{"id":196751,"date":"2025-03-05T19:42:59","date_gmt":"2025-03-05T19:42:59","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196751"},"modified":"2025-03-05T19:43:02","modified_gmt":"2025-03-05T19:43:02","slug":"the-figure-shows-a-conical-pendulum-in-which-the-bob-the-small-object-at-the-lower-end-of-the-cord-moves-in-a-horizontal-circle-at-constant-speed","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/05\/the-figure-shows-a-conical-pendulum-in-which-the-bob-the-small-object-at-the-lower-end-of-the-cord-moves-in-a-horizontal-circle-at-constant-speed\/","title":{"rendered":"The figure shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed"},"content":{"rendered":"\n

The figure shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.040 kg, the string has length L = 1.1 m and negligible mass, and the bob follows a circular path of circumference 1.0 m. What are (a) the tension in the string and (b) the period of the motion?<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

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

A conical pendulum consists of a mass (bob) attached to the end of a string, moving in a horizontal circular path while the string traces out a cone. In this scenario, the bob has a mass of 0.040 kg, the string length (L) is 1.1 m, and the bob follows a circular path with a circumference of 1.0 m. We aim to determine (a) the tension in the string and (b) the period of the motion.\ue206<\/p>\n\n\n\n

(a) Tension in the String<\/strong><\/p>\n\n\n\n

First, we calculate the radius (r) of the circular path:<\/p>\n\n\n\n

[ \\text{Circumference} = 2\\pi r ]\ue206<\/p>\n\n\n\n

[ 1.0\\, \\text{m} = 2\\pi r ]\ue206<\/p>\n\n\n\n

[ r = \\frac{1.0\\, \\text{m}}{2\\pi} \\approx 0.159\\, \\text{m} ]\ue206<\/p>\n\n\n\n

The angle \u03b8 between the string and the vertical can be found using:\ue206<\/p>\n\n\n\n

[ \\sin\\theta = \\frac{r}{L} = \\frac{0.159\\, \\text{m}}{1.1\\, \\text{m}} \\approx 0.145 ]\ue206<\/p>\n\n\n\n

[ \\theta = \\arcsin(0.145) \\approx 8.33^\\circ ]\ue206<\/p>\n\n\n\n

In the vertical direction, the vertical component of the tension (T) balances the gravitational force:\ue206<\/p>\n\n\n\n

[ T\\cos\\theta = mg ]\ue206<\/p>\n\n\n\n

[ T = \\frac{mg}{\\cos\\theta} ]\ue206<\/p>\n\n\n\n

Substituting the known values:\ue206<\/p>\n\n\n\n

[ T = \\frac{0.040\\, \\text{kg} \\times 9.8\\, \\text{m\/s}^2}{\\cos(8.33^\\circ)} \\approx 0.393\\, \\text{N} ]\ue206<\/p>\n\n\n\n

(b) Period of the Motion<\/strong><\/p>\n\n\n\n

The centripetal force required for circular motion is provided by the horizontal component of the tension:\ue206<\/p>\n\n\n\n

[ T\\sin\\theta = \\frac{mv^2}{r} ]\ue206<\/p>\n\n\n\n

Using the previously calculated tension and angle:\ue206<\/p>\n\n\n\n

[ 0.393\\, \\text{N} \\times \\sin(8.33^\\circ) = \\frac{0.040\\, \\text{kg} \\times v^2}{0.159\\, \\text{m}} ]\ue206<\/p>\n\n\n\n

[ 0.393\\, \\text{N} \\times 0.145 = \\frac{0.040\\, \\text{kg} \\times v^2}{0.159\\, \\text{m}} ]\ue206<\/p>\n\n\n\n

[ 0.057\\, \\text{N} = \\frac{0.040\\, \\text{kg} \\times v^2}{0.159\\, \\text{m}} ]\ue206<\/p>\n\n\n\n

Solving for v:\ue206<\/p>\n\n\n\n

[ v^2 = \\frac{0.057\\, \\text{N} \\times 0.159\\, \\text{m}}{0.040\\, \\text{kg}} ]\ue206<\/p>\n\n\n\n

[ v^2 \\approx 0.226\\, \\text{m}^2\/\\text{s}^2 ]\ue206<\/p>\n\n\n\n

[ v \\approx 0.475\\, \\text{m\/s} ]\ue206<\/p>\n\n\n\n

The period (T) is the time taken for one complete revolution:\ue206<\/p>\n\n\n\n

[ T = \\frac{\\text{Circumference}}{v} = \\frac{1.0\\, \\text{m}}{0.475\\, \\text{m\/s}} \\approx 2.11\\, \\text{s} ]\ue206<\/p>\n\n\n\n

Thus, the tension in the string is approximately 0.393 N, and the period of the motion is approximately 2.11 seconds.\ue206<\/p>\n","protected":false},"excerpt":{"rendered":"

The figure shows a conical pendulum, in which the bob (the small object at the lower end of the cord) moves in a horizontal circle at constant speed. (The cord sweeps out a cone as the bob rotates.) The bob has a mass of 0.040 kg, the string has length L = 1.1 m and […]<\/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-196751","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196751","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=196751"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196751\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}