{"id":5047,"date":"2021-11-05T17:23:05","date_gmt":"2021-11-05T11:53:05","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=5047"},"modified":"2024-09-30T08:10:39","modified_gmt":"2024-09-30T02:40:39","slug":"simple-harmonic-motion-spring-mass-system","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/simple-harmonic-motion-spring-mass-system\/","title":{"rendered":"Simple Harmonic Motion | Spring Mass System"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5047\" class=\"elementor elementor-5047\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-22b517e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"22b517e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-cc49603\" data-id=\"cc49603\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d194b27 elementor-widget elementor-widget-text-editor\" data-id=\"d194b27\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Top_of_the_page\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Let&#8217;s explore different spring mass systems, with mass under linear or rotational motion, and determine angular frequency or time period of simple harmonic motion\/oscillations of such a system. More specifically we will look at the following scenarios<\/p>\n<ul>\n<li><a href=\"#Spring_Mass_System_Linear SHM Springs_in_Series\">Linear SHM | Springs in Series<\/a><\/li>\n<li><a href=\"#Spring_Mass_System_Linear_SHM_Springs_in_Parallel\">Linear SHM | Springs in Parallel<\/a><\/li>\n<li><a href=\"#Spring_Mass_System_Linear_SHM_Multiple_Springs_in_2D_Arrangement\">Linear SHM | Multiple Springs in 2D Arrangement<\/a><\/li>\n<li><a href=\"#Spring_Mass_System_2D_SHM\">Spring Mass System | 2D SHM<\/a><\/li>\n<li><a href=\"#Spring_Mass_System_Linear_SHM_Pulley_Spring_Mass_System\">Linear SHM | Pulley &#8211; Spring &#8211; Mass System<\/a><\/li>\n<li><a href=\"#Spring_Mass_System_Linear_SHM_Constant_Force\">Linear SHM | Constant Force<\/a><\/li>\n<li><a href=\"#Linear_SHM_Two_particles_executing_SHM\">Linear SHM | Two particles executing SHM<\/a><\/li>\n<li><a href=\"#Angular_SHM_Scenario_1\">Angular SHM | Scenario 1<\/a><\/li>\n<li><a href=\"#Angular_SHM_Scenario_2\">Angular SHM | Scenario 2<\/a><\/li>\n<li><a href=\"#Simple_Harmonic_Motion_Linear_Rotational_Motion\">Simple Harmonic Motion | Linear + Rotational Motion<\/a><\/li>\n<\/ul>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e40d644 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e40d644\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6641aa8\" data-id=\"6641aa8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54ebc9a elementor-widget elementor-widget-heading\" data-id=\"54ebc9a\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_Linear SHM Springs_in_Series\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Spring Mass System | Linear SHM | Springs in Series<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-85cafdb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"85cafdb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7066a47\" data-id=\"7066a47\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-374be58 elementor-widget elementor-widget-image\" data-id=\"374be58\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"300\" height=\"90\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1-300x90.png\" class=\"attachment-medium size-medium wp-image-5072\" alt=\"Spring Mass System - Linear SHM - Springs in Series\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1-300x90.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1-1024x308.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1-768x231.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1-1536x461.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Series-rev1.png 1631w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5875884 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5875884\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-def24ce\" data-id=\"def24ce\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a97a485 elementor-widget elementor-widget-text-editor\" data-id=\"a97a485\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>For springs in series, tension or restoring force exerted by two springs in series is same, i.e. $F = -k_1 x_1 = -k_2 x_2$ or $x_1 + x_2 = x = -F(\\cfrac{1}{k_1}+\\cfrac{1}{k_2})$<br>So, equivalent stiffness, $k_{eq} =-\\cfrac{F}{x}= \\cfrac{k_1 k_2}{k_1+k_2}$<br>and, angular frequency, $\\omega = \\sqrt{\\cfrac{k_{eq}}{m}}$<br>and, time period $T = \\cfrac{2 \\pi}{\\omega}$&nbsp;<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0b90e22 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0b90e22\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d01bd2b\" data-id=\"d01bd2b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-eacca46 elementor-widget elementor-widget-text-editor\" data-id=\"eacca46\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-db12423 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"db12423\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-75f808b\" data-id=\"75f808b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-18d7a56 elementor-widget elementor-widget-heading\" data-id=\"18d7a56\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_Linear_SHM_Springs_in_Parallel\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Spring Mass System | Linear SHM | Springs in Parallel<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5170742 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5170742\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-089194d\" data-id=\"089194d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6d74de6 elementor-widget elementor-widget-image\" data-id=\"6d74de6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"300\" height=\"169\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2-300x169.png\" class=\"attachment-medium size-medium wp-image-5102\" alt=\"Spring Mass System - Linear SHM - Springs in Parallel\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2-300x169.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2-1024x577.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2-768x433.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2-1536x865.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Springs-in-Parallel-rev2.png 1626w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2938f0b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2938f0b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-eb6cd4d\" data-id=\"eb6cd4d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b6b362e elementor-widget elementor-widget-text-editor\" data-id=\"b6b362e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>For springs in parallel, the amount of extension, $x$, is the same<br \/>and equivalent stiffness $k_{eq} = k_1 + k_2$<br \/>so, angular frequency, $\\omega = \\sqrt{\\cfrac{k_{eq}}{m}}$\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b57ce77 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b57ce77\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-07882ea\" data-id=\"07882ea\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a9b3583 elementor-widget elementor-widget-text-editor\" data-id=\"a9b3583\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cfef3d4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cfef3d4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4ead2b3\" data-id=\"4ead2b3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8a5abc1 elementor-widget elementor-widget-heading\" data-id=\"8a5abc1\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_Linear_SHM_Multiple_Springs_in_2D_Arrangement\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Spring Mass System | Linear SHM | Multiple Springs in 2D Arrangement<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-158710e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"158710e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c923176\" data-id=\"c923176\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-adf13ee elementor-widget elementor-widget-image\" data-id=\"adf13ee\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"207\" height=\"300\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Multiple-Springs-in-2D-Arrangement-rev2-207x300.png\" class=\"attachment-medium size-medium wp-image-5124\" alt=\"Spring Mass System - Linear SHM - Multiple Springs in 2D Arrangement\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Multiple-Springs-in-2D-Arrangement-rev2-207x300.png 207w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Multiple-Springs-in-2D-Arrangement-rev2-706x1024.png 706w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Multiple-Springs-in-2D-Arrangement-rev2-768x1114.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Multiple-Springs-in-2D-Arrangement-rev2.png 820w\" sizes=\"(max-width: 207px) 100vw, 207px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7513005 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7513005\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4140c10\" data-id=\"4140c10\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a143d88 elementor-widget elementor-widget-text-editor\" data-id=\"a143d88\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Given that the mass is allowed to move along the $y-$axis (dotted line),<br \/>For a small displacement, $\\delta$, of mass $m$ (from its mean position) along say positive $y$-axis, the restoring force, $F$ will be $=(k_1 + 2 k_2 \\cos^2 \\theta) \\delta$<br \/>So, $k_{eq} = k_1 + 2 k_2 \\cos^2 \\theta$<br \/>and angular frequency, $\\omega = \\sqrt{\\cfrac{k_{eq}}{m}}$\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-153b107 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"153b107\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7df2b23\" data-id=\"7df2b23\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f27b893 elementor-widget elementor-widget-text-editor\" data-id=\"f27b893\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4a277fc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4a277fc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d77a655\" data-id=\"d77a655\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7d360bc elementor-widget elementor-widget-heading\" data-id=\"7d360bc\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_2D_SHM\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Spring Mass System | 2D SHM<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c21da0d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c21da0d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-146b695\" data-id=\"146b695\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5cc5271 elementor-widget elementor-widget-image\" data-id=\"5cc5271\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"260\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-2D-SHM\u200b-300x260.png\" class=\"attachment-medium size-medium wp-image-5075\" alt=\"Spring Mass System - 2D SHM\u200b\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-2D-SHM\u200b-300x260.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-2D-SHM\u200b-768x666.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-2D-SHM\u200b.png 859w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6305965 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6305965\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-afe24b2\" data-id=\"afe24b2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e8b5e04 elementor-widget elementor-widget-text-editor\" data-id=\"e8b5e04\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>If $k_1 = k_2$, then the mass will move along a straight line<\/p><p>If $k_1 = 4k_2$, meaning $\\omega_1 = 2 \\omega_2$ then the particle will trace a horizontal 8 as shown. Note that as the particle completes 1 oscillation in the horizontal direction, it completes two oscillations in the vertical direction<br \/>(and if $4k_1 = k_2$, then the particle will trace the standing or vertical 8)<\/p><p>You can use similar approach to determine the resulting trajectory of the particle for other combinations of $k_1$ and $k_2$.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7e41b8c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7e41b8c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-79b9251\" data-id=\"79b9251\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7983def elementor-widget elementor-widget-text-editor\" data-id=\"7983def\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-06c50f8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"06c50f8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0c23b30\" data-id=\"0c23b30\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7bc20ee elementor-widget elementor-widget-heading\" data-id=\"7bc20ee\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_Linear_SHM_Pulley_Spring_Mass_System\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Spring Mass System | Linear SHM | Pulley - Spring - Mass System<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4faebe5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4faebe5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8119f6c\" data-id=\"8119f6c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4e33890 elementor-widget elementor-widget-image\" data-id=\"4e33890\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"231\" height=\"300\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Pulley-Spring-Mass-System-231x300.png\" class=\"attachment-medium size-medium wp-image-5076\" alt=\"Spring Mass System - Linear SHM - Pulley - Spring - Mass System\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Pulley-Spring-Mass-System-231x300.png 231w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Pulley-Spring-Mass-System-789x1024.png 789w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Pulley-Spring-Mass-System-768x997.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Pulley-Spring-Mass-System.png 1055w\" sizes=\"(max-width: 231px) 100vw, 231px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bf971ab elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bf971ab\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0e91ca2\" data-id=\"0e91ca2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c98cf0f elementor-widget elementor-widget-text-editor\" data-id=\"c98cf0f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Assuming that the pulleys are massless, let&#8217;s say that we displace the mass $m$ by a small distance $\\delta$ in the downward direction (from its mean or equilibrium position), let&#8217;s figure out the increase (or change) in tension.<br \/>so, let&#8217;s say that the increase in tension is $T$,<br \/>then<br \/>Increase in stretching of spring $1$, $\\delta_1 = \\cfrac{2T}{k_1}$<br \/>Similarly, increase in stretching of spring $2$ will be $\\delta_2 = \\cfrac{2T}{k_2}$<br \/>$\\delta_3 = \\cfrac{2T}{k_3}$<br \/>$\\delta_4 = \\cfrac{2T}{k_4}$<\/p><p>And $2 \\delta_1 + 2 \\delta_2 + 2 \\delta_3 + 2 \\delta_4 = \\delta$\u00a0<\/p><p>So, $4T (\\cfrac{1}{k_1} + \\cfrac{1}{k_2} + \\cfrac{1}{k_3} + \\cfrac{1}{k_4}) = \\delta$<\/p><p>Now, $k_{eq} = \\cfrac{T}{\\delta}$<br \/>i.e. $k_{eq} = \\cfrac{1}{4(\\cfrac{1}{k_1} + \\cfrac{1}{k_2} + \\cfrac{1}{k_3} + \\cfrac{1}{k_4})}$<\/p><p>&amp; angular frequency $\\omega = \\sqrt{\\cfrac{k_{eq}}{m}}$<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-dfb89ee elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"dfb89ee\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a1b5c66\" data-id=\"a1b5c66\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-57c4145 elementor-widget elementor-widget-text-editor\" data-id=\"57c4145\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-da3b755 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"da3b755\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1197dd6\" data-id=\"1197dd6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-66b3495 elementor-widget elementor-widget-heading\" data-id=\"66b3495\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Spring_Mass_System_Linear_SHM_Constant_Force\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Spring Mass System | Linear SHM | Constant Force<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-8d38d88 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"8d38d88\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-322f85e\" data-id=\"322f85e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3f063f0 elementor-widget elementor-widget-image\" data-id=\"3f063f0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"139\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Constant-Force-300x139.png\" class=\"attachment-medium size-medium wp-image-5077\" alt=\"Spring Mass System - Linear SHM - Constant Force\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Constant-Force-300x139.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Constant-Force-1024x476.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Constant-Force-768x357.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Spring-Mass-System-Linear-SHM-Constant-Force.png 1026w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0a5a639 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0a5a639\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fd8e5c7\" data-id=\"fd8e5c7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-42f9576 elementor-widget elementor-widget-text-editor\" data-id=\"42f9576\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Note that, in the equilibrium position spring is compressed by $\\cfrac{mg \\sin \\theta}{k}$ where $\\theta$ is the angle of incline, assuming the incline surface is frictionless<\/p><p>Now if we displace mass $m$ slightly from its mean or equilibrium position, it will oscillate with angular frequency $\\omega = \\sqrt{\\cfrac{k}{m}}$\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7522925 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7522925\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-70fab95\" data-id=\"70fab95\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-abb3b55 elementor-widget elementor-widget-text-editor\" data-id=\"abb3b55\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-06e2078 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"06e2078\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d064b1a\" data-id=\"d064b1a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-126d698 elementor-widget elementor-widget-heading\" data-id=\"126d698\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Linear_SHM_Two_particles_executing_SHM\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Linear SHM | Two particles executing SHM<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fb655db elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fb655db\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-34887e8\" data-id=\"34887e8\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-90b564b elementor-widget elementor-widget-image\" data-id=\"90b564b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"58\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM-300x58.png\" class=\"attachment-medium size-medium wp-image-5078\" alt=\"Linear SHM - Two particles executing SHM\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM-300x58.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM-1024x200.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM-768x150.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM-1536x299.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Linear-SHM-Two-particles-executing-SHM.png 2001w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6eb7b36 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6eb7b36\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f1164a7\" data-id=\"f1164a7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-da25668 elementor-widget elementor-widget-text-editor\" data-id=\"da25668\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>If we displace masses $m_1$ and $m_2$ (connected by spring of spring constant $k_{eq}$) towards each other and then release them, they will undergo simple harmonic motion (with their center of mass at rest)<\/p><p>So, let&#8217;s write the newton&#8217;s law $F = ma = m \\cfrac{d^2 x}{dt^2}$ for mass $m_1$, when it is a distance $\\delta_1$ from its mean position towards left<\/p><p>Well to determine the restoring spring force $F$, we need to know the total extension of the spring when $m_1$ is at $\\delta_1$ to the left of its mean position.<\/p><p>Well recall that $x_{cm} = \\cfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$<\/p><p>Now if you take origin at center of mass then $m_1 x_1 + m_2 x_2 = 0$ or $m_1 \\delta_1 + m_2 \\delta_2 = 0$ or $\\delta_2 = &#8211; \\cfrac{m_1}{m_2} \\delta_1$ with minus sign indicating that if $\\delta_1$ was towards left $\\delta_2$ will be towards right<\/p><p>so net extension of the spring $=\\delta_1 + \\delta_2 = \\cfrac{m_1 + m_2}{m_2} \\delta_1$<\/p><p>or, $-k_{eq} \\cfrac{m_1 + m_2}{m_2} \\delta_1$ $=m_1 \\cfrac{d^2 \\delta_1}{dt^2}$<\/p><p>or $\\cfrac{d^2 \\delta_1}{dt^2} = &#8211; \\cfrac{k_{eq}}{\\cfrac{m_1 m_2}{m_1+m_2}} \\delta_1$<\/p><p>i.e. $\\omega^2 = \\cfrac{k_{eq}}{\\cfrac{m_1 m_2}{m_1+m_2}}$\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b42f7e6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b42f7e6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-df54223\" data-id=\"df54223\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1beea6e elementor-widget elementor-widget-text-editor\" data-id=\"1beea6e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-af351ee elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"af351ee\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-82e2fc1\" data-id=\"82e2fc1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0d5a2e4 elementor-widget elementor-widget-heading\" data-id=\"0d5a2e4\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Angular_SHM_Scenario_1\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Angular SHM | Scenario 1<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-baba20f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"baba20f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-303e5eb\" data-id=\"303e5eb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a5e2ca5 elementor-widget elementor-widget-image\" data-id=\"a5e2ca5\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"218\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-1-rev2-300x218.png\" class=\"attachment-medium size-medium wp-image-5156\" alt=\"Angular SHM - Scenario 1\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-1-rev2-300x218.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-1-rev2-1024x745.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-1-rev2-768x559.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-1-rev2.png 1528w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-09c4dca elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"09c4dca\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-599ebf4\" data-id=\"599ebf4\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4aefca3 elementor-widget elementor-widget-text-editor\" data-id=\"4aefca3\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>If we rotate the rod by a small angle $\\theta$ about its center, the springs will be stretched by $\\cfrac{L}{2} \\theta$<\/p><p>So, the restoring torque about the pivot point will be $\\tau = \\cfrac{L}{2}$$(k_1 \\cfrac{L}{2} \\theta +$ $k_2 \\cfrac{L}{2} \\theta)$ $=\\cfrac{L^2}{4} (k_1 + k_2) \\theta$<\/p><p>Now, as per newton&#8217;s 2nd law $\\tau = I \\alpha = I \\cfrac{d^2 \\theta}{dt^2}$<\/p><p>So, $\\cfrac{d^2 \\theta}{dt^2}=-\\cfrac{L^2}{4I} (k_1 + k_2) \\theta$, where $I = \\cfrac{1}{12} ML^2$ for the rod<\/p><p>So angular frequency, $\\omega^2 = \\cfrac{L^2}{4I} (k_1 + k_2)$<\/p><p>Note that above is true only for small angular displacements. For larger angular displacements the motion will not be simple harmonic\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-beded08 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"beded08\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f718c8a\" data-id=\"f718c8a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-727f507 elementor-widget elementor-widget-text-editor\" data-id=\"727f507\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-48ba018 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"48ba018\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-539b882\" data-id=\"539b882\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d2946d9 elementor-widget elementor-widget-heading\" data-id=\"d2946d9\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Angular_SHM_Scenario_2\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Angular SHM | Scenario 2<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1a6393d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1a6393d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5fa58c1\" data-id=\"5fa58c1\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dc45453 elementor-widget elementor-widget-image\" data-id=\"dc45453\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"217\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3-300x217.png\" class=\"attachment-medium size-medium wp-image-5171\" alt=\"Angular SHM - Scenario 2\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3-300x217.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3-1024x739.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3-768x554.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3-1536x1109.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Angular-SHM-Scenario-2-rev3.png 1978w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b866d15 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b866d15\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ee1a163\" data-id=\"ee1a163\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-857cada elementor-widget elementor-widget-text-editor\" data-id=\"857cada\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Note that assuming that the springs are ideal springs, motion will be simple harmonic ir-respective of how large the angular displacement, $\\theta$, is from its mean equilibrium position of $PQ$<\/p><p>Here the restoring force on each mass is $2 k_1 R \\theta + 2 k_2 R \\theta$ $=2(k_1 + k_2) R \\theta$<\/p><p>So, restoring torque on mass $m$ is $2(k_1 + k_2) R^2 \\theta$<\/p><p>Now, writing netwon&#8217;s 2nd law for mass $m$, $\\tau = I \\alpha$ $= I \\cfrac{d^2 \\theta}{dt^2}$ $mR^2 \\cfrac{d^2 \\theta}{dt^2}$<\/p><p>i.e. $\\cfrac{d^2 \\theta}{dt^2} = &#8211; \\cfrac{2(k_1+k_2)}{m} \\theta$\u00a0<\/p><p>or, angular frequency $\\omega = \\sqrt{\\cfrac{2(k_1+k_2)}{m}}$<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e4f3bb elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0e4f3bb\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-caba6b9\" data-id=\"caba6b9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e2b7524 elementor-widget elementor-widget-text-editor\" data-id=\"e2b7524\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#Top_of_the_page\">Top<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-191a473 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"191a473\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8904548\" data-id=\"8904548\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f3f90ec elementor-widget elementor-widget-heading\" data-id=\"f3f90ec\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"Simple_Harmonic_Motion_Linear_Rotational_Motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Simple Harmonic Motion | Linear + Rotational Motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7486f5e elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7486f5e\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0476575\" data-id=\"0476575\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c08b2d6 elementor-widget elementor-widget-image\" data-id=\"c08b2d6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"215\" height=\"300\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2-215x300.png\" class=\"attachment-medium size-medium wp-image-6399\" alt=\"Simple Harmonic Motion - Linear + Rotational Motion\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2-215x300.png 215w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2-735x1024.png 735w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2-768x1071.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2-1102x1536.png 1102w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/11\/Simple-Harmonic-Motion-Linear-Rotational-Motion-rev2.png 1358w\" sizes=\"(max-width: 215px) 100vw, 215px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ee48ad0 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ee48ad0\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e23b011\" data-id=\"e23b011\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-beb6db6 elementor-widget elementor-widget-text-editor\" data-id=\"beb6db6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Assuming that cylinder is in pure rolling at all times, let&#8217;s say that the center of cylinder is displaced by a distance $x$ to the right as shown, then the net restoring force acting on it is<\/p>\n<p>$F_{net} = -(3kx &#8211; f_s)$ (assuming that the friction force is acting to the right)<\/p>\n<p>Newton&#8217;s 2nd law ($F_{net}=Ma_{cm}$) gives us<br>\n$-(3kx &#8211; f_s) = M a_{cm}$ $\\cdots$(i)<\/p>\n<p>Now torque on cylinder about its center of mass $\\tau_{net} =-(2kx+f_s)R$<\/p>\n<p>Using newton&#8217;s 2nd law $\\tau_{net} = I \\alpha$ we get<br>\n$-(2kx+f_s)R = I \\alpha$<\/p>\n<p>Now in pure rolling $a_{cm} = \\alpha R$<br>\ni.e. $-(2kx+f_s) = a_{cm} \\cfrac{I}{R^2}$ $\\cdots$(ii)<\/p>\n<p>Solving for $a_{cm}$ in terms of $x$ we get<br>\n$a_{cm} = \\cfrac{d^2x}{dt^2}=$ $-\\cfrac{5k}{M + \\cfrac{I}{R^2}} x$<\/p>\n<p>or angular frequency $\\omega = \\sqrt{\\cfrac{5k}{M + \\cfrac{I}{R^2}}}$&nbsp;<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Let&#8217;s explore different spring mass systems and determine angular frequency \/ time period of the simple harmonic motion (includes JEE Main &#038; Advanced questions)<\/p>\n","protected":false},"author":1,"featured_media":5049,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"0","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"on","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"off","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","cybocfi_hide_featured_image":"yes","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"off","ocean_gallery_id":[],"footnotes":""},"categories":[16],"tags":[],"class_list":["post-5047","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-iit-jee-physics-chapter-wise-approach","entry","has-media"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/5047","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/comments?post=5047"}],"version-history":[{"count":1,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/5047\/revisions"}],"predecessor-version":[{"id":9435,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/5047\/revisions\/9435"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media\/5049"}],"wp:attachment":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media?parent=5047"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/categories?post=5047"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/tags?post=5047"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}