{"id":9110,"date":"2023-10-03T16:21:34","date_gmt":"2023-10-03T10:51:34","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=9110"},"modified":"2024-09-30T08:10:10","modified_gmt":"2024-09-30T02:40:10","slug":"center-of-mass-of-a-uniform-circular-arc","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/center-of-mass-of-a-uniform-circular-arc\/","title":{"rendered":"Center of Mass of a Uniform Circular Arc"},"content":{"rendered":"\n<figure class=\"wp-block-image size-large\"><img fetchpriority=\"high\" decoding=\"async\" width=\"1024\" height=\"590\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-1024x590.jpg\" alt=\"Center of mass of uniform circular arc\" class=\"wp-image-4609\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-1024x590.jpg 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-300x173.jpg 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-768x442.jpg 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-1536x885.jpg 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/09\/Arc-Disc-2048x1179.jpg 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Center of mass of uniform circular arc<\/figcaption><\/figure>\n\n\n\n<p>$x_{cm} = 0$<br>$y_{cm} =$ $\\cfrac{2}{3} R \\cfrac{ \\sin ( \\alpha \/ 2)}{( \\alpha \/ 2)}$<\/p>\n\n\n<style>.kt-accordion-id9110_18c65a-10 .kt-accordion-inner-wrap{column-gap:var(--global-kb-gap-md, 2rem);row-gap:10px;}.kt-accordion-id9110_18c65a-10 .kt-accordion-panel-inner{border-top:0px solid transparent;border-right:1px solid transparent;border-bottom:1px solid transparent;border-left:1px solid transparent;background:#ffffff;padding-top:var(--global-kb-spacing-sm, 1.5rem);padding-right:var(--global-kb-spacing-sm, 1.5rem);padding-bottom:var(--global-kb-spacing-sm, 1.5rem);padding-left:var(--global-kb-spacing-sm, 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.kt-blocks-accordion-icon-trigger:before{background:#ffffff;}.kt-accordion-id9110_18c65a-10 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{color:#444444;background:#ffffff;border-top-color:#eeeeee;border-top-style:solid;border-right-color:#eeeeee;border-right-style:solid;border-bottom-color:#eeeeee;border-bottom-style:solid;border-left-color:#0e9cd1;border-left-style:solid;}.kt-accordion-id9110_18c65a-10:not( .kt-accodion-icon-style-basiccircle ):not( .kt-accodion-icon-style-xclosecircle ):not( .kt-accodion-icon-style-arrowcircle )  > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active .kt-blocks-accordion-icon-trigger:after, .kt-accordion-id9110_18c65a-10:not( .kt-accodion-icon-style-basiccircle ):not( 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.kt-blocks-accordion-icon-trigger:before{background:#ffffff;}@media all and (max-width: 1024px){.kt-accordion-id9110_18c65a-10 .kt-accordion-panel-inner{border-top:0px solid transparent;border-right:1px solid transparent;border-bottom:1px solid transparent;border-left:1px solid transparent;}}@media all and (max-width: 1024px){.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top:1px solid #eeeeee;border-right:1px solid #eeeeee;border-bottom:1px solid #eeeeee;border-left:2px solid #eeeeee;}}@media all and (max-width: 1024px){.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id9110_18c65a-10 .kt-blocks-accordion-header:focus-visible{border-top-color:#d4d4d4;border-top-style:solid;border-right-color:#d4d4d4;border-right-style:solid;border-bottom-color:#d4d4d4;border-bottom-style:solid;border-left-color:#d4d4d4;border-left-style:solid;}}@media all and (max-width: 1024px){.kt-accordion-id9110_18c65a-10 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{border-top-color:#eeeeee;border-top-style:solid;border-right-color:#eeeeee;border-right-style:solid;border-bottom-color:#eeeeee;border-bottom-style:solid;border-left-color:#0e9cd1;border-left-style:solid;}}@media all and (max-width: 767px){.kt-accordion-id9110_18c65a-10 .kt-accordion-inner-wrap{display:block;}.kt-accordion-id9110_18c65a-10 .kt-accordion-inner-wrap .kt-accordion-pane:not(:first-child){margin-top:10px;}.kt-accordion-id9110_18c65a-10 .kt-accordion-panel-inner{border-top:0px solid transparent;border-right:1px solid transparent;border-bottom:1px solid transparent;border-left:1px solid transparent;}.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header{border-top:1px solid #eeeeee;border-right:1px solid #eeeeee;border-bottom:1px solid #eeeeee;border-left:2px solid #eeeeee;}.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header:hover, \n\t\t\t\tbody:not(.hide-focus-outline) .kt-accordion-id9110_18c65a-10 .kt-blocks-accordion-header:focus-visible{border-top-color:#d4d4d4;border-top-style:solid;border-right-color:#d4d4d4;border-right-style:solid;border-bottom-color:#d4d4d4;border-bottom-style:solid;border-left-color:#d4d4d4;border-left-style:solid;}.kt-accordion-id9110_18c65a-10 .kt-accordion-header-wrap .kt-blocks-accordion-header:focus-visible,\n\t\t\t\t.kt-accordion-id9110_18c65a-10 > .kt-accordion-inner-wrap > .wp-block-kadence-pane > .kt-accordion-header-wrap > .kt-blocks-accordion-header.kt-accordion-panel-active{border-top-color:#eeeeee;border-top-style:solid;border-right-color:#eeeeee;border-right-style:solid;border-bottom-color:#eeeeee;border-bottom-style:solid;border-left-color:#0e9cd1;border-left-style:solid;}}<\/style>\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id9110_18c65a-10 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-arrow kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"0\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane9110_37ef48-81\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">Derivation<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p>$x_{cm} = 0$ because of symmetry about the $y-$axis<\/p>\n\n\n\n<p>$y_{cm} = \\cfrac{\\int ydm}{\\int dm}$<\/p>\n\n\n\n<p>$dm = \\sigma r \\alpha dr$<\/p>\n\n\n\n<p>$y_{cm} = \\cfrac{\\int_0^R r \\cfrac{\\sin (\\alpha \/2)}{\\alpha \/2} r \\alpha dr}{\\int_0^R r \\alpha dr}$<\/p>\n\n\n\n<p>$=\\cfrac{2}{3} R \\cfrac{\\sin (\\alpha \/2)}{(\\alpha \/2)}$<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\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><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-two-or-more-particle-systems\" title=\"\">Two Particle System<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/acejee.com\/blog\/center-of-mass-of-two-or-more-particle-systems\" title=\"\">System of Three Particles<\/a><\/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>Uniform Circular arc<\/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>Hollow cone<\/li>\n\n\n\n<li>Solid cone<\/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 a uniform circular arc along with its derivation. <\/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":"","ocean_both_sidebars_content_width":0,"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":"default","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-9110","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\/9110","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=9110"}],"version-history":[{"count":5,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/9110\/revisions"}],"predecessor-version":[{"id":9404,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/9110\/revisions\/9404"}],"wp:attachment":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media?parent=9110"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/categories?post=9110"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/tags?post=9110"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}