{"id":270517,"date":"2025-07-25T08:00:24","date_gmt":"2025-07-25T08:00:24","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=270517"},"modified":"2025-07-25T08:00:28","modified_gmt":"2025-07-25T08:00:28","slug":"to-double-the-total-energy-of-a-mass-spring-system-oscillating-in-simple-harmonic-motion-the-amplitude-must-increase-by-a-factor-of-2-times-the-square-root-of-2-square-root-of-2-4-2","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/25\/to-double-the-total-energy-of-a-mass-spring-system-oscillating-in-simple-harmonic-motion-the-amplitude-must-increase-by-a-factor-of-2-times-the-square-root-of-2-square-root-of-2-4-2\/","title":{"rendered":"To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of 2 times the square root of 2 square root of 2 4 2"},"content":{"rendered":"\n

To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of 2 times the square root of 2 square root of 2 4 2<\/p>\n\n\n\n

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

The correct factor by which the amplitude must increase to double the total energy of a mass-spring system oscillating in simple harmonic motion is \u221a2<\/strong> (square root of 2).<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

In simple harmonic motion (SHM), the total mechanical energy EEE of the system is given by the sum of its potential and kinetic energies, and it remains constant throughout the oscillation. The total energy is related to the amplitude AAA of the motion.<\/p>\n\n\n\n

The formula for the total energy in SHM is:E=12kA2E = \\frac{1}{2} k A^2E=21\u200bkA2<\/p>\n\n\n\n

Where:<\/p>\n\n\n\n

    \n
  • EEE is the total mechanical energy,<\/li>\n\n\n\n
  • kkk is the spring constant,<\/li>\n\n\n\n
  • AAA is the amplitude of the oscillation.<\/li>\n<\/ul>\n\n\n\n

    From this, it\u2019s clear that the energy is directly proportional to the square of the amplitude.<\/p>\n\n\n\n

    Now, if you want to double the total energy, you set the new energy E\u2032E’E\u2032 equal to twice the original energy EEE:E\u2032=2EE’ = 2EE\u2032=2E<\/p>\n\n\n\n

    Using the equation for energy, we have:12kA\u20322=2\u00d712kA2\\frac{1}{2} k A’^2 = 2 \\times \\frac{1}{2} k A^221\u200bkA\u20322=2\u00d721\u200bkA2<\/p>\n\n\n\n

    Canceling the common terms:A\u20322=2A2A’^2 = 2A^2A\u20322=2A2<\/p>\n\n\n\n

    Taking the square root of both sides:A\u2032=A2A’ = A \\sqrt{2}A\u2032=A2\u200b<\/p>\n\n\n\n

    So, the amplitude must increase by a factor of 2\\sqrt{2}2\u200b, or approximately 1.414, in order to double the total energy.<\/p>\n\n\n\n

    Thus, the correct answer is \u221a2<\/strong>.<\/p>\n\n\n\n

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

    To double the total energy of a mass-spring system oscillating in simple harmonic motion, the amplitude must increase by a factor of 2 times the square root of 2 square root of 2 4 2 The Correct Answer and Explanation is: The correct factor by which the amplitude must increase to double the total energy […]<\/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-270517","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270517","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=270517"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/270517\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=270517"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=270517"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=270517"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}