{"id":550,"date":"2020-12-01T12:10:42","date_gmt":"2020-12-01T12:10:42","guid":{"rendered":"https:\/\/pickedshares.com\/?p=550"},"modified":"2021-05-08T11:30:53","modified_gmt":"2021-05-08T11:30:53","slug":"engineering-mechanics-1-exercise-8-calculate-the-resulting-moment-force-under-angle","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-8-calculate-the-resulting-moment-force-under-angle\/","title":{"rendered":"Calculate the resulting moment, force under angle"},"content":{"rendered":"\n<p>This exercise addresses the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>What moment arises when a force is shifted parallel to its effective direction?<\/li><li>How can you calculate the moment for a force acting at an angle?<\/li><li>How do you calculate the lever arm of an attacking force?<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>The force F shall be shifted into point P. Which moment results?<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"568\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205242-1024x568.jpg\" alt=\"Frame with force acting at an angle\" class=\"wp-image-400\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205242-1024x568.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205242-300x167.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205242-768x426.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/20201104_205242.jpg 1200w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Frame with force acting at an angle<\/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 and describes the first (more complicated) way of solution.<\/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, schr&auml;g angreifende Kraft - Technische Mechanik 1, &Uuml;bung 8\" style=\"width:640px;max-width:100%;margin:5px auto;\"><div class=\"lyMe hidef\" id=\"WYL_qp64gKasMpY\" 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%2Fqp64gKasMpY%2Fmaxresdefault.jpg\" \/><meta itemprop=\"embedURL\" content=\"https:\/\/www.youtube.com\/embed\/qp64gKasMpY\" \/><meta itemprop=\"duration\" content=\"PT8M25S\" \/><meta itemprop=\"uploadDate\" content=\"2020-10-14T08:52:10Z\" \/><\/div><meta itemprop=\"accessibilityFeature\" content=\"captions\" \/><div id=\"lyte_qp64gKasMpY\" data-src=\"https:\/\/pickedshares.com\/wp-content\/plugins\/wp-youtube-lyte\/lyteCache.php?origThumbUrl=https%3A%2F%2Fi.ytimg.com%2Fvi%2Fqp64gKasMpY%2Fmaxresdefault.jpg\" class=\"pL\"><div class=\"tC\"><div class=\"tT\" itemprop=\"name\">Resultierendes Moment berechnen, schr\u00e4g angreifende Kraft - Technische Mechanik 1, \u00dcbung 8<\/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\/qp64gKasMpY\" 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%2Fqp64gKasMpY%2F0.jpg\" alt=\"Resultierendes Moment berechnen, schr&auml;g angreifende Kraft - Technische Mechanik 1, &Uuml;bung 8\" width=\"640\" height=\"340\" \/><br \/>Watch this video on YouTube<\/a><\/noscript><meta itemprop=\"description\" content=\"Die einfachere L\u00f6sung dieser Aufgabe gibt es als Variante 2 hier: https:\/\/pickedshares.com\/technische-mechanik-1-uebung-9-resultierendes-moment-berechnen-schraeg-angreifende-kraft\/ Die Kraft F soll in den Punkt P verschoben werden. Welches Moment entsteht dabei? Das besondere an dieser Aufgabe ist, dass auf den ersten Blick der Satz des Pythagoras gen\u00fcgt, um den senkrechten Abstand zwischen Kraft und Punkt P zu ermitteln. Das ist aber ein Trugschluss, denn diese L\u00f6sung gilt nur f\u00fcr einen ganz speziellen Winkel Alpha. Da hier aber eine allgemeine L\u00f6sung gefragt ist, muss Alpha in Berechnung des senkrechten Abstands mit einflie\u00dfen. The force F shall be transposed into point P. Which torque results? The special character of this exercise is that, at first, it seems quite easy to calculate the perpendicular distance between the force and point P. But this is only valid for one specific value of the angle Alpha. To come to a general solution, Alpha has to be involved in the calculation of the perpendicular distance.\"><\/div><\/div><div class=\"lL\" style=\"max-width:100%;width:640px;margin:5px auto;\"><\/div><figcaption><\/figcaption><\/figure>\n\n\n<p>Shifting a force parallel to its effective direction creates a moment, also known as an <a href=\"http:\/\/web.mit.edu\/4.441\/1_lectures\/1_lecture5\/1_lecture5.html\" target=\"_blank\" rel=\"noreferrer noopener\">offset moment<\/a>. The lever arm of the moment is the amount of displacement.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution approach 1<\/h3>\n\n\n\n<p>In approach 1, calculations are carried out without the decomposition of forces. It is more complex than the second approach.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Geometrical conditions for the calculation<\/h4>\n\n\n\n<p>The way in which the sketch for the task is carried out suggests that the lever arm for the force F can be calculated directly using the Pythagorean theorem and the edge lengths 3a and 4a-2a = 2a. But this is a fallacy, because as soon as you make the sketch for a changed angle, you can see that the lever arm must be determined as in the following sketch.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"666\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-1024x666.jpg\" alt=\"Geometrical conditions for the calculation of the resulting moment\" class=\"wp-image-484\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-1024x666.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-300x195.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-768x499.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-400x260.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved-800x520.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/11\/tm1-8solved.jpg 1460w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Geometrical conditions for the calculation of the resulting moment<\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">The math<\/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.<\/script>\n\\[ \\require{cancel} \\]\n\\[ \\newcommand{\\myvec}[1]{{\\begin{pmatrix}#1\\end{pmatrix}}} \\]\n<p>Since the solution has to be found for an arbitrary angle \u03b1, the forces F' and \u03b1' have been added in the solution sketch to clarify the geometrical relationships. In the following formulas, however, the values \u200b\u200bF and \u03b1 are used.\nIn the solution described here, right-turning moments were defined as positive (in the solution video, left-turning moments were assumed to be positive). The moment in point P is calculated from<\/p>\n\n<p>\\[\\tag{1} M_P = F \\cdot x \\]<\/p>\n\n<p>The lever arm x is made up<\/p>\n\n<p>\\[\\tag{2} x = c \\cdot \\cos \\gamma \\]<\/p>\n\n<p>Side c can be calculated using the Pythagorean Theorem.<\/p>\n\n<p>\\[\\tag{3} c = \\sqrt{3^2a^2+2^2a^2} \\]<\/p>\n\n<p>\\[\\tag{4} c = \\sqrt{13}a \\]<\/p>\n\n<p>The angles are related as follows:<\/p>\n\n<p>\\[\\tag{5} \\gamma = 90\u00b0 - \\delta \\]<\/p>\n\n<p>\\[\\tag{6} \\delta = 90\u00b0 - \\alpha + \\beta \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{7} \\beta = arctan \\left( \\frac{2\\bcancel{a}}{3\\bcancel{a}} \\right) = 33.7\u00b0 \\]<\/p>\n<\/div>\n<p>\\[\\tag{8} \\delta = 123.7\u00b0 - \\alpha \\]<\/p>\n\n<p>\\[\\tag{9} \\gamma = \\alpha - 33.7\u00b0 \\]<\/p>\n\n<p>The lever arm x can now be determined<\/p>\n\n<div style=\"overflow:auto;\">                                           \n<p>\\[\\tag{10} x = \\sqrt{13}a \\cdot \\cos (\\alpha - 33.7\u00b0) \\]<\/p>\n<\/div>\n<p>And finally the resulting moment<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{11} M_P = F \\cdot \\sqrt{13}a \\cdot \\cos (\\alpha - 33.7\u00b0) \\]<\/p>\n<\/div>\n\n\n\n<p>The resulting moment can also be determined by decomposition of the force into its components:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution approach 2<\/h3>\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<p>This solution is based on an x-y coordinate system and the forces are broken down into their components. Right turning moments are positive again. The two components of the acting force F are:<\/p>\n<p>\\[\\tag{1} F_x = F \\cdot cos \\alpha \\]<\/p>\n<p>\\[\\tag{2} F_y = F \\cdot sin \\alpha \\]<\/p>\n<p>The resulting moment follows from:<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{3} M_P = F \\cdot cos \\alpha \\cdot 3a + F \\cdot sin \\alpha \\cdot 2a \\]<\/p>\n<p>\\[\\tag{4} M_P = F \\cdot a \\cdot (3 \\cdot cos \\alpha + 2 \\cdot sin \\alpha) \\]<\/p>\n<\/div>\n<p>The second way to solve this problem is much easier than the first one.<\/p>\n\n\n\n<p>Here we have some more <a href=\"https:\/\/pickedshares.com\/en\/tag\/resulting-moments\/\">exercises about the determination of resulting moments<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Calculate the resulting moment, force under angle\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-8-calculate-the-resulting-moment-force-under-angle\/\" aria-label=\"Read more about Calculate the resulting moment, force under angle\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":401,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[48,29],"class_list":["post-550","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-resulting-moments","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>How to calculate the resulting moment if the force is acting under an angle<\/title>\n<meta name=\"description\" content=\"This exercise shows two different approaches how to calculate the resulting moment for a force acting under an angle.\" \/>\n<meta name=\"robots\" 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