{"id":717,"date":"2020-12-07T14:20:19","date_gmt":"2020-12-07T14:20:19","guid":{"rendered":"https:\/\/pickedshares.com\/?p=717"},"modified":"2021-05-08T11:46:10","modified_gmt":"2021-05-08T11:46:10","slug":"engineering-mechanics-1-exercise-17-determination-of-the-reaction-forces","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-17-determination-of-the-reaction-forces\/","title":{"rendered":"Determination of the reaction forces"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A railway wagon stands on a sloping track on a (frictionless) buffer stop. Which reaction forces occur on the wheels and on the buffer stop?<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"876\" height=\"560\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17.jpg\" alt=\"Railway wagon on a sloping track at the buffer stop\" class=\"wp-image-430\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17.jpg 876w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17-300x192.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17-768x491.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17-400x256.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-17-800x511.jpg 800w\" sizes=\"auto, (max-width: 876px) 100vw, 876px\" \/><figcaption>Railway wagon on a sloping track at the buffer stop<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>The following video is in german language.<\/p>\n\n\n<figure class=\"wp-block-embed-youtube wp-block-embed is-type-video is-provider-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"lyte-wrapper\" title=\"Reaktionskr&auml;fte ermitteln - Technische Mechanik 1, &Uuml;bung 17\" style=\"width:640px;max-width:100%;margin:5px auto;\"><div class=\"lyMe hidef\" id=\"WYL_VJpvBMG1Mos\" itemprop=\"video\" itemscope itemtype=\"https:\/\/schema.org\/VideoObject\"><div><meta itemprop=\"thumbnailUrl\" content=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FVJpvBMG1Mos%2Fmaxresdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/VJpvBMG1Mos\" \/><meta itemprop=\"duration\" content=\"PT9M\" \/><meta itemprop=\"uploadDate\" content=\"2020-11-09T23:00:04Z\" \/><\/div><div id=\"lyte_VJpvBMG1Mos\" data-src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FVJpvBMG1Mos%2Fmaxresdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">Reaktionskr\u00e4fte ermitteln - Technische Mechanik 1, \u00dcbung 17<\/div><\/div><div class=\"play\"><\/div><div class=\"ctrl\"><div class=\"Lctrl\"><\/div><div class=\"Rctrl\"><\/div><\/div><\/div><noscript><a href=\"https:\/\/youtu.be\/VJpvBMG1Mos\" rel=\"nofollow\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FVJpvBMG1Mos%2F0.jpg\" alt=\"Reaktionskr&auml;fte ermitteln - Technische Mechanik 1, &Uuml;bung 17\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"Ein Eisenbahnwaggon steht auf einer absch\u00fcssigen Strecke an einem (reibungsfreien) Prellbock. Welche Reaktionskr\u00e4fte treten an den R\u00e4dern und am Prellbock auf?\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px auto;\"><\/div><figcaption><\/figcaption><\/figure>\n\n\n<h4 class=\"wp-block-heading\">Solution sketch<\/h4>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1011\" height=\"552\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch.jpg\" alt=\"Reaction forces on the track and buffer stop\" class=\"wp-image-596\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch.jpg 1011w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch-300x164.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch-768x419.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch-400x218.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/Screenshot_20201202-121745_Sketch-800x437.jpg 800w\" sizes=\"auto, (max-width: 1011px) 100vw, 1011px\" \/><figcaption>Reaction forces on the track and buffer stop<\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">Solution approach<\/h4>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n\\[ \\newcommand{\\myvec}[1]{{\\begin{pmatrix}#1\\end{pmatrix}}} \\]\n\n<p>The coordinate system is rotated by the angle \u03b1 to solve the problem. Left turning moments are positive.<\/p>\n\n<p>The equilibrium of forces in the x-direction is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{1} \\sum F_x = 0 = F_3 - G \\cdot \\sin \\alpha \\]<\/p>\n<\/div>\n<p>\\[\\tag{2} F_3 = G \\cdot \\sin \\alpha \\]<\/p>\n\n<p>The equilibrium of forces in the y-direction is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{3} \\sum F_y = 0 = F_1 - G \\cdot \\cos \\alpha + F_2 \\]<\/p>\n<\/div>\n\n<p>Due to the geometric specifications, the equilibrium of moments can only be considered around G and leads to<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{4} \\sum M(G) = 0 = F_3 \\cdot h - F_1 \\cdot l + F_2 \\cdot l \\]<\/p>\n\n<p>\\[\\tag{5} 0 = G \\cdot \\sin \\alpha \\cdot h - F_1 \\cdot l + F_2 \\cdot l \\]<\/p>\n<\/div>\n\n<p>\\[\\tag{6} F_2 = \\frac{F_1 \\cdot l - G \\cdot \\sin \\alpha \\cdot h}{l}  \\]<\/p>\n\n<p>From equation 3 follows<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{7} 0 = F_1 - G \\cdot \\cos \\alpha + \\frac{F_1 \\cdot l - G \\cdot \\sin \\alpha \\cdot h}{l} \\]<\/p>\n<p>\\[\\tag{8}  G \\cdot \\cos \\alpha = F_1  + \\frac{F_1 \\ \\cdot \\bcancel{l}}{\\bcancel{l}} - \\frac{G \\cdot \\sin \\alpha \\cdot h}{l} \\]<\/p>\n<p>\\[\\tag{9}  F_1 = \\frac{G}{2} \\cdot \\left( \\cos \\alpha + \\frac{\\sin \\alpha \\cdot h}{l} \\right) \\]<\/p>\n<\/div>\n\n<p>F<sub>2<\/sub> follows from equation 6<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{10} F_2 = \\frac{\\frac{G}{2} \\cdot \\left( \\cos \\alpha + \\frac{\\sin \\alpha \\cdot h}{l} \\right) \\cdot l - G \\cdot \\sin \\alpha \\cdot h}{l}  \\]<\/p>\n<p>\\[\\tag{11} F_2 = \\frac{ \\frac{G}{2} \\cdot \\cos \\alpha \\cdot l + \\frac{G}{2} \\cdot \\frac{\\sin \\alpha \\cdot h}{\\bcancel{l}} \\cdot \\bcancel{l} - G \\cdot \\sin \\alpha \\cdot h}{l}  \\]<\/p>\n<p>\\[\\tag{12} F_2 = \\frac{ G \\cdot \\cos \\alpha \\cdot l - G \\cdot \\sin \\alpha \\cdot h}{2 \\cdot l}  \\]<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Determination of the reaction forces\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-17-determination-of-the-reaction-forces\/\" aria-label=\"Read more about Determination of the reaction forces\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":430,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[45,46,29],"class_list":["post-717","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-bearing-loads","tag-reaction-forces","tag-statics","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Determination of the reaction forces &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"A railway wagon stands on a sloping track at a buffer stop. 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