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Calculated Fields with Dax - 4
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Calculated Fields with Dax - 1Calculated Fields with Dax - 2Calculated Fields with Dax - 3What is Electric Force? Electric force is just one of many types of forces in the world of physics. Forces are how and why things move, and can be explained by Newton's Laws of Motion. On the smallest scale, electric force is the resulting interaction between two charged particles. These charges can be either positive or negative. Larger objects can be charged by having an abundance of either of these particles, and therefore can create an electric force on a larger scale. Electric force is the reason why hair will sometimes stand up on its own and is also why we have electricity, allowing us to live in the modern world with lights and technology. Even out in nature electric force is present, as electric force causes lightning to strike. Electric force is fundamental to our everyday way of living. Reviewing Newton's Laws of Motion Newton's Laws of motion are the basic principles or ground rules that are applied all across physics. They describe how objects move and can be used to describe the interaction of charges. They are the following: An object in motion will stay in motion unless an external force is applied The force exerted on an object is equal to the mass times the acceleration of the object. ( ) Every force has an equal and opposite force Newton's laws explain how and why charged particles move. Since there is a force involved (e.g. electric force), particles will move around, which is explained by the first law. The second law describes how acceleration of charges can be calculated once the electric force is known. The third law explains how attractive and repulsive forces between charged objects are equal and opposite. Electric Force Examples and Types of Charge As previously mentioned, there are only two types of charges; positive and negative. Two like charges will repel (or move away from) each other, and two opposite charges will attract (or move towards) each other. In other words, two positive or two negative charges will repel, while a positive and a negative charge will attract. Opposite charges will attract while like charges will repel. Attraction versus Repelling Forces Notice how the forces acting upon each other are equal and opposite, as Newton's third law states. Both charges are exerting forces onto each other. Charges in Atoms An atom is made up of three types of particles; protons, neutrons, and electrons. Protons have a positive charge, neutrons have no charge, and electrons have a negative charge. There are no positive or negative charges smaller than protons and electrons. Objects on a larger scale result in an overall positive or negative charged due to an uneven distribution of protons to electrons. An atom consisting of more protons than electrons would be considered positive, and an atom with more electrons than protons would be considered negative. Protons are held close to the nucleus and are tightly bound to an atom, so it's difficult for protons to escape an atom. Electrons, on the other hand, are much further away from the nucleus of an atom. This makes it much easier for them to be removed from an atom. Electrons can leave or join atoms, making them positive or negative depending on the amount of protons. Similarly, for the bigger picture, overall materials and objects with more electrons than protons would be considered negative, and vice versa. Electric Force Examples Hair standing up: When hair is brushed, the hairbrush can strip electrons from hair strands, resulting in the hair being positively charged. This addition of electrons to the hairbrush in turn makes the hairbrush negatively charged. Since the hair is now positively charged, and like forces repel, hair strands will move away from each other, resulting in the hair standing up. Current electricity: All of our everyday technology is powered through current electricity, which is the consistent flow of electrons through conductive materials. This flow is caused by the electric force, as the electrons flow from a negative source to a positive source. Lightning: During a storm, it is common for an abundance of electrons to build up on the bottom of a cloud, making that part of the cloud negatively charged. Positive charges in the ground start to gather on the surface or even on tall objects such as trees as they are attracted towards the negatively charged undersides of clouds. Lightning strikes as a result of these charges becoming extremely built up. Lightning is caused by electric force Lightning Electric Force Equation: Coulomb's Law The magnitude of the electric force, or the amount of force in which objects repel or attract, depends on the distance between the two charged objects and the amount of charge each object carries. The electric force is stronger the closer together the two charges are, and weaker as the two charges move apart. Electric force is also stronger with more charge, and weaker with less charge. This effect on electric force is predictable, and is known as Coulomb's Law. It can be calculated using a mathematical equation, and the resulting magnitude of electric force is measured in Newtons. Coulomb's Law Electric force can be calculated using the following equation known as Coulomb's Law: In this equation, F is the electric force measured in newtons, K is a constant known as the electrostatic constant, and are charges one and two measured in coulombs, and is the radial distance in meters between the two charges. Since the distance is squared and on the denominator, the electric force drops off exponentially as charges move away from each other. This means that the Electric force is inversely proportional to distance. As charges move away from each other, the electric force between them gets smaller and smaller, until the force is negligible. The amount of charges are in the numerator of this equation, making the magnitude of the force larger with more charge. This means that the force is directly proportional to the amount of charge. When the charges are smaller, the amount of force will be smaller. When there is a lot of charge, the force will be much greater. When calculating the electric force using Coulomb's law, the resulting answer only gives the magnitude of the force and not the direction. In order to know the direction, you must know the types of charges. Once again, like forces repel, and unlike forces attract. It helps to draw a visual representation, or a free-body diagram, of the charges and forces acting upon them in order to understand the resulting force direction. Electric Field versus Electric Force An electric field is a direct result of an electric force. Its pure definition is electric force per unit charge, and can be thought of as a mapping of the force vectors. An electric field is present anytime there is an electric force. Therefore, when there are two or more charged particles, there is a surrounding electric field. The direction of the electric field is the direction a positive charge would flow if it were placed within the field. The electric field moves out from a positive charge and goes into a negative charge. Particles with unlike charges move towards each other, and their corresponding electric field lines move out from the positive charge and into the negative charge. The strength of the force at any given point can be seen through the spacing of the electric field lines. The electric force is strongest where the electric field lines are closest together, and weaker as these lines move apart. Like Coulomb's law expresses, electric field lines show how the electric force is strongest with a minimum distance between the two charges. Unlike charges will result in a repelling force, and the resulting electric field is a visual representation of this effect. Electric fields of two positive charges have the electric field moving out away from both of them. As with two negative charges, the field lines move in towards each negative. Lesson Summary An electric force is created when there are two or more charged particles or objects. These charges can be either positive or negative. Like charges will attract (move towards each other) while unlike charges will repel (move away from each other). As Newton's third law suggests, the forces acting upon each other are both equal and opposite. Electrons and protons within an atom are the two smallest types of charges there are. Electrons carry a negative charge while protons carry a positive charge. Electrons can be easily removed or added to atoms, making the overall charge positive or negative. Objects with more electrons than protons are negatively charged. Electric force is strengthened with increased charge and a shorter distance between the charges. This effect is known as Coulomb's law and can be calculated with the Coulomb's law equation. The magnitude of the force is measured in Newtons, and the direction can be determined by knowing whether the charges are attracting or repelling each other. An electric field is present wherever there is an electric force. The direction of this electric field is the direction a positive charge would flow if it where to be dropped in the field, which is from the positive to the negative.Electrostatics The section of CBSE Class 12 Physics electrostatic potential and capacitance notes mainly deals with the in-depth analysis of electromagnetic phenomena when they are not performing any movements. Additionally, it is divided into ten further sub-topics to study the companion processes of reaching the state. These are - 1. Electric charge In this section of Physics ch 2 Class 12 notes, you get to learn about the basic features of electric charge and its expression in Physics. Along with its basics, the sections help to understand the full potential of charge. Different aspects of Charge included in Class 12 Physics Chapter 2 notes are - Definition Type: Positive and Negative Charge Unit and dimensional formula Point Charge Properties of Charge Comparison of Charge and Mass Methods of Charging Electroscope 2. Coulomb's Law Force is created when charges of opposite signs attract each other, and they repulse if the signs are the same. Coulomb's law tries to define this phenomenon through a mathematical formula, explicitly mentioned in Physics Class 12 notes Chapter 2. Moreover, there is key information about the variation of the constant k and its effect on a medium. Coulomb's law's vector form and the principle of superimposition are also explained in ch 2 Physics Class 12 notes. (Image will be uploaded soon) 3. Electric Field As stated in Class 12 Physics Chapter 2 notes, every positively or negatively charged particle has their respective electric fields. It feels a force at the time of interaction which might be attraction or repulsion. As it arises from electric charge, it is crucial to know about its different parts like - Electric field intensity Relation between electric force and electric field Super imposition of electric field Point charge Continuous charge distributions Properties of Electric Field Lines Motion of Charged Particles in an Electric field Learning more about the electric field from electric potential and capacitance notes Class 12 helps a student to get a grasp of upcoming chapters. 4. Electric Potential Energy When energy helps a charge to move from an electric field, it is known as the Electric Potential Energy. This section of electrostatic chapter Class 12 notes requires a student to study the Electron volt (eV), and the potential energy that an n number of charges can hold. 5. Electric Potential This section of Class 12 Physics Chapter 2 notes focuses on in-depth learning of Electric Potential or Voltage. Basically, it defines the potential movement of energy. 6. Relation between Electric Field and Potential Apart from knowing more about the relationship between the two values, Physics Class 12 Chapter 2 notes also discuss equipotential surfaces. 7. Electric Dipole Essentially, 'Dipoles' are two opposite points of charge represented with q and –q, with their distance between each other being 2a. Electric Dipoles are crucial in your study of Physics Class 12 Chapter 2 notes to learn more about electric fields and their potential. Additionally, Class 12 Physics Chapter 2 notes focus on the influence of electric dipoles on a uniform electric field mainly through Force and Torque, Work, and Potential Energy. In the last part of Electrostatics, further focus is on using the formulas to their fullest potential. It includes subsections of Electric Field, Electric Potential Energy, Electric Potential, and Electric Dipole. In the notes for electrostatic potential and capacitance, you will find proper solutions accompanied by clear and crisp diagrams for better understanding. 8. Gauss's Law Apart from just discussing the Gauss's Law, in Physics Class 12 ch 2 notes there is a thorough explanation of its properties and applications. The Gauss' Law states that net electric flux passing through a hypothetical closed surface is equal to the net electric charge present within the same closed surface. Being a broad part of the whole chapter, you may need to spend a little more time on it. Moving forward, it starts discussing the properties of conductors in relation to Gauss's Law. The Class 12 Physics notes Chapter 2 perfectly defines the journey to Gauss' Law from Coulomb's Law. Here is the Gauss's Law present in the Class 12 Physics ch 2 notes, (image will be uploaded soon) 9. Capacitors There is a dedicated section about Capacitors in the Class 12 Physics Chapter 2 notes elucidating its functions and importance as storage of potential electric energy. After explaining the structure of a capacitor, it points out the different types, parallel plate, spherical and cylindrical. The section of Chapter 2 notes of Physics Class 12 is further divided into subheads like: Properties of an ideal battery Grouping of capacitors Simple circuits (Series and Parallel) Dielectric Van de Graaff generator Combination of drops Charge distribution method Wheatstone Bridge-based circuit Extended Wheatstone Bridge Infinite network of capacitors Redistribution of charge between two capacitors Vedantu prepares the Class 12 Physics Chapter 2 notes with help from subject matter experts. In the PDF, you get a comprehensive idea of the topic along with potential answers to the most asked questions. Furthermore, the detailed explanation on each section and subsections are written in a simple language allows a student to ace their exams with wholesome knowledge. These Physics Chapter 2 Class 12 notes are going to be one of the best supplementary study materials besides a student’s textbooks. Visit the Vedantu website or download the app to get your hands on all important notes! Important Questions A charge of 4 × 10–8C is uniformly distributed on the surface of a spherical conductor, having a radius of 15 cm. Determine the electric field just outside this sphere at a point that is 15 cm from the centre of this sphere. Determine the capacitance given that the distance between the two plates has been reduced by half and the parallel plate capacitor holds a capacitance of 20 pF (where 1pF = 10-12 F) having air between the two plates. What will be the total capacitance of a combination where three capacitors, each having a capacitance of 20 pF, are connected in series. A square having a side of 10 cm has a 500 µC charge at its centre. Determine the work done to move a charge of 10 µC between two points that are diagonally opposite each other on the square. At an equatorial point, what will be the electrostatic potential because of an electric dipole? Calculate the work done to move a test charge, q, through a length of 1 cm along the equatorial axis of an electric dipole? Polarisation A capacitor has its plates enclosed in a medium that can be filled by insulating substances. A net dipole moment is then induced by an electric field in the dielectric. This event causes the field in an opposite direction. Equipotential Surface An equipotential surface is a type of surface where the potential always has a constant value. If considered as a point charge, the concentric spheres that are centred at a particular area of this charge are basically equipotential surfaces. Advantages of Vedantu's Revision Notes: A Comprehensive Resource for Effective Learning There are several reasons why one may refer to Vedantu's revision notes for studying a subject like Electrostatic Potential and Capacitance. Here are some key points: Comprehensive Coverage: Vedantu's revision notes provide a comprehensive coverage of the entire topic, ensuring that all important concepts and subtopics are included. Concise and Organized: The notes are designed to be concise, focusing on the key points and core ideas. They are organized in a structured manner, making it easy for students to navigate and revise the content. Simplified Explanation: The revision notes offer simplified explanations of complex concepts, making them more accessible and easier to understand. This helps students grasp the material more effectively. Key Formulas and Equations: The notes highlight the key formulas and equations relevant to the topic, ensuring that students have a clear understanding of the mathematical aspects of Electrostatic Potential and Capacitance. Examples and Illustrations: Vedantu's revision notes often include examples and illustrations that help clarify concepts and provide practical applications, enabling students to better relate theory to real-world scenarios. Quick Recap: The revision notes serve as a quick recap of the important points, allowing students to review the material efficiently before exams or assessments. Exam-Oriented Approach: Vedantu's revision notes are designed with an exam-oriented approach, focusing on the topics and concepts that are frequently asked in examinations. This helps students prepare effectively and increase their chances of scoring well. Accessible Anytime: Vedantu's revision notes are easily accessible online, allowing students to study at their convenience and revise the material anytime, anywhere.She went by the name of Belisa Crepusculario, not because she had been baptized with that name or given it by her mother, but because she herself had searched until she found the poetry of "beauty" and "twilight" and cloaked herself in it. She made her living selling words. She journeyed through the country from the high cold mountains to the burning coasts, stopping at fairs and in markets where she set up four poles covered by a canvas awning under which she took refuge from the sun and rain to minister to her customers. She did not have to peddle her merchandise because from having wandered far and near, everyone knew who she was. Some people waited for her from one year to the next, and when she appeared in the village with her bundle beneath her arm, they would form a line in front of her stall. Her prices were fair. For five centavos she delivered verses from memory, for seven she improved the quality of dreams, for nine she wrote love letters, for twelve she invented insults for irreconcilable enemies. She also sold stories, not fantasies but long, true stories she recited at one telling, never skipping a word. This is how she carried news from one town to another. People paid her to add a line or two: our son was born, so-and-so died, our children got married, the crops burned in the field. Wherever she went a small crowd gathered around to listen as she began to speak, and that was how they learned about each others' doings, about distant relatives, about what was going on in the civil war. To anyone who paid her fifty centavos in trade, she gave the gift of a secret word to drive away melancholy. It was not the same word for everyone, naturally, because that would have been collective dece it. Each person received his or her own word, with the assurance that no one else would use it that way in this universe or the Beyond. Belisa Crepusculario had been born into a family so poor they did not even have names to give their children. She came into the world and grew up in an inhospitable land where some years the rains became avalanches of water that bore everything away before them and others when not a drop fell from the sky and the sun swelled to fill the horizon and the world became a desert. Until she was twelve, Belisa had no occupation or virtue other than having withstood hunger and the exhaustion of centuries. During one interminable drought, it fell to her to bury four younger brothers and sisters, when she realized that her turn was next, she decided to set out across the 2 plains in the direction of the sea, in hopes that she might trick death along the way. The land was eroded, split with deep cracks, strewn with rocks, fossils of trees and thorny bushes, and skeletons of animals bleached by the sun. From time to time she ran into families who, like her, were heading south, following the mirage of water. Some had begun the march carrying their belongings on their back or in small carts, but they could barely move their own bones, and after a while they had to abandon their possessions. They dragged themselves along painfully, their skin turned to lizard hide and their eyes burned by the reverberating glare. Belisa greeted them with a wave as she passed, but she did not stop, because she had no strength to waste in acts of compassion. Many people fell by the wayside, but she was so stubborn that she survived to cross through that hell and at long last reach the first trickles of water, fine, almost invisible threads that fed spindly vegetation and farther down widened into small streams and marshes. Belisa Crepusculario saved her life and in the process accidentally discovered writing. In a village near the coast, the wind blew a page of newspaper at her feet. She picked up the brittle yellow paper and stood a long while looking at it, unable to determine its purpose, until curiosity overcame her shyness. She walked over to a man who was washing his horse in the muddy pool where she had quenched her thirst. "What is this?" she asked. "The sports page of the newspaper," the man replied, concealing his surprise at her ignorance. The answer astounded the girl, but she did not want to seem rude, so she merely inquired about the significance of the fly tracks scattered across the page. "Those are words, child. Here it says that Fulgencio Barba knocked out El Negro Tiznao in the third round." That was the day Belisa Crepusculario found out that words make their way in the world without a master, and that anyone with a little cleverness can appropriate them and do business with them. She made a quick assessment of her situation and concluded that aside from becoming a prostitute or working as a servant in the kitchens of the rich there were few occupations she was qualified for. It seemed to her that selling words would be an honorable alternative. From that moment on, she worked at that profession, and was never tempted by any other. At the beginning, she offered her merchandise unaware that words could be written outside of newspapers. When she learned otherwise, she calculated the infinite possibilities of her trade and with her savings paid a priest twenty pesos to teach her to read and write, with her three 3 remaining coins she bought a dictionary. She poured over it from A to Z and then threw it into the sea, because it was not her intention to defraud her customers with packaged words. One August morning several years later, Belisa Crepusculario was sitting in her tent in the middle of a plaza, surrounded by the uproar of market day, selling legal arguments to an old man who had been trying for sixteen years to get his pension. Suddenly she heard yelling and thudding hoofbeats. She looked up from her writing and saw, first, a cloud of dust, and then a band of horsemen come galloping into the plaza. They were the Colonel's men, sent under orders of El Mulato, a giant known throughout the land for the speed of his knife and his loyalty to his chief. Both the Colonel and El Mulato had spent their lives fighting in the civil war, and their names were ineradicably linked to devastation and calamity. The rebels swept into town like a stampeding herd, wrapped in noise, bathed in sweat, and leaving a hurricane of fear in their trail. Chickens took wing, dogs ran for their lives, women and children scurried out of sight, until the only living soul left in the market was Belisa Crepusculario. She had never seen El Mulato and was surprised to see him walking toward her. "I'm looking for you," he shouted, pointing his coiled whip at her, even before the words were out, two men rushed her -- knocking over her canopy and shattering her inkwell -- bound her hand and foot, and threw her like a sea bag across the rump of El Mulato's mount. Then they thundered off toward the hills. Hours later, just as Belisa Crepusculario was near death, her heart ground to sand by the pounding of the horse, they stopped, and four strong hands set her down. She tried to stand on her feet and hold her head high, but her strength failed her and she slumped to the ground, sinking into a confused dream. She awakened several hours later to the murmur of night in the camp, but before she had time to sort out the sounds, she opened her eyes and found herself staring into the impatient glare of El Mulato, kneeling beside her. "Well, woman, at last you've come to," he said. To speed her to her senses, he tipped his canteen and offered her a sip of liquor laced with gunpowder. She demanded to know the reason for such rough treatment, and El Mulato explained that the Colonel needed her services. He allowed her to splash water on her face, and then led her to the far end of the camp where the most feared man in all the land was lazing in a hammock strung between two trees. She could not see his face, because he lay in the deceptive shadow of the leaves and the indelible shadow of all his years as a bandit, but she imagined from the way his 4 gigantic aide addressed him with such humility that he must have a very menacing expression. She was surprised by the Colonel's voice, as soft and well-modulated as a professor's. "Are you the woman who sells words?" he asked. "At your service," she stammered, peering into the dark and trying to see him better. The Colonel stood up, and turned straight toward her. She saw dark skin and the eyes of a ferocious puma, and she knew immediately that she was standing before the loneliest man in the world. "I want to be President," he announced. The Colonel was weary of riding across that godforsaken land, waging useless wars and suffering defeats that no subterfuge could transform into victories. For years he had been sleeping in the open air, bitten by mosquitoes, eating iguanas and snake soup, but those minor inconveniences were not why he wanted to change his destiny. What truly troubled him was the terror he saw in people's eyes. He longed to ride into a town beneath a triumphal arch with bright flags and flowers everywhere, he wanted to be cheered, and be given newly laid eggs and freshly baked bread. Men fled at the sight of him, children trembled, and women miscarried from fright, he had had enough, and so he had decided to become President. El Mulato had suggested that they ride to the capital, gallop up to the Palace, and take over the government, the way they had taken so many other things without anyone's permission. The Colonel, however, did not want to be just another tyrant, there had been enough of those before him and, besides, if he did that, he would never win people's hearts. It was his aspiration to win the popular vote in the December elections. "To do that, I have to talk like a candidate. Can you sell me the words for a speech?" the Colonel asked Belisa Crepusculario. She had accepted many assignments, but none like this. She did not dare refuse, fearing that El Mulato would shoot her between the eyes, or worse still, that the Colonel would burst into tears. There was more to it than that, however, she felt the urge to help him because she felt a throbbing warmth beneath her skin, a powerful desire to touch that man, to fondle him, to clasp him in her arms. All night and a good part of the following day, Belisa Crepusculario searched her repertory for words adequate for a presidential speech, closely watched by El Mulato, who could not take his eyes from her firm wanderer's legs and virginal breasts. She discarded harsh, cold words, words 5 that were too flowery, words worn from abuse, words that offered improbable promises, untruthful and confusing words, until all she had left were words sure to touch the minds of men and women's intuition. Calling upon the knowledge she had purchased from the priest for twenty pesos, she wrote the speech on a sheet of paper and then signaled El Mulato to untie the rope that bound her ankles to a tree. He led her once more to the Colonel, and again she felt the throbbing anxiety that had seized her when she first saw him. She handed him the paper and waited while he looked at it, holding it gingerly between thumbs and fingertips. "What the shit does this say," he asked finally. "Don't you know how to read?" "War's what I know," he replied. She read the speech aloud. She read it three times, so her client could engrave it on his memory. When she finished, she saw the emotion in the faces of the soldiers who had gathered round to listen, and saw that the Colonel's eyes glittered with enthusiasm, convinced that with those words the presidential chair would be his. "If after they've heard it three times, the boys are still standing there with their mouths hanging open, it must mean the thing's damn good, Colonel" was El Mulato's approval. "All right, woman. How much do I owe you?" the leader asked. "One peso, Colonel." "That's not much," he said, opening the pouch he wore at his belt, heavy with proceeds from the last foray. "The peso entitles you to a bonus. I'm going to give you two secret words," said Belisa Crepusculario. "What for?" She explained that for every fifty centavos a client paid, she gave him the gift of a word for his exclusive use. The Colonel shrugged. He had no interest at all in her offer, but he did not want to be impolite to someone who had served him so well. She walked slowly to the leather stool where he was sitting, and bent down to give him her gift. The man smelled the scent of a mountain cat issuing from the woman, a fiery heat radiating from her hips, he heard the terrible whisper of her hair, and a breath of sweetmint murmured into his ear the two secret words that were his alone. "They are yours, Colonel," she said as she stepped back. "You may use them as much as you 6 please." El Mulato accompanied Belisa to the roadside, his eyes as entreating as a stray dog's, but when he reached out to touch her, he was stopped by an avalanche of words he had never heard before; believing them to be an irrevocable curse, the flame of his desire was extinguished. During the months of September, October, and November the Colonel delivered his speech so many times that had it not been crafted from glowing and durable words it would have turned to ash as he spoke. He travelled up and down and across the country, riding into cities with a triumphal air, stopping in even the most forgotten villages where only the dump heap betrayed a human presence, to convince his fellow citizens to vote for him. While he spoke from a platform erected in the middle of the plaza, El Mulato and his men handed out sweets and painted his name on all the walls in gold frost. No one paid the least attention to those advertising ploys; they were dazzled by the clarity of the Colonel's proposals and the poetic lucidity of his arguments, infected by his powerful wish to right the wrongs of history, happy for the first time in their lives. When the Candidate had finished his speech, his soldiers would fire their pistols into the air and set off firecrackers, and when finally they rode off, they left behind a wake of hope that lingered for days on the air, like the splendid memory of a comet's tail. Soon the Colonel was the favorite. No one had ever witnessed such a phenomenon: a man who surfaced from the civil war, covered with scars and speaking like a professor, a man whose fame spread to every corner of the land and captured the nation's heart. The press focused their attention on him. Newspapermen came from far away to interview him and repeat his phrases, and the number of his followers and enemies continued to grow. "We're doing great, Colonel," said El Mulato, after twelve successful weeks of campaigning. But the Candidate did not hear. He was repeating his secret words, as he did more and more obsessively. He said them when he was mellow with nostalgia; he murmured them in his sleep; he carried them with him on horseback; he thought them before delivering his famous speech; and he caught himself savoring them in his leisure time. And every time he thought of those two words, he thought of Belisa Crepusculario, and his senses were inflamed with the memory of her feral scent, her fiery heat, the whisper of her hair, and her sweetmint breath in his ear, until he began to go around like a sleepwalker, and his men realized that he might die before he ever sat in the presidential chair. "What's got hold of you, Colonel," El Mulato asked so often that finally one day his chief broke 7 down and told him the source of his befuddlement: those two words that were buried like two daggers in his gut. "Tell me what they are and maybe they'll lose their magic," his faithful aide suggested. "I can't tell them, they're for me alone," the Colonel replied. Saddened by watching his chief decline like a man with a death sentence on his head, El Mulato slung his rifle over his shoulder and set out to find Belisa Crepusculario. He followed her trail through all that vast country, until he found her in a village in the far south, sitting under her tent reciting her rosary of news. He planted himself, spraddle-legged, before her, weapon in hand. "You! You're coming with me," he ordered. She had been waiting. She picked up her inkwell, folded the canvas of her small stall, arranged her shawl around her shoulders, and without a word took her place behind El Mulato's saddle. They did not exchange so much as a word in all the trip; El Mulato's desire for her had turned into rage, and only his fear of her tongue prevented his cutting her to shreds with his whip. Nor was he inclined to tell her that the Colonel was in a fog, and that a spell whispered into his ear had done what years of battle had not been able to do. Three days later they arrived at the encampment, and immediately, in view of all the troops, El Mulato led his prisoner before the Candidate. "I brought this witch here so you can give her back her words, Colonel," El Mulato said, pointing the barrel of his rifle at the woman's head. "And then she can give you back your manhood." The Colonel and Belisa Crepusculario stared at each other, measuring one another from a distance. The men knew then that their leader would never undo the witchcraft of those accursed words, because the whole world could see the voracious-puma eyes soften as the woman walked to him and took his hand in hers. Copyright © 1989 by Isabel Allende From The Stories of Eva Luna, Translated by Margaret Sayers PedenIntroduction to Free Fall A free-falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects: • Free-falling objects do not encounter air resistance. • All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s for back-of-the-envelope calculations) Because free-falling objects are accelerating downwards at a rate of 9.8 m/s/s, a ticker tape trace or dot diagram of its motion would depict an acceleration. The dot diagram at the right depicts the acceleration of a free-falling object. The position of the object at regular time intervals - say, every 0.1 second - is shown. The fact that the distance that the object travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. Recall from an earlier lesson, that if an object travels downward and speeds up, then its acceleration is downward. Free-fall acceleration is often witnessed in a physics classroom by means of an ever-popular strobe light demonstration. The room is darkened and a jug full of water is connected by a tube to a medicine dropper. The dropper drips water and the strobe illuminate the falling droplets at a regular rate - say once every 0.2 seconds. Instead of seeing a stream of water free-falling from the medicine dropper, several consecutive drops with increasing separation distance are seen. The pattern of drops resembles the dot diagram shown in the graphic at the right. The Acceleration of Gravity It was learned in the previous part of this lesson that a free-falling object is an object that is falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the acceleration of gravity - the acceleration for any object moving under the sole influence of gravity. A matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it - the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s2. There are slight variations in this numerical value (to the second decimal place) that are dependent primarily upon on altitude. We will occasionally use the approximated value of 10 m/s2 in order to reduce the complexity of the many mathematical tasks that we will perform with this number. By so doing, we will be able to better focus on the conceptual nature of physics without too much of a sacrifice in numerical accuracy. g = 9.8 m/s2, downward Look It Up! Even on the surface of the Earth, there are local variations in the value of the acceleration of gravity (g). These variations are due to latitude, altitude and the local geological structure of the region. Recall from an earlier lesson that acceleration is the rate at which an object changes its velocity. It is the ratio of velocity change to time between any two points in an object's path. To accelerate at 9.8 m/s2 means to change the velocity by 9.8 m/s each second. If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, then one would note the following pattern. Time (s) Velocity (m/s) 0 0 1 - 9.8 2 - 19.6 3 - 29.4 4 - 39.2 5 - 49.0 . Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8 m/s each consecutive second. That is, the free-falling object has an acceleration of approximately 9.8 m/s2. Another way to represent this acceleration of 9.8 m/s2 is to add numbers to our dot diagram that we saw earlier in this lesson. The velocity of the ball is seen to increase as depicted in the diagram at the right. (NOTE: The diagram is not drawn to scale - in two seconds, the object would drop considerably further than the distance from shoulder to toes.) Representing Free Fall by Graphs • Early in Lesson 1 it was mentioned that there are a variety of means of describing the motion of objects. One such means of describing the motion of objects is through the use of graphs - position versus time and velocity vs. time graphs. In this part of Lesson 5, the motion of a free-falling motion will be represented using these two basic types of graphs. Representing Free Fall by Position-Time Graphs A position versus time graph for a free-falling object is shown below. Observe that the line on the graph curves. As learned earlier, a curved line on a position versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s/s), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object (as learned in Lesson 3), the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity. Representing Free Fall by Velocity-Time Graphs A velocity versus time graph for a free-falling object is shown below. Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9,8 m/s/s, downward), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object that is moving in the negative direction and speeding up is said to have a negative acceleration (if necessary, review the vector nature of acceleration). Since the slope of any velocity versus time graph is the acceleration of the object (as learned in Lesson 4), the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 9.8 m/s/s in the downward direction. The Kinematic Equations The goal of this first unit has been to investigate the variety of means by which the motion of objects can be described. The variety of representations that we have investigated includes verbal representations, pictorial representations, numerical representations, and graphical representations (position-time graphs and velocity-time graphs). In Lesson 6, we will investigate the use of equations to describe and represent the motion of objects. These equations are known as kinematic equations. There are a variety of quantities associated with the motion of objects - displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object's motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, North for 12.0 seconds for a northward displacement of 264 meters, then the motion of the car is fully described. And if a second car is known to accelerate from a rest position with an eastward acceleration of 3.0 m/s2 for a time of 8.0 seconds, providing a final velocity of 24 m/s, East and an eastward displacement of 96 meters, then the motion of this car is fully described. These two statements provide a complete description of the motion of an object. However, such completeness is not always known. It is often the case that only a few parameters of an object's motion are known, while the rest are unknown. For example as you approach the stoplight, you might know that your car has a velocity of 22 m/s, East and is capable of a skidding acceleration of 8.0 m/s2, West. However you do not know the displacement that your car would experience if you were to slam on your brakes and skid to a stop; and you do not know the time required to skid to a stop. In such an instance as this, the unknown parameters can be determined using physics principles and mathematical equations (the kinematic equations). The BIG 4 The kinematic equations are a set of four equations that can be utilized to predict unknown information about an object's motion if other information is known. The equations can be utilized for any motion that can be described as being either a constant velocity motion (an acceleration of 0 m/s/s) or a constant acceleration motion. They can never be used over any time period during which the acceleration is changing. Each of the kinematic equations include four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object's motion if other information is known. For example, if the acceleration value and the initial and final velocity values of a skidding car is known, then the displacement of the car and the time can be predicted using the kinematic equations. Lesson 6 of this unit will focus upon the use of the kinematic equations to predict the numerical values of unknown quantities for an object's motion. The four kinematic equations that describe an object's motion are: There are a variety of symbols used in the above equations. Each symbol has its own specific meaning. The symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the velocity of the object; a subscript of i after the v (as in vi) indicates that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value. Each of these four equations appropriately describes the mathematical relationship between the parameters of an object's motion. As such, they can be used to predict unknown information about an object's motion if other information is known. In the next part of Lesson 6 we will investigate the process of doing this. Kinematic Equations and Problem-Solving The four kinematic equations that describe the mathematical relationship between the parameters that describe an object's motion were introduced in the previous part of Lesson 6. The four kinematic equations are: In the above equations, the symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stand for the acceleration of the object. And the symbol v stands for the instantaneous velocity of the object; a subscript of i after the v (as in vi) indicates that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value. Problem-Solving Strategy In this part of Lesson 6 we will investigate the process of using the equations to determine unknown information about an object's motion. The process involves the use of a problem-solving strategy that will be used throughout the course. The strategy involves the following steps: 1. Construct an informative diagram of the physical situation. 2. Identify and list the given information in variable form. 3. Identify and list the unknown information in variable form. 4. Identify and list the equation that will be used to determine unknown information from known information. 5. Substitute known values into the equation and use appropriate algebraic steps to solve for the unknown information. 6. Check your answer to ensure that it is reasonable and mathematically correct. The use of this problem-solving strategy in the solution of the following problem is modeled in Examples A and B below. Example Problem A . Ima Hurryin is approaching a stoplight moving with a velocity of +30.0 m/s. The light turns yellow, and Ima applies the brakes and skids to a stop. If Ima's acceleration is -8.00 m/s2, then determine the displacement of the car during the skidding process. (Note that the direction of the velocity and the acceleration vectors are denoted by a + and a - sign.) The solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. Note that the vf value can be inferred to be 0 m/s since Ima's car comes to a stop. The initial velocity (vi) of the car is +30.0 m/s since this is the velocity at the beginning of the motion (the skidding motion). And the acceleration (a) of the car is given as - 8.00 m/s2. (Always pay careful attention to the + and - signs for the given quantities.) The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown quantity. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = +30.0 m/s vf = 0 m/s a = - 8.00 m/s2 d = ?? The next step of the strategy involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation that contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are vf, vi, a, and d. Thus, you will look for an equation that has these four variables listed in it. An inspection of the four equations above reveals that the equation on the top right contains all four variables. vf2 = vi2 + 2 • a • d Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. (0 m/s)2 = (30.0 m/s)2 + 2 • (-8.00 m/s2) • d 0 m2/s2 = 900 m2/s2 + (-16.0 m/s2) • d (16.0 m/s2) • d = 900 m2/s2 - 0 m2/s2 (16.0 m/s2)*d = 900 m2/s2 d = (900 m2/s2)/ (16.0 m/s2) d = (900 m2/s2)/ (16.0 m/s2) d = 56.3 m The solution above reveals that the car will skid a distance of 56.3 meters. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. It takes a car a considerable distance to skid from 30.0 m/s (approximately 65 mi/hr) to a stop. The calculated distance is approximately one-half a football field, making this a very reasonable skidding distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is! Example Problem B Ben Rushin is waiting at a stoplight. When it finally turns green, Ben accelerated from rest at a rate of a 6.00 m/s2 for a time of 4.10 seconds. Determine the displacement of Ben's car during this time period. Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step of the strategy involves the identification and listing of known information in variable form. Note that the vi value can be inferred to be 0 m/s since Ben's car is initially at rest. The acceleration (a) of the car is 6.00 m/s2. And the time (t) is given as 4.10 s. The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown information. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 0 m/s t = 4.10 s a = 6.00 m/s2 d = ?? The next step of the strategy involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation that contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are t, vi, a, and d. An inspection of the four equations above reveals that the equation on the top left contains all four variables. d = vi • t + ½ • a • t2 Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. d = (0 m/s) • (4.1 s) + ½ • (6.00 m/s2) • (4.10 s)2 d = (0 m) + ½ • (6.00 m/s2) • (16.81 s2) d = 0 m + 50.43 m d = 50.4 m The solution above reveals that the car will travel a distance of 50.4 meters. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. A car with an acceleration of 6.00 m/s/s will reach a speed of approximately 24 m/s (approximately 50 mi/hr) in 4.10 s. The distance over which such a car would be displaced during this time period would be approximately one-half a football field, making this a very reasonable distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed, it is! The two example problems above illustrate how the kinematic equations can be combined with a simple problem-solving strategy to predict unknown motion parameters for a moving object. Provided that three motion parameters are known, any of the remaining values can be determined. In the next part of Lesson 6, we will see how this strategy can be applied to free fall situations. Or if interested, you can try some practice problems and check your answer against the given solutions. Kinematic Equations and Free Fall As mentioned in Lesson 5, a free-falling object is an object that is falling under the sole influence of gravity. That is to say that any object that is moving and being acted upon only be the force of gravity is said to be "in a state of free fall." Such an object will experience a downward acceleration of 9.8 m/s/s. Whether the object is falling downward or rising upward towards its peak, if it is under the sole influence of gravity, then its acceleration value is 9.8 m/s/s. Like any moving object, the motion of an object in free fall can be described by four kinematic equations. The kinematic equations that describe any object's motion are: The symbols in the above equation have a specific meaning: the symbol d stands for the displacement; the symbol t stands for the time; the symbol a stands for the acceleration of the object; the symbol vi stands for the initial velocity value; and the symbol vf stands for the final velocity. Applying Free Fall Concepts to Problem-Solving There are a few conceptual characteristics of free fall motion that will be of value when using the equations to analyze free fall motion. These concepts are described as follows: • An object in free fall experiences an acceleration of -9.8 m/s/s. (The - sign indicates a downward acceleration.) Whether explicitly stated or not, the value of the acceleration in the kinematic equations is -9.8 m/s/s for any freely falling object. • If an object is merely dropped (as opposed to being thrown) from an elevated height, then the initial velocity of the object is 0 m/s. • If an object is projected upwards in a perfectly vertical direction, then it will slow down as it rises upward. The instant at which it reaches the peak of its trajectory, its velocity is 0 m/s. This value can be used as one of the motion parameters in the kinematic equations; for example, the final velocity (vf) after traveling to the peak would be assigned a value of 0 m/s. • If an object is projected upwards in a perfectly vertical direction, then the velocity at which it is projected is equal in magnitude and opposite in sign to the velocity that it has when it returns to the same height. That is, a ball projected vertically with an upward velocity of +30 m/s will have a downward velocity of -30 m/s when it returns to the same height. These four principles and the four kinematic equations can be combined to solve problems involving the motion of free-falling objects. The two examples below illustrate application of free fall principles to kinematic problem-solving. In each example, the problem solving strategy that was introduced earlier in this lesson will be utilized. Example Problem A Luke Autbeloe drops a pile of roof shingles from the top of a roof located 8.52 meters above the ground. Determine the time required for the shingles to reach the ground. The solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. You might note that in the statement of the problem, there is only one piece of numerical information explicitly stated: 8.52 meters. The displacement (d) of the shingles is -8.52 m. (The - sign indicates that the displacement is downward). The remaining information must be extracted from the problem statement based upon your understanding of the above principles. For example, the vi value can be inferred to be 0 m/s since the shingles are dropped (released from rest; see note above). And the acceleration (a) of the shingles can be inferred to be -9.8 m/s2 since the shingles are free-falling (see note above). (Always pay careful attention to the + and - signs for the given quantities.) The next step of the solution involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the time of fall. So t is the unknown quantity. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 0.0 m/s d = -8.52 m a = - 9.8 m/s2 t = ?? The next step involves identifying a kinematic equation that allows you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation that contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are d, vi, a, and t. Thus, you will look for an equation that has these four variables listed in it. An inspection of the four equations above reveals that the equation on the top left contains all four variables. d = vi • t + ½ • a • t2 Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. -8.52 m = (0 m/s) • (t) + ½ • (-9.8 m/s2) • (t)2 -8.52 m = (0 m) *(t) + (-4.9 m/s2) • (t)2 -8.52 m = (-4.9 m/s2) • (t)2 (-8.52 m)/(-4.9 m/s2) = t2 1.739 s2 = t2 t = 1.32 s The solution above reveals that the shingles will fall for a time of 1.32 seconds before hitting the ground. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. The shingles are falling a distance of approximately 10 yards (1 meter is pretty close to 1 yard); it seems that an answer between 1 and 2 seconds would be highly reasonable. The calculated time easily falls within this range of reasonability. Checking for accuracy involves substituting the calculated value back into the equation for time and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is! Example Problem B Rex Things throws his mother's crystal vase vertically upwards with an initial velocity of 26.2 m/s. Determine the height to which the vase will rise above its initial height. Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. You might note that in the statement of the problem, there is only one piece of numerical information explicitly stated: 26.2 m/s. The initial velocity (vi) of the vase is +26.2 m/s. (The + sign indicates that the initial velocity is an upwards velocity). The remaining information must be extracted from the problem statement based upon your understanding of the above principles. Note that the vf value can be inferred to be 0 m/s since the final state of the vase is the peak of its trajectory (see note above). The acceleration (a) of the vase is -9.8 m/s2 (see note above). The next step involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the vase (the height to which it rises above its starting height). So d is the unknown information. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 26.2 m/s vf = 0 m/s a = -9.8 m/s2 d = ?? The next step involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation that contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are vi, vf, a, and d. An inspection of the four equations above reveals that the equation on the top right contains all four variables. vf2 = vi2 + 2 • a • d Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. (0 m/s)2 = (26.2 m/s)2 + 2 •(-9.8m/s2) •d 0 m2/s2 = 686.44 m2/s2 + (-19.6 m/s2) •d (-19.6 m/s2) • d = 0 m2/s2 -686.44 m2/s2 (-19.6 m/s2) • d = -686.44 m2/s2 d = (-686.44 m2/s2)/ (-19.6 m/s2) d = 35.0 m The solution above reveals that the vase will travel upwards for a displacement of 35.0 meters before reaching its peak. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. The vase is thrown with a speed of approximately 50 mi/hr (merely approximate 1 m/s to be equivalent to 2 mi/hr). Such a throw will never make it further than one football field in height (approximately 100 m), yet will surely make it past the 10-yard line (approximately 10 meters). The calculated answer certainly falls within this range of reasonability. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed, it is! Kinematic equations provide a useful means of determining the value of an unknown motion parameter if three motion parameters are known. In the case of a free-fall motion, the acceleration is often known. And in many cases, another motion parameter can be inferred through a solid knowledge of some basic kinematic principles.Creating Calculated Fields In Power BI | How To Create Calculated Columns In Power BI