{"id":238840,"date":"2025-06-18T12:31:24","date_gmt":"2025-06-18T12:31:24","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238840"},"modified":"2025-06-18T12:31:25","modified_gmt":"2025-06-18T12:31:25","slug":"an-adventurous-archaeologist-crosses-between-two-rock-cliffs-by-slowly-going-hand-over-hand-along-rope-stretched-between-the-cliffs","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/18\/an-adventurous-archaeologist-crosses-between-two-rock-cliffs-by-slowly-going-hand-over-hand-along-rope-stretched-between-the-cliffs\/","title":{"rendered":"An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along rope stretched between the cliffs."},"content":{"rendered":"\n

An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig: P5.56). The rope will break if the tension in it exceeds 2.50 X 104N, and our hero’$ mass is 90.0 kg: (a) If the angle 0 is 10.09 . find the tension in the rope. (b) What is the smallest value the angle 0 can have if the rope is not t0 break? Figure P5.56<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Given:<\/strong><\/h3>\n\n\n\n
    \n
  • Mass of the archaeologist: m=90.0\u2009kgm = 90.0 \\, \\text{kg}m=90.0kg<\/li>\n\n\n\n
  • Gravitational acceleration: g=9.8\u2009m\/s2g = 9.8 \\, \\text{m\/s}^2g=9.8m\/s2<\/li>\n\n\n\n
  • Weight: W=mg=90.0\u00d79.8=882\u2009NW = mg = 90.0 \\times 9.8 = 882 \\, \\text{N}W=mg=90.0\u00d79.8=882N<\/li>\n\n\n\n
  • Maximum tension: Tmax=2.50\u00d7104\u2009NT_{\\text{max}} = 2.50 \\times 10^4 \\, \\text{N}Tmax\u200b=2.50\u00d7104N<\/li>\n\n\n\n
  • He is at the midpoint \u2192 rope is symmetric<\/li>\n\n\n\n
  • The vertical component of the tension supports the weight.<\/li>\n\n\n\n
  • \u03b8\\theta\u03b8: angle between rope and horizontal.<\/li>\n<\/ul>\n\n\n\n
    \n\n\n\n

    (a) Find tension when \u03b8=10.0\u2218\\theta = 10.0^\\circ\u03b8=10.0\u2218<\/strong><\/h2>\n\n\n\n

    Since the system is symmetric, the vertical components of the tension in the two rope segments must add up to balance the weight: 2Tsin\u2061(\u03b8)=mg2T \\sin(\\theta) = mg2Tsin(\u03b8)=mg T=mg2sin\u2061(\u03b8)T = \\frac{mg}{2 \\sin(\\theta)}T=2sin(\u03b8)mg\u200b<\/p>\n\n\n\n

    Substitute known values: T=8822sin\u2061(10\u2218)=8822\u00d70.1736=8820.3472\u22482541\u2009NT = \\frac{882}{2 \\sin(10^\\circ)} = \\frac{882}{2 \\times 0.1736} = \\frac{882}{0.3472} \\approx 2541 \\, \\text{N}T=2sin(10\u2218)882\u200b=2\u00d70.1736882\u200b=0.3472882\u200b\u22482541N<\/p>\n\n\n\n

    \u2705 Answer (a):<\/strong> The tension is approximately 2541 N<\/strong> when the angle is 10\u221810^\\circ10\u2218.<\/p>\n\n\n\n

    \n\n\n\n

    (b) What is the smallest angle \u03b8\\theta\u03b8 so the rope does not break?<\/strong><\/h2>\n\n\n\n

    We solve for \u03b8\\theta\u03b8 in the formula: T=mg2sin\u2061(\u03b8)\u22642.50\u00d7104T = \\frac{mg}{2 \\sin(\\theta)} \\leq 2.50 \\times 10^4T=2sin(\u03b8)mg\u200b\u22642.50\u00d7104 sin\u2061(\u03b8)\u2265mg2\u00d72.50\u00d7104=88250000=0.01764\\sin(\\theta) \\geq \\frac{mg}{2 \\times 2.50 \\times 10^4} = \\frac{882}{50000} = 0.01764sin(\u03b8)\u22652\u00d72.50\u00d7104mg\u200b=50000882\u200b=0.01764 \u03b8\u2265sin\u2061\u22121(0.01764)\u22481.01\u2218\\theta \\geq \\sin^{-1}(0.01764) \\approx 1.01^\\circ\u03b8\u2265sin\u22121(0.01764)\u22481.01\u2218<\/p>\n\n\n\n

    \u2705 Answer (b):<\/strong> The smallest angle \u03b8\\theta\u03b8 can have is approximately 1.01\u00b0<\/strong> to prevent the rope from breaking.<\/p>\n\n\n\n

    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig: P5.56). The rope will break if the tension in it exceeds 2.50 X 104N, and our hero’$ mass is 90.0 kg: (a) If the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-238840","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238840","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=238840"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238840\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238840"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238840"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238840"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}