{"id":519,"date":"2020-11-30T11:46:15","date_gmt":"2020-11-30T11:46:15","guid":{"rendered":"https:\/\/pickedshares.com\/?p=519"},"modified":"2021-05-08T11:29:24","modified_gmt":"2021-05-08T11:29:24","slug":"engineering-mechanics-1-exercise-7-calculate-the-resulting-moment","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-7-calculate-the-resulting-moment\/","title":{"rendered":"Calculate the resulting moment"},"content":{"rendered":"\n<p>This exercise is about the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\" id=\"block-a6f63cc0-a8f0-46a5-b9d5-9bd90d8b7bbc\"><li>How to calculate the resulting moment?<\/li><li>How to calculate the cross product of location vector and force vector?<\/li><li>How to apply the rule of Sarrus?<\/li><\/ul>\n\n\n\n<p>If you are looking for the calculation of resulting moments without vectors and the rule of sarrus, take a look into our other <a href=\"https:\/\/pickedshares.com\/en\/tag\/resulting-moments\/\">exercises about moments<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Task<\/h3>\n\n\n\n<p>The forces F<sub>1<\/sub>, F<sub>2<\/sub> and F<sub>3<\/sub> are acting on a frame. Which moment results in point P?<\/p>\n\n\n\n<p>given: F<sub>1<\/sub> = F<sub>2<\/sub> = 2F, F<sub>3<\/sub> = F<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"750\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205155-1024x750.jpg\" alt=\"frame with acting forces\" class=\"wp-image-406\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205155-1024x750.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205155-300x220.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205155-768x562.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205155.jpg 1123w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>frame with acting forces<\/figcaption><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\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=\"Resultierendes Moment berechnen, mehrere Kr&auml;fte, Vektorprodukt - Technische Mechanik 1, &Uuml;bung 7\" style=\"width:640px;max-width:100%;margin:5px auto;\"><div class=\"lyMe hidef\" id=\"WYL_jVEp0SOuJpI\" 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%2FjVEp0SOuJpI%2Fmaxresdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/jVEp0SOuJpI\" \/><meta itemprop=\"duration\" content=\"PT8M33S\" \/><meta itemprop=\"uploadDate\" content=\"2020-11-30T21:31:12Z\" \/><\/div><meta itemprop=\"accessibilityFeature\" content=\"captions\" \/><div id=\"lyte_jVEp0SOuJpI\" data-src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2FjVEp0SOuJpI%2Fmaxresdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">Resultierendes Moment berechnen, mehrere Kr\u00e4fte, Vektorprodukt - Technische Mechanik 1, \u00dcbung 7<\/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\/jVEp0SOuJpI\" 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%2FjVEp0SOuJpI%2F0.jpg\" alt=\"Resultierendes Moment berechnen, mehrere Kr&auml;fte, Vektorprodukt - Technische Mechanik 1, &Uuml;bung 7\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"An einem Rahmen greifen die Kr\u00e4fte F1, F2 und F3 an. Welches resultierende Moment entsteht in Punkt P? Die \u00dcbung zum Nachlesen findet man hier: https:\/\/pickedshares.com\/technische-mechanik-1-uebung-7-resultierendes-moment-berechnen-mehrere-kraefte-vektorprodukt\/ Obwohl die Kr\u00e4fte in einer Ebene liegen, wird die Aufgabe in diesem Beispiel mithilfe der Vektorrechnung im dreidimensionalen Koordinatensystem gel\u00f6st. Dabei werden die Kreuzprodukte aus den Richtungsvektoren und Ortsvektoren der Kr\u00e4fte berechnet und addiert.\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px auto;\"><\/div><figcaption>Solved exercise<\/figcaption><\/figure>\n\n\n<p>In the solution shown, the resulting moment is determined using the cross product. For the Sarrus rule to be applied for the calculation, the position vectors and the force vectors are each supplemented by the z-coordinate, even if this is a flat problem.<\/p>\n\n\n\n<p>The coordinate direction is y upwards, x to the right.<\/p>\n\n\n\n<p>The moment in point P results from the following equation: the sum of moments is the sum of the cross products of each force vector with its respective position vector.<\/p>\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>\\[\\tag{1}  \\vec{M(P)} = \\sum_{i=1}^3 \\vec{r_i} \\times \\vec{F_i}  \\]<\/p>\n\n<p>The force vectors are<\/p>\n\n<p>\\[\\tag{2}  \\vec{F_1} = \\myvec{2F\\\\0\\\\0} \\]<\/p>\n<p>\\[\\tag{3}  \\vec{F_2} = \\myvec{0\\\\2F\\\\0} \\]<\/p>\n<p>\\[\\tag{4}  \\vec{F_3} = \\myvec{F\\\\0\\\\0} \\]<\/p>\n\n<p>The position vectors are<\/p>\n<p>\\[\\tag{5}  \\vec{r_1} = \\myvec{2a\\\\3a\\\\0} \\]<\/p>\n<p>\\[\\tag{6}  \\vec{r_2} = \\myvec{2a\\\\a\\\\0} \\]<\/p>\n<p>\\[\\tag{7}  \\vec{r_3} = \\myvec{0\\\\a\\\\0} \\]<\/p>\n\n<p>The detailed procedure for calculating the cross product with the Sarrus rule is described in the solution video above. The resulting moment is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{8}  \\vec{M(P)} = \\vec{r_1} \\times \\vec{F_1}  + \\vec{r_2} \\times \\vec{F_2}  + \\vec{r_3} \\times \\vec{F_3} \\]<\/p>\n\n<p>\\[\\tag{9}  \\vec{M(P)} = \\myvec{2a\\\\3a\\\\0} \\times \\myvec{2F\\\\0\\\\0}  + \\myvec{2a\\\\a\\\\0} \\times \\myvec{0\\\\2F\\\\0}  + \\myvec{0\\\\a\\\\0} \\times \\myvec{F\\\\0\\\\0} \\]<\/p>\n<p>\\[\\tag{10}  \\vec{M(P)} = \\myvec{0\\\\0\\\\-6Fa} + \\myvec{0\\\\0\\\\4Fa} + \\myvec{0\\\\0\\\\-Fa} = \\myvec{0\\\\0\\\\-3Fa}\\]<\/p>\n<\/div>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Calculate the resulting moment\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-7-calculate-the-resulting-moment\/\" aria-label=\"Read more about Calculate the resulting moment\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":407,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[26,48,29,31],"class_list":["post-519","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-cross-product","tag-resulting-moments","tag-statics","tag-vector-calculation","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>Calculate the resulting moment &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"In this exercise the resulting moment for three forces has to be calculated. 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