{"id":9078,"date":"2023-10-03T15:22:33","date_gmt":"2023-10-03T09:52:33","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=9078"},"modified":"2024-09-30T08:10:10","modified_gmt":"2024-09-30T02:40:10","slug":"center-of-mass-of-two-or-more-particle-systems","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/center-of-mass-of-two-or-more-particle-systems\/","title":{"rendered":"Center of Mass of Two or More Particle Systems"},"content":{"rendered":"\n<div class=\"wp-block-group custom-roboto-font\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>Here is the center of mass of two (or more) particle systems<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"Two_Particle_System\">Two Particle System<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"313\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-1024x313.jpg\" alt=\"Moment of inertia of two particle system\" class=\"wp-image-4577\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-1024x313.jpg 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-300x92.jpg 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-768x235.jpg 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-1536x470.jpg 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Two-Particle-System-rev1-2048x627.jpg 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Moment of inertia of two particle system<\/figcaption><\/figure>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<p>For the scenario on the left:<br>$x_{CM} = \\cfrac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$ $=x_1 + \\cfrac{m_2 d}{m_1 + m_2}$<\/p>\n\n\n\n<p>For the scenario on the right<br>$x_{CM} = \\cfrac{m_2 d}{m_1 + m_2}$<\/p>\n\n\n\n<p>Note:<br>(i) If $m_1 &gt; m_2$, $x_{CM} &lt; \\cfrac{d}{2}$ (i.e. $x_{CM}$ is located closer to $m_1$)<br>(ii) If $m_1 &lt; m_2$, $x_{CM} &gt; \\cfrac{d}{2}$ (i.e. $x_{CM}$ is located closer to $m_2$)&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"System_Of_Three_Particles\">Multiple Particle System<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" width=\"1024\" height=\"580\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-1024x580.jpg\" alt=\"Moment of inertia of multi particle system\" class=\"wp-image-4581\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-1024x580.jpg 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-300x170.jpg 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-768x435.jpg 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-1536x870.jpg 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Multi-Particle-System-rev1-2048x1159.jpg 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Moment of inertia of multi particle system<\/figcaption><\/figure>\n\n\n\n<p>For a system comprising of two or more simple rigid bodies (with the known center of masses), let&#8217;s determine the center of mass of the system. OR for a simple rigid body, such as a disc, with a cut out of a standard shape (for example circle or square, etc.), let&#8217;s determine its center of mass<\/p>\n\n\n\n<p>$\\color{blue}{\\text{Solution Steps}}$<br>A : $x_{cm} =$ $\\cfrac{\\sum m_i x_i}{\\sum m_i}$<br>$=\\cfrac{\\sum_A m_ix_i + \\sum_B m_ix_i}{m_A+m_B}$<br>$= \\cfrac{m_Ax_{cm,A} + \\sum_B m_B x_{cm,B}}{m_A+m_B}$<br>$y_{cm} = \\cfrac{\\sum m_i y_i}{\\sum m_i}$<br>$=\\cfrac{\\sum_A m_iy_i + \\sum_B m_iy_i}{m_A+m_B}$<br>$= \\cfrac{m_Ay_{cm,A} + \\sum_B m_B y_{cm,B}}{m_A+m_B}$<\/p>\n\n\n\n<p>B) : $\\tan \\theta = \\cfrac{x_{cm}}{L-y_{cm}}$<br>$x_{cm} = \\cfrac{m.0+m.L\/2}{2m}, \\quad y_{cm}$<br>$= \\cfrac{m.L\/2 + m.0}{2m}$<br>$x_{cm} = y_{cm} = \\cfrac{L}{4}$<br>$\\theta = \\tan^{-1} \\cfrac{1}{3}$<\/p>\n\n\n\n<p>C) : $(m_1 + m_2)x_{cm}$&nbsp;$= m_1x_{cm,1} + m_2 x_{cm,2}$<br>$x_{cm} = 0$<br>$x_{cm,2} = &#8211; \\cfrac{m_1x_{cm,1}}{m_2}$<\/p>\n\n\n\n<p>To continue to explore center of mass of other commonly encountered shapes, <a href=\"https:\/\/acejee.com\/blog\/center-of-mass-formula-list-for-different-shapes-jee-main\/\" title=\"\">click here<\/a>&#8230; <\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Two Particle System<\/li>\n\n\n\n<li>System of Three Particles<\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-uniform-i-shaped-lamina\/\" title=\"\">A group of simple rigid bodies (for example a uniform $I$ shaped lamina<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-non-uniform-rod-of-length-l\/\" title=\"\">Non uniform rod of length $L$<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-an-arc-of-a-ring\/\" title=\"\">Arc<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-uniform-circular-arc\/\" title=\"\">Uniform Circular arc<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-an-arc-of-a-ring\/\" title=\"\">Semicircular ring<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-semicircular-and-annular-semicircular-discs\/\" title=\"\">Semicircular disc<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-semicircular-and-annular-semicircular-discs\/\" title=\"\">Annular semicircular disc<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-triangle-triangular-lamina\/\" title=\"\">Triangle<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-solid-and-hollow-hemispheres\/\" title=\"\">Solid hemisphere<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-solid-and-hollow-hemispheres\/\" title=\"\">Hollow hemisphere<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-solid-and-a-hollow-cone\/\" title=\"\">Hollow cone<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-a-solid-and-a-hollow-cone\/\" title=\"\">Solid cone<\/a><\/li>\n<\/ul>\n\n\n\n<a href=\"https:\/\/acejee.com\/jee-main-advanced-physics-online-classes\/\" class=\"floating-button-JEE-Phy-Online-Classes-GutenbergPosts\">JEE Physics Online Classes<\/a>\n","protected":false},"excerpt":{"rendered":"<p>Here is the center of mass of two or more particle systems that you can expect to come across in JEE Main and JEE Advanced<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"full-width","ocean_both_sidebars_style":"css-style","ocean_both_sidebars_content_width":100,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","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":"","ocean_custom_header_template":"","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":"","ocean_menu_typo_font_family":"","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":"off","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","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":"on","ocean_gallery_id":[],"footnotes":""},"categories":[1],"tags":[],"class_list":["post-9078","post","type-post","status-publish","format-standard","hentry","category-uncategorized","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/9078","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=9078"}],"version-history":[{"count":5,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/9078\/revisions"}],"predecessor-version":[{"id":9408,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/9078\/revisions\/9408"}],"wp:attachment":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media?parent=9078"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/categories?post=9078"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/tags?post=9078"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}