{"id":837,"date":"2020-12-11T13:20:37","date_gmt":"2020-12-11T13:20:37","guid":{"rendered":"https:\/\/pickedshares.com\/?p=837"},"modified":"2021-05-08T12:20:44","modified_gmt":"2021-05-08T12:20:44","slug":"engineering-mechanics-1-exercise-30-ladder-on-a-wall","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-30-ladder-on-a-wall\/","title":{"rendered":"Ladder on a wall"},"content":{"rendered":"\n<p>This exercise shows the solution for the classical mechanics exercise \"ladder on a wall\" and addresses the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Which forces occur on a ladder leaning against a wall?<\/li><li>When does the ladder slip?<\/li><\/ul>\n\n\n\n<p>At the bottom of the page is an online calculator for testing various combinations of <a href=\"https:\/\/pickedshares.com\/en\/static-and-sliding-friction-values\/\">coefficients of static friction<\/a>, lengths and angles.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>A ladder is leaning against a wall. There are different materials on the floor and wall, so that the static friction values \u200b\u200bdiffer. How high (distance l) can a person with the weight G climb the ladder without slipping?<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-1-662x1024.png\" alt=\"Ladder on the wall\" class=\"wp-image-2429\" width=\"331\" height=\"512\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-1-662x1024.png 662w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-1-194x300.png 194w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-1-400x619.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-1.png 679w\" sizes=\"auto, (max-width: 331px) 100vw, 331px\" \/><figcaption>Ladder on the wall<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>To solve the task, the ladder is cut free in the first step. The coordinate system is set to x-y and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Right-hand_rule\" target=\"_blank\" rel=\"noreferrer noopener\">left turning moments are positive<\/a>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-818x1024.png\" alt=\"Free body diagram for the ladder on the wall\" class=\"wp-image-2435\" width=\"409\" height=\"512\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-818x1024.png 818w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-240x300.png 240w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-768x961.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-400x501.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2-800x1002.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/01\/tm1-30-2.png 905w\" sizes=\"auto, (max-width: 409px) 100vw, 409px\" \/><figcaption>Free body diagram for the ladder on the wall<\/figcaption><\/figure>\n\n\n\n<p>Next, the balance of forces in x- and y-direction and the balance of moments around B are noted.<\/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\n<p>The balance of forces in x-direction is<\/p>\n<p>\\[ \\tag{1} \\sum F_x = 0 = F_{Ax} - F_{Bx} \\]<\/p>\n\n<p>The balance of forces in y-direction is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{2} \\sum F_y = 0 = F_{Ay} + F_{By} - G \\]<\/p>\n<\/div>\n\n<p>The balance of moments around B is<\/p>\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{3} \\sum M(B) = 0 = G \\cdot l \\cdot \\cos \\alpha - F_{Ay} \\cdot L \\cdot \\cos \\alpha - F_{Ax} \\cdot L \\cdot \\sin \\alpha \\]<\/p>\n<\/div>\n\n\n\n<p>The forces F<sub>Ay<\/sub> and F<sub>Bx<\/sub> are each dependent on their normal forces and coefficients of static friction. <\/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\n\n<p>\\[ \\tag{4} F_{Ay}  = F_{Ax} \\cdot \u00b5_A  \\]<\/p>\n\n<p>\\[ \\tag{5} F_{Bx} = F_{By} \\cdot \u00b5_B \\]<\/p>\n\n\n\n<p>Equation (1) is solved for F<sub>Ax<\/sub> and F<sub>Bx<\/sub> from equation (5) is inserted:<\/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\n\n<p>\\[ \\tag{6} F_{Ax} = F_{By} \\cdot \u00b5_B \\]<\/p>\n\n<p>The result obtained is inserted directly into equation (4).<\/p>\n\n<p>\\[ \\tag{7} F_{Ay} = F_{By} \\cdot \u00b5_A \\cdot \u00b5_B \\]<\/p>\n\n\n\n<p>F<sub>Ay<\/sub> is used in the balance of forces in the y-direction, equation (2), and the equation is solved for F<sub>By<\/sub>:<\/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\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{8} 0 = F_{By} + F_{By} \\cdot \u00b5_A \\cdot \u00b5_B - G \\]<\/p>\n<\/div>\n<p>\\[ \\tag{9} F_{By} = \\frac{G}{\u00b5_A \\cdot \u00b5_B + 1} \\]<\/p>\n\n<p>This means that F<sub>Ax<\/sub> and F<sub>Ay<\/sub> can now also be calculated.<\/p>\n\n<p>\\[ \\tag{10} F_{Ax} = \\frac{G \\cdot \u00b5_A}{\u00b5_A \\cdot \u00b5_B + 1} \\]<\/p>\n<p>\\[ \\tag{11} F_{Ay} = \\frac{G \\cdot \u00b5_A \\cdot \u00b5_B}{\u00b5_A \\cdot \u00b5_B + 1} \\]<\/p>\n\n\n\n<p>The moment balance (3) is converted according to the length l and the calculated values \u200b\u200bare used.<\/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\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{12} l=\\frac{{F_{\\mathit{Ax}}} \\cdot L \\cdot \\sin{\\left( \\alpha \\right) }+{F_{\\mathit{Ay}}} \\cdot  L \\cdot \\cos{\\left( \\alpha \\right) }}{G \\cdot \\cos{\\left( \\alpha \\right) }} \\]<\/p>\n<\/div>\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{13} l=\\frac{\\frac{G  \\cdot L \\cdot  {\u00b5_B}  \\cdot \\sin{\\left( \\alpha \\right) }}{{\u00b5_A} \\cdot  {\u00b5_B}+1}+\\frac{G \\cdot  L \\cdot  {\u00b5_A} \\cdot  {\u00b5_B}  \\cdot \\cos{\\left( \\alpha \\right) }}{{\u00b5_A}  \\cdot {\u00b5_B}+1}}{G  \\cdot \\cos{\\left( \\alpha \\right) }}\\]<\/p>\n<\/div>\n\n<p>or shortened and reshaped:<\/p>\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{14} l=\\frac{L  \\cdot  {\u00b5_B}  \\cdot \\sin{\\left( \\alpha \\right) }+L \\cdot  {\u00b5_A}  \\cdot  {\u00b5_B}  \\cdot \\cos{\\left( \\alpha \\right) }}{\\left( {\u00b5_A} \\cdot  {\u00b5_B}+1\\right)  \\cdot  \\cos{\\left( \\alpha \\right) }}\\]  <\/p>\n<\/div>\n\n\n\n<p>The length you are looking for is now determined. As expected, the weight force could be reduced for the calculation, so it has no influence on the result.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Online calculator<\/h2>\n\n\n\n<p>The result can be tested here. <\/p>\n\n\n\n<p>If you enter the values, you will find that the function calculates a length for an angle of 0\u00b0 (or if you leave the field empty) or fot very small angles.<\/p>\n\n\n\n<p>What does this mean?<\/p>\n\n\n\n<p>This may seem wrong at first glance, but it is correct. The calculated length is below the center of gravity of the ladder up to a certain angle. Any weight acting on the ladder above this length would cause the ladder to slip away. And because of the center of gravity in the middle of the ladder, this is the case up to a certain angle. As an example: a ladder with a <a href=\"https:\/\/pickedshares.com\/en\/static-and-sliding-friction-values\/\">coefficient of static friction<\/a> of 0.5 in both A and B would only stand alone at an angle of approx. 37\u00b0.<\/p>\n\n\n\n<p>And now try it for yourself:<\/p>\n\n\n\n<div style=\"display:grid\">\n<label for=\"laenge\">Length L in m<\/label>\n<p><input class=\"calc\" id=\"laenge\" type=\"text\"><\/p>\n<label for=\"mya\">Coefficient of static friction \u03bc<sub>A<\/sub><\/label>\n<p><input class=\"calc\" id=\"mya\" type=\"text\"><\/p>\n<label for=\"myb\">Coefficient of static friction \u03bc<sub>B<\/sub><\/label>\n<p><input class=\"calc\" id=\"myb\" type=\"text\"><\/p>\n<label for=\"alpha\">Angle \u03b1 in \u00b0<\/label>\n<p><input class=\"calc\" id=\"alpha\" type=\"text\"><\/p>\n<p id=\"result\"><\/p><p>\n\n<script src=\"https:\/\/ajax.googleapis.com\/ajax\/libs\/jquery\/3.5.1\/jquery.min.js\"><\/script>\n<script>\n$( document ).ready(function() {\n    console.log('ready');\n\n    $( \".calc\" ).keyup(function() {\n    \n    var L = $('#laenge').val();\n    var mya = $('#mya').val();\n    var myb = $('#myb').val();\n    var alpha = $('#alpha').val();\n    \n    L = L.replace( \/,\/,\".\" );\n    mya = mya.replace( \/,\/,\".\" );\n    myb = myb.replace( \/,\/,\".\" );\n    alpha = alpha.replace( \/,\/,\".\" );\n    \n    \n    \n    try {\n    \n    alpha = alpha\/180*Math.PI;\n    \n    l = (L * myb * Math.sin(alpha)+L*mya*myb*Math.cos(alpha))\/((mya*myb+1)*Math.cos(alpha));\n    \n    l=l*100;\n    l=Math.round(l);\n    l=l\/100;\n    \n    console.log(l);\n    \n    document.getElementById('result').innerHTML = 'The calculated length is l ='+l +' m.';    \n    }\n    catch (ex) {\n    console.log(ex);\n    }\n    finally {\n    }\n    \n});\n});\n\n<\/script>\n\n<\/p><\/div>\n\n\n\n<p>We have a lot more <a href=\"https:\/\/pickedshares.com\/en\/category\/engineering-mechanics-ii\/\">interesting exercises regarding statics<\/a> here.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Ladder on a wall\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-30-ladder-on-a-wall\/\" aria-label=\"Read more about Ladder on a wall\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":2431,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[27,46,29],"class_list":["post-837","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-friction","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>Ladder on a wall &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"A ladder leans against a wall. 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