{"id":89033,"date":"2025-06-01T07:28:06","date_gmt":"2025-06-01T07:28:06","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=89033"},"modified":"2025-06-01T07:28:06","modified_gmt":"2025-06-01T07:28:06","slug":"shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/","title":{"rendered":"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass"},"content":{"rendered":"

Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass partially filled (depicted by darkened area) with liquids of densities \u03c1\u2081 and \u03c1\u2082, respectively, floating in water (density \u03c1w). The relationship between \u03c1\u2081, \u03c1\u2082 and \u03c1w is<\/p>\n

[amp_mcq option1=”\u03c1\u2082 < \u03c1w < \u03c1\u2081\" option2=\"\u03c1\u2082 < \u03c1\u2081 < \u03c1w\" option3=\"\u03c1\u2081 < \u03c1\u2082 < \u03c1w\" option4=\"\u03c1\u2081 < \u03c1w < \u03c1\u2082\" correct=\"option4\"]\n\n\n\n

\n
\n
This question was previously asked in<\/div>\n
UPSC NDA-2 – 2024<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082.\n<\/section>\n
\nBoth cubes are floating in water. According to Archimedes’ principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside.
\nFor Cube C\u2081:
\nWeight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g.
\nBuoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g.
\nSince it’s floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w.
\nFrom the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w.\n\nFor Cube C\u2082:\nWeight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g.\nBuoyant force = (A * h\u2082) * \u03c1w * g.\nSince it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w.\nFrom the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.
\n<\/section>\n
\nCombining the results, we have \u03c1\u2081 < \u03c1w and \u03c1\u2082 > \u03c1w. This means \u03c1\u2081 is less than the density of water, while \u03c1\u2082 is greater than the density of water. Therefore, the relationship between the densities is \u03c1\u2081 < \u03c1w < \u03c1\u2082.\n<\/section>\n","protected":false},"excerpt":{"rendered":"

Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass partially filled (depicted by darkened area) with liquids of densities \u03c1\u2081 and \u03c1\u2082, respectively, floating in water (density \u03c1w). The relationship between \u03c1\u2081, \u03c1\u2082 and \u03c1w is [amp_mcq option1=”\u03c1\u2082 < \u03c1w < \u03c1\u2081\" option2=\"\u03c1\u2082 < \u03c1\u2081 < \u03c1w\" option3=\"\u03c1\u2081 < \u03c1\u2082 ... \n\n

Detailed SolutionShown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass<\/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":[1094],"tags":[1103,1129,1128],"class_list":["post-89033","post","type-post","status-publish","format-standard","hentry","category-upsc-nda-2","tag-1103","tag-mechanics","tag-physics","no-featured-image-padding"],"yoast_head":"\nShown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass<\/title>\n<meta name=\"description\" content=\"The correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082. Both cubes are floating in water. According to Archimedes' principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside. For Cube C\u2081: Weight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g. Buoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g. Since it's floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w. From the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w. For Cube C\u2082: Weight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g. Buoyant force = (A * h\u2082) * \u03c1w * g. Since it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w. From the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.\" \/>\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\/shown-in-the-figure-are-two-hollow-cubes-c\u2081-and-c\u2082-of-negligible-mass\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass\" \/>\n<meta property=\"og:description\" content=\"The correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082. Both cubes are floating in water. According to Archimedes' principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside. For Cube C\u2081: Weight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g. Buoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g. Since it's floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w. From the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w. For Cube C\u2082: Weight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g. Buoyant force = (A * h\u2082) * \u03c1w * g. Since it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w. From the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c\u2081-and-c\u2082-of-negligible-mass\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T07:28:06+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=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass","description":"The correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082. Both cubes are floating in water. According to Archimedes' principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside. For Cube C\u2081: Weight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g. Buoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g. Since it's floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w. From the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w. For Cube C\u2082: Weight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g. Buoyant force = (A * h\u2082) * \u03c1w * g. Since it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w. From the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.","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\/shown-in-the-figure-are-two-hollow-cubes-c\u2081-and-c\u2082-of-negligible-mass\/","og_locale":"en_US","og_type":"article","og_title":"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass","og_description":"The correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082. Both cubes are floating in water. According to Archimedes' principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside. For Cube C\u2081: Weight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g. Buoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g. Since it's floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w. From the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w. For Cube C\u2082: Weight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g. Buoyant force = (A * h\u2082) * \u03c1w * g. Since it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w. From the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c\u2081-and-c\u2082-of-negligible-mass\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T07:28:06+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/","url":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/","name":"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T07:28:06+00:00","dateModified":"2025-06-01T07:28:06+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is D) \u03c1\u2081 < \u03c1w < \u03c1\u2082. Both cubes are floating in water. According to Archimedes' principle, a floating object displaces a volume of fluid whose weight is equal to the weight of the object. The weight of each hollow cube is solely due to the liquid inside, as the cube itself has negligible mass. Let A be the base area and H be the total height of the cubes (assuming they are identical). Let h\u2081 and h\u2082 be the submerged heights in water, and H\u2081_liquid and H\u2082_liquid be the heights of the liquid inside. For Cube C\u2081: Weight of liquid inside = (Volume of liquid inside) * \u03c1\u2081 * g = (A * H\u2081_liquid) * \u03c1\u2081 * g. Buoyant force = (Volume submerged) * \u03c1w * g = (A * h\u2081) * \u03c1w * g. Since it's floating, (A * H\u2081_liquid) * \u03c1\u2081 * g = (A * h\u2081) * \u03c1w * g, which simplifies to H\u2081_liquid * \u03c1\u2081 = h\u2081 * \u03c1w, or \u03c1\u2081 = (h\u2081 \/ H\u2081_liquid) * \u03c1w. From the figure, the submerged height h\u2081 is significantly less than the height of the liquid inside H\u2081_liquid. Therefore, (h\u2081 \/ H\u2081_liquid) < 1, which implies \u03c1\u2081 < \u03c1w. For Cube C\u2082: Weight of liquid inside = (A * H\u2082_liquid) * \u03c1\u2082 * g. Buoyant force = (A * h\u2082) * \u03c1w * g. Since it's floating, (A * H\u2082_liquid) * \u03c1\u2082 * g = (A * h\u2082) * \u03c1w * g, which simplifies to H\u2082_liquid * \u03c1\u2082 = h\u2082 * \u03c1w, or \u03c1\u2082 = (h\u2082 \/ H\u2082_liquid) * \u03c1w. From the figure, the submerged height h\u2082 is significantly greater than the height of the liquid inside H\u2082_liquid. Therefore, (h\u2082 \/ H\u2082_liquid) > 1, which implies \u03c1\u2082 > \u03c1w.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/shown-in-the-figure-are-two-hollow-cubes-c%e2%82%81-and-c%e2%82%82-of-negligible-mass\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC NDA-2","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-nda-2\/"},{"@type":"ListItem","position":3,"name":"Shown in the figure are two hollow cubes C\u2081 and C\u2082 of negligible mass"}]},{"@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\/89033","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=89033"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/89033\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=89033"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=89033"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=89033"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}