{"id":156762,"date":"2022-03-27T11:49:38","date_gmt":"2022-03-27T06:19:38","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=156762"},"modified":"2025-02-10T15:27:01","modified_gmt":"2025-02-10T09:57:01","slug":"blog-iit-jee-moment-of-inertia-of-a-square","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/blog\/iit-jee\/moment-of-inertia-of-a-square\/","title":{"rendered":"Moment of Inertia of a Square"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/blog\/iit-jee\/moment-of-inertia-of-a-square\/#Moment_of_Inertia_of_a_Square_Plate\" title=\"Moment of Inertia of a Square Plate\">Moment of Inertia of a Square Plate<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/blog\/iit-jee\/moment-of-inertia-of-a-square\/#FAQs\" title=\"FAQs\">FAQs<\/a><ul class='ez-toc-list-level-5'><li class='ez-toc-heading-level-5'><ul class='ez-toc-list-level-5'><li class='ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/blog\/iit-jee\/moment-of-inertia-of-a-square\/#What_is_the_use_of_moment_of_inertia\" title=\"What is the use of moment of inertia?\">What is the use of moment of inertia?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-5'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/blog\/iit-jee\/moment-of-inertia-of-a-square\/#How_do_we_find_the_Area_of_a_Hollow_Square\" title=\"How do we find the Area of a Hollow Square?\">How do we find the Area of a Hollow Square?<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>The formula for the moment of inertia of a square is:<\/p>\n<p style=\"text-align: center;\"><strong>I = a<sup>4 <\/sup>\/ 12<\/strong><\/p>\n<p>In this case, a = the square section&#8217;s sides. This would be the equation for a solid square with its centre of mass along the x-axis.<\/p>\n<p>A square&#8217;s diagonal moment of inertia can also be calculated as;<\/p>\n<p style=\"text-align: center;\"><strong>I<sub>x<\/sub> = I<sub>y<\/sub> = a<sup>4<\/sup> \/ 12<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156766 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/1-17.png\" alt=\"Moment of Inertia of a Square\" width=\"643\" height=\"543\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/1-17.png?v=1648305780 643w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/1-17-300x253.png?v=1648305780 300w\" sizes=\"(max-width: 643px) 100vw, 643px\" \/><\/p>\n<p>If indeed the centre of mass (cm) is moved to a certain distance (d) from the x-axis, we will use a different expression to calculate the moment of inertia of the same square.<\/p>\n<p style=\"text-align: center;\"><strong>I = a<sup>4 <\/sup>\/ 3<\/strong><\/p>\n<div class=\"card\" style=\"text-align: center;\"><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/test-series\/jee-12-aits?utm_source=surge&amp;utm_medium=Org&amp;utm_campaign=AITS&amp;mrk_coupon_code=AITS30&amp;tracking_source=il_surge\"><br \/>\n<img loading=\"lazy\" class=\" lazyloaded\" style=\"width: 100%; border-radius: 10px;\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/12\/aitsclass12.png\" alt=\"class 12 jee test series\" width=\"1201\" height=\"636\" data-src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/12\/aitsclass12.png\" \/><\/a><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/test-series\/jee-12-aits?utm_source=surge&amp;utm_medium=Org&amp;utm_campaign=AITS&amp;mrk_coupon_code=AITS30&amp;tracking_source=il_surge\"><button><strong>Get 30 Off! Use Code: AITS30<\/strong><\/button><\/a><\/p>\n<div style=\"text-align: center;\">Boost Your Preparation With Our All India Test Series for Class 12 JEE Main &amp; Advanced 2025<\/div>\n<\/div>\n<p>The parallel axis theorem states that the moment of inertia can be easily calculated.<\/p>\n<p style=\"text-align: center;\"><strong>I = I<sub>cm<\/sub> + Ad<sup>2<\/sup><\/strong><\/p>\n<p style=\"text-align: center;\">Icm = centre of mass<\/p>\n<p>Even so, in this lesson, we will replace mass (M) with the area (A). In addition to integration, we will use a rectangle as a reference to find the M.O.I. during the derivation.<\/p>\n<p>Remember that the moment of inertia of a rectangle is given as;<\/p>\n<p>I<sub>X<\/sub> = \u2153 WH<sup>3<\/sup><\/p>\n<p style=\"text-align: center;\">W = width and H = height<\/p>\n<p style=\"text-align: center;\">I<sub>X<\/sub> = \u2153 (WH)H<sup>2<\/sup><\/p>\n<p style=\"text-align: center;\">I<sub>X<\/sub> = \u2153 (A)H<sup>2<\/sup><\/p>\n<p>(1) When we look at the square with its centre of mass passing through the x-axis, we see that it is made up of two equal-sized rectangles.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156769 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/2-13.png\" alt=\"Moment of Inertia of a Square Derivation\" width=\"660\" height=\"412\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/2-13.png?v=1648305839 660w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/2-13-300x187.png?v=1648305839 300w\" sizes=\"(max-width: 660px) 100vw, 660px\" \/><\/p>\n<p>Now, we can express it as;<\/p>\n<p style=\"text-align: center;\"><strong>I<sub>x<\/sub> = 2 [\u2153 a (a \/ 2)<sup>3<\/sup> ]<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>I<sub>x<\/sub> = [\u2154 a ( a<sup>3<\/sup> \/ 8) ]<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>I<sub>x<\/sub> = (1\/12)a<sup>4<\/sup><\/strong><\/p>\n<p style=\"text-align: center;\"><strong>I<sub>Xcm<\/sub> = a<sup>4<\/sup> \/ 12<\/strong><\/p>\n<p>(2) The following derivation is for a square when the centre of mass is moved a certain distance (d).<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156770 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/3-6.png\" alt=\"Moment of Inertia of a Square Derivation using parallel axis theorem\" width=\"542\" height=\"417\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/3-6.png?v=1648305864 542w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/3-6-300x231.png?v=1648305864 300w\" sizes=\"(max-width: 542px) 100vw, 542px\" \/><\/p>\n<p>By using the parallel axis theorem we can now state;<\/p>\n<p style=\"text-align: center;\">I<sub>x<\/sub> = I<sub>cm<\/sub> + Ad<sup>2<\/sup><\/p>\n<p style=\"text-align: center;\">I<sub>x<\/sub> = (1\/12) a<sup>4<\/sup> + a<sup>2<\/sup> (a \/ 2)<sup>2<\/sup><\/p>\n<p style=\"text-align: center;\">I<sub>x<\/sub> = (1\/12) a<sup>4<\/sup> + (1 \/ 4) a<sup>4<\/sup><\/p>\n<p style=\"text-align: center;\">I<sub>x<\/sub> = (1\/12) a<sup>4<\/sup> + (3 \/ 12) a<sup>4<\/sup><\/p>\n<p style=\"text-align: center;\">I<sub>x<\/sub> = (\u2153) a<sup>4<\/sup><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Moment_of_Inertia_of_a_Square_Plate\"><\/span>Moment of Inertia of a Square Plate<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156772 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/4-4.png\" alt=\"Moment of Inertia of a Square Plate\" width=\"742\" height=\"386\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/4-4.png?v=1648305921 742w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/4-4-300x156.png?v=1648305921 300w\" sizes=\"(max-width: 742px) 100vw, 742px\" \/><\/p>\n<p>A few factors must be considered when calculating the moment of inertia of a square plate.<\/p>\n<p>To begin, we will assume that the plate has mass (M) and length sides (L).<\/p>\n<p>That is, Surface area A = L X L = L<sup>2<\/sup><\/p>\n<p>Now, we will define the mass per unit area as;<\/p>\n<p>That is, Surface density, \u03c1 = M \/ A = M \/ L<sup>2<\/sup><\/p>\n<p>By using integration;<\/p>\n<p>I<sub>plate<\/sub> = \u222b dI = \u222b (dI<sub>com<\/sub> + dI<sub>parallel axis<\/sub>)<\/p>\n<p>I<sub>plate<\/sub> = <sub>x=-L\/2<\/sub>\u222b<sup>x=L\/2<\/sup> (1\/12) \u03c1 L<sup>3<\/sup>dx + \u03c1 Lx<sup>2<\/sup>dx<\/p>\n<p>I<sub>plate<\/sub> = \u03c1 (L<sup>3<\/sup> \/ 12) [x |<sub>-L\/2<\/sub><sup>L\/2<\/sup> + \u03c1 L [ \u2153 x<sup>3<\/sup> |<sub>-L\/2<\/sub><sup>L\/2<\/sup><\/p>\n<p>I<sub>plate<\/sub> = \u03c1 (L<sup>3<\/sup> \/ 12) [ L \/ 2 \u2013 (-L \/ 2)] + \u03c1 L [(\u2153 L<sup>3<\/sup> \/ 8) \u2013 (- \u2153 L<sup>3<\/sup> \/ 8)]\n<p>I<sub>plate<\/sub> = \u03c1 (L<sup>3<\/sup> \/ 12) (L) + \u03c1 L (\u2154 L<sup>3<\/sup> \/ 8)<\/p>\n<p>I<sub>plate<\/sub> = (\u03c1 \/ 12) L<sup>4<\/sup> + (\u03c1 \/ 12) L<sup>4<\/sup><\/p>\n<p>I<sub>plate<\/sub> = (1 \/ 6) \u03c1 L<sup>4<\/sup><\/p>\n<p>I<sub>plate<\/sub> = (1 \/ 6) (M \/ L<sup>2<\/sup>) L<sup>4<\/sup><\/p>\n<p><strong>I<sub>plate<\/sub> = (1 \/ 6) M L<sup>2<\/sup><\/strong><\/p>\n<div class=\"card\" style=\"text-align: center;\"><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/test-series\/jee-13-aits?utm_source=surge&amp;utm_medium=Org&amp;utm_campaign=AITS&amp;mrk_coupon_code=AITS30&amp;tracking_source=il_surge\"><br \/>\n<img loading=\"lazy\" class=\" lazyloaded\" style=\"width: 100%; border-radius: 10px;\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/12\/aitsclass12.png\" alt=\"class 12 jee test series\" width=\"1201\" height=\"636\" data-src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2024\/12\/aitsclass12.png\" \/><\/a><a class=\"btn btn-primary\" href=\"https:\/\/infinitylearn.com\/test-series\/jee-12-aits?utm_source=surge&amp;utm_medium=Org&amp;utm_campaign=AITS&amp;mrk_coupon_code=AITS30&amp;tracking_source=il_surge\"><button><strong>Get 30 Off! Use Code: AITS30<\/strong><\/button><\/a><\/p>\n<div style=\"text-align: center;\">Boost Your Preparation With Our All India Test Series for Class 13 JEE Main &amp; Advanced 2025<\/div>\n<\/div>\n<h3><span class=\"ez-toc-section\" id=\"FAQs\"><\/span><strong>FAQs<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h5><span class=\"ez-toc-section\" id=\"What_is_the_use_of_moment_of_inertia\"><\/span>What is the use of moment of inertia?<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe MOI of an entity helps determine how much torque is required to achieve a given angular acceleration. Once calculating torque or rotational force, the mass MOI must be known.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h5><span class=\"ez-toc-section\" id=\"How_do_we_find_the_Area_of_a_Hollow_Square\"><\/span>How do we find the Area of a Hollow Square?<span class=\"ez-toc-section-end\"><\/span><\/h5>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tOnly when P.x is the first moment of area of a specific section, then (Px). X denotes the section's Moment of Inertia (second moment of area). The hollow section's moment of inertia can be calculated by first determining the inertia of a larger rectangle and then subtracting the hollow section from that large rectangle.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the use of moment of inertia?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The MOI of an entity helps determine how much torque is required to achieve a given angular acceleration. Once calculating torque or rotational force, the mass MOI must be known.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How do we find the Area of a Hollow Square?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Only when P.x is the first moment of area of a specific section, then (Px). X denotes the section's Moment of Inertia (second moment of area). The hollow section's moment of inertia can be calculated by first determining the inertia of a larger rectangle and then subtracting the hollow section from that large rectangle.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>The formula for the moment of inertia of a square is: I = a4 \/ 12 In this case, a [&hellip;]<\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Moment of Inertia of a Square","_yoast_wpseo_title":"(Moment of Inertia of a Square)","_yoast_wpseo_metadesc":"The formula for the (moment of inertia of a square) is: I\u00a0= a4\u00a0\/ 12 In this case, a = the square section's sides.","custom_permalink":"blog\/iit-jee\/moment-of-inertia-of-a-square\/"},"categories":[53,57,4],"tags":[7767,7766],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the 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