chơi xổ số keno trực tuyến

{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2025-01-31T04:01:09+00:00"},"categoryId":33769,"data":{"title":"Physics","slug":"physics","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"parentCategory":{"categoryId":33756,"title":"Science","slug":"science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"}},"childCategories":[],"description":"Hundreds of articles spanning the basic laws of motion, advanced physics calculations, handy reference material, and even string theory.","relatedArticles":{"self":"//dummies-api.coursofppt.com/v2/articles?category=33769&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":459,"bookCount":7},"_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":460,"items":[{"headers":{"creationTime":"2017-03-26T21:38:56+00:00","modifiedTime":"2024-09-14T16:47:58+00:00","timestamp":"2024-09-14T18:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"The Theory of Parallel Universes","strippedTitle":"the theory of parallel universes","slug":"the-theory-of-parallel-universes","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"Some physicists suggest the possibility of a multiverse, or parallel universes that exist in adition to our own.","noIndex":0,"noFollow":0},"content":"The <i>multiverse</i> is a theory that suggests our universe is not the only one, and that many universes exist parallel to each other. These distinct universes within the multiverse theory are called <i>parallel universes.</i> A variety of different theories lend themselves to a multiverse viewpoint.\r\n\r\nNot all physicists really believe that these universes exist. Even fewer believe that it would ever be possible to contact these parallel universes. Following, are descriptions of different levels, or types of parallel universes, scientists have discussed.\r\n<h2 id=\"tab1\" >Level 1: If you go far enough, you’ll get back home</h2>\r\nThe idea of Level 1 parallel universes basically says that space is so big that the rules of probability imply that surely, somewhere else out there, are other planets exactly like Earth. In fact, an infinite universe would have infinitely many planets, and on some of them, the events that play out would be virtually identical to those on our own Earth.\r\n\r\nWe don’t see these other universes because our cosmic vision is limited by the speed of light — the ultimate speed limit. Light started traveling at the moment of the big bang, about 14 billion years ago, and so we can’t see any further than about 14 billion light-years (a bit farther, since space is expanding). This volume of space is called the <i>Hubble volume</i> and represents our observable universe.\r\n\r\nThe existence of Level 1 parallel universes depends on two assumptions:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The universe is infinite (or virtually so).</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Within an infinite universe, every single possible configuration of particles in a Hubble volume takes place multiple times.</p>\r\n</li>\r\n</ul>\r\nIf Level 1 parallel universes do exist, reaching one is virtually (but not entirely) impossible. For one thing, we wouldn’t know where to look for one because, by definition, a Level 1 parallel universe is so far away that no message can ever get from us to them, or them to us. (Remember, we can only get messages from within our own Hubble volume.)\r\n<h2 id=\"tab2\" >Level 2: If you go far enough, you’ll fall into wonderland</h2>\r\nIn a Level 2 parallel universe, regions of space are continuing to undergo an inflation phase. Because of the continuing inflationary phase in these universes, space between us and the other universes is literally expanding faster than the speed of light — and they are, therefore, completely unreachable.\r\n\r\nTwo possible theories present reasons to believe that Level 2 parallel universes may exist: eternal inflation and ekpyrotic theory.\r\n\r\nIn eternal inflation, recall that the quantum fluctuations in the early universe’s vacuum energy caused bubble universes to be created all over the place, expanding through their inflation stages at different rates. The initial condition of these universes is assumed to be at a maximum energy level, although at least one variant, <i>chaotic inflation,</i> predicts that the initial condition can be chaotically chosen as any energy level, which may have no maximum, and the results will be the same.\r\n\r\nThe findings of eternal inflation mean that when inflation starts, it produces not just one universe, but an infinite number of universes.\r\n\r\nRight now, the only noninflationary model that carries any kind of weight is the ekpyrotic model, which is so new that it’s still highly speculative.\r\n<p class=\"Tip\">In the ekpyrotic theory picture, if the universe is the region that results when two branes collide, then the branes could actually collide in multiple locations. Consider flapping a sheet up and down rapidly onto the surface of a bed. The sheet doesn’t touch the bed only in one location, but rather touches it in multiple locations. If the sheet were a brane, then each point of collision would create its own universe with its own initial conditions.</p>\r\nThere’s no reason to expect that branes collide in only one place, so the ekpyrotic theory makes it very probable that there are other universes in other locations, expanding even as you consider this possibility.\r\n<h2 id=\"tab3\" >Level 3: If you stay where you are, you’ll run into yourself</h2>\r\nA Level 3 parallel universe is a consequence of the many worlds interpretation (MWI) from quantum physics in which every single quantum possibility inherent in the quantum wavefunction becomes a real possibility in some reality. When the average person (especially a science fiction fan) thinks of a “parallel universe,” he’s probably thinking of Level 3 parallel universes.\r\n\r\nLevel 3 parallel universes are different from the others posed because they take place in the same space and time as our own universe, but you still have no way to access them. You have never had and will never have contact with any Level 1 or Level 2 universe (we assume), but you’re continually in contact with Level 3 universes — every moment of your life, every decision you make, is causing a split of your “now” self into an infinite number of future selves, all of which are unaware of each other.\r\n<p class=\"Tip\">Though we talk of the universe “splitting,” this isn’t precisely true. From a mathematical standpoint, there’s only one wavefunction, and it evolves over time. The <i>superpositions</i> of different universes all coexist simultaneously in the same infinite-dimensional Hilbert space. These separate, coexisting universes interfere with each other, yielding the bizarre quantum behaviors.</p>\r\nOf the four types of universes, Level 3 parallel universes have the least to do with string theory directly.\r\n<h2 id=\"tab4\" >Level 4: Somewhere over the rainbow, there’s a magical land</h2>\r\nA Level 4 parallel universe is the strangest place (and most controversial prediction) of all, because it would follow fundamentally different mathematical laws of nature than our universe. In short, any universe that physicists can get to work out on paper would exist, based on the <i>mathematical democracy principle:</i> Any universe that is mathematically possible has equal possibility of actually existing.","description":"The <i>multiverse</i> is a theory that suggests our universe is not the only one, and that many universes exist parallel to each other. These distinct universes within the multiverse theory are called <i>parallel universes.</i> A variety of different theories lend themselves to a multiverse viewpoint.\r\n\r\nNot all physicists really believe that these universes exist. Even fewer believe that it would ever be possible to contact these parallel universes. Following, are descriptions of different levels, or types of parallel universes, scientists have discussed.\r\n<h2 id=\"tab1\" >Level 1: If you go far enough, you’ll get back home</h2>\r\nThe idea of Level 1 parallel universes basically says that space is so big that the rules of probability imply that surely, somewhere else out there, are other planets exactly like Earth. In fact, an infinite universe would have infinitely many planets, and on some of them, the events that play out would be virtually identical to those on our own Earth.\r\n\r\nWe don’t see these other universes because our cosmic vision is limited by the speed of light — the ultimate speed limit. Light started traveling at the moment of the big bang, about 14 billion years ago, and so we can’t see any further than about 14 billion light-years (a bit farther, since space is expanding). This volume of space is called the <i>Hubble volume</i> and represents our observable universe.\r\n\r\nThe existence of Level 1 parallel universes depends on two assumptions:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The universe is infinite (or virtually so).</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Within an infinite universe, every single possible configuration of particles in a Hubble volume takes place multiple times.</p>\r\n</li>\r\n</ul>\r\nIf Level 1 parallel universes do exist, reaching one is virtually (but not entirely) impossible. For one thing, we wouldn’t know where to look for one because, by definition, a Level 1 parallel universe is so far away that no message can ever get from us to them, or them to us. (Remember, we can only get messages from within our own Hubble volume.)\r\n<h2 id=\"tab2\" >Level 2: If you go far enough, you’ll fall into wonderland</h2>\r\nIn a Level 2 parallel universe, regions of space are continuing to undergo an inflation phase. Because of the continuing inflationary phase in these universes, space between us and the other universes is literally expanding faster than the speed of light — and they are, therefore, completely unreachable.\r\n\r\nTwo possible theories present reasons to believe that Level 2 parallel universes may exist: eternal inflation and ekpyrotic theory.\r\n\r\nIn eternal inflation, recall that the quantum fluctuations in the early universe’s vacuum energy caused bubble universes to be created all over the place, expanding through their inflation stages at different rates. The initial condition of these universes is assumed to be at a maximum energy level, although at least one variant, <i>chaotic inflation,</i> predicts that the initial condition can be chaotically chosen as any energy level, which may have no maximum, and the results will be the same.\r\n\r\nThe findings of eternal inflation mean that when inflation starts, it produces not just one universe, but an infinite number of universes.\r\n\r\nRight now, the only noninflationary model that carries any kind of weight is the ekpyrotic model, which is so new that it’s still highly speculative.\r\n<p class=\"Tip\">In the ekpyrotic theory picture, if the universe is the region that results when two branes collide, then the branes could actually collide in multiple locations. Consider flapping a sheet up and down rapidly onto the surface of a bed. The sheet doesn’t touch the bed only in one location, but rather touches it in multiple locations. If the sheet were a brane, then each point of collision would create its own universe with its own initial conditions.</p>\r\nThere’s no reason to expect that branes collide in only one place, so the ekpyrotic theory makes it very probable that there are other universes in other locations, expanding even as you consider this possibility.\r\n<h2 id=\"tab3\" >Level 3: If you stay where you are, you’ll run into yourself</h2>\r\nA Level 3 parallel universe is a consequence of the many worlds interpretation (MWI) from quantum physics in which every single quantum possibility inherent in the quantum wavefunction becomes a real possibility in some reality. When the average person (especially a science fiction fan) thinks of a “parallel universe,” he’s probably thinking of Level 3 parallel universes.\r\n\r\nLevel 3 parallel universes are different from the others posed because they take place in the same space and time as our own universe, but you still have no way to access them. You have never had and will never have contact with any Level 1 or Level 2 universe (we assume), but you’re continually in contact with Level 3 universes — every moment of your life, every decision you make, is causing a split of your “now” self into an infinite number of future selves, all of which are unaware of each other.\r\n<p class=\"Tip\">Though we talk of the universe “splitting,” this isn’t precisely true. From a mathematical standpoint, there’s only one wavefunction, and it evolves over time. The <i>superpositions</i> of different universes all coexist simultaneously in the same infinite-dimensional Hilbert space. These separate, coexisting universes interfere with each other, yielding the bizarre quantum behaviors.</p>\r\nOf the four types of universes, Level 3 parallel universes have the least to do with string theory directly.\r\n<h2 id=\"tab4\" >Level 4: Somewhere over the rainbow, there’s a magical land</h2>\r\nA Level 4 parallel universe is the strangest place (and most controversial prediction) of all, because it would follow fundamentally different mathematical laws of nature than our universe. In short, any universe that physicists can get to work out on paper would exist, based on the <i>mathematical democracy principle:</i> Any universe that is mathematically possible has equal possibility of actually existing.","blurb":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":9500,"name":"Daniel Robbins","slug":"daniel-robbins","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9500"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Level 1: If you go far enough, you’ll get back home","target":"#tab1"},{"label":"Level 2: If you go far enough, you’ll fall into wonderland","target":"#tab2"},{"label":"Level 3: If you stay where you are, you’ll run into yourself","target":"#tab3"},{"label":"Level 4: Somewhere over the rainbow, there’s a magical land","target":"#tab4"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[{"title":"Contemplating the Cosmos","slug":"theres-something-about-space","collectionId":294090}],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-65034a5f1c775\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" 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Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"Conservative and Nonconservative Forces in Physics","strippedTitle":"conservative and nonconservative forces in physics","slug":"conservative-and-nonconservative-forces-in-physics","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"In physics, it’s important to know the difference between conservative and nonconservative forces. The work a conservative force does on an object is path-indep","noIndex":0,"noFollow":0},"content":"<p>In physics, it’s important to know the difference between conservative and nonconservative forces. The work a <i>conservative force</i> does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That’s different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a <i>nonconservative force.</i></p>\r\n<p>For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises <i>h</i> meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is <i>mgh</i> greater.”</p>\r\n<p>“Mine, too,” you say, looking out over the view. You pull out this equation:</p>\r\n<pre>Δ<i>PE</i> = <i>mg</i>(<i>h<sub>f</sub></i> - <i>h<sub>i</sub></i>)</pre>\r\n<!--<img src=\"//coursofppt.com/wp-content/uploads/331450.image0.png\" width=\"128\" height=\"27\" alt=\"image0.png\"/>-->\r\n<p>This equation basically states that the actual path you take when going vertically from <i>h</i><i><sub>i</sub></i> to<i> h</i><i><sub>f</sub></i><i> </i>doesn’t matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn’t matter, gravity is a conservative force.</p>\r\n<p>Here’s another way of looking at conservative and nonconservative forces. Say you’re vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton.</p>\r\n<p>What’s the change in your gravitational potential energy? In other words, how much net work did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you’ve experienced no net change in your gravitational potential energy, gravity did no net work on you during the day.</p>\r\n<p>The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it didn’t do any work, either.</p>\r\n<p>Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you’re dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that’s twice as long will involve twice as much work to overcome friction.</p>\r\n<p>What’s really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is <i>mechanical energy.</i> When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy doesn’t change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it isn’t recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy.</p>","description":"<p>In physics, it’s important to know the difference between conservative and nonconservative forces. The work a <i>conservative force</i> does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That’s different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a <i>nonconservative force.</i></p>\r\n<p>For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises <i>h</i> meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is <i>mgh</i> greater.”</p>\r\n<p>“Mine, too,” you say, looking out over the view. You pull out this equation:</p>\r\n<pre>Δ<i>PE</i> = <i>mg</i>(<i>h<sub>f</sub></i> - <i>h<sub>i</sub></i>)</pre>\r\n<!--<img src=\"//coursofppt.com/wp-content/uploads/331450.image0.png\" width=\"128\" height=\"27\" alt=\"image0.png\"/>-->\r\n<p>This equation basically states that the actual path you take when going vertically from <i>h</i><i><sub>i</sub></i> to<i> h</i><i><sub>f</sub></i><i> </i>doesn’t matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn’t matter, gravity is a conservative force.</p>\r\n<p>Here’s another way of looking at conservative and nonconservative forces. Say you’re vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton.</p>\r\n<p>What’s the change in your gravitational potential energy? In other words, how much net work did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you’ve experienced no net change in your gravitational potential energy, gravity did no net work on you during the day.</p>\r\n<p>The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it didn’t do any work, either.</p>\r\n<p>Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you’re dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that’s twice as long will involve twice as much work to overcome friction.</p>\r\n<p>What’s really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is <i>mechanical energy.</i> When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy doesn’t change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it isn’t recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy.</p>","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> taught physics at Cornell University for more than 10 years. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}},{"articleId":194225,"title":"How Does Nuclear Fusion Work?","slug":"nuclear-fusion-the-hope-for-our-energy-future","categoryList":["academics-the-arts","science","environmental-science"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194225"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/174308"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"8967\">Steven Holzner</b>, PhD,</b> taught physics at Cornell University for more than 10 years.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> taught physics at Cornell University for more than 10 years. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-64c7ccb046188\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-64c7ccb0466b8\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-07-31T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":174040},{"headers":{"creationTime":"2017-03-26T17:21:03+00:00","modifiedTime":"2024-06-28T21:02:23+00:00","timestamp":"2024-06-29T00:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"Physics: How to Find the Final Height of a Moving Object","strippedTitle":"physics: how to find the final height of a moving object","slug":"use-the-principle-of-conservation-of-mechanical-energy-to-find-the-final-height-of-a-moving-object","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"You can use the physics principle of conservation of mechanical energy to determine the final height of a moving object. Here's how.","noIndex":0,"noFollow":0},"content":"Thanks to the principle of conservation of mechanical energy, you can use physics to determine the final height of a moving object. At this very moment, for example, suppose Tarzan is swinging on a vine over a crocodile-infested river at a speed of 13.0 meters/second. He needs to reach the opposite river bank 9.0 meters above his present position in order to be safe. Can he swing it? The principle of conservation of mechanical energy gives you the answer:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331468.image0.png\" alt=\"image0.png\" width=\"201\" height=\"37\" />\r\n\r\nAt Tarzan’s maximum height at the end of the swing, his speed, <i>v</i><sub>2</sub>, will be 0 meters/second, and assuming <i>h</i><sub>1</sub> = 0 meters — meaning that he started swinging from the same height as the tree branch he's swinging to — you can relate <i>h</i><sub>2</sub> to <i>v</i><sub>1</sub> like this:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331469.image1.png\" alt=\"image1.png\" width=\"73\" height=\"37\" />\r\n\r\nSolving for <i>h</i><sub>2</sub>, this means that\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331470.image2.png\" alt=\"image2.png\" width=\"180\" height=\"120\" />\r\n\r\nTarzan will come up 0.4 meters short of the 9.0 meters he needs to be safe, so he needs some help.","description":"Thanks to the principle of conservation of mechanical energy, you can use physics to determine the final height of a moving object. At this very moment, for example, suppose Tarzan is swinging on a vine over a crocodile-infested river at a speed of 13.0 meters/second. He needs to reach the opposite river bank 9.0 meters above his present position in order to be safe. Can he swing it? The principle of conservation of mechanical energy gives you the answer:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331468.image0.png\" alt=\"image0.png\" width=\"201\" height=\"37\" />\r\n\r\nAt Tarzan’s maximum height at the end of the swing, his speed, <i>v</i><sub>2</sub>, will be 0 meters/second, and assuming <i>h</i><sub>1</sub> = 0 meters — meaning that he started swinging from the same height as the tree branch he's swinging to — you can relate <i>h</i><sub>2</sub> to <i>v</i><sub>1</sub> like this:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331469.image1.png\" alt=\"image1.png\" width=\"73\" height=\"37\" />\r\n\r\nSolving for <i>h</i><sub>2</sub>, this means that\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/331470.image2.png\" alt=\"image2.png\" width=\"180\" height=\"120\" />\r\n\r\nTarzan will come up 0.4 meters short of the 9.0 meters he needs to be safe, so he needs some help.","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> taught physics at Cornell University for more than 10 years. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}},{"articleId":194225,"title":"How Does Nuclear Fusion Work?","slug":"nuclear-fusion-the-hope-for-our-energy-future","categoryList":["academics-the-arts","science","environmental-science"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194225"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/174308"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"8967\">Steven Holzner</b>, PhD,</b> taught physics at Cornell University for more than 10 years.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> taught physics at Cornell University for more than 10 years. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-649cc9bf70046\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-649cc9bf712a2\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2023-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":174029},{"headers":{"creationTime":"2017-03-26T17:23:57+00:00","modifiedTime":"2024-06-06T17:25:41+00:00","timestamp":"2024-06-06T18:01:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"Calculating Torque Perpendicular to the Applied Force","strippedTitle":"calculating torque perpendicular to the applied force","slug":"how-to-calculate-torque-perpendicular-to-where-force-is-applied","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"In physics, how much torque you exert on an object depends on two things: the force you exert, F ; and the lever arm. Also called the moment arm , the lever arm","noIndex":0,"noFollow":0},"content":"In physics, how much torque you exert on an object depends on two things: the force you exert, <i>F</i>; and the <i>lever arm.</i> Also called the <i>moment</i><i>
</i><i>arm</i>, the lever arm is the perpendicular distance from the pivot point to the point at which you exert your force and is related to the distance from the axis, <i>r</i>, by\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329710.image0.png\" alt=\"image0.png\" width=\"121\" height=\"19\" />\r\n\r\nis the angle between the force and a line from the axis to the point where the force is applied.\r\n<div class=\"imageBlock\" style=\"width: 204px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329711.image1.jpg\" alt=\"The torque you exert on a door depends on where you push it.\" width=\"204\" height=\"400\" />\r\n<div class=\"imageCaption\">The torque you exert on a door depends on where you push it.</div>\r\n</div>\r\nAssume that you’re trying to open a door, as in the various scenarios in the figure. You know that if you push on the hinge, as in diagram A, the door won’t open; if you push the middle of the door, as in diagram B, the door will open; but if you push the edge of the door, as in diagram C, the door will open more easily.\r\n\r\nIn the figure, the lever arm, <i>l,</i> is distance <i>r </i>from the hinge to the point at which you exert your force. The torque is the product of the magnitude of the perpendicular force multiplied by the lever arm. It has a special symbol, the Greek letter tau:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329712.image2.png\" alt=\"image2.png\" width=\"41\" height=\"17\" />\r\n\r\nThe units of torque are force units multiplied by distance units, which are newton-meters in the MKS (meter-kilogram-second) system and foot-pounds in the foot-pound-second system.\r\n\r\nFor example, the lever arm in the figure is distance <i>r</i> (because this lever arm is perpendicular to the force), so\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329713.image3.png\" alt=\"image3.png\" width=\"47\" height=\"17\" />\r\n\r\nIf you push with a force of 200 newtons and <i>r</i> is 0.5 meters, what’s the torque you see in the figure? In diagram A, you push on the hinge, so your distance from the pivot point is zero, which means the lever arm is zero. Therefore, the magnitude of the torque is zero. In diagram B, you exert the 200 newtons of force at a distance of 0.5 meters perpendicular to the hinge, so\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329714.image4.png\" alt=\"image4.png\" width=\"231\" height=\"27\" />\r\n\r\nThe magnitude of the torque here is 100 newton-meters. But now take a look at diagram C. You push with 200 newtons of force at a distance of 2<i>r</i> perpendicular to the hinge, which makes the lever arm 2<i>r</i> or 1.0 meter, so you get this torque:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329715.image5.png\" alt=\"image5.png\" width=\"235\" height=\"27\" />\r\n\r\nNow you have 200 newton-meters of torque, because you push at a point twice as far away from the pivot point. In other words, you double the magnitude of your torque. But what would happen if, say, the door were partially open when you exerted your force? Well, you would calculate the torque easily, if you have lever-arm mastery.","description":"In physics, how much torque you exert on an object depends on two things: the force you exert, <i>F</i>; and the <i>lever arm.</i> Also called the <i>moment</i><i>
</i><i>arm</i>, the lever arm is the perpendicular distance from the pivot point to the point at which you exert your force and is related to the distance from the axis, <i>r</i>, by\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329710.image0.png\" alt=\"image0.png\" width=\"121\" height=\"19\" />\r\n\r\nis the angle between the force and a line from the axis to the point where the force is applied.\r\n<div class=\"imageBlock\" style=\"width: 204px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329711.image1.jpg\" alt=\"The torque you exert on a door depends on where you push it.\" width=\"204\" height=\"400\" />\r\n<div class=\"imageCaption\">The torque you exert on a door depends on where you push it.</div>\r\n</div>\r\nAssume that you’re trying to open a door, as in the various scenarios in the figure. You know that if you push on the hinge, as in diagram A, the door won’t open; if you push the middle of the door, as in diagram B, the door will open; but if you push the edge of the door, as in diagram C, the door will open more easily.\r\n\r\nIn the figure, the lever arm, <i>l,</i> is distance <i>r </i>from the hinge to the point at which you exert your force. The torque is the product of the magnitude of the perpendicular force multiplied by the lever arm. It has a special symbol, the Greek letter tau:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329712.image2.png\" alt=\"image2.png\" width=\"41\" height=\"17\" />\r\n\r\nThe units of torque are force units multiplied by distance units, which are newton-meters in the MKS (meter-kilogram-second) system and foot-pounds in the foot-pound-second system.\r\n\r\nFor example, the lever arm in the figure is distance <i>r</i> (because this lever arm is perpendicular to the force), so\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329713.image3.png\" alt=\"image3.png\" width=\"47\" height=\"17\" />\r\n\r\nIf you push with a force of 200 newtons and <i>r</i> is 0.5 meters, what’s the torque you see in the figure? In diagram A, you push on the hinge, so your distance from the pivot point is zero, which means the lever arm is zero. Therefore, the magnitude of the torque is zero. In diagram B, you exert the 200 newtons of force at a distance of 0.5 meters perpendicular to the hinge, so\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329714.image4.png\" alt=\"image4.png\" width=\"231\" height=\"27\" />\r\n\r\nThe magnitude of the torque here is 100 newton-meters. But now take a look at diagram C. You push with 200 newtons of force at a distance of 2<i>r</i> perpendicular to the hinge, which makes the lever arm 2<i>r</i> or 1.0 meter, so you get this torque:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/329715.image5.png\" alt=\"image5.png\" width=\"235\" height=\"27\" />\r\n\r\nNow you have 200 newton-meters of torque, because you push at a point twice as far away from the pivot point. In other words, you double the magnitude of your torque. But what would happen if, say, the door were partially open when you exerted your force? Well, you would calculate the torque easily, if you have lever-arm mastery.","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> was a contributing editor at PC Magazine and was on the faculty of both MIT and Cornell University. He wrote <i>Physics For Dummies, Physics II For Dummies,</i> and <i>Quantum Physics For Dummies.</i> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}},{"articleId":194225,"title":"How Does Nuclear Fusion Work?","slug":"nuclear-fusion-the-hope-for-our-energy-future","categoryList":["academics-the-arts","science","environmental-science"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194225"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/174308"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"8967\">Steven Holzner</b>, PhD,</b> was a contributing editor at PC Magazine and was on the faculty of both MIT and Cornell University. He wrote <i>Physics For Dummies, Physics II For Dummies,</i> and <i>Quantum Physics For Dummies.</i></p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b>Steven Holzner, PhD,</b> was a contributing editor at PC Magazine and was on the faculty of both MIT and Cornell University. He wrote <i>Physics For Dummies, Physics II For Dummies,</i> and <i>Quantum Physics For Dummies.</i> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-647f74608ef17\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-647f74608f9d3\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":174265},{"headers":{"creationTime":"2017-03-26T17:23:27+00:00","modifiedTime":"2024-05-03T20:01:06+00:00","timestamp":"2024-05-03T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"Calculating a Spring’s Potential & Kinetic Energy","strippedTitle":"calculating a spring’s potential & kinetic energy","slug":"how-to-calculate-the-potential-and-kinetic-energy-in-a-spring","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretchi","noIndex":0,"noFollow":0},"content":"In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called <i>elastic potential energy</i> and is equal to the force,<i> F, </i>times the distance, <i>s:</i>\r\n\r\n<i>W</i> = <i>Fs</i>\r\n\r\nAs you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement).\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330075.image0.png\" alt=\"image0.png\" width=\"447\" height=\"61\" />\r\n\r\nThe distance (or displacement),<i> s,</i> is just the difference in position, <i>x</i><i><sub>f</sub></i> – <i>x</i><i><sub>i</sub></i>, and the average force is (1/2)(<i>F</i><i><sub>f</sub></i> + <i>F</i><i><sub>i</sub></i>). Therefore, you can rewrite the equation as follows:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330076.image1.png\" alt=\"image1.png\" width=\"172\" height=\"40\" />\r\n\r\nHooke’s law says that <i>F</i> = –<i>kx</i>. Therefore, you can substitute –<i>kx</i><i><sub>f</sub></i> and –<i>kx</i><i><sub>i</sub></i> for <i>F</i><i><sub>f</sub></i> and <i>F</i><i><sub>i</sub></i><i>:</i>\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330077.image2.png\" alt=\"image2.png\" width=\"197\" height=\"40\" />\r\n\r\nDistributing and simplifying the equation gives you the equation for work in terms of the spring constant and position:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330078.image3.png\" alt=\"image3.png\" width=\"121\" height=\"37\" />\r\n\r\nThe work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330079.image4.png\" alt=\"image4.png\" width=\"76\" height=\"37\" />\r\n\r\nFor example, suppose a spring is elastic and has a spring constant, <i>k,</i> of\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330080.image5.png\" alt=\"image5.png\" width=\"195\" height=\"27\" />\r\n\r\nand you compress the spring by 10.0 centimeters. You store the following amount of energy in it:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330081.image6.png\" alt=\"image6.png\" width=\"361\" height=\"39\" />\r\n\r\nYou can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved:\r\n\r\n<i>PE</i><sub>1</sub> + <i>KE</i><sub>1</sub> = <i>PE</i><sub>2</sub> + <i>KE</i><sub>2</sub>\r\n\r\nWhen you compress the spring 10.0 centimeters, you know that you have\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330082.image7.png\" alt=\"image7.png\" width=\"112\" height=\"27\" />\r\n\r\nof energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330083.image8.png\" alt=\"image8.png\" width=\"115\" height=\"27\" />\r\n\r\nby the conservation of mechanical energy.","description":"In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called <i>elastic potential energy</i> and is equal to the force,<i> F, </i>times the distance, <i>s:</i>\r\n\r\n<i>W</i> = <i>Fs</i>\r\n\r\nAs you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement).\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330075.image0.png\" alt=\"image0.png\" width=\"447\" height=\"61\" />\r\n\r\nThe distance (or displacement),<i> s,</i> is just the difference in position, <i>x</i><i><sub>f</sub></i> – <i>x</i><i><sub>i</sub></i>, and the average force is (1/2)(<i>F</i><i><sub>f</sub></i> + <i>F</i><i><sub>i</sub></i>). Therefore, you can rewrite the equation as follows:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330076.image1.png\" alt=\"image1.png\" width=\"172\" height=\"40\" />\r\n\r\nHooke’s law says that <i>F</i> = –<i>kx</i>. Therefore, you can substitute –<i>kx</i><i><sub>f</sub></i> and –<i>kx</i><i><sub>i</sub></i> for <i>F</i><i><sub>f</sub></i> and <i>F</i><i><sub>i</sub></i><i>:</i>\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330077.image2.png\" alt=\"image2.png\" width=\"197\" height=\"40\" />\r\n\r\nDistributing and simplifying the equation gives you the equation for work in terms of the spring constant and position:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330078.image3.png\" alt=\"image3.png\" width=\"121\" height=\"37\" />\r\n\r\nThe work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330079.image4.png\" alt=\"image4.png\" width=\"76\" height=\"37\" />\r\n\r\nFor example, suppose a spring is elastic and has a spring constant, <i>k,</i> of\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330080.image5.png\" alt=\"image5.png\" width=\"195\" height=\"27\" />\r\n\r\nand you compress the spring by 10.0 centimeters. You store the following amount of energy in it:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330081.image6.png\" alt=\"image6.png\" width=\"361\" height=\"39\" />\r\n\r\nYou can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved:\r\n\r\n<i>PE</i><sub>1</sub> + <i>KE</i><sub>1</sub> = <i>PE</i><sub>2</sub> + <i>KE</i><sub>2</sub>\r\n\r\nWhen you compress the spring 10.0 centimeters, you know that you have\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330082.image7.png\" alt=\"image7.png\" width=\"112\" height=\"27\" />\r\n\r\nof energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330083.image8.png\" alt=\"image8.png\" width=\"115\" height=\"27\" />\r\n\r\nby the conservation of mechanical energy.","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}},{"articleId":194225,"title":"How Does Nuclear Fusion Work?","slug":"nuclear-fusion-the-hope-for-our-energy-future","categoryList":["academics-the-arts","science","environmental-science"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194225"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/174308"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b> Dr. <b data-author-id=\"8967\">Steven Holzner</b></b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-6452cb8fb1c8f\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-6452cb8fb238b\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-05-03T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":174223},{"headers":{"creationTime":"2017-03-26T21:39:13+00:00","modifiedTime":"2024-04-27T18:07:03+00:00","timestamp":"2024-04-27T21:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"Acceleration & Einstein's Relativity Theory","strippedTitle":"acceleration & einstein's relativity theory","slug":"einsteins-general-relativity-theory-gravity-as-acceleration","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"Learn the basic concepts behind Einstein's general theory of relativity and and how the theory relates to gravity as acceleration.","noIndex":0,"noFollow":0},"content":"<i>General relativity</i> was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account <i>non-inertial frames of reference</i> — areas that are accelerating with respect to each other.\r\n\r\nGeneral relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity.\r\n\r\nEinstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve?\r\n\r\nEinstein came to realize the <i>principle of equivalence, </i>and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field.\r\n\r\nAs Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle.\r\n\r\nAs with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in this figure.\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"//coursofppt.com/wp-content/uploads/183863.image0.jpg\" alt=\"(Left) Scientists performing experiments in an accelerating spaceship. (Right) The scientists get t\" width=\"535\" height=\"394\" /> (Left) Scientists performing experiments in an accelerating spaceship. (Right) The scientists get the same results after landing on a planet.[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nEinstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of.\r\n\r\nFor example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of the following figure. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve.\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"//coursofppt.com/wp-content/uploads/183864.image1.jpg\" alt=\"Both acceleration and gravity bend a beam of light.\" width=\"535\" height=\"394\" /> Both acceleration and gravity bend a beam of light.[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nBy the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of the figure above. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support.","description":"<i>General relativity</i> was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account <i>non-inertial frames of reference</i> — areas that are accelerating with respect to each other.\r\n\r\nGeneral relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity.\r\n\r\nEinstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve?\r\n\r\nEinstein came to realize the <i>principle of equivalence, </i>and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field.\r\n\r\nAs Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle.\r\n\r\nAs with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in this figure.\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"//coursofppt.com/wp-content/uploads/183863.image0.jpg\" alt=\"(Left) Scientists performing experiments in an accelerating spaceship. (Right) The scientists get t\" width=\"535\" height=\"394\" /> (Left) Scientists performing experiments in an accelerating spaceship. (Right) The scientists get the same results after landing on a planet.[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nEinstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of.\r\n\r\nFor example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of the following figure. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve.\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"535\"]<img src=\"//coursofppt.com/wp-content/uploads/183864.image1.jpg\" alt=\"Both acceleration and gravity bend a beam of light.\" width=\"535\" height=\"394\" /> Both acceleration and gravity bend a beam of light.[/caption]\r\n\r\n<div class=\"imageCaption\"></div>\r\n</div>\r\nBy the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of the figure above. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support.","blurb":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":9500,"name":"Daniel Robbins","slug":"daniel-robbins","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9500"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-644ae28e8c62c\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-644ae28e8d2c0\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Explore","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2023-08-17T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":193330},{"headers":{"creationTime":"2017-03-26T21:39:12+00:00","modifiedTime":"2024-04-14T18:50:01+00:00","timestamp":"2024-04-14T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"What Is Einstein's General Relativity Theory?","strippedTitle":"what is einstein's general relativity theory?","slug":"einsteins-general-relativity-theory-gravity-as-geometry","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"Learn the basic concepts behind Einstein's general theory of relativity and how the theory relates to gravity as geometry.","noIndex":0,"noFollow":0},"content":"<i>General relativity</i> was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account <i>non-inertial frames of reference</i> — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity.\r\n\r\nThe theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of <i>field equations</i><i>,</i> which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity.\r\n\r\nAs Einstein developed his general theory of relativity, he had to refine the accepted notion of the space-time continuum into a more precise mathematical framework. He also introduced another principle, <i>the principle of covariance</i><i>.</i> This principle states that the laws of physics must take the same form in all coordinate systems.\r\n\r\nIn other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case.\r\n<p class=\"Remember\">Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations.</p>\r\nApplying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of this figure), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of the figure).\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/183867.image0.jpg\" alt=\"Without matter, space-time is flat (left), but it curves when matter is present (right).\" width=\"535\" height=\"165\" />\r\n<div class=\"imageCaption\">Without matter, space-time is flat (left), but it curves when matter is present (right).</div>\r\n</div>\r\nIn other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself.\r\n\r\nBecause space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it.\r\n\r\nAgain, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would <i>not</i> be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to.\r\nSimilarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe.\r\n\r\nIn 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse.\r\n\r\nWhen he published his complete theory of general relativity in 1915, Einstein had corrected a couple of errors and in 1919, an expedition set out to observe the deflection of light by the sun during an eclipse, in to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein.\r\n\r\nEddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation.\r\n\r\nAlbert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it. Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s.","description":"<i>General relativity</i> was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account <i>non-inertial frames of reference</i> — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity.\r\n\r\nThe theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of <i>field equations</i><i>,</i> which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity.\r\n\r\nAs Einstein developed his general theory of relativity, he had to refine the accepted notion of the space-time continuum into a more precise mathematical framework. He also introduced another principle, <i>the principle of covariance</i><i>.</i> This principle states that the laws of physics must take the same form in all coordinate systems.\r\n\r\nIn other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case.\r\n<p class=\"Remember\">Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations.</p>\r\nApplying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of this figure), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of the figure).\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/183867.image0.jpg\" alt=\"Without matter, space-time is flat (left), but it curves when matter is present (right).\" width=\"535\" height=\"165\" />\r\n<div class=\"imageCaption\">Without matter, space-time is flat (left), but it curves when matter is present (right).</div>\r\n</div>\r\nIn other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself.\r\n\r\nBecause space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it.\r\n\r\nAgain, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would <i>not</i> be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to.\r\nSimilarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe.\r\n\r\nIn 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse.\r\n\r\nWhen he published his complete theory of general relativity in 1915, Einstein had corrected a couple of errors and in 1919, an expedition set out to observe the deflection of light by the sun during an eclipse, in to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein.\r\n\r\nEddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation.\r\n\r\nAlbert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it. Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s.","blurb":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":9500,"name":"Daniel Robbins","slug":"daniel-robbins","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9500"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6439bf0f6477a\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6439bf0f65202\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Explore","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2023-08-17T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":193329},{"headers":{"creationTime":"2017-03-26T18:14:29+00:00","modifiedTime":"2024-02-07T16:00:49+00:00","timestamp":"2024-02-07T18:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"String Theory and The Hierarchy Problem in Physics","strippedTitle":"string theory and the hierarchy problem in physics","slug":"string-theory-and-the-hierarchy-problem-in-physics","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"Many physicists feel that string theory will ultimately be successful at resolving the hierarchy problem of the Standard Model of particle physics. Although it ","noIndex":0,"noFollow":0},"content":"Many physicists feel that string theory will ultimately be successful at resolving the hierarchy problem of the Standard Model of particle physics. Although it is an astounding success, the Standard Model hasn’t answered every question that physics hands to it. One of the major questions that remains is the <em>hierarchy problem</em>, which seeks an explanation for the diverse values that the Standard Model lets physicists work with.\r\n\r\nFor example, if you count the theoretical Higgs boson (and both types of W bosons), the Standard Model of particle physics has 18 elementary particles. The masses of these particles aren’t predicted by the Standard Model. Physicists had to find these by experiment and plug them into the equations to get everything to work out right.\r\n\r\nYou notice three families of particles among the fermions, which seems like a lot of unnecessary duplication. If we already have an electron, why does nature need to have a muon that’s 200 times as heavy? Why do we have so many types of quarks?\r\n\r\nBeyond that, when you look at the energy scales associated with the quantum field theories of the Standard Model, as shown in this figure, even more questions may occur to you. Why is there a gap of 16 orders of magnitude (16 zeroes!) between the intensity of the Planck scale energy and the weak scale?\r\n\r\n[caption id=\"attachment_16789\" align=\"aligncenter\" width=\"226\"]<img src=\"//coursofppt.com/wp-content/uploads/315460.image0.jpg\" alt=\"\" width=\"226\" height=\"400\" class=\"size-full wp-image-16789\" /> Credit: � 2005 by Lisa Randall, reprinted by permission of HarperCollins Publishers.[/caption]\r\n\r\nAt the bottom of this scale is the <em>vacuum energy,</em> which is the energy generated by all the strange quantum behavior in empty space — virtual particles exploding into existence and quantum fields fluctuating wildly due to the uncertainty principle.\r\n\r\nThe hierarchy problem occurs because the fundamental parameters of the Standard Model don’t reveal anything about these scales of energy. Just as physicists have to put the particles and their masses into the theory by hand, so too have they had to construct the energy scales by hand. Fundamental principles of physics don’t tell scientists how to transition smoothly from talking about the weak scale to talking about the Planck scale.\r\n\r\nTrying to understand the “gap” between the weak scale and the Planck scale is one of the major motivating factors behind trying to search for a quantum gravity theory in general, and string theory in particular.\r\n\r\nMany physicists would like a single theory that could be applied at all scales, without the need for renormalization (the mathematical process of removing infinities), or at least to understand what properties of nature determine the rules that work for different scales. Others are perfectly happy with renormalization, which has been a major tool of physics for nearly 40 years and works in virtually every problem that physicists run into.","description":"Many physicists feel that string theory will ultimately be successful at resolving the hierarchy problem of the Standard Model of particle physics. Although it is an astounding success, the Standard Model hasn’t answered every question that physics hands to it. One of the major questions that remains is the <em>hierarchy problem</em>, which seeks an explanation for the diverse values that the Standard Model lets physicists work with.\r\n\r\nFor example, if you count the theoretical Higgs boson (and both types of W bosons), the Standard Model of particle physics has 18 elementary particles. The masses of these particles aren’t predicted by the Standard Model. Physicists had to find these by experiment and plug them into the equations to get everything to work out right.\r\n\r\nYou notice three families of particles among the fermions, which seems like a lot of unnecessary duplication. If we already have an electron, why does nature need to have a muon that’s 200 times as heavy? Why do we have so many types of quarks?\r\n\r\nBeyond that, when you look at the energy scales associated with the quantum field theories of the Standard Model, as shown in this figure, even more questions may occur to you. Why is there a gap of 16 orders of magnitude (16 zeroes!) between the intensity of the Planck scale energy and the weak scale?\r\n\r\n[caption id=\"attachment_16789\" align=\"aligncenter\" width=\"226\"]<img src=\"//coursofppt.com/wp-content/uploads/315460.image0.jpg\" alt=\"\" width=\"226\" height=\"400\" class=\"size-full wp-image-16789\" /> Credit: � 2005 by Lisa Randall, reprinted by permission of HarperCollins Publishers.[/caption]\r\n\r\nAt the bottom of this scale is the <em>vacuum energy,</em> which is the energy generated by all the strange quantum behavior in empty space — virtual particles exploding into existence and quantum fields fluctuating wildly due to the uncertainty principle.\r\n\r\nThe hierarchy problem occurs because the fundamental parameters of the Standard Model don’t reveal anything about these scales of energy. Just as physicists have to put the particles and their masses into the theory by hand, so too have they had to construct the energy scales by hand. Fundamental principles of physics don’t tell scientists how to transition smoothly from talking about the weak scale to talking about the Planck scale.\r\n\r\nTrying to understand the “gap” between the weak scale and the Planck scale is one of the major motivating factors behind trying to search for a quantum gravity theory in general, and string theory in particular.\r\n\r\nMany physicists would like a single theory that could be applied at all scales, without the need for renormalization (the mathematical process of removing infinities), or at least to understand what properties of nature determine the rules that work for different scales. Others are perfectly happy with renormalization, which has been a major tool of physics for nearly 40 years and works in virtually every problem that physicists run into.","blurb":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":9500,"name":"Daniel Robbins","slug":"daniel-robbins","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9500"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":178042,"title":"More Dimensions Make String Theory Work","slug":"more-dimensions-make-string-theory-work","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/178042"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":294138,"slug":"string-theory-for-dummies-2","isbn":"9781119888970","categoryList":["academics-the-arts","science","quantum-physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119888972/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119888972/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119888972-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119888972/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119888972/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/string-theory-for-dummies-cover-9781119888970-203x255.jpg","width":203,"height":255},"title":"String Theory For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":35165,"name":"Alessandro Sfrondrini","slug":"alessandro-sfrondrini","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/35165"}},{"authorId":35186,"name":"Alessandro Sfondrini","slug":"alessandro-sfondrini","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/35186"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119888970&quot;]}]\" id=\"du-slot-63e291df91643\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119888970&quot;]}]\" id=\"du-slot-63e291df91cb2\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"One year","lifeExpectancySetFrom":"2024-02-07T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":178179},{"headers":{"creationTime":"2017-03-26T17:23:25+00:00","modifiedTime":"2023-12-23T15:45:58+00:00","timestamp":"2023-12-23T18:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"How to Calculate a Spring Constant Using Hooke's Law","strippedTitle":"how to calculate a spring constant using hooke's law","slug":"how-to-calculate-a-spring-constant-using-hookes-law","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"Learn about Hooke's law and how to calculate the spring constant, including the formula and insight on a spring's impact on force.","noIndex":0,"noFollow":0},"content":"Any physicist knows that if an object applies a force to a spring, then the spring applies an equal and opposite force to the object. <a href=\"//coursofppt.com/education/science/using-a-generalized-hookes-law-for-stress-and-strain/\" target=\"_blank\" rel=\"noopener\">Hooke’s law</a> gives the force a spring exerts on an object attached to it with the following equation:\r\n\r\n<i>F</i> = –<i>kx</i>\r\n\r\nThe minus sign shows that this force is in the opposite direction of the force that’s stretching or compressing the spring. The variables of the equation are <em>F,</em> which represents force, <em>k, </em>which is called the spring constant and measures how stiff and strong the spring is, and <em>x,</em> the distance the spring is stretched or compressed away from its equilibrium or rest position.\r\n\r\nThe force exerted by a spring is called a <i>restoring force;</i> it always acts to restore the spring toward equilibrium.\r\n\r\nIn Hooke’s law, the negative sign on the spring’s force means that the force exerted by the spring opposes the spring’s displacement.\r\n<h2 id=\"tab1\" >Understanding springs and their direction of force</h2>\r\n<img class=\"aligncenter wp-image-239155 size-full\" src=\"//coursofppt.com/wp-content/uploads/direction-of-force-in-springs.jpg\" alt=\"direction of force exzerted in springs\" width=\"535\" height=\"285\" />\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n<div class=\"imageCaption\">The direction of force exerted by a spring</div>\r\n</div>\r\nThe preceding figure shows a ball attached to a spring. You can see that if the spring isn’t stretched or compressed, it exerts no force on the ball. If you push the spring, however, it pushes back, and if you pull the spring, it pulls back.\r\n<p class=\"Remember\">Hooke’s law is valid as long as the elastic material you’re dealing with stays elastic — that is, it stays within its <i><a href=\"//coursofppt.com/education/science/physics/how-to-calculate-the-potential-and-kinetic-energy-in-a-spring/\" target=\"_blank\" rel=\"noopener\">elastic limit</a>.</i> If you pull a spring too far, it loses its stretchy ability. As long as a spring stays within its elastic limit, you can say that <i>F</i> = –<i>kx</i>.</p>\r\n<p class=\"Remember\">When a spring stays within its elastic limit and obeys Hooke’s law, the spring is called an <i>ideal spring.</i></p>\r\n\r\n<h2 id=\"tab2\" >How to find the spring constant (example problem)</h2>\r\nSuppose that a group of car designers knocks on your door and asks whether you can help design a suspension system. “Sure,” you say. They inform you that the car will have a mass of 1,000 kilograms, and you have four shock absorbers, each 0.5 meters long, to work with. How strong do the springs have to be? Assuming these shock absorbers use springs, each one has to support a mass of at least 250 kilograms, which weighs the following:\r\n\r\n<i>F</i> = <i>mg</i> = (250 kg)(9.8 m/s<sup>2</sup>) = 2,450 N\r\n\r\nwhere <i>F</i> equals force, <i>m</i> equals the mass of the object, and <i>g</i> equals the acceleration due to gravity, 9.8 meters per second<sup>2</sup>. The spring in the shock absorber will, at a minimum, have to give you 2,450 newtons of force at the maximum compression of 0.5 meters. What does this mean the spring constant should be?\r\n\r\nIn order to figure out <strong>how to calculate the spring constant</strong>, we must remember what Hooke’s law says:\r\n\r\n<i>F</i> = –<i>kx</i>\r\n\r\nNow, we need to rework the equation so that we are calculating for the missing metric, which is the spring constant, or <em>k</em>. Looking only at the magnitudes and therefore omitting the negative sign, you get\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330102.image1.png\" alt=\"image1.png\" width=\"41\" height=\"37\" />\r\n\r\nTime to plug in the numbers:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330103.image2.png\" alt=\"image2.png\" width=\"195\" height=\"39\" />\r\n\r\nThe springs used in the shock absorbers must have spring constants of at least 4,900 newtons per meter. The car designers rush out, ecstatic, but you call after them, “Don’t forget, you need to at least double that if you actually want your car to be able to handle potholes.”","description":"Any physicist knows that if an object applies a force to a spring, then the spring applies an equal and opposite force to the object. <a href=\"//coursofppt.com/education/science/using-a-generalized-hookes-law-for-stress-and-strain/\" target=\"_blank\" rel=\"noopener\">Hooke’s law</a> gives the force a spring exerts on an object attached to it with the following equation:\r\n\r\n<i>F</i> = –<i>kx</i>\r\n\r\nThe minus sign shows that this force is in the opposite direction of the force that’s stretching or compressing the spring. The variables of the equation are <em>F,</em> which represents force, <em>k, </em>which is called the spring constant and measures how stiff and strong the spring is, and <em>x,</em> the distance the spring is stretched or compressed away from its equilibrium or rest position.\r\n\r\nThe force exerted by a spring is called a <i>restoring force;</i> it always acts to restore the spring toward equilibrium.\r\n\r\nIn Hooke’s law, the negative sign on the spring’s force means that the force exerted by the spring opposes the spring’s displacement.\r\n<h2 id=\"tab1\" >Understanding springs and their direction of force</h2>\r\n<img class=\"aligncenter wp-image-239155 size-full\" src=\"//coursofppt.com/wp-content/uploads/direction-of-force-in-springs.jpg\" alt=\"direction of force exzerted in springs\" width=\"535\" height=\"285\" />\r\n<div class=\"imageBlock\" style=\"width: 535px;\">\r\n<div class=\"imageCaption\">The direction of force exerted by a spring</div>\r\n</div>\r\nThe preceding figure shows a ball attached to a spring. You can see that if the spring isn’t stretched or compressed, it exerts no force on the ball. If you push the spring, however, it pushes back, and if you pull the spring, it pulls back.\r\n<p class=\"Remember\">Hooke’s law is valid as long as the elastic material you’re dealing with stays elastic — that is, it stays within its <i><a href=\"//coursofppt.com/education/science/physics/how-to-calculate-the-potential-and-kinetic-energy-in-a-spring/\" target=\"_blank\" rel=\"noopener\">elastic limit</a>.</i> If you pull a spring too far, it loses its stretchy ability. As long as a spring stays within its elastic limit, you can say that <i>F</i> = –<i>kx</i>.</p>\r\n<p class=\"Remember\">When a spring stays within its elastic limit and obeys Hooke’s law, the spring is called an <i>ideal spring.</i></p>\r\n\r\n<h2 id=\"tab2\" >How to find the spring constant (example problem)</h2>\r\nSuppose that a group of car designers knocks on your door and asks whether you can help design a suspension system. “Sure,” you say. They inform you that the car will have a mass of 1,000 kilograms, and you have four shock absorbers, each 0.5 meters long, to work with. How strong do the springs have to be? Assuming these shock absorbers use springs, each one has to support a mass of at least 250 kilograms, which weighs the following:\r\n\r\n<i>F</i> = <i>mg</i> = (250 kg)(9.8 m/s<sup>2</sup>) = 2,450 N\r\n\r\nwhere <i>F</i> equals force, <i>m</i> equals the mass of the object, and <i>g</i> equals the acceleration due to gravity, 9.8 meters per second<sup>2</sup>. The spring in the shock absorber will, at a minimum, have to give you 2,450 newtons of force at the maximum compression of 0.5 meters. What does this mean the spring constant should be?\r\n\r\nIn order to figure out <strong>how to calculate the spring constant</strong>, we must remember what Hooke’s law says:\r\n\r\n<i>F</i> = –<i>kx</i>\r\n\r\nNow, we need to rework the equation so that we are calculating for the missing metric, which is the spring constant, or <em>k</em>. Looking only at the magnitudes and therefore omitting the negative sign, you get\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330102.image1.png\" alt=\"image1.png\" width=\"41\" height=\"37\" />\r\n\r\nTime to plug in the numbers:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/330103.image2.png\" alt=\"image2.png\" width=\"195\" height=\"39\" />\r\n\r\nThe springs used in the shock absorbers must have spring constants of at least 4,900 newtons per meter. The car designers rush out, ecstatic, but you call after them, “Don’t forget, you need to at least double that if you actually want your car to be able to handle potholes.”","blurb":"","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"primaryCategoryTaxonomy":{"categoryId":33769,"title":"Physics","slug":"physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Understanding springs and their direction of force","target":"#tab1"},{"label":"How to find the spring constant (example problem)","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}},{"articleId":194225,"title":"How Does Nuclear Fusion Work?","slug":"nuclear-fusion-the-hope-for-our-energy-future","categoryList":["academics-the-arts","science","environmental-science"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194225"}},{"articleId":184049,"title":"A List of Physics Constants","slug":"a-list-of-physics-constants","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184049"}},{"articleId":184043,"title":"Physics Equations and Formulas","slug":"physics-equations-and-formulas","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/184043"}},{"articleId":174308,"title":"Calculating Tangential Velocity on a Curve","slug":"calculating-tangential-velocity-on-a-curve","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/174308"}}],"fromCategory":[{"articleId":209405,"title":"String Theory For Dummies Cheat Sheet","slug":"string-theory-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209405"}},{"articleId":209012,"title":"Physics II For Dummies Cheat Sheet","slug":"physics-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/209012"}},{"articleId":208592,"title":"Thermodynamics For Dummies Cheat Sheet","slug":"thermodynamics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208592"}},{"articleId":208578,"title":"Optics For Dummies Cheat Sheet","slug":"optics-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208578"}},{"articleId":208460,"title":"Physics I For Dummies Cheat Sheet","slug":"physics-i-for-dummies-cheat-sheet","categoryList":["academics-the-arts","science","physics"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208460"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282467,"slug":"physics-i-for-dummies","isbn":"9781119872221","categoryList":["academics-the-arts","science","physics"],"amazon":{"default":"//www.amazon.com/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119872227-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119872227/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/9781119872221-203x255.jpg","width":203,"height":255},"title":"Physics I For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b> Dr. <b data-author-id=\"8967\">Steven Holzner</b></b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p>","authors":[{"authorId":8967,"name":"Steven Holzner","slug":"steven-holzner","description":" <p><b> Dr. Steven Holzner</b> has written more than 40 books about physics and programming. He was a contributing editor at <i>PC Magazine</i> and was on the faculty at both MIT and Cornell. He has authored Dummies titles including <i>Physics For Dummies</i> and <i>Physics Essentials For Dummies.</i> Dr. Holzner received his PhD at Cornell.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8967"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[{"title":"For the Spring Term Student","slug":"for-the-spring-term-student","collectionId":296450}],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-63a5ecde9d57a\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;science&quot;,&quot;physics&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119872221&quot;]}]\" id=\"du-slot-63a5ecde9daa9\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-09-10T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":174221},{"headers":{"creationTime":"2017-03-26T18:02:35+00:00","modifiedTime":"2023-12-14T14:49:07+00:00","timestamp":"2023-12-14T15:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Science","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33756"},"slug":"science","categoryId":33756},{"name":"Physics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33769"},"slug":"physics","categoryId":33769}],"title":"More Dimensions Make String Theory Work","strippedTitle":"more dimensions make string theory work","slug":"more-dimensions-make-string-theory-work","canonicalUrl":"","网络搜目录擎调整":{"metaDescription":"For most interpretations, superstring theory requires a large number of extra space dimensions to be mathematically consistent: M-theory requires ten space dime","noIndex":0,"noFollow":0},"content":"For most interpretations, superstring theory requires a large number of extra space dimensions to be mathematically consistent: M-theory requires ten space dimensions. With the introduction of branes as multidimensional objects in string theory, it becomes possible to construct and imagine wildly creative geometries for space that correspond to different possible particles and forces. It’s unclear, at present, whether those extra dimensions exist or are just mathematical artifacts.\r\n<p class=\"Remember\">The reason string theory requires extra dimensions is that trying to eliminate them results in much more complicated mathematical equations. It’s not impossible, but most physicists haven’t pursued these concepts in a great deal of depth, leaving science (perhaps by default) with a theory that requires many extra dimensions.</p>\r\nFrom the time of Descartes, mathematicians have been able to translate between geometric and physical representations. Mathematicians can tackle their equations in virtually any number of dimensions that they choose, even if they can’t visually picture what they’re talking about.\r\n<p class=\"Tip\">One of the tools mathematicians use in exploring higher dimensions is analogy. If you start with a zero-dimensional point and extend it through space, you get a one-dimensional line. If you take that line and extend it into a second dimension, you end up with a square.</p>\r\nIf you extend a square through a third dimension, you end up with a cube. If you then were to take a cube and extend into a fourth dimension, you’d get a shape called a <i>hypercube.</i>\r\n\r\nA line has two “corners” but extending it to a square gives four corners, while a cube has eight corners. By continuing to extend this algebraic relationship, a hypercube would be a four-dimensional object with 16 corners, and a similar relationship can be used to create analogous objects in additional dimensions. Such objects are obviously well outside of what our minds can picture.\r\n<p class=\"Tip\">Humans aren’t psychologically wired to be able to picture more than three space dimensions. A handful of mathematicians (and possibly some physicists) have devoted their lives to the study of extra dimensions so fully that they may be able to actually picture a four-dimensional object, such as a hypercube. Most mathematicians can’t (so don’t feel bad if you can’t).</p>\r\nWhole fields of mathematics — linear algebra, abstract algebra, topology, knot theory, complex analysis, and others — exist with the sole purpose of trying to take abstract concepts, frequently with large numbers of possible variables, degrees of freedom, or dimensions, and make sense of them.\r\n\r\nThese sorts of mathematical tools are at the heart of string theory. Regardless of the ultimate success or failure of string theory as a physical model of reality, it has motivated mathematics to grow and explore new questions in new ways, and for that alone, it has proved useful.","description":"For most interpretations, superstring theory requires a large number of extra space dimensions to be mathematically consistent: M-theory requires ten space dimensions. With the introduction of branes as multidimensional objects in string theory, it becomes possible to construct and imagine wildly creative geometries for space that correspond to different possible particles and forces. It’s unclear, at present, whether those extra dimensions exist or are just mathematical artifacts.\r\n<p class=\"Remember\">The reason string theory requires extra dimensions is that trying to eliminate them results in much more complicated mathematical equations. It’s not impossible, but most physicists haven’t pursued these concepts in a great deal of depth, leaving science (perhaps by default) with a theory that requires many extra dimensions.</p>\r\nFrom the time of Descartes, mathematicians have been able to translate between geometric and physical representations. Mathematicians can tackle their equations in virtually any number of dimensions that they choose, even if they can’t visually picture what they’re talking about.\r\n<p class=\"Tip\">One of the tools mathematicians use in exploring higher dimensions is analogy. If you start with a zero-dimensional point and extend it through space, you get a one-dimensional line. If you take that line and extend it into a second dimension, you end up with a square.</p>\r\nIf you extend a square through a third dimension, you end up with a cube. If you then were to take a cube and extend into a fourth dimension, you’d get a shape called a <i>hypercube.</i>\r\n\r\nA line has two “corners” but extending it to a square gives four corners, while a cube has eight corners. By continuing to extend this algebraic relationship, a hypercube would be a four-dimensional object with 16 corners, and a similar relationship can be used to create analogous objects in additional dimensions. Such objects are obviously well outside of what our minds can picture.\r\n<p class=\"Tip\">Humans aren’t psychologically wired to be able to picture more than three space dimensions. A handful of mathematicians (and possibly some physicists) have devoted their lives to the study of extra dimensions so fully that they may be able to actually picture a four-dimensional object, such as a hypercube. Most mathematicians can’t (so don’t feel bad if you can’t).</p>\r\nWhole fields of mathematics — linear algebra, abstract algebra, topology, knot theory, complex analysis, and others — exist with the sole purpose of trying to take abstract concepts, frequently with large numbers of possible variables, degrees of freedom, or dimensions, and make sense of them.\r\n\r\nThese sorts of mathematical tools are at the heart of string theory. Regardless of the ultimate success or failure of string theory as a physical model of reality, it has motivated mathematics to grow and explore new questions in new ways, and for that alone, it has proved useful.","blurb":"","authors":[{"authorId":9499,"name":"Andrew Zimmerman Jones","slug":"andrew-zimmerman-jones","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9499"}},{"authorId":9500,"name":"Daniel 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