{"id":88041,"date":"2025-06-01T07:00:16","date_gmt":"2025-06-01T07:00:16","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=88041"},"modified":"2025-06-01T07:00:16","modified_gmt":"2025-06-01T07:00:16","slug":"a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/","title":{"rendered":"A spherical shell of outer radius R and inner radius R\/2 contains a so"},"content":{"rendered":"

A spherical shell of outer radius R and inner radius R\/2 contains a solid sphere of radius R\/2 (see figure). The density of the material of the solid sphere is \u03c1 and that of the shell is \u03c1\/2. What is the average mass density of the larger sphere thus formed?<\/p>\n

[amp_mcq option1=”3\u03c1\/4″ option2=”9\u03c1\/16″ option3=”7\u03c1\/8″ option4=”5\u03c1\/8″ correct=”option2″]<\/p>\n

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This question was previously asked in<\/div>\n
UPSC NDA-1 – 2024<\/div>\n<\/div>\n
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\nThe average mass density of the larger sphere is the total mass divided by the total volume.
\n<\/section>\n
\nLet R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3.
\nMass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1.
\nVolume of the shell material (V_shell) = Volume of sphere with radius R – Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 – (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 – (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3.
\nMass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1.
\nTotal mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.
\n<\/section>\n
\nAverage density = M_total \/ V_total = ((3\/4)\u03c0R\u00b3\u03c1) \/ ((4\/3)\u03c0R\u00b3) = (3\/4) * (3\/4) * \u03c1 = 9\u03c1\/16. This calculation represents the overall density if the entire composite structure were considered a single sphere of radius R with uniform density.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

A spherical shell of outer radius R and inner radius R\/2 contains a solid sphere of radius R\/2 (see figure). The density of the material of the solid sphere is \u03c1 and that of the shell is \u03c1\/2. What is the average mass density of the larger sphere thus formed? [amp_mcq option1=”3\u03c1\/4″ option2=”9\u03c1\/16″ option3=”7\u03c1\/8″ option4=”5\u03c1\/8″ … <\/p>\n

Detailed SolutionA spherical shell of outer radius R and inner radius R\/2 contains a so<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1093],"tags":[1103,1160,1128],"class_list":["post-88041","post","type-post","status-publish","format-standard","hentry","category-upsc-nda-1","tag-1103","tag-physical-properties-of-materials","tag-physics","no-featured-image-padding"],"yoast_head":"\nA spherical shell of outer radius R and inner radius R\/2 contains a so<\/title>\n<meta name=\"description\" content=\"The average mass density of the larger sphere is the total mass divided by the total volume. Let R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3. Mass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1. Volume of the shell material (V_shell) = Volume of sphere with radius R - Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 - (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 - (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3. Mass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1. Total mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A spherical shell of outer radius R and inner radius R\/2 contains a so\" \/>\n<meta property=\"og:description\" content=\"The average mass density of the larger sphere is the total mass divided by the total volume. Let R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3. Mass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1. Volume of the shell material (V_shell) = Volume of sphere with radius R - Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 - (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 - (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3. Mass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1. Total mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T07:00:16+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"A spherical shell of outer radius R and inner radius R\/2 contains a so","description":"The average mass density of the larger sphere is the total mass divided by the total volume. Let R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3. Mass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1. Volume of the shell material (V_shell) = Volume of sphere with radius R - Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 - (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 - (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3. Mass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1. Total mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/","og_locale":"en_US","og_type":"article","og_title":"A spherical shell of outer radius R and inner radius R\/2 contains a so","og_description":"The average mass density of the larger sphere is the total mass divided by the total volume. Let R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3. Mass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1. Volume of the shell material (V_shell) = Volume of sphere with radius R - Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 - (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 - (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3. Mass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1. Total mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T07:00:16+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/","url":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/","name":"A spherical shell of outer radius R and inner radius R\/2 contains a so","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T07:00:16+00:00","dateModified":"2025-06-01T07:00:16+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The average mass density of the larger sphere is the total mass divided by the total volume. Let R be the outer radius. The solid sphere has radius R\/2 and density \u03c1. The shell has outer radius R, inner radius R\/2, and density \u03c1\/2. The total volume of the larger sphere is (4\/3)\u03c0R\u00b3. Mass of solid sphere (M_sphere) = Density * Volume = \u03c1 * (4\/3)\u03c0(R\/2)\u00b3 = \u03c1 * (4\/3)\u03c0(R\u00b3\/8) = (1\/6)\u03c0R\u00b3\u03c1. Volume of the shell material (V_shell) = Volume of sphere with radius R - Volume of sphere with radius R\/2 = (4\/3)\u03c0R\u00b3 - (4\/3)\u03c0(R\/2)\u00b3 = (4\/3)\u03c0R\u00b3 - (1\/6)\u03c0R\u00b3 = (7\/6)\u03c0R\u00b3. Mass of the shell (M_shell) = Density * Volume = (\u03c1\/2) * (7\/6)\u03c0R\u00b3 = (7\/12)\u03c0R\u00b3\u03c1. Total mass (M_total) = M_sphere + M_shell = (1\/6)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (2\/12)\u03c0R\u00b3\u03c1 + (7\/12)\u03c0R\u00b3\u03c1 = (9\/12)\u03c0R\u00b3\u03c1 = (3\/4)\u03c0R\u00b3\u03c1.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/a-spherical-shell-of-outer-radius-r-and-inner-radius-r-2-contains-a-so\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC NDA-1","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-nda-1\/"},{"@type":"ListItem","position":3,"name":"A spherical shell of outer radius R and inner radius R\/2 contains a so"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/88041","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=88041"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/88041\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=88041"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=88041"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=88041"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}