{"id":778,"date":"2020-12-09T09:17:13","date_gmt":"2020-12-09T09:17:13","guid":{"rendered":"https:\/\/pickedshares.com\/?p=778"},"modified":"2021-06-09T13:34:13","modified_gmt":"2021-06-09T13:34:13","slug":"engineering-mechanics-1-exercise-28-required-angle-for-state-of-equilibrium","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-28-required-angle-for-state-of-equilibrium\/","title":{"rendered":"Required angle for state of rest"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>Two bodies are connected with a massless rope. The <a href=\"https:\/\/pickedshares.com\/en\/static-and-sliding-friction-values\/\">coefficient of static friction<\/a> for the upper body is \u00b5<sub>0<\/sub>, the contact between the hanging body and the wall is frictionless. How big must the angle \u03b1 be so that the two bodies are at rest?<\/p>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image\"><img loading=\"lazy\" decoding=\"async\" width=\"679\" height=\"707\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-1.png\" alt=\"Two bodies at rest\" class=\"wp-image-3731\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-1.png 679w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-1-288x300.png 288w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-1-400x416.png 400w\" sizes=\"auto, (max-width: 679px) 100vw, 679px\" \/><figcaption>Two bodies at rest<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>An x-y coordinate system is defined and the two bodies are cut free. The rope is replaced by the rope force F<sub>S<\/sub>. The balance of forces in the x and y directions are considered separately for both bodies.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image\"><img loading=\"lazy\" decoding=\"async\" width=\"725\" height=\"669\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-2.png\" alt=\"Free body diagram\" class=\"wp-image-3735\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-2.png 725w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-2-300x277.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/03\/tm1-28-2-400x369.png 400w\" sizes=\"auto, (max-width: 725px) 100vw, 725px\" \/><figcaption>Free body diagram<\/figcaption><\/figure>\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><strong>Upper body<\/strong><\/p>\n<p>Forces in x-direction<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{1} \\sum F_x = 0 = -F_{Ax} + F_S \\cdot sin \\alpha \\]<\/p>\n<\/div>\n<p>Forces in y-direction<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{2} \\sum F_y = 0 = F_{Ay} - F_S \\cdot cos \\alpha - G \\]<\/p>\n<\/div>\n\n<p><strong>Hanging body<\/strong><\/p>\n<p>Forces in x-direction<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{3} \\sum F_x = 0 = F_{B} - F_S \\cdot sin \\alpha \\]<\/p>\n<\/div>\n<p>Forces in y-direction<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4} \\sum F_y = 0 = -G + F_S \\cdot cos \\alpha \\]<\/p>\n<\/div>\n\n\n<p>The relationship between normal force and frictional force means for the forces on the upper body<\/p>\n<p>\\[ \\tag{5} F_{Ax} = F_{Ay} \\cdot \u00b5_0 \\]<\/p>\n\n<p>The two bodies are in a state of rest as long as the x component of the rope force F<sub>S<\/sub> is smaller or exactly as large as the frictional force F<sub>Ax<\/sub>.<\/p>\n\n<p>\\[ \\tag{6} F_S \\cdot sin \\alpha \\leq F_{Ax} \\]<\/p>\n\n<p>For the rope force F<sub>S<\/sub> can be used from equation (4), the friction force F<sub>Ax<\/sub> follows from equation (5)<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{7} \\frac{G}{cos \\alpha} \\cdot sin \\alpha \\leq F_{Ay} \\cdot \u00b5_0 \\]<\/p>\n<\/div>\n<p>F<sub>Ay<\/sub> is determined from equation (2) and leads to<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{8} \\frac{G}{cos \\alpha} \\cdot sin \\alpha \\leq \\left( \\frac{G}{cos \\alpha} \\cdot cos \\alpha + G  \\right) \\cdot \u00b5_0 \\]<\/p>\n<\/div>\n<p>\\[ \\tag{9} G \\cdot tan \\alpha \\leq 2 \\cdot G \\cdot \u00b5_0 \\]<\/p>\n<p>\\[ \\tag{10} tan \\alpha \\leq 2 \\cdot  \u00b5_0 \\]<\/p>\n<p>\\[ \\tag{11} \\alpha \\leq arctan \\left( 2 \\cdot  \u00b5_0 \\right) \\]<\/p>\n\n<p><strong>A test with two different friction values:<\/strong> Let's assume \u00b5<sub>0<\/sub> = 0.3<\/p>\n<p>\\[ \\tag{12} \\alpha \\leq arctan (0.6) \\approx 31\u00b0 \\]<\/p>\n<p>With a smaller coefficient of friction, the angle would also have to be smaller. Assume that the coefficient of friction is \u00b5<sub>0<\/sub> = 0.2<\/p>\n<p>\\[ \\tag{13} \\alpha \\leq arctan (0.4) \\approx 22\u00b0 \\]<\/p>\n<p>The result seems plausible.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Required angle for state of rest\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-28-required-angle-for-state-of-equilibrium\/\" aria-label=\"Read more about Required angle for state of rest\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":3731,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[27,29],"class_list":["post-778","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-friction","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>Required angle for state of rest &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"Two bodies are linked with a weightless rope. 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