A Study On Hookes Law Mechanics Essay

A Study On Hookes Law Mechanics Essay HOOKES LAW: Hookes lawof elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hookes law is a useful approximation are known as linear elasticor Hookean materials. If a metal is lightly stressed, a temporary deformation permitted by an elastic displacement of atoms in space takes place. Removal of stress results in a gradual return of metal to its original shape. Mathematically, Hookes law states that Where, xis the displacement of the end of the spring from its equilibrium position; Fis the restoring force exerted by the material; and kis theforce constant(orspring constant). DIAGRAMATICALLY:- When no weight is applied to the spring, the strain is zero, And, we can measure its length,. and when we apply a force F to the spring It stretches And it extends length,x, that is, the strain, caused by the stress is F = mg. Also, In terms of mechanics hooks state that:- â€Å"For an elastic material stress applied on a body is directly proportional to the strain produced† That is, à Ã†â€™ ÃŽÂ ± e Or à Ã†â€™ = E e Where, à Ã†â€™ is the stress applied e is the strain developed E is the YOUNGS MODULUS OF ELASTICITY Now STRESS it is the force causing the deformation. It is measured in units of force per unit area of cross-section (N.m-2) denoted byà Ã†â€™(sigma). That is à Ã†â€™= F/A Units of stress are Pascal Strain is the deformation that takes place in the body. It is the ratio of the increase in length,DLto the original length (L), Represented by symbolÃŽÂ µ(epsilon) or e. That is e=DL/L It is dimensionless. And according to hooks law: à Ã†â€™ = E e Or, E = à Ã†â€™/e Putting values of stress and strain in above equation we get:- E = FÃÆ'-L/AÃÆ'-DL Youngs modulus of elasticity (E) is defined as the ratio of unit stress to unit strain . GENERALIZED HOOKS LAW: The generalized Hookes Law can be used to predict the deformations caused in a given material by an arbitrary combination of stresses. The linear relationship between stress and strain applies for The generalized Hookes Law also reveals that strain can exist without stress. For example, if the member is experiencing a load in the y-direction (which in turn causes a stress in the y-direction), the Hookes Law shows that strain in the x-direction does not equal to zero. This is because as material is being pulled outward by the y-plane, the material in the x-plane moves inward to fill in the space once occupied, just like an elastic band becomes thinner as you try to pull it apart. In this situation, the x-plane does not have any external force acting on them but they experience a change in length. Therefore, it is valid to say that strain exist without stress in the x-plane. STRESS-STRAIN CURVE:- Thestress-straincurve is a graphical representation of the relationship betweenstress, derived from measuring the load applied on the sample, andstrain, derived from measuring thedeformationof the sample, i.e. elongation, compression, or distortion. The nature of the curve varies from material to material. ELASTIC LIMIT: The elastic limit is where the graph departs from a straight line. If we go past it, the spring wont go back to its original length. When we remove the force, were left with apermanent extension. Below the elastic limit, we say that the spring is showing elastic behaviour: the extension is proportional to the force, and itll go back to its original length when we remove the force. Beyond the elastic limit, we say that it shows plastic behaviour. This means that when a force is applied to deform the shape, it stays deformed when the force is removed. YIELD POINT: Theyield strengthoryield pointof a materialis defined in engineering and material science as the stress at which a material begins to deform plastically . Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed some fraction of the deformation will be permanent and non-reversible. True elastic limit: The lowest stress at whichdislocationsmove. This definition is rarely used, since dislocations move at very low stresses, and detecting such movement is very difficult. Proportionality limit: Up to this amount of stress, stress is proportional to strain hookes law so the stress-strain graph is a straight line, and the gradient will be equal to the elastic modulus of the material. Elastic limit (yield strength): Beyond the elastic limit, permanent deformation will occur. The lowest stress at which permanent deformation can be measured. This requires a manual load-unload procedure, and the accuracy is critically dependent on equipment and operator skill. For elastomers such as rubber the elastic limit is much larger than the proportionality limit. Also, precise strain measurements have shown that plastic strain begins at low stresses. Offset yield point (proof stress) This is the most widely used strength measure of metals, and is found from the stress-strain curve. A plastic strain of 0.2% is usually used to define the offset yield stress, although other values may be used depending on the material and the application. The offset value is given as a subscript, e.g., Rp0.2=310 MPa. In some materials there is essentially no linear region and so a certain value of strain is defined instead. Although somewhat arbitrary, this method does allow for a consistent comparison of materials. Upper yield point and lower yield point Some metals, such as mild steel reach an upper yield point before dropping rapidly to a lower yield point. The material response is linear up until the upper yield point, but the lower yield point is used in structural engineering as a conservative value. If a metal is only stressed to the upper yield point, and beyond rubber band can develop. NUMERICALS:- Q1) When a 13.2-kg mass is placed on top of a vertical spring, the spring compresses 5.93 cm. Find the force constant of the spring. Solution: Mass = 13.2 kg Weight = 13.2ÃÆ'-9.8 = 129 Compression (x) = 5.93 = 0.0593 m From Hookes Law: F = kx The force on the spring is the weight of the object, i.e.(F) = 129 N Putting values of force and compression in above equation; 129 = (0.0593) ÃÆ'- k Or, k = 2181 N/m Answer Q2) A 3340 N ball is supported vertically by a 2m diameter steel cable assuming cable has a length of 10m, determine stress and strain in the cable. Youngs modulus for steel is 200N/sq.m. Solution: Force (F) = 3340N Diameter = 2m Radius (r) = 1m Length of cable = 10m Youngs modulus (E) = 200N/sq.m Now we know, Stress (à Ã†â€™) = F/A Area = = 3.14ÃÆ'-1ÃÆ'-1 = 3.14 So, à Ã†â€™ = 3340/3.14 à Ã†â€™ = 1063.69N/m.sq Also, strain (e) = à Ã†â€™/E Putting values e = 1063.69/200 e =5.3184 Answer Q3) If a spring has a spring constant of 400 N/m, how much work is required to compress the spring 25.0 m from its undisturbed position? Solution: spring constant (K) = 400 N/m compression (x) = 25m we know, force required for compression:- F = kx i.e. F = 40025 = 10000N and work done = force x compression w = F x X w = 10000 x 25 w = 25,000 Joules Answer Q4) On a of steel rod of length 15m and diameter 6m a force of 60N is applied. Calculate the extension and new length of the rod. Youngs modulus of steel is 250N/m.sq. Solution: : Force (F) = 60 N Diameter = 6m So, Radius (r) = 3m Length (L) = 15 m Youngs modulus (E) = 250N/m.sq. Now, Area (A) A = 3.14 x 3 x 3 A = 28.26 sq.m Also, , E = FÃÆ'-L/AÃÆ'-DL Or, DL = FÃÆ'-L/AÃÆ'-E DL = 60ÃÆ'-15/28.26ÃÆ'-250 DL = 0.127m SO, new length = 15+0.127 L = 15.127m ANSWER REFERENCES:- 1) www.physicsworld.com 2) www.wikipedia.org 3) www.123iitjee.com 4) www.physicsforum.com 5) www.matter.org.content/HookesLaw 6) www.webphysics.davidson.edu/hook 7) www.scienceworld.com

Wind Power Essay -- Energy Physics Essays

Wind Power In the society we live in today, many people have the misconception that there is a never-ending supply of energy available for our wasteful use. However, people must realize that the fossil fuels that we have come to depend on for our energy are quickly being depleted. There are several renewable energy resources, most of which come from the sun, that are available for our use. One of these abundant energy resources is the wind. By taking advantage of the wind, and harnessing its power to supply useful energy, people can ensure that they will have energy for as long as the sun continues to heat the earth. The wind is a direct product of the sun. The earth receives 1.74 x 10^17 Watts of power (per hour) from the sun (Tour 1). The sun causes differential heating of earth’s surface and atmosphere, inducing vertical and horizontal air currents that are affected by the rotation of the earth and the contours of the land (Ristinen 133). As the land is heated, the warm air near the surface rises into the cooler atmosphere, causing a pressure gradient between the surface and the upper atmosphere. The lower pressure near the surface causes an inward current of air (wind) from the higher-pressure surroundings. A great example of this is the Land Sea Breeze Cycle, which we feel when we step on the beach (Tour 1). As the warmer air rises into the atmosphere, the cooler air over the ocean rushes onto the shore, and this is the refreshing sea breeze we have come to expect whenever we step on the beach. About one to two percent of the energy that the earth receives from the sun is converted to wind energy (Tour 1). The amount of energy that the wind transfers to the rotors of a wind turbine depends on the dens... ... long as the sun continues to heat the earth. It causes no pollution and has little effect on the environment. Therefore, due to the availability and advantages of wind power, the decreasing cost of wind power, and the growing interest in renewable energy resources, one can be assured that wind power will soon become a feasible energy source in the United States, as well as throughout the world. Works Cited Ristinen, Robert A., Jack J. Kraushaar.. Energy and the Environment. New York: John Wiley and Sons, Inc., 1999. Urone, Paul P.. College Physics: Second Edition. California: Brooks/Cole, 2001. Guided Tour on Wind Energy. 16 Oct. 2002 . The National Wind Technology Center (NWTC). 16 Oct. 2002 . Wind Energy. 16 Oct. 2002 .

Philosophy of Education for Foundations of Education Essay

Introduction Education is an ongoing process based upon experience. The old adage you learn something new everyday is very true, and nothing fascinates me more than simply talking to other people; you can learn so much from them. Education is the foundation of our American society, and the children of today are the future of our country, but educating them is not enough; we must be good role models and present a system of morals and values in our classrooms. Our objective in education is directly related to the social sciences in that the classroom is a microcosm of our society. As teachers we try to prepare our students for real-world situations. Interacting socially, communicating effectively and understanding other peoples’ emotions, feelings and points of view will help our students blossom into productive and understanding adults. Howard Gardner wrote about multiple intelligences, which I think is a huge step in improving the classroom environment and lesson plans to include ways that everyone can learn (Tomlinson, 2002). He identified eight different ways to be â€Å"smart† that traditional IQ tests would not show. This model allows students to excel in these categories through different types of instruction, such as verbal ability, referred to as the linguistic intelligence, or the ability to play an instrument, referred to as the musical intelligence (Johnson et al. , 2005). My classroom will be entertaining, first and foremost. I understand the material I am teaching elementary school children, but delivering that material effectively depends on how they feel about school. Motivating children and getting them excited about learning is not always easy, but it is a key ingredient to a successful classroom. As a male entering the elementary education field I understand my role as a possible father figure (Kindlon and Thompson, 1999) and recognize the importance. I want to make a difference in the world, and I see no better way than through the efforts of education. Definition of Philosophy From the perspective of a future educator, I see myself identifying with the qualities of existentialism the most. This theory can be hard to explain at times because it relies so heavily on the meaning we impose on our lives through education, an idea that cannot be expressed in a limited amount of words and one that differs from individual to individual. Existentialism is routed in the fact that our lives are meaningless; we live in a meaningless world and a meaningless period of time no different than any other. In essence, the quest to find meaning in our lives defines our life. A large part of this view is the idea that we are all free, an idea embraced in America but not typically recognized throughout the world or throughout history. Given this, our freedom allows us to make choices and these choices define us. The goal of the teacher through this philosophy is not to teach freedom, but to embrace it. Tracking, measurement and standardization homogenize the classroom, whereas our goal as teachers is to individualize the classroom and foster a healthy student-teacher relationship. The students’ feelings are important and we should not compare ourselves to an ideal self but who we actually are, and education is the tool that fills the gaps of understanding this. Because I have yet to become a teacher, I am not quite sure how and in what ways measurement and standardization negatively impact the classroom, but I am aware of the separation and animosity created my tracking. Getting the students to think positively about themselves will foster healthy learning and create a positive self-esteem that will drive them to learn both in the classroom and through questioning on their own. Sparking that fire is the idea behind existentialism, and in a world where we are essentially meaningless, it is up to us as teachers to inspire our students to grasp the freedom they have, bring meaning into their life, and use that excitement to find meaning in all that they do. Also, in sparking critical thinking, students will engage in a thoughtful and reflective process similar to Bloom’s taxonomy where a number of levels of learning are happening, dependent on how information in the classroom affects them and their existence as well as essence. Another important tool we can use to define our philosophy as a teacher is the Ways of Knowing. I feel like I identify most with the Eastern Way of Knowing, particularly the Chinese thought because of the emphasis on moral development. Understanding facts and theories is important in education, but sometimes it is hard to distinguish what is actually necessary from what is trivial. Preaching morals allows us to respect others, particularly elders, and gain from their knowledge, something that cannot always be taught in the classroom. Also, it will foster a more productive and peaceful society, one that is the best to learn in. Confucianism and Taoism play a large role in this type of knowing as well, and the reliance on harmony can be compared to the need for smooth transitions in the field of education. Although Confucius implemented the need for rules and standards, contradictory to my belief in existentialism, he has influenced society and education in particular for thousands of years. These rules help keep life orderly and efficient, and the success of Chinese government, business and family life can be attributed to this as well as the harmony between the three. Taoism affects the Chinese thought by suggesting that we should leave things alone and not force personal desires onto the way things will naturally occur. This eliminates the need for competition and is more consistent with the existentialist view. Competition can be good in many ways, but when comparing students we must be careful and considerate. There is no place in the classroom for making anyone feel inferior, and I feel more strongly about that than any other aspect of teaching right now. In relation to morals and respect, I will never allow a student to think they are better than any other student, and hopefully using this method will prepare them for life in the â€Å"real world,† where manners and morals can actually take you pretty far. Even if it’s not for achievement, preaching this will promote self-improvement and a genuine compassion for other students, which is what I strive for in the classroom. Based on the work of Jean-Jacques Rousseau, and rooted in existentialism, humanism is the educational approach that I fell closest too. Humanism makes the assumption that we are essentially born good and that children enter the world not as a blank slate, but with innate qualities and dispositions. Consistent and nearly identical to existentialism, humanism stresses the role of individualism in the classroom and says that society turns people evil. Group-oriented education is rejected because it weakens the student-teacher relationship, limits self-actualization (based on the student’s feelings), and measures students as objects, effectively doing away with the individual self. Teachers involving humanism in the classroom look for student interests as guidelines for instruction and assessment and even go as far as letting the students pick the material and activities. Students discover their own opinions through divergent thinking and should not be influenced by the teachers’ views. Eastern influences on Humanism are also prevalent, as people should be looked at as valuable individuals that deserve respect, not as objects to be manipulated, described by the I-Thou and I-It example, respectively. I am an optimist; the glass is always half full. Therefore, I strongly believe that people are created equally, good and free. Society can have a negative impact on individuals, but it can also have a positive impact. Our goal as teachers is to become that positive force and to let every student know that we care about them as individuals. The book cites how a college classroom can consist of more than one-hundred students, leaving know room for a personal relationship with the professor. One of the main reasons I came to W and J was for that small school feel where I could have a personal relationship with my professors: I feel like I do better in my classes and that I also learn more and feel more confident about my abilities. If we can achieve this goal, rather than making our students just a number, we will truly succeed as educators. References Johnson, J. A. , Musial, D. , Hall, G. E. , Gollnick, D. M. , & Dupuis, V. L. (2005). Introduction to the foundations of American education (pp. 448). Boston: Allyn& Bacon. Kindlon, D. , & Thompson M. (1999). Raising Cain: Protecting the emotional life of boys (pp. 333). New York: Ballentine. Tomlinson, C. A. (2002). Different learners, different lessons. Scholastic Instructor, 9, 21, 24-26, 91.

Using German Participles as Adjectives and Adverbs

As in English, the past participle of a German verb may be used as an adjective or adverb. In English,  stolen  is the past participle of the verb  to steal. The word  stolen  can be used as an adjective, as in: â€Å"Thats a  stolen  car.† Similarly, in German the past participle  gestohlen  (fromstehlen, to steal) can also be used as an adjective: â€Å"Das ist ein  gestohlenes  Auto.† The only significant difference between the ways that English and German use the past participle as an adjective is the fact that, unlike English adjectives, German adjectives must have an appropriate ending if they precede a noun. (Notice the -es  ending in the example above. More about adjective endings in  Lesson 5  and  Adjective Endings.) Of course, it also helps if you know the correct past participle forms to use. A past participle such as  interessiert  (interested) can also be used as an adverb: â€Å"Wir saheninteressiert  zu.† (â€Å"We watched  interestedly/with interest.†) Present Participles Unlike its English equivalent, the present participle in German is used almost exclusively as an adjective or adverb. For other uses, German present participles are usually replaced by nominalized verbs (verbs used as nouns) —  das Lesen  (reading),  das Schwimmen  (swimming) — to function like English gerunds, for instance. In English, the present participle has an -ingending. In German the present participle ends in -end:  weinend  (crying),  pfeifend  (whistling),schlafend  (sleeping). In German, â€Å"a  sleeping  child† is â€Å"ein  schlafendes  Kind.† As with any adjective in German, the ending must fit the grammatical context, in this case an -es  ending (neuter/das). Many present participle adjective phrases in German are translated with a relative clause or an appositive phrase in English. For example, â€Å"Der  schnell vorbeifahrende  Zug machte großen Là ¤rm,† would be, â€Å"The train, which was  quickly passing by, made a tremendous noise,† rather than the literal, â€Å"The quickly passing by train...† When used as adverbs, German present participles are treated like any other adverb, and the English translation usually places the adverb or adverbial phrase at the end: â€Å"Er kam  pfeifend  ins Zimmer.† â€Å"He came into the room  whistling.† Present participles are used more often in writing than in spoken German. Youll run across them a lot when reading books, magazines, or newspapers.