{"id":193343,"date":"2025-02-19T10:28:31","date_gmt":"2025-02-19T10:28:31","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=193343"},"modified":"2025-02-19T10:28:33","modified_gmt":"2025-02-19T10:28:33","slug":"a-toy-chest-and-its-contents-have-a-combined-weight-of-180-n","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/19\/a-toy-chest-and-its-contents-have-a-combined-weight-of-180-n\/","title":{"rendered":"A toy chest and its contents have a combined weight of 180 N"},"content":{"rendered":"\n

A toy chest and its contents have a combined weight of 180 N. The coefficient of static friction between toy chest and floor is 0.42.The child in Fig. 6-38 attempts to move the chest across the floor by pulling on an attached rope. (a) If \u03b8 is 42\u00b0, what is the magnitude of the force F that the child must exert on the rope to put the chest on the verge of moving? (b) Write an expression for the magnitude F required to put the chest on the verge of moving as a function of the angle \u03b8. Determine (c) the value of \u03b8 for which F is a minimum and (d) that minimum magnitude.<\/p>\n\n\n\n

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

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

Given:<\/h3>\n\n\n\n
    \n
  • The combined weight of the toy chest and its contents = 180 N.<\/li>\n\n\n\n
  • The coefficient of static friction, ( \\mu_s = 0.42 ).<\/li>\n\n\n\n
  • The angle at which the child pulls the rope, ( \\theta = 42^\\circ ).<\/li>\n<\/ul>\n\n\n\n

    (a) Magnitude of the force ( F ) to put the chest on the verge of moving:<\/h3>\n\n\n\n

    The force ( F ) that the child exerts can be broken into two components:<\/p>\n\n\n\n

      \n
    • The horizontal component: ( F_{\\text{horizontal}} = F \\cos \\theta ).<\/li>\n\n\n\n
    • The vertical component: ( F_{\\text{vertical}} = F \\sin \\theta ).<\/li>\n<\/ul>\n\n\n\n

      The weight of the chest, ( W = 180 \\, \\text{N} ), creates a normal force. The normal force, ( N ), is affected by both the weight of the chest and the vertical component of the applied force:
      [
      N = W – F \\sin \\theta.
      ]<\/p>\n\n\n\n

      The force of static friction is given by:
      [
      f_{\\text{friction}} = \\mu_s N = \\mu_s (W – F \\sin \\theta).
      ]<\/p>\n\n\n\n

      For the chest to be on the verge of moving, the applied horizontal force must balance the static friction:
      [
      F \\cos \\theta = \\mu_s (W – F \\sin \\theta).
      ]<\/p>\n\n\n\n

      Solve for ( F ):
      [
      F \\cos \\theta + F \\mu_s \\sin \\theta = \\mu_s W.
      ]
      [
      F (\\cos \\theta + \\mu_s \\sin \\theta) = \\mu_s W.
      ]
      [
      F = \\frac{\\mu_s W}{\\cos \\theta + \\mu_s \\sin \\theta}.
      ]<\/p>\n\n\n\n

      Substituting ( \\mu_s = 0.42 ), ( W = 180 \\, \\text{N} ), and ( \\theta = 42^\\circ ), we can calculate the magnitude of ( F ).<\/p>\n\n\n\n

      (b) Expression for ( F ) as a function of ( \\theta ):<\/h3>\n\n\n\n

      From the above, the general expression for the magnitude of the force is:
      [
      F(\\theta) = \\frac{\\mu_s W}{\\cos \\theta + \\mu_s \\sin \\theta}.
      ]<\/p>\n\n\n\n

      (c) Angle ( \\theta ) for which ( F ) is minimum:<\/h3>\n\n\n\n

      To minimize ( F ), we differentiate ( F(\\theta) ) with respect to ( \\theta ) and set the derivative equal to zero:
      [
      \\frac{dF}{d\\theta} = 0.
      ]
      This gives us the value of ( \\theta ) where ( F ) is minimized.<\/p>\n\n\n\n

      (d) Minimum magnitude of ( F ):<\/h3>\n\n\n\n

      Substitute the value of ( \\theta ) from part (c) into the expression for ( F(\\theta) ) to find the minimum force required.<\/p>\n\n\n\n

      Let me know if you’d like me to walk through the specific calculations!<\/p>\n","protected":false},"excerpt":{"rendered":"

      A toy chest and its contents have a combined weight of 180 N. The coefficient of static friction between toy chest and floor is 0.42.The child in Fig. 6-38 attempts to move the chest across the floor by pulling on an attached rope. (a) If \u03b8 is 42\u00b0, what is the magnitude of the force […]<\/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-193343","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193343","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=193343"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193343\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=193343"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=193343"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=193343"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}