{"id":268168,"date":"2025-07-24T01:21:59","date_gmt":"2025-07-24T01:21:59","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=268168"},"modified":"2025-07-24T01:22:01","modified_gmt":"2025-07-24T01:22:01","slug":"find-the-tension-in-the-lower-portions-of-the-string","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/24\/find-the-tension-in-the-lower-portions-of-the-string\/","title":{"rendered":"Find the tension in the lower portions of the string."},"content":{"rendered":"\n

Find the tension in the lower portions of the string. Verify that your answers agree with part B if you set mp. Express your answer in terms of some or all of the variables m, g, and constant g.<\/p>\n\n\n\n

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

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

Correct Answer:<\/strong><\/p>\n\n\n\n

T_lower = (m2*g*(2*m1 + mp)) \/ (2*m1 + 2*m2 + mp)<\/p>\n\n\n\n

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

This problem involves a dynamic system with two masses and a massive pulley. Based on the variable T_lower, the most probable physical setup is a mass m1 on a frictionless horizontal surface connected by a string to a hanging mass m2. The string passes over a pulley with mass mp. The term T_lower refers to the tension in the vertical portion of the string attached to the hanging mass m2.<\/p>\n\n\n\n

To find this tension, we apply Newton’s second law to each component of the system. Let a be the acceleration of the masses and T_upper be the tension in the horizontal part of the string.<\/p>\n\n\n\n

    \n
  1. For the hanging mass\u00a0m2:<\/strong>\u00a0The net force is the gravitational force minus the upward tension.
    m2*g – T_lower = m2*a<\/li>\n\n\n\n
  2. For the mass\u00a0m1\u00a0on the table:<\/strong>\u00a0The net force is the tension in the horizontal part of the string.
    T_upper = m1*a<\/li>\n\n\n\n
  3. For the massive pulley\u00a0mp:<\/strong>\u00a0The pulley is modeled as a solid disk with a moment of inertia\u00a0I = (1\/2)mp*R^2. The two different tensions create a net torque, causing it to rotate with angular acceleration\u00a0\u03b1. The string does not slip, so\u00a0a = \u03b1R.
    The net torque is\u00a0\u03c4 = (T_lower – T_upper)R.
    Using\u00a0\u03c4 = I\u03b1, we get\u00a0(T_lower – T_upper)R = (1\/2)mp*R^2 * (a\/R), which simplifies to\u00a0T_lower – T_upper = (1\/2)mp*a.<\/li>\n<\/ol>\n\n\n\n

    Now we solve this system of equations. We substitute the expressions for T_lower (from m2’s equation) and T_upper (from m1’s equation) into the pulley’s torque equation, but it is simpler to first substitute T_upper = m1*a into the torque equation to relate the tensions: T_lower – m1*a = (1\/2)mp*a.<\/p>\n\n\n\n

    Next, we solve for acceleration by substituting this into the first equation. Rearranging the torque equation gives T_lower = m1*a + (1\/2)mp*a.
    Now, set the two expressions for T_lower equal:
    m2*g – m2*a = m1*a + (1\/2)mp*a
    m2*g = a * (m1 + m2 + mp\/2)
    a = (m2*g) \/ (m1 + m2 + mp\/2)<\/p>\n\n\n\n

    Finally, we find T_lower by substituting a back into the equation for mass m2:
    T_lower = m2 * (g – a) = m2*g * (1 – m2 \/ (m1 + m2 + mp\/2))
    T_lower = m2*g * ((m1 + mp\/2) \/ (m1 + m2 + mp\/2))
    Multiplying the numerator and denominator by 2 gives the final form:
    T_lower = (m2*g*(2*m1 + mp)) \/ (2*m1 + 2*m2 + mp)<\/p>\n\n\n\n

    To verify, if we set mp = 0, the expression becomes T_lower = (m2*g*2*m1) \/ (2*m1 + 2*m2) = (m1*m2*g) \/ (m1 + m2), which is the correct tension for a system with a massless pulley.<\/p>\n\n\n\n

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

    Find the tension in the lower portions of the string. Verify that your answers agree with part B if you set mp. Express your answer in terms of some or all of the variables m, g, and constant g. The Correct Answer and Explanation is: Correct Answer: T_lower = (m2*g*(2*m1 + mp)) \/ (2*m1 + […]<\/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-268168","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268168","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=268168"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/268168\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=268168"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=268168"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=268168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}