{"id":6755,"date":"2024-04-15T02:33:41","date_gmt":"2024-04-15T02:33:41","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=6755"},"modified":"2024-04-15T02:33:41","modified_gmt":"2024-04-15T02:33:41","slug":"m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/","title":{"rendered":"M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} &#8211; {{\\text{d}}^4}} \\right)$$ C. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ D. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} \\times {{\\text{d}}^4}} \\right)$$"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a9d6f87a2876\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    $$\frac{pi }{{64}}left( {{{\text{D}}^4} + {{\text{d}}^4}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    $$\frac{pi }{{32}}left( {{{\text{D}}^4} - {{\text{d}}^4}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    $$\frac{pi }{{32}}left( {{{\text{D}}^4} + {{\text{d}}^4}} \right)$$                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    $$\frac{pi }{{32}}left( {{{\text{D}}^4} \times {{\text{d}}^4}} \right)$$                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is $\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$.<\/p>\n<p>The moment of inertia of a thin ring about an axis perpendicular to the plane of the ring is given by:<\/p>\n<p>$$I = \\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$<\/p>\n<p>where $D$ is the external diameter of the ring and $d$ is the internal diameter of the <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> ring.<\/p>\n<p>The moment of inertia is a measure of how difficult it is to change the rotation of an object. The larger the moment of inertia, the more difficult it is to rotate the object.<\/p>\n<p>The moment of inertia of a thin ring is calculated by considering the mass of each infinitesimal element of the ring and its distance from the axis of rotation. The mass of each element is given by:<\/p>\n<p>$$dm = \\pi r \\, dr$$<\/p>\n<p>where $r$ is the distance from the element to the axis of rotation.<\/p>\n<p>The moment of inertia of each element is given by:<\/p>\n<p>$$I_d = mr^2$$<\/p>\n<p>The total moment of inertia is then <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> given by:<\/p>\n<p>$$I = \\int_0^D I_d \\, dr = \\int_0^D \\pi r^3 \\, dr = \\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[645],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} - {{\\text{d}}^4}} \\right)$$ C. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ D. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} \\times {{\\text{d}}^4}} \\right)$$<\/title>\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\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} - {{\\text{d}}^4}} \\right)$$ C. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ D. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} \\times {{\\text{d}}^4}} \\right)$$\" \/>\n<meta property=\"og:description\" content=\"Join Our Telegram Channel Subscribe on YouTube\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T02:33:41+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":"M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} - {{\\text{d}}^4}} \\right)$$ C. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ D. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} \\times {{\\text{d}}^4}} \\right)$$","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\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/","og_locale":"en_US","og_type":"article","og_title":"M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} - {{\\text{d}}^4}} \\right)$$ C. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ D. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} \\times {{\\text{d}}^4}} \\right)$$","og_description":"Join Our Telegram Channel Subscribe on YouTube","og_url":"https:\/\/exam.pscnotes.com\/mcq\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T02:33:41+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\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/","url":"https:\/\/exam.pscnotes.com\/mcq\/m-i-of-a-thin-ring-external-diameter-d-internal-diameter-d-about-an-axis-perpendicular-to-the-plane-of-the-ring-is-a-fracpi-64left-textd4-textd4-right-b\/","name":"M.I. of a thin ring (external diameter D, internal diameter d) about an axis perpendicular to the plane of the ring, is A. $$\\frac{\\pi }{{64}}\\left( {{{\\text{D}}^4} + {{\\text{d}}^4}} \\right)$$ B. $$\\frac{\\pi }{{32}}\\left( {{{\\text{D}}^4} - 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