under the F-x graph {May need to â€Å"count the squares†} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? k x2 Forces on Masses in Gravitational Fields: A region of space in which a mass experiences an (attractive) force due to the presence of another mass. Forces on Charge in Electric Fields: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the upper and lower surfaces of the object. Archimedes’ Principle: Upthrust = weight of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a maximum value {called the static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * Only exists when the re is (relative) motion * Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act. A couple is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple: The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of Moments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Work, Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ? Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by â€Å"counting the squares†. } By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction} Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation: 2 = u2 +2as, as = 12 (v2 – u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 – u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them. Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = – dU / dx. Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h| Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| Useful Output Energy| x100% =| Useful Output Power| x100%| | Input Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total Time| | t| Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object moving at const speed: F = Total resistive force {equilibrium condition} * for object beginning to accelerate: F = Total resistive force + ma Wave Motion Displacement (y): Position of an oscillating particle from its equilibrium position. Amplitude (y0 or A): The maximum magnitude of the displacement of an oscillating particle from its equilibrium position. Period (T): Time taken for a particle to undergo one complete cycle of oscillation. Frequency (f): Number of oscillations performed by a particle per unit time. Wavelength (? ): For a progressive wave, it is the distance between any two successive particles that are in phase, e. g. it is the distance between 2 consecutive crests or 2 troughs. Wave speed (v): The speed at which the waveform travels in the direction of the propagation of the wave. Wave front: A line or surface joining points which are at the same state of oscillation, i. e. in phase, e. g. a line joining crest to crest in a wave. Ray: The path taken by the wave. This is used to indicate the direction of wave propagation. Rays are always at right angles to the wave fronts (i. e. wave fronts are always perpendicular to the direction of propagation). From the definition of speed, Speed = Distance / Time A wave travels a distance of one wavelength, ? , in a time interval of one period, T. The frequency, f, of a wave is equal to 1 / T Therefore, speed, v = ? / T = (1 / T)? f? v = f? Example 1: A wave travelling in the positive x direction is showed in the figure. Find the amplitude, wavelength, period, and speed of the wave if it has a frequency of 8. 0 Hz. Amplitude (A) = 0. 15 mWavelength (? ) = 0. 40 mPeriod (T) = 1f = 18. 0 ? 0. 125 sSpeed (v) =f? = 8. 0 x 0. 40 = 3. 20 m s-1A wave which results in a net transfer of energy from one place to another is known as a progressive wave. | | Intensity {of a wave}: is defined as the rate of energy flow per unit time {power} per unit cross-sectional area perpendicular to the direction of wave propagation. Intensity = Power / Area = Energy / (Time x Area) For a point source (which would emit spherical wavefronts), Intensity = (? m? 2xo2) / (t x 4? r2) where x0: amplitude ; r: distance from the point source. Therefore, I ? xo2 / r2 (Pt Source) For all wave sources, I ? (Amplitude)2 Transverse wave: A wave in which the oscillations of the wave particles {NOT: movement} are perpendicular to the direction of the propagation of the wave. Longitudinal wave: A wave in which the oscillations of the wave particles are parallel to the direction of the propagation of the wave. Polarisation is said to occur when oscillations are in one direction in a plane, {NOT just â€Å"in one direction†} normal to the direction of propagation. {Only transverse waves can be polarized; longitudinal waves can’t. }Example 2: The following stationary wave pattern is obtained using a C. R. O. whose screen is graduated in centimetre squares. Given that the time-base is adjusted such that 1 unit on the horizontal axis of the screen corresponds to a time of 1. 0 ms, find the period and frequency of the wave. Period, T = (4 units) x 1. 0 = 4. 0 ms = 4. 0 x 10-3 sf = 1 / T = 14 x 10-3 250 Hz| | Superposition Principle of Superposition: When two or more waves of the same type meet at a point, the resultant displacement of the waves is equal to the vector sum of their individual displacements at that point. Stretched String A horizontal rope with one end fixed and another attached to a vertical oscillator. Stationary waves will be produced by the direct and reflected w aves in the string. Or we can have the string stopped at one end with a pulley as shown below. Microwaves A microwave emitter placed a distance away from a metal plate that reflects the emitted wave. By moving a detector along the path of the wave, the nodes and antinodes could be detected. Air column A tuning fork held at the mouth of a open tube projects a sound wave into the column of air in the tube. The length of the tube can be changed by varying the water level. At certain lengths of the tube, the air column resonates with the tuning fork. This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface. Stationary (Standing) Wave) is one * whose waveform/wave profile does not advance {move}, where there is no net transport of energy, and * where the positions of antinodes and nodes do not change (with time). A stationary wave is formed when two progressive waves of the same frequency, amplitude and speed, travelling in opposite directions are superposed. {Assume boundary conditions are met} | Stationary waves| Stationary Waves Progressive Waves| Amplitude| Varies from maximum at the anti-nodes to zero at the nodes. | Same for all particles in the wave (provided no energy is lost). | Wavelength| Twice the distance between a pair of adjacent nodes or anti-nodes. The distance between two consecutive points on a wave, that are in phase. | Phase| Particles in the same segment/ between 2 adjacent nodes, are in phase. Particles in adjacent segments are in anti-phase. | All particles within one wavelength have different phases. | Wave Profile| The wave profile does not advance. | The wave profile advances. | Energy| No energy is transported by the wave. | Energy is transported in the direction of the wave. | Node is a region of destructive superposition where the waves always meet out of phase by ? radians. Hence displacement here is permanently zero {or minimum}. Antinode is a region of constructive superposition where the waves always meet in phase. Hence a particle here vibrates with maximum amplitude {but it is NOT a pt with a permanent large displacement! } Dist between 2 successive nodes / antinodes = ? / 2 Max pressure change occurs at the nodes {NOT the antinodes} because every node changes fr being a pt of compression to become a pt of rarefaction {half a period later} Diffraction: refers to the spreading {or bending} of waves when they pass through an opening {gap}, or round an obstacle (into the â€Å"shadow† region). Illustrate with diag} For significant diffraction to occur, the size of the gap ? ? of the wave For a diffraction grating, d sin ? = n ? , d = dist between successive slits {grating spacing} = reciprocal of number of lines per metre When a â€Å"white light† passes through a diffraction grating, for each order of diffraction, a longer wavelength {red} diffracts more than a shorter wavelength {violet} {as sin ? ? ? }. Diffraction refers to the spreading of waves as they pass through a narrow slit or near an obstacle. For diffraction to occur, the size of the gap should approximately be equal to the wavelength of the wave. Coherent waves: Waves having a constant phase difference {not: zero phase difference / in phase} Interference may be described as the superposition of waves from 2 coherent sources. For an observable / well-defined interference pattern, the waves must be coherent, have about the same amplitude, be unpolarised or polarised in the same direction, ; be of the same type. Two-source interference using: 1. Water Waves Interference patterns could be observed when two dippers are attached to the vibrator of the ripple tank. The ripples produce constructive and destructive interference. The dippers are coherent sources because they are fixed to the same vibrator. 2. Microwaves Microwave emitted from a transmitter through 2 slits on a metal plate would also produce interference patterns. By moving a detector on the opposite side of the metal plate, a series of rise and fall in amplitude of the wave would be registered. 3. Light Waves (Young? s double slit experiment) Since light is emitted from a bulb randomly, the way to obtain two coherent light sources is by splitting light from a single slit. The 2 beams from the double slit would then interfere with each other, creating a pattern of alternate bright and dark fringes (or high and low intensities) at regular intervals, which is also known as our interference pattern. Condition for Constructive Interference at a pt P: Phase difference of the 2 waves at P = 0 {or 2? , 4? , etc} Thus, with 2 in-phase sources, * implies path difference = n? ; with 2 antiphase sources: path difference = (n + ? )? Condition for Destructive Interference at a pt P: Phase difference of the 2 waves at P = ? { or 3? , 5? , etc } With 2 in-phase sources, + implies path difference = (n+ ? ), with 2 antiphase sources: path difference = n ? Fringe separation x = ? D / a, if a;;D {applies only to Young’s Double Slit interference of light, ie, NOT for microwaves, sound waves, water waves} Phase difference betw the 2 waves at any pt X {betw the central 1st maxima) is (approx) proportional to the dist of X from the central maxima. Using 2 sources of equal amplitude x0, the resultant amplitude of a bright fringe would be doubled {2Ãâ€"0}, the resultant intensity increases by 4 times {not 2 times}. { IResultant ? (2 x0)2 } Electric Fields Electric field strength / intensity at a point is defined as the force per unit positive charge acting at that point {a vector; Unit: N C-1 or V m-1} E = F / q F = qE * The electric force on a positive charge in an electric field is in the direction of E, while * The electric force on a negative charge is opposite to the direction of E. * Hence a +ve charge placed in an electric field will accelerate in the direction of E and gain KE { simultaneously lose EPE}, while a negative charge caused to move (projected) in the direction of E will decelerate, ie lose KE, { gain EPE}. Representation of electric fields by field lines | | | | | Coulomb’s law: The (mutual) electric force F acting between 2 point charges Q1 and Q2 separated by a distance r is given by: F = Q1Q2 / 4 or2 where ? 0: permittivity of free space or, the (mutual) electric force between two point charges is proportional to the product of their charges ; inversely proportional to the square of their separation. Exa mple 1: Two positive charges, each 4. 18 ? C, and a negative charge, -6. 36 ? C, are fixed at the vertices of an equilateral triangle of side 13. 0 cm. Find the electrostatic force on the negative charge. | F = Q1Q2 / 4 or2= (8. 99 x 109) [(4. 18 x 10-6)(6. 6 x 10-6) / (13. 0 x 10-2)2]= 14. 1 N (Note: negative sign for -6. 36 ? C has been ignored in the calculation)FR = 2 x Fcos300= 24. 4 N, vertically upwards| Electric field strength due to a Point Charge Q : E = Q / 4 or2 {NB: Do NOT substitute a negative Q with its negative sign in calculations! } Example 2: In the figure below, determine the point (other than at infinity) at which the total electric field strength is zero. From the diagram, it can be observed that the point where E is zero lies on a straight line where the charges lie, to the left of the -2. 5 ? C charge. Let this point be a distance r from the left charge. Since the total electric field strength is zero, E6? = E-2? [6? / (1 + r)2] / 4 or2 = [2. 5? / r2] / 4 or2 (Note: negative sign for -2. 5 ? C has been ignored here) 6 / (1 + r)2 = 2. 5 / r2 v(6r) = 2. 5 (1 + r) r = 1. 82 m The point lies on a straight line where the charges lie, 1. 82 m to the left of the -2. 5 ? C charge. Uniform electric field between 2 Charged Parallel Plates: E = Vd, d: perpendicular dist between the plates, V: potential difference between plates Path of charge moving at 90 ° to electric field: parabolic. Beyond the pt where it exits the field, the path is a straight line, at a tangent to the parabola at exit. Example 3: An electron (m = 9. 11 x 10-31 kg; q = -1. 6 x 10-19 C) moving with a speed of 1. 5 x 107 ms-1, enters a region between 2 parallel plates, which are 20 mm apart and 60 mm long. The top plate is at a potential of 80 V relative to the lower plate. Determine the angle through which the electron has been deflected as a result of passing through the plates. Time taken for the electron to travel 60 mm horizontally = Distance / Speed = 60 x 10-3 / 1. 5 x 107 = 4 x 10-9 s E = V / d = 80 / 20 x 10-3 = 4000 V m-1 a = F / m = eE / m = (1. 6 x 10-19)(4000) / (9. 1 x 10-31) = 7. 0 x 1014 ms-2 vy = uy + at = 0 + (7. x 1014)( 4 x 10-9) = 2. 8 x 106 ms-1 tan ? = vy / vx = 2. 8 x 106 / 1. 5 x 107 = 0. 187 Therefore ? = 10. 6 ° Effect of a uniform electric field on the motion of charged particles * Equipotential surface: a surface where the electric potential is constant * Potential gradient = 0, ie E along surface = 0 } * Hence no work is done when a charge is moved along this surface. { W=QV, V=0 } * Electric field lines must meet this surface at right angles. * {If the field lines are not at 90 ° to it, it would imply that there is a non-zero component of E along the surface. This would contradict the fact that E along an equipotential = 0. Electric potential at a point: is defined as the work done in moving a unit positive charge from infinity to that point, { a scalar; unit: V } ie V = W / Q The electric potential at infinity is defined as zero. At any other point, it may be positive or negative depending on the sign of Q that sets up the field. {Contrast gravitational potential. } Relation between E and V: E = – dV / dr i. e. The electric field strength at a pt is numerically equal to the potential gradient at that pt. NB: Electric field lines point in direction of decreasing potential {ie from high to low pot}. Electric potential energy U of a charge Q at a pt where the potential is V: U = QV Work done W on a charge Q in moving it across a pd ? V: W = Q ? V Electric Potential due to a point charge Q : V = Q / 4 or {NB: Substitute Q with its sign} Electromagnetism When a conductor carrying a current is placed in a magnetic field, it experiences a magnetic force. The figure above shows a wire of length L carrying a current I and lying in a magnetic field of flux density B. Suppose the angle between the current I and the field B is ? , the magnitude of the force F on the conductor is iven by F = BILsin? The direction of the force can be found using Fleming? s Left Hand Rule (see figure above). Note that the force is always perpendicular to the plane containing both the current I and the magnetic field B. * If the wire is parallel to the field lines, then ? = 0 °, and F = 0. (No magnetic force acts on the wire) * If the wire is at right angles to the field lines, then ? = 90 °, and the magn etic force acting on the wire would be maximum (F = BIL) Example The 3 diagrams below each show a magnetic field of flux density 2 T that lies in the plane of the page. In each case, a current I of 10 A is directed as shown. Use Fleming’s Left Hand Rule to predict the directions of the forces and work out the magnitude of the forces on a 0. 5 m length of wire that carries the current. (Assume the horizontal is the current) | | | F = BIL sin? = 2 x 10 x 0. 5 x sin90 = 10 N| F = BIL sin? = 2 x 10 x 0. 5 x sin60 = 8. 66 N| F = BIL sin ? = 2 x 10 x 0. 5 x sin180 = 0 N| Magnetic flux density B is defined as the force acting per unit current in a wire of unit length at right-angles to the field B = F / ILsin ? F = B I L sin ? {? Angle between the B and L} {NB: write down the above defining equation define each symbol if you’re not able to give the â€Å"statement form†. } Direction of the magnetic force is always perpendicular to the plane containing the current I and B {even if ? ? 0} The Tesla is defined as the magnetic flux density of a magnetic field that causes a force of one newton to act on a current of one ampere in a wire o f length one metre which is perpendicular to the magnetic field. By the Principle of moments, Clockwise moments = Anticlockwise moments mg †¢ x = F †¢ y = BILsin90 †¢ y B = mgx / ILy Example A 100-turn rectangular coil 6. 0 cm by 4. 0 cm is pivoted about a horizontal axis as shown below. A horizontal uniform magnetic field of direction perpendicular to the axis of the coil passes through the coil. Initially, no mass is placed on the pan and the arm is kept horizontal by adjusting the counter-weight. When a current of 0. 50 A flows through the coil, equilibrium is restored by placing a 50 mg mass on the pan, 8. 0 cm from the pivot. Determine the magnitude of the magnetic flux density and the direction of the current in the coil. Taking moments about the pivot, sum of Anti-clockwise moments = Clockwise moment (2 x n)(FB) x P = W x Q (2 x n)(B I L) x P = m g x Q, where n: no. of wires on each side of the coil (2 x 100)(B x 0. 5 x 0. 06) x 0. 02 = 50 x 10 How to cite Physics Notes, Papers

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string(244) " lie in the equatorial plane of the earth because it must accelerate in a plane where the centre of Earth lies since the net orce exerted on the satellite is the Earth’s gravitational force, which is directed towards the centre of Earth\." Gravitation Gravitational field strength at a point is defined as the gravitational force per unit mass at that point. Newton’s law of gravitation: The (mutual) gravitational force F between two point masses M and m separated by a distance r is given by F =| GMm| (where G: Universal gravitational constant)| | r2| | or, the gravitational force of between two point masses is proportional to the product of their masses ; inversely proportional to the square of their separation. Gravitational field strength at a point is the gravitational force per unit mass at that point. We will write a custom essay sample on Physics Notes or any similar topic only for you Order Now It is a vector and its S. I. unit is N kg-1. By definition, g = F / m By Newton Law of Gravitation, F = GMm / r2 Combining, magnitude of g = GM / r2 Therefore g = GM / r2, M = Mass of object â€Å"creating† the field Example 1: Assuming that the Earth is a uniform sphere of radius 6. 4 x 106 m and mass 6. 0 x 1024 kg, find the gravitational field strength g at a point: (a) on the surface, g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / (6. 4 x 106)2 = 9. 77ms-2 (b) at height 0. 50 times the radius of above the Earth’s surface. g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / ( (1. 5 ? 6. 4 x 106)2 = 4. 34ms-2 Example 2: The acceleration due to gravity at the Earth’s surface is 9. 0ms-2. Calculate the acceleration due to gravity on a planet which has the same density but twice the radius of Earth. g = GM / r2 gP / gE = MPrE2 / MErP2 = (4/3) ? rP3rE2? P / (4/3) ? rE3rP2? E = rP / rE = 2 Hence gP = 2 x 9. 81 = 19. 6ms-2 Assuming that Earth is a uniform sphere of mass M. The magnitude of the gravitational force from E arth on a particle of mass m, located outside Earth a distance r from the centre of the Earth is F = GMm / r2. When a particle is released, it will fall towards the centre of the Earth, as a result of the gravitational force with an acceleration ag. FG = mag ag = GM / r2 Hence ag = g Thus gravitational field strength g is also numerically equal to the acceleration of free fall. Example 1: A ship is at rest on the Earth’s equator. Assuming the earth to be a perfect sphere of radius R and the acceleration due to gravity at the poles is go, express its apparent weight, N, of a body of mass m in terms of m, go, R and T (the period of the earth’s rotation about its axis, which is one day). At the North Pole, the gravitational attraction is F = GMEm / R2 = mgo At the equator, Normal Reaction Force on ship by Earth = Gravitational attraction – centripetal force N = mgo – mR? = mgo – mR (2? / T)2 Gravitational potential at a point is defined as the work done (by an external agent) in bringing a unit mass from infinity to that point (without changing its kinetic energy). ? = W / m = -GM / r Why gravitational potential values are always negative? As the gravitational force on the mass is attractive, the work done by an ext ag ent in bringing unit mass from infinity to any point in the field will be negative work {as the force exerted by the ext agent is opposite in direction to the displacement to ensure that ? KE = 0} Hence by the definition of negative work, all values of ? re negative. g = -| d? | = – gradient of ? -r graph {Analogy: E = -dV/dx}| | dr| | Gravitational potential energy U of a mass m at a point in the gravitational field of another mass M, is the work done in bringing that mass m {NOT: unit mass, or a mass} from infinity to that point. ; U = m ? = -GMm / r Change in GPE, ? U = mgh only if g is constant over the distance h; {; h;; radius of planet} otherwise, must use: ? U = m? f-m? i | Aspects| Electric Field| Gravitational Field| 1. | Quantity interacting with or producing the field| Charge Q| Mass M| 2. Definition of Field Strength| Force per unit positive charge E = F / q| Force per unit mass g = F / M| 3. | Force between two Point Charges or Masses| Coulomb’s Law: Fe = Q1Q2 / 4 or2| Newton’s Law of Gravitation: Fg = G (GMm / r2)| 4. | Field Strength of isolated Point Charge or Mass| E = Q / 4 or2| g = G (GM / r2)| 5. | Definition of Potential| Work done in bringing a unit positive charge from infinity to the point; V = W /Q| Work done in bringing a unit mass from infinity to the point; ? = W / M| 6. | Potential of isolated Point Charge or Mass| V = Q / 4 or| ? -G (M / r)| 7. | Change in Potential Energy| ? U = q ? V| ? U = m | Total Energy of a Satellite = GPE + KE = (-GMm / r) + ? (GMm / r) Escape Speed of a Satellite By Conservation of Energy, Initial KE| +| Initial GPE| =| Final KE| +| Final GPE| (? mvE2)| +| (-GMm / r)| =| (0)| +| (0)| Thus escape speed, vE = v(2GM / R) Note : Escape speed of an object is independent of its mass For a satellite in circular orbit, â€Å"the centripetal force is provided by the gravitational force† {Must always state what force is providing the centripetal force before following eqn is used! Henc e GMm / r2 = mv2 / r = mr? 2 = mr (2? / T)2 A satellite does not move in the direction of the gravitational force {ie it stays in its circular orbit} because: the gravitational force exerted by the Earth on the satellite is just sufficient to cause the centripetal acceleration but not enough to also pull it down towards the Earth. {This explains also why the Moon does not fall towards the Earth} Geostationary satellite is one which is always above a certain point on the Earth (as the Earth rotates about its axis. For a geostationary orbit: T = 24 hrs, orbital radius (; height) are fixed values from the centre of the Earth, ang velocity w is also a fixed value; rotates fr west to east. However, the mass of the satellite is NOT a particular value ; hence the ke, gpe, ; the centripetal force are also not fixed values {ie their values depend on the mass of the geostationary satellite. } A geostationary orbit must lie in the equatorial plane of the earth because it must accelerate in a p lane where the centre of Earth lies since the net orce exerted on the satellite is the Earth’s gravitational force, which is directed towards the centre of Earth. You read "Physics Notes" in category "Papers" {Alternatively, may explain by showing why it’s impossible for a satellite in a non-equatorial plane to be geostationary. } Thermal Physics Internal Energy: is the sum of the kinetic energy of the molecules due to its random motion ; the potential energy of the molecules due to the intermolecular forces. Internal energy is determined by the values of the current state and is independent of how the state is arrived at. You can read also Thin Film Solar Cell Thus if a system undergoes a series of changes from one state A to another state B, its change in internal energy is the same, regardless of which path {the changes in the p ; V} it has taken to get from A to B. Since Kinetic Energy proportional to temp, and internal energy of the system = sum of its Kinetic Energy and Potential Energy, a rise in temperature will cause a rise in Kinetic Energy and thus an increase in internal energy. If two bodies are in thermal equilibrium, there is no net flow of heat energy between them and they have the same temperature. NB: this does not imply they must have the same internal energy as internal energy depends also on the number of molecules in the 2 bodies, which is unknown here} Thermodynamic (Kelvin) scale of temperature: theoretical scale that is independent of the properties of any particular substance. An absolute scale of temp is a temp scale which does not depend on the property of any particular substance (ie the thermodynamic scale) Abs olute zero: Temperature at which all substances have a minimum internal energy {NOT: zero internal energy. } T/K = T/ °C + 273. 15, by definition of the Celsius scale. Specific heat capacity is defined as the amount of heat energy needed to produce unit temperature change {NOT: by 1 K} for unit mass {NOT: 1 kg} of a substance, without causing a change in state. c = Q / m? T Specific latent heat of vaporisation is defined as the amount of heat energy needed to change unit mass of a substance from liquid phase to gaseous phase without a change of temperature. Specific latent heat of fusion is defined as the amount of heat energy needed to change unit mass of a substance from solid phase to liquid phase without a change of temperature L = Q / m {for both cases of vaporisation ; melting} The specific latent heat of vaporisation is greater than the specific latent heat of fusion for a given substance because * During vaporisation, there is a greater increase in volume than in fusion, * Thus more work is done against atmospheric pressure during vaporisation, * The increase in vol also means the INCREASE IN THE (MOLECULAR) POTENTIAL ENERGY, ; hence, internal energy, during vaporisation more than that during melting, * Hence by 1st Law of Thermodynamics, heat supplied during vaporisation more than that during melting; hence lv ; lf {since Q = ml = ? U – W}. Note: 1. the use of comparative terms: greater, more, and; 2. the increase in internal energy is due to an increase in the PE, NOT KE of molecules 3. the system here is NOT to be considered as an ideal gas system Similarly, you need to explain why, when a liq is boiling, thermal energy is being supplied, and yet, the temp of the liq does not change. | Melting| Boiling| Evaporation| Occurrence| Throughout the substance, at fixed temperature and pressure| On the surface, at all temperatures| Spacing(vol) ; PE of molecules| Increase slightly| Increase significantly| | Temperature ; hence KE of molecules| Remains constant during process| Decrease for remaining liquid| First Law of Thermodynamics: The increase in internal energy of a system is equal to the sum of the heat supplied to the system and the work done on the system. ?U = W + Q| ? U: Increase in internal energy of the system Q: Heat supplied to the system W: work done on the system| {Need to recall the sign convention for all 3 terms} Work is done by a gas when it expands; work is done on a gas when it is ompressed. W = area under pressure – volume graph. For constant pressure {isobaric process}, Work done = pressure x ? Volume Isothermal process: a process where T = const {? U = 0 for ideal gas} ? U for a cycle = 0 {since U ? T, ; ? T = 0 for a cycle } Equation of state for an ideal gas: p V = n R T, where T is in Kelvin {NOT:  °C}, n: no. of moles. p V = N k T, where N: no. of molecules, k:Boltzmann con st Ideal Gas: a gas which obeys the ideal gas equation pV = nRT FOR ALL VALUES OF P, V ; T Avogadro constant: defined as the number of atoms in 12g of carbon-12. It is thus the number of particles (atoms or molecules) in one mole of substance. For an ideal gas, internal energy U = Sum of the KE of the molecules only {since PE = 0 for ideal gas} U = N x? m ;c2; = N x (3/2)kT {for monatomic gas} * U depends on T and number of molecules N * U ? T for a given number of molecules Ave KE of a molecule, ? m ;c2; ? T {T in K: not  °C} Dynamics Newton’s laws of motion: Newton’s First Law Every body continues in a state of rest or uniform motion in a straight line unless a net (external) force acts on it. Newton’s Second Law The rate of change of momentum of a body is directly proportional to the net force acting on the body, and the momentum change takes place in the direction of the net force. Newton’s Third Law When object X exerts a force on object Y, object Y exerts a force of the same type that is equal in magnitude and opposite in direction on object X. The two forces ALWAYS act on different objects and they form an action-reaction pair. Linear momentum and its conservation: Mass: is a measure of the amount of matter in a body, ; is the property of a body which resists change in motion. Weight: is the force of gravitational attraction (exerted by the Earth) on a body. Linear momentum: of a body is defined as the product of its mass and velocity ie p = m v Impulse of a force (I): is defined as the product of the force and the time ? t during which it acts ie I = F x ? t {for force which is const over the duration ? t} For a variable force, the impulse I = Area under the F-t graph { ? Fdt; may need to â€Å"count squares†} Impulse is equal in magnitude to the change in momentum of the body acted on by the force. Hence the change in momentum of the body is equal in mag to the area under a (net) force-time graph. {Incorrect to define impulse as change in momentum} Force: is defined as the rate of change of momentum, ie F = [ m (v – u) ] / t = ma or F = v dm / dt The {one} Newton: is defined as the force needed to accelerate a mass of 1 kg by 1 m s-2. Principle of Conservation of Linear Momentum: When objects of a system interact, their total momentum before and after interaction are equal if no net (external) force acts on the system. * The total momentum of an isolated system is constant m1 u1 + m2 u2 = m1 v1 + m2 v2 if net F = 0 {for all collisions } NB: Total momentum DURING the interaction/collision is also conserved. (Perfectly) elastic collision: Both momentum ; kinetic energy of the system are conserved. Inelastic collision: Only momentum is conserved, total kinetic energy is not conserved. Perfectly inelastic collision: Only momentum is conserved, and the particles stick togethe r after collision. (i. e. move with the same velocity. ) For all elastic collisions, u1 – u2 = v2 – v1 ie. relative speed of approach = relative speed of separation or, ? m1u12 + ? m2u22 = ? m1v12 + ? 2v22 In inelastic collisions, total energy is conserved but Kinetic Energy may be converted into other forms of energy such as sound and heat energy. Current of Electricity Electric current is the rate of flow of charge. {NOT: charged particles} Electric charge Q passing a point is defined as the product of the (steady) current at that point and the time for which the current flows, Q = I t One coulomb is defined as the charge flowing per second pass a point at which the current is one ampere. Example 1: An ion beam of singly-charged Na+ and K+ ions is passing through vacuum. If the beam current is 20 ? A, calculate the total number of ions passing any fixed point in the beam per second. (The charge on each ion is 1. 6 x 10-19 C. ) Current, I = Q / t = Ne / t where N is the no. of ions and e is the charge on one ion. No. of ions per second = N / t = I / e = (20 x 10-6) / (1. 6 x 10-19) = 1. 25 x 10-14 Potential difference is defined as the energy transferred from electrical energy to other forms of energy when unit charge passes through an electrical device, V = W / Q P. D. = Energy Transferred / Charge = Power / Current or, is the ratio of the power supplied to the device to the current flowing, V = P / I The volt: is defined as the potential difference between 2 pts in a circuit in which one joule of energy is converted from electrical to non-electrical energy when one coulomb passes from 1 pt to the other, ie 1 volt = One joule per coulomb Difference between Potential and Potential Difference (PD): The potential at a point of the circuit is due to the amount of charge present along with the energy of the charges. Thus, the potential along circuit drops from the positive terminal to negative terminal, and potential differs from points to points. Potential Difference refers to the difference in potential between any given two points. For example, if the potential of point A is 1 V and the potential at point B is 5 V, the PD across AB, or VAB , is 4 V. In addition, when there is no energy loss between two points of the circuit, the potential of these points is same and thus the PD across is 0 V. Example 2: A current of 5 mA passes through a bulb for 1 minute. The potential difference across the bulb is 4 V. Calculate: (a) The amount of charge passing through the bulb in 1 minute. Charge Q = I t = 5 x 10-3 x 60 = 0. 3 C (b) The work done to operate the bulb for 1 minute. Potential difference across the bulb = W / Q 4 = W / 0. Work done to operate the bulb for 1 minute = 0. 3 x 4 = 1. 2 J Electrical Power, P = V I = I2 / R = V2 / R {Brightness of a lamp is determined by the power dissipated, NOT: by V, or I or R alone} Example 3: A high-voltage transmission line with a resistance of 0. 4 ? km-1 carries a current of 500 A. The line is at a potential of 1200 kV at the power station and carries the current to a city lo cated 160 km from the power station. Calculate (a) the power loss in the line. The power loss in the line P = I2 R = 5002 x 0. 4 x 160 = 16 MW (b) the fraction of the transmitted power that is lost. The total power transmitted = I V = 500 x 1200 x 103 = 600 MW The fraction of power loss = 16 / 600 = 0. 267 Resistance is defined as the ratio of the potential difference across a component to the current flowing through it , R = VI {It is NOT defined as the gradient of a V-I graph; however for an ohmic conductor, its resistance equals the gradient of its V-I graph as this graph is a straight line which passes through the origin} The Ohm: is the resistance of a resistor if there is a current of 1 A flowing through it when the pd across it is 1 V, ie, 1 ? = One volt per ampere Example 4: In the circuit below, the voltmeter reading is 8. 00 V and the ammeter reading is 2. 00 A. Calculate the resistance of R. Resistance of R = V / I = 8 / 2 = 4. 0 ? | | Temperature characteristics of thermistors: The resistance (i. e. the ratio V / I) is constant because metallic conductors at constant temperature obey Ohm’s Law. | As V increases, the temperature increases, resulting in an increase in the amplitude of vibration of ions and the collision frequency of electrons with the lattice ions. Hence the resistance of the filament increases with V. | A thermistor is made from semi-conductors. As V increases, temperature increases. This releases more charge carriers (electrons and holes) from the lattice, thus reducing the resistance of the thermistor. Hence, resistance decreases as temperature increases. | In forward bias, a diode has low resistance. In reverse bias, the diode has high resistance until the breakdown voltage is reached. | Ohm’s law: The current in a component is proportional to the potential difference across it provided physical conditions (eg temp) stay constant. R = ? L / A {for a conductor of length l, uniform x-sect area A and resistivity ? Resistivity is defined as the resistance of a material of unit cross-sectional area and unit length. {From R = ? l / A , ? = RA / L} Example 5: Calculate the resistance of a nichrome wire of length 500 mm and diameter 1. 0 mm, given that the resistivity of nichrome is 1. 1 x 10-6 ? m. Resistance, R = ? l / A = [(1. 1 x 10-6)(500 x 10-3)] / ? (1 x 10-3 / 2)2 = 0. 70 ? Electromotive force (Emf) is defined as t he energy transferred / converted from non-electrical forms of energy into electrical energy when unit charge is moved round a complete circuit. ie EMF = Energy Transferred per unit charge E = WQ EMF refers to the electrical energy generated from non-electrical energy forms, whereas PD refers to electrical energy being changed into non-electrical energy. For example, EMF Sources| Energy Change| PD across| Energy Change| Chemical Cell| Chem ; Elec| Bulb| Elec ; Light| Generator| Mech ; Elec| Fan| Elec ; Mech| Thermocouple| Thermal ; Elec| Door Bell| Elec ; Sound| Solar Cell| Solar ; Elec| Heating element| Elec ; Thermal| Effects of the internal resistance of a source of EMF: Internal resistance is the resistance to current flow within the power source. It reduces the potential difference (not EMF) across the terminal of the power supply when it is delivering a current. Consider the circuit below: The voltage across the resistor, V = IR, The voltage lost to internal resistance = Ir Thus, the EMF of the cell, E = IR + Ir = V + Ir Therefore If I = 0A or if r = 0? , V = E Motion in a Circle Kinematics of uniform circular motion Radian (rad) is the S. I. unit for angle, ? and it can be related to degrees in the following way. In one complete revolution, an object rotates through 360 ° , or 2? rad. As the object moves through an angle ? , with respect to the centre of rotation, this angle ? s known as the angular displacement. Angular velocity (? ) of the object is the rate of change of angular displacement with respect to time. ? = ? / t = 2? / T (for one complete revolution) Linear velocity, v, of an object is its instantaneous velocity at any point in its circular path. v = arc length / time taken = r? / t = r? * The direction of th e linear velocity is at a tangent to the circle described at that point. Hence it is sometimes referred to as the tangential velocity * ? is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis. A body moving in a circle at a constant speed changes velocity {since its direction changes}. Thus, it always experiences an acceleration, a force and a change in momentum. Centripetal acceleration a = r? 2 = v2 / r {in magnitude} Centripetal force Centripetal force is the resultant of all the forces that act on a system in circular motion. {It is not a particular force; â€Å"centripetal† means â€Å"centre-seeking†. Also, when asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as â€Å"centripetal force†. } Centripetal force, F = m r ? 2 = mv2 / r {in magnitude} A person in a satellite orbiting the Earth experiences â€Å"weightlessness† although the gravi field strength at that height is not zero because the person and the satellite would both have the same acceleration; hence the contact force between man ; satellite / normal reaction on the person is zero {Not because the field strength is negligible}. D. C. Circuits Circuit Symbols: Open Switch| Closed Switch| Lamp| Cell| Battery| Voltmeter| Resistor| Fuse| Ammeter| Variable resistor| Thermistor| Light dependent resistor (LDR)| Resistors in Series: R = R1 + R2 + †¦ Resistors in Parallel: 1/R = 1/R1 + 1/R2 + †¦ Example 1: Three resistors of resistance 2 ? , 3 ? and 4 ? respectively are used to make the combinations X, Y and Z shown in the diagrams. List the combinations in order of increasing resistance. Resistance for X = [1/2 + 1/(4+3)]-1 = 1. 56 ? Resistance for Y = 2 + (1/4 + 1/3)-1 = 3. 71 ? Resistance for Z = (1/3 + 1/2 + 1/4)-1 = 0. 923 ? Therefore, the combination of resistors in order of increasing resistance is Z X Y. Example: Referring to the circuit drawn, determine the value of I1, I and R, the combined resistance in the circuit. E = I1 (160) = I2 (4000) = I3 (32000) I1 = 2 / 160 = 0. 0125 A I2 = 2 / 4000 = 5 x 10-4 A I3 = 2 / 32000 = 6. 25 x 10-5 ASince I = I1 + I2 + I3, I = 13. 1 mAApplying Ohm’s Law, R = 213. 1 x 10-3 = 153 ? | | Example: A battery with an EMF of 20 V and an internal resistance of 2. 0 ? is connected to resistors R1 and R2 as shown in the diagram. A total current of 4. 0 A is supplied by the battery and R2 has a resistance of 12 ?. Calculate the resistance of R1 and the power supplied to each circuit component. E – I r = I2 R2 20 – 4 (2) = I2 (12) I2 = 1A Therefore, I1 = 4 – 1 = 3 AE – I r = I1 R1 12 = 3 R1 Therefore, R1 = 4Power supplied to R1 = (I1)2 R1 = 36 W Power supplied to R2 = (I2)2 R2 = 12 W| | For potential divider with 2 resistors in series, Potential drop across R1, V1 = R1 / (R1 + R2) x PD across R1 ; R2 Potential drop across R2, V1 = R2 / (R1 + R2) x PD across R1 ; R2 Example: Two resistors, of resistance 300 k? and 500 k? respectively, form a potential divider with outer junctions maintained at potentials of +3 V and -15 V. Determine the potential at the junction X between the resistors. The potential difference across the 300 k? resistor = 300 / (300 + 500) [3 – (-15)] = 6. 75 V The potential at X = 3 – 6. 75 = -3. 75 V A thermistor is a resistor whose resistance varies greatly with temperature. Its resistance decreases with increasing temperature. It can be used in potential divider circuits to monitor and control temperatures. Example: In the figure on the right, the thermistor has a resistance of 800 ? when hot, and a resistance of 5000 ? when cold. Determine the potential at W when the temperature is hot. When thermistor is hot, potential difference across it = [800 / (800 + 1700)] x (7 – 2) = 1. 6 VThe potential at W = 2 + 1. 6 V = 3. 6 V| | A Light dependent resistor (LDR) is a resistor whose resistance varies with the intensity of light falling on it. Its resistance decreases with increasing light intensity. It can be used in a potential divider circuit to monitor light intensity. Example: In the figure below, the resistance of the LDR is 6. 0 M in the dark but then drops to 2. 0 k in the light Determine the potential at point P when the LDR is in the light. In the light the potential difference across the LDR= [2k / (3k + 2k)] x (18 – 3) = 6 VThe potential at P = 18 – 6= 12 V| | The potential difference along the wire is proportional to the length of the wire. The sliding contact will move along wire AB until it finds a point along the wire such that the galvanometer shows a zero reading. When the galvanometer shows a zero reading, the current through the galvanometer (and the device that is being tested) is zero and the potentiometer is said to be â€Å"balanced†. If the cell has negligible internal resistance, and if the potentiometer is balanced, EMF / PD of the unknown source, V = [L1 / (L1 + L2)] x E Example: In the circuit shown, the potentiometer wire has a resistance of 60 ?. Determine the EMF of the unknown cell if the balanced point is at B. Resistance of wire AB= [0. 65 / (0. 65 + 0. 35)] x 60 = 39 ? EMF of the test cell= [39 / (60 + 20)] x 12| Work, Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ? Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by â€Å"counting the squares†. } By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction} Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation: v2 = u2 +2as, as = 12 (v2 – u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 – u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them. Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = – dU / dx. Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h| Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| Useful Output Energy| x100% =| Useful Output Power| x100%| | Input Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total Time| | t| Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object moving at const speed: F = Total resistive force {equilibrium condition} * for object beginning to accelerate: F = Total resistive force + ma Forces Hooke’s Law: Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain energy = Area under the F-x graph {May need to â€Å"count the squares†} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? x2 Forces on Masses in Gravitational Fields: A region of space in which a mass experiences an (attractive) force due to the presence of another mass. Forces on Charge in Electric Fields: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the upper and lower surfaces of the object. Archimedes’ Principle: Upthrust = weight of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: * The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a maximum value {called the static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * Only exists when there is (relative) motion Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act. A couple is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple: The produce of one of the force s of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of Moments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Measurement Base quantities and their units; mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol): Base Quantities| SI Units| | Name| Symbol| Length| metre| m| Mass| kilogram| kg| Time| second| s| Amount of substance| mole| mol| Temperature| Kelvin| K| Current| ampere| A| Luminous intensity| candela| cd| Derived units as products or quotients of the base units: Derived| Quantities Equation| Derived Units| Area (A)| A = L2| m2| Volume (V)| V = L3| m3| Density (? )| ? = m / V| kg m-3| Velocity (v)| v = L / t| ms-1| Acceleration (a)| a = ? v / t| ms-1 / s = ms-2| Momentum (p)| p = m x v| (kg)(ms-1) = kg m s-1| Derived Quantities| Equation| Derived Unit| Derived Units| | | Special Name| Symbol| | Force (F)| F = ? p / t| Newton| N| [(kg m s-1) / s = kg m s-2| Pressure (p)| p = F / A| Pascal| Pa| (kg m s-2) / m2 = kg m-1 s-2| Energy (E)| E = F x d| joule| J| (kg m s-2)(m) = kg m2 s-2| Power (P)| P = E / t| watt| W| (kg m2 s-2) / s = kg m2 s-3| Frequency (f)| f = 1 / t| hertz| Hz| 1 / s = s-1| Charge (Q)| Q = I x t| coulomb| C| A s| Potential Difference (V)| V = E / Q| volt| V| (kg m2 s-2) / A s = kg m2 s-3 A-1| Resistance (R)| R = V / I| ohm| ? (kg m2 s-3 A-1) / A = kg m2 s-3 A-2| Prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: Multiplying Factor| Prefix| Symbol| 10-12| pico| p| 10-9| nano| n| 10-6| micro| ? | 10-3| milli| m| 10-2| centi| c| 10-1| decid| d| 103| kilo| k| 106| mega| M| 109| giga| G| 1012| tera| T| Estimates of physical quantities: When making an estimate, it is only reason able to give the figure to 1 or at most 2 significant figures since an estimate is not very precise. Physical Quantity| Reasonable Estimate| Mass of 3 cans (330 ml) of Coke| 1 kg| Mass of a medium-sized car| 1000 kg| Length of a football field| 100 m| Reaction time of a young man| 0. 2 s| * Occasionally, students are asked to estimate the area under a graph. The usual method of counting squares within the enclosed area is used. (eg. Topic 3 (Dynamics), N94P2Q1c) * Often, when making an estimate, a formula and a simple calculation may be involved. EXAMPLE 1: Estimate the average running speed of a typical 17-year-old? s 2. 4-km run. velocity = distance / time = 2400 / (12. 5 x 60) = 3. 2 ? 3 ms-1 EXAMPLE 2: Which estimate is realistic? | Option| Explanation| A| The kinetic energy of a bus travelling on an expressway is 30000J| A bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13. 8 to 22. 2 ms-1. Thus, its KE will be approximately ? m(182) = 162m. Thus, for its KE to be 30000J: 162m = 30000. Thus, m = 185kg, which is an absurd weight for a bus; ie. This is not a realistic estimate. | B| The power of a domestic light is 300W. | A single light bulb in the house usually runs at about 20W to 60W. Thus, a domestic light is unlikely to run at more than 200W; this estimate is rather high. | C| The temperature of a hot oven is 300 K. 300K = 27 0C. Not very hot. | D| The volume of air in a car tyre is 0. 03 m3. | | Estimating the width of a tyre, t, is 15 cm or 0. 15 m, and estimating R to be 40 cm and r to be 30 cm,volume of air in a car tyre is = ? (R2 – r2)t = ? (0. 42 – 0. 32)(0. 15) = 0. 033 m3 ? 0. 03 m3 (to one sig. fig. )| Distinction between systematic errors (including zero errors) an d random errors and between precision and accuracy: Random error: is the type of error which causes readings to scatter about the true value. Systematic error: is the type of error which causes readings to deviate in one direction from the true value. Precision: refers to the degree of agreement (scatter, spread) of repeated measurements of the same quantity. {NB: regardless of whether or not they are correct. } Accuracy: refers to the degree of agreement between the result of a measurement and the true value of the quantity. | ; ; R Error Higher ; ; ; ; ; ; Less Precise ; ; ;| v v vS Error HigherLess Accuratev v v| | | | | | Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required). For a quantity x = (2. 0  ± 0. 1) mm, Actual/ Absolute uncertainty, ? x =  ± 0. 1 mm Fractional uncertainty, ? xx = 0. 05 Percentage uncertainty, ? xx 100% = 5 % If p = (2x + y) / 3 or p = (2x – y) / 3, ? p = (2? x + ? y) / 3 If r = 2xy3 or r = 2x / y3, ? r / r = ? x / x + 3? y / y Actual error must be recorded to only 1 significant figure, ; The number of decimal places a calculated quantity should have is determined by its actual error. For eg, suppose g has been initially calculated to be 9. 80645 ms-2 ; ? g has been initially calculated to be 0. 04848 ms-2. The final value of ? g must be recorded as 0. 5 ms-2 {1 sf }, and the appropriate recording of g is (9. 81  ± 0. 05) ms-2. Distinction between scalar and vector quantities: | Scalar| Vector| Definition| A scalar quantity has a magnitude only. It is completely described by a certain number and a unit. | A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-hea d represents the direction of the vector. | Examples| Distance, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc. Common Error: Students tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar. | Displacement, velocity, moments (or torque), momentum, force, electric field etc. | Representation of vector as two perpendicular components: In the diagram below, XY represents a flat kite of weight 4. 0 N. At a certain instant, XY is inclined at 30 ° to the horizontal and the wind exerts a steady force of 6. 0 N at right angles to XY so that the kite flies freely. By accurate scale drawing| By calculations using sine and cosine rules, or Pythagoras? theorem| Draw a scale diagram to find the magnitude and direction of the resultant force acting on the kite. R = 3. 2 N (? 3. 2 cm) at ? = 112 ° to the 4 N vector. | Using cosine rule, a2 = b2 + c2 – 2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30 °) R = 3. 23 NUsing sine rule: a / sin A = b / sin B 6 / sin ? = 3. 23 / sin 30 ° ? = 68 ° or 112 ° = 112 ° to the 4 N vector| Summing Vector Components| | Fx = – 6 sin 30 ° = – 3 NFy = 6 cos 30 ° – 4 = 1. 2 NR = v(-32 + 1. 22) = 3. 23 Ntan ? = 1. 2 / 3 = 22 °R is at an angle 112 ° to the 4 N vector. (90 ° + 22 °)| Kinematics Displacement, speed, velocity and acceleration: Distance: Total length covered irrespective of the direction of motion. Displacement: Distance moved in a certain direction. Speed: Distance travelled per unit time. Velocity: is defined as the rate of change of displacement, or, displacement per unit time {NOT: displacement over time, nor, displacement per second, nor, rate of change of displacement per unit time} Acceleration: is defined as the rate of change of velocity. Using graphs to find displacement, velocity and acceleration: * The area under a velocity-time graph is the change in displacement. The gradient of a displacement-time graph is the {instantaneous} velocity. * The gradient of a velocity-time graph is the acceleration. The ‘SUVAT’ Equations of Motion The most important word for this chapter is SUVAT, which stands for: * S (displacement), * U (initial velocity), * V (final velocity), * A (acceleration) and * T (time) of a particle that is in moti on. Below is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly. 1. v = u +at| derived from definition of acceleration: a = (v – u) / t| 2. | s = ? (u + v) t| derived from the area under the v-t graph| 3. | v2 = u2 + 2as| derived from equations (1) and (2)| 4. | s = ut + ? at2| derived from equations (1) and (2)| These equations apply only if the motion takes place along a straight line and the acceleration is constant; {hence, for eg. , air resistance must be negligible. } Motion of bodies falling in a uniform gravitational field with air resistance: Consider a body moving in a uniform gravitational field under 2 different conditions: Without Air Resistance: Assuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free fall. With Air Resistance: If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate greater than . 81 ms-2 . This causes the time taken to reach its maximum height reached to be lower than in the case with no air resistance. The max height reached is also reduced. At the highest point, the body is momentarily at rest; air resistance becomes zero and hence the only force acting on it is the weight. The acceleration is thus 9. 81 ms-2 at this point. As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9. 81 ms-2. As air resistance increases with speed, it eventually equals its weight (but in opposite direction). From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity. Equations for the horizontal and vertical motion: | x direction (horizontal – axis)| y direction (vertical – axis)| s (displacement)| sx = ux t sx = ux t + ? ax t2| sy = uy t + ? ay t2 (Note: If projectile ends at same level as the start, then sy = 0)| u (initial velocity)| ux| uy| v (final velocity)| vx = ux + axt (Note: At max height, vx = 0)| vy = uy + at vy2 = uy2 + 2asy| a (acceleration)| ax (Note: Exists when a force in x direction present)| ay (Note: If object is falling, then ay = -g)| (time)| t| t| Parabolic Motion: tan ? = vy / vx ?: direction of tangential velocity {NOT: tan ? = sy / sx } Forces Hooke’s Law: Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain ener gy = Area under the F-x graph {May need to â€Å"count the squares†} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? k x2 Forces on Masses in Gravitational Fields: A region of space in which a mass experiences an (attractive) force due to the presence of another mass. Forces on Charge in Electric Fields: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the upper and lower surfaces of the object. Archimedes’ Principle: Upthrust = weight of the fluid displaced by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a maximum value {called the static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * Only exists when the re is (relative) motion * Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the entire weight of the object may be considered to act. A couple is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple: The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of Moments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Work, Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ? Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by â€Å"counting the squares†. } By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction} Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation: 2 = u2 +2as, as = 12 (v2 – u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 – u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them. Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = – dU / dx. Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h| Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| Useful Output Energy| x100% =| Useful Output Power| x100%| | Input Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total Time| | t| Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object moving at const speed: F = Total resistive force {equilibrium condition} * for object beginning to accelerate: F = Total resistive force + ma Wave Motion Displacement (y): Position of an oscillating particle from its equilibrium position. Amplitude (y0 or A): The maximum magnitude of the displacement of an oscillating particle from its equilibrium position. Period (T): Time taken for a particle to undergo one complete cycle of oscillation. Frequency (f): Number of oscillations performed by a particle per unit time. Wavelength (? ): For a progressive wave, it is the distance between any two successive particles that are in phase, e. g. it is the distance between 2 consecutive crests or 2 troughs. Wave speed (v): The speed at which the waveform travels in the direction of the propagation of the wave. Wave front: A line or surface joining points which are at the same state of oscillation, i. e. in phase, e. g. a line joining crest to crest in a wave. Ray: The path taken by the wave. This is used to indicate the direction of wave propagation. Rays are always at right angles to the wave fronts (i. e. wave fronts are always perpendicular to the direction of propagation). From the definition of speed, Speed = Distance / Time A wave travels a distance of one wavelength, ? , in a time interval of one period, T. The frequency, f, of a wave is equal to 1 / T Therefore, speed, v = ? / T = (1 / T)? f? v = f? Example 1: A wave travelling in the positive x direction is showed in the figure. Find the amplitude, wavelength, period, and speed of the wave if it has a frequency of 8. 0 Hz. Amplitude (A) = 0. 15 mWavelength (? ) = 0. 40 mPeriod (T) = 1f = 18. 0 ? 0. 125 sSpeed (v) =f? = 8. 0 x 0. 40 = 3. 20 m s-1A wave which results in a net transfer of energy from one place to another is known as a progressive wave. | | Intensity {of a wave}: is defined as the rate of energy flow per unit time {power} per unit cross-sectional area perpendicular to the direction of wave propagation. Intensity = Power / Area = Energy / (Time x Area) For a point source (which would emit spherical wavefronts), Intensity = (? m? 2xo2) / (t x 4? r2) where x0: amplitude ; r: distance from the point source. Therefore, I ? xo2 / r2 (Pt Source) For all wave sources, I ? (Amplitude)2 Transverse wave: A wave in which the oscillations of the wave particles {NOT: movement} are perpendicular to the direction of the propagation of the wave. Longitudinal wave: A wave in which the oscillations of the wave particles are parallel to the direction of the propagation of the wave. Polarisation is said to occur when oscillations are in one direction in a plane, {NOT just â€Å"in one direction†} normal to the direction of propagation. {Only transverse waves can be polarized; longitudinal waves can’t. }Example 2: The following stationary wave pattern is obtained using a C. R. O. whose screen is graduated in centimetre squares. Given that the time-base is adjusted such that 1 unit on the horizontal axis of the screen corresponds to a time of 1. 0 ms, find the period and frequency of the wave. Period, T = (4 units) x 1. 0 = 4. 0 ms = 4. 0 x 10-3 sf = 1 / T = 14 x 10-3 250 Hz| | Superposition Principle of Superposition: When two or more waves of the same type meet at a point, the resultant displacement of the waves is equal to the vector sum of their individual displacements at that point. Stretched String A horizontal rope with one end fixed and another attached to a vertical oscillator. Stationary waves will be produced by the direct and reflected w aves in the string. Or we can have the string stopped at one end with a pulley as shown below. Microwaves A microwave emitter placed a distance away from a metal plate that reflects the emitted wave. By moving a detector along the path of the wave, the nodes and antinodes could be detected. Air column A tuning fork held at the mouth of a open tube projects a sound wave into the column of air in the tube. The length of the tube can be changed by varying the water level. At certain lengths of the tube, the air column resonates with the tuning fork. This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface. Stationary (Standing) Wave) is one * whose waveform/wave profile does not advance {move}, where there is no net transport of energy, and * where the positions of antinodes and nodes do not change (with time). A stationary wave is formed when two progressive waves of the same frequency, amplitude and speed, travelling in opposite directions are superposed. {Assume boundary conditions are met} | Stationary waves| Stationary Waves Progressive Waves| Amplitude| Varies from maximum at the anti-nodes to zero at the nodes. | Same for all particles in the wave (provided no energy is lost). | Wavelength| Twice the distance between a pair of adjacent nodes or anti-nodes. The distance between two consecutive points on a wave, that are in phase. | Phase| Particles in the same segment/ between 2 adjacent nodes, are in phase. Particles in adjacent segments are in anti-phase. | All particles within one wavelength have different phases. | Wave Profile| The wave profile does not advance. | The wave profile advances. | Energy| No energy is transported by the wave. | Energy is transported in the direction of the wave. | Node is a region of destructive superposition where the waves always meet out of phase by ? radians. Hence displacement here is permanently zero {or minimum}. Antinode is a region of constructive superposition where the waves always meet in phase. Hence a particle here vibrates with maximum amplitude {but it is NOT a pt with a permanent large displacement! } Dist between 2 successive nodes / antinodes = ? / 2 Max pressure change occurs at the nodes {NOT the antinodes} because every node changes fr being a pt of compression to become a pt of rarefaction {half a period later} Diffraction: refers to the spreading {or bending} of waves when they pass through an opening {gap}, or round an obstacle (into the â€Å"shadow† region). Illustrate with diag} For significant diffraction to occur, the size of the gap ? ? of the wave For a diffraction grating, d sin ? = n ? , d = dist between successive slits {grating spacing} = reciprocal of number of lines per metre When a â€Å"white light† passes through a diffraction grating, for each order of diffraction, a longer wavelength {red} diffracts more than a shorter wavelength {violet} {as sin ? ? ? }. Diffraction refers to the spreading of waves as they pass through a narrow slit or near an obstacle. For diffraction to occur, the size of the gap should approximately be equal to the wavelength of the wave. Coherent waves: Waves having a constant phase difference {not: zero phase difference / in phase} Interference may be described as the superposition of waves from 2 coherent sources. For an observable / well-defined interference pattern, the waves must be coherent, have about the same amplitude, be unpolarised or polarised in the same direction, ; be of the same type. Two-source interference using: 1. Water Waves Interference patterns could be observed when two dippers are attached to the vibrator of the ripple tank. The ripples produce constructive and destructive interference. The dippers are coherent sources because they are fixed to the same vibrator. 2. Microwaves Microwave emitted from a transmitter through 2 slits on a metal plate would also produce interference patterns. By moving a detector on the opposite side of the metal plate, a series of rise and fall in amplitude of the wave would be registered. 3. Light Waves (Young? s double slit experiment) Since light is emitted from a bulb randomly, the way to obtain two coherent light sources is by splitting light from a single slit. The 2 beams from the double slit would then interfere with each other, creating a pattern of alternate bright and dark fringes (or high and low intensities) at regular intervals, which is also known as our interference pattern. Condition for Constructive Interference at a pt P: Phase difference of the 2 waves at P = 0 {or 2? , 4? , etc} Thus, with 2 in-phase sources, * implies path difference = n? ; with 2 antiphase sources: path difference = (n + ? )? Condition for Destructive Interference at a pt P: Phase difference of the 2 waves at P = ? { or 3? , 5? , etc } With 2 in-phase sources, + implies path difference = (n+ ? ), with 2 antiphase sources: path difference = n ? Fringe separation x = ? D / a, if a;;D {applies only to Young’s Double Slit interference of light, ie, NOT for microwaves, sound waves, water waves} Phase difference betw the 2 waves at any pt X {betw the central 1st maxima) is (approx) proportional to the dist of X from the central maxima. Using 2 sources of equal amplitude x0, the resultant amplitude of a bright fringe would be doubled {2Ãâ€"0}, the resultant intensity increases by 4 times {not 2 times}. { IResultant ? (2 x0)2 } Electric Fields Electric field strength / intensity at a point is defined as the force per unit positive charge acting at that point {a vector; Unit: N C-1 or V m-1} E = F / q F = qE * The electric force on a positive charge in an electric field is in the direction of E, while * The electric force on a negative charge is opposite to the direction of E. * Hence a +ve charge placed in an electric field will accelerate in the direction of E and gain KE { simultaneously lose EPE}, while a negative charge caused to move (projected) in the direction of E will decelerate, ie lose KE, { gain EPE}. Representation of electric fields by field lines | | | | | Coulomb’s law: The (mutual) electric force F acting between 2 point charges Q1 and Q2 separated by a distance r is given by: F = Q1Q2 / 4 or2 where ? 0: permittivity of free space or, the (mutual) electric force between two point charges is proportional to the product of their charges ; inversely proportional to the square of their separation. Exa mple 1: Two positive charges, each 4. 18 ? C, and a negative charge, -6. 36 ? C, are fixed at the vertices of an equilateral triangle of side 13. 0 cm. Find the electrostatic force on the negative charge. | F = Q1Q2 / 4 or2= (8. 99 x 109) [(4. 18 x 10-6)(6. 6 x 10-6) / (13. 0 x 10-2)2]= 14. 1 N (Note: negative sign for -6. 36 ? C has been ignored in the calculation)FR = 2 x Fcos300= 24. 4 N, vertically upwards| Electric field strength due to a Point Charge Q : E = Q / 4 or2 {NB: Do NOT substitute a negative Q with its negative sign in calculations! } Example 2: In the figure below, determine the point (other than at infinity) at which the total electric field strength is zero. From the diagram, it can be observed that the point where E is zero lies on a straight line where the charges lie, to the left of the -2. 5 ? C charge. Let this point be a distance r from the left charge. Since the total electric field strength is zero, E6? = E-2? [6? / (1 + r)2] / 4 or2 = [2. 5? / r2] / 4 or2 (Note: negative sign for -2. 5 ? C has been ignored here) 6 / (1 + r)2 = 2. 5 / r2 v(6r) = 2. 5 (1 + r) r = 1. 82 m The point lies on a straight line where the charges lie, 1. 82 m to the left of the -2. 5 ? C charge. Uniform electric field between 2 Charged Parallel Plates: E = Vd, d: perpendicular dist between the plates, V: potential difference between plates Path of charge moving at 90 ° to electric field: parabolic. Beyond the pt where it exits the field, the path is a straight line, at a tangent to the parabola at exit. Example 3: An electron (m = 9. 11 x 10-31 kg; q = -1. 6 x 10-19 C) moving with a speed of 1. 5 x 107 ms-1, enters a region between 2 parallel plates, which are 20 mm apart and 60 mm long. The top plate is at a potential of 80 V relative to the lower plate. Determine the angle through which the electron has been deflected as a result of passing through the plates. Time taken for the electron to travel 60 mm horizontally = Distance / Speed = 60 x 10-3 / 1. 5 x 107 = 4 x 10-9 s E = V / d = 80 / 20 x 10-3 = 4000 V m-1 a = F / m = eE / m = (1. 6 x 10-19)(4000) / (9. 1 x 10-31) = 7. 0 x 1014 ms-2 vy = uy + at = 0 + (7. x 1014)( 4 x 10-9) = 2. 8 x 106 ms-1 tan ? = vy / vx = 2. 8 x 106 / 1. 5 x 107 = 0. 187 Therefore ? = 10. 6 ° Effect of a uniform electric field on the motion of charged particles * Equipotential surface: a surface where the electric potential is constant * Potential gradient = 0, ie E along surface = 0 } * Hence no work is done when a charge is moved along this surface. { W=QV, V=0 } * Electric field lines must meet this surface at right angles. * {If the field lines are not at 90 ° to it, it would imply that there is a non-zero component of E along the surface. This would contradict the fact that E along an equipotential = 0. Electric potential at a point: is defined as the work done in moving a unit positive charge from infinity to that point, { a scalar; unit: V } ie V = W / Q The electric potential at infinity is defined as zero. At any other point, it may be positive or negative depending on the sign of Q that sets up the field. {Contrast gravitational potential. } Relation between E and V: E = – dV / dr i. e. The electric field strength at a pt is numerically equal to the potential gradient at that pt. NB: Electric field lines point in direction of decreasing potential {ie from high to low pot}. Electric potential energy U of a charge Q at a pt where the potential is V: U = QV Work done W on a charge Q in moving it across a pd ? V: W = Q ? V Electric Potential due to a point charge Q : V = Q / 4 or {NB: Substitute Q with its sign} Electromagnetism When a conductor carrying a current is placed in a magnetic field, it experiences a magnetic force. The figure above shows a wire of length L carrying a current I and lying in a magnetic field of flux density B. Suppose the angle between the current I and the field B is ? , the magnitude of the force F on the conductor is iven by F = BILsin? The direction of the force can be found using Fleming? s Left Hand Rule (see figure above). Note that the force is always perpendicular to the plane containing both the current I and the magnetic field B. * If the wire is parallel to the field lines, then ? = 0 °, and F = 0. (No magnetic force acts on the wire) * If the wire is at right angles to the field lines, then ? = 90 °, and the magn etic force acting on the wire would be maximum (F = BIL) Example The 3 diagrams below each show a magnetic field of flux density 2 T that lies in the plane of the page. In each case, a current I of 10 A is directed as shown. Use Fleming’s Left Hand Rule to predict the directions of the forces and work out the magnitude of the forces on a 0. 5 m length of wire that carries the current. (Assume the horizontal is the current) | | | F = BIL sin? = 2 x 10 x 0. 5 x sin90 = 10 N| F = BIL sin? = 2 x 10 x 0. 5 x sin60 = 8. 66 N| F = BIL sin ? = 2 x 10 x 0. 5 x sin180 = 0 N| Magnetic flux density B is defined as the force acting per unit current in a wire of unit length at right-angles to the field B = F / ILsin ? F = B I L sin ? {? Angle between the B and L} {NB: write down the above defining equation define each symbol if you’re not able to give the â€Å"statement form†. } Direction of the magnetic force is always perpendicular to the plane containing the current I and B {even if ? ? 0} The Tesla is defined as the magnetic flux density of a magnetic field that causes a force of one newton to act on a current of one ampere in a wire o f length one metre which is perpendicular to the magnetic field. By the Principle of moments, Clockwise moments = Anticlockwise moments mg †¢ x = F †¢ y = BILsin90 †¢ y B = mgx / ILy Example A 100-turn rectangular coil 6. 0 cm by 4. 0 cm is pivoted about a horizontal axis as shown below. A horizontal uniform magnetic field of direction perpendicular to the axis of the coil passes through the coil. Initially, no mass is placed on the pan and the arm is kept horizontal by adjusting the counter-weight. When a current of 0. 50 A flows through the coil, equilibrium is restored by placing a 50 mg mass on the pan, 8. 0 cm from the pivot. Determine the magnitude of the magnetic flux density and the direction of the current in the coil. Taking moments about the pivot, sum of Anti-clockwise moments = Clockwise moment (2 x n)(FB) x P = W x Q (2 x n)(B I L) x P = m g x Q, where n: no. of wires on each side of the coil (2 x 100)(B x 0. 5 x 0. 06) x 0. 02 = 50 x 10 How to cite Physics Notes, Papers