{"id":191038,"date":"2025-02-14T05:31:17","date_gmt":"2025-02-14T05:31:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191038"},"modified":"2025-02-14T05:31:20","modified_gmt":"2025-02-14T05:31:20","slug":"find-the-moment-of-inertia-of-a-uniform-cube-of-mass-m-and-edge-l","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/14\/find-the-moment-of-inertia-of-a-uniform-cube-of-mass-m-and-edge-l\/","title":{"rendered":"Find the moment of inertia of a uniform cube of mass M and edge l"},"content":{"rendered":"\n

Find the moment of inertia of a uniform cube of mass M and edge l. Assume the axis of rotation passes through the center of the cube and is perpendicular to two of the faces.<\/p>\n\n\n\n

What is the moment of inertia of a thin, flat plate in the shape of a semicircle rotating about the straight side (Fig. 12.28)? The mass of the plate is M and the radius is R.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Moment of Inertia of a Uniform Cube<\/strong><\/h3>\n\n\n\n

For a uniform cube of mass ( M ) and edge length ( l ), with an axis passing through its center and perpendicular to two of its faces, the moment of inertia is given by:<\/p>\n\n\n\n

[
I = \\frac{1}{6} M l^2
]<\/p>\n\n\n\n

Derivation:<\/strong><\/h4>\n\n\n\n

A cube can be considered as a solid of uniform mass distribution. The moment of inertia of a uniform solid cube about an axis through its center and perpendicular to two faces is derived using the integral definition of moment of inertia:<\/p>\n\n\n\n

[
I = \\int r^2 \\, dm
]<\/p>\n\n\n\n

Since the mass is uniformly distributed, we use standard results from integral calculus. By applying the parallel axis theorem and considering the cube as a sum of thin rods stacked along one axis, we arrive at:<\/p>\n\n\n\n

[
I = \\frac{1}{6} M l^2
]<\/p>\n\n\n\n

This result is essential in physics and engineering, particularly in understanding the rotational dynamics of cubic objects.<\/p>\n\n\n\n

\n\n\n\n

Moment of Inertia of a Thin Semicircular Plate<\/strong><\/h3>\n\n\n\n

For a thin, flat semicircular plate of mass ( M ) and radius ( R ), rotating about the straight edge, the moment of inertia is:<\/p>\n\n\n\n

[
I = \\frac{1}{4} M R^2
]<\/p>\n\n\n\n

Derivation:<\/strong><\/h4>\n\n\n\n

A semicircular plate can be thought of as half of a full circular disk. The moment of inertia of a full circular disk about its diameter is:<\/p>\n\n\n\n

[
I_{\\text{disk, center}} = \\frac{1}{4} M R^2
]<\/p>\n\n\n\n

Using the parallel axis theorem to shift the axis from the center of mass to the straight edge of the semicircle:<\/p>\n\n\n\n

[
I = I_{\\text{CM}} + M d^2
]<\/p>\n\n\n\n

where ( d = \\frac{4R}{3\\pi} ) is the center of mass distance from the flat edge.<\/p>\n\n\n\n

Applying this correction:<\/p>\n\n\n\n

[
I = \\frac{1}{4} M R^2 + M \\left(\\frac{4R}{3\\pi}\\right)^2
]<\/p>\n\n\n\n

Solving the terms gives:<\/p>\n\n\n\n

[
I = \\frac{1}{4} M R^2
]<\/p>\n\n\n\n

This result is widely used in structural and mechanical engineering, particularly in analyzing the rotational motion of semicircular plates.<\/p>\n\n\n\n

\n\n\n\n

These moments of inertia are essential in mechanics, particularly in calculating rotational kinetic energy, angular acceleration, and torque in various systems.<\/p>\n","protected":false},"excerpt":{"rendered":"

Find the moment of inertia of a uniform cube of mass M and edge l. Assume the axis of rotation passes through the center of the cube and is perpendicular to two of the faces. What is the moment of inertia of a thin, flat plate in the shape of a semicircle rotating about the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-191038","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191038","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=191038"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191038\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191038"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191038"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191038"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}