Calculate the total rotational inertia of a ring sandwiched betwenn two discs. Use a mass of 1.50 kg for all three objects. The two discs have a radius of 0.333 m, and the ring has inner and outer radii of 0.125m and 0.225 m respectively<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n The total rotational inertia of the system comprising a ring sandwiched between two discs is approximately 0.216 kg\u00b7m\u00b2<\/strong>.<\/p>\n\n\n\n Rotational inertia, also known as the moment of inertia, is a measure of an object’s resistance to angular acceleration about a rotational axis. It depends on the mass distribution relative to that axis. In this scenario, we have two discs and one ring, each with a mass of 1.50 kg, and we need to calculate their combined moment of inertia.<\/p>\n\n\n\n In summary, the combined rotational inertia of the system is approximately 0.216 kg\u00b7m\u00b2<\/strong>, reflecting the overall resistance to rotational acceleration based on the masses and geometries of the components.<\/p>\n","protected":false},"excerpt":{"rendered":" Calculate the total rotational inertia of a ring sandwiched betwenn two discs. Use a mass of 1.50 kg for all three objects. The two discs have a radius of 0.333 m, and the ring has inner and outer radii of 0.125m and 0.225 m respectively The Correct Answer and Explanation is: The total rotational inertia […]<\/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-157464","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157464","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=157464"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/157464\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=157464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=157464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=157464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Explanation<\/h3>\n\n\n\n
\n
Each disc is treated as a solid cylinder. The formula for the moment of inertia ( I ) of a disc about its central axis is given by:
[
I_{\\text{disc}} = \\frac{1}{2} m r^2
]
where ( m ) is the mass of the disc and ( r ) is its radius. Substituting the values:
[
I_{\\text{disc}} = \\frac{1}{2} \\times 1.50 \\, \\text{kg} \\times (0.333 \\, \\text{m})^2 \\approx 0.083 \\, \\text{kg\u00b7m}^2
]
Since there are two identical discs, the total moment of inertia from the discs is:
[
I_{\\text{total discs}} = 2 \\times I_{\\text{disc}} \\approx 0.166 \\, \\text{kg\u00b7m}^2
]<\/li>\n\n\n\n
The ring has an inner radius ( R_{\\text{inner}} = 0.125 \\, \\text{m} ) and an outer radius ( R_{\\text{outer}} = 0.225 \\, \\text{m} ). The moment of inertia of a ring is given by:
[
I_{\\text{ring}} = \\frac{1}{2} m (R_{\\text{outer}}^2 + R_{\\text{inner}}^2)
]
Substituting the values:
[
I_{\\text{ring}} = \\frac{1}{2} \\times 1.50 \\, \\text{kg} \\times ((0.225 \\, \\text{m})^2 + (0.125 \\, \\text{m})^2) \\approx 0.050 \\, \\text{kg\u00b7m}^2
]<\/li>\n\n\n\n
Adding the contributions from both the discs and the ring, we find:
[
I_{\\text{total}} = I_{\\text{total discs}} + I_{\\text{ring}} \\approx 0.166 + 0.050 \\approx 0.216 \\, \\text{kg\u00b7m}^2
]<\/li>\n<\/ol>\n\n\n\n