{"id":6250,"date":"2020-10-13T15:25:42","date_gmt":"2020-10-13T09:55:42","guid":{"rendered":"http:\/\/astan.lk\/al_virtualclassroom\/?p=6250"},"modified":"2020-10-13T15:25:52","modified_gmt":"2020-10-13T09:55:52","slug":"rotational-motion","status":"publish","type":"post","link":"https:\/\/astan.lk\/al_virtualclassroom\/rotational-motion\/","title":{"rendered":"Rotational motion"},"content":{"rendered":"<p>A body is said to perform a pure rotational motion if every particle in the body moves in a circular path such that the centers of all those circles lie on a single straight line called as the axis of rotation.<\/p>\n<p><strong>Angular acceleration<\/strong><\/p>\n<p>When an object rotates its angular velocity changes with time.Angular acceleration is defined as rate of change of angular velocity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/encrypted-tbn2.gstatic.com\/images?q=tbn:ANd9GcSEB_zOYE5LoAFfPcVgwrAqKq2za33Rihr5os27H42rnfLg__AvuA\" alt=\"Image result for angular acceleration formula\" width=\"111\" height=\"83\" \/><\/p>\n<p>[\u03b1]=T<sup>-2<\/sup> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Unit &#8211; \u00a0rads<sup>-2<\/sup><\/p>\n<p>The translational acceleration of a point on the object rotating is given by<\/p>\n<h2><strong>a<sub>T<\/sub>=r\u03b1<\/strong><\/h2>\n<p>where <i>r<\/i> is the radius or distance from the axis of rotation. This is also the tangential component of acceleration: it is tangential to the direction of motion of the point.<\/p>\n<p>When the angular acceleration is constant, the five quantities angular displacement <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af\" alt=\"\\theta \" \/><\/span>, initial angular velocity <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e174c191a5ba3889c66597461ef260811cce0481\" alt=\"\\omega _{i}\" \/><\/span>, final angular velocity <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/abf83b9ffeab32fed5a3f8dad3709ffc80e8d0fe\" alt=\"\\omega _{f}\" \/><\/span>, angular acceleration <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b79333175c8b3f0840bfb4ec41b8072c83ea88d3\" alt=\"\\alpha \" \/><\/span>, and time <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/65658b7b223af9e1acc877d848888ecdb4466560\" alt=\"t\" \/><\/span> can be related by four equations of kinematics:<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec6fa3353ef122d037e444b7e6ea5ca6b6b07209\" alt=\"\\omega _{f}=\\omega _{i}+\\alpha t\\;\\!\" \/><\/span><\/dd>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddb6d77a488f47c78f5505d42271757769785c29\" alt=\"\\theta =\\omega _{i}t+{\\begin{matrix}{\\frac {1}{2}}\\end{matrix}}\\alpha t^{2}\" \/><\/span><\/dd>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c05d3bade89a8038748e7d805dc8596ed0716d9f\" alt=\"\\omega _{f}^{2}=\\omega _{i}^{2}+2\\alpha \\theta \" \/><\/span><\/dd>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e1888aafb8ba2e3b4465fae90651b61b8132f7eb\" alt=\"\\theta ={\\tfrac {1}{2}}\\left(\\omega _{f}+\\omega _{i}\\right)t\" \/><\/span><\/dd>\n<\/dl>\n<p><span style=\"text-decoration: underline;\"><strong>Moment of inertia<\/strong><\/span><\/p>\n<p>The moment of inertia of an object\u00a0is a measure of the object&#8217;s resistance to changes to its rotation. The moment of inertia is measured in kg m\u00b2. It depends on the object&#8217;s mass: increasing the mass of an object increases the moment of inertia. It also depends on the distribution of the mass: distributing the mass further from the centre of rotation increases the moment of inertia by a greater degree.<\/p>\n<h3>For a single particle of mass m\u00a0a distance r\u00a0from the axis of rotation, the moment of inertia is given by \u00a0 \u00a0 \u00a0 <strong>I=mr\u00b2<\/strong><\/h3>\n<p><span style=\"text-decoration: underline;\"><strong>Torque<\/strong><\/span><\/p>\n<p>Torque <span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/beb1359149db88a28c86f0a3030894b71610a224\" alt=\"{\\boldsymbol {\\tau }}\" \/><\/span> is the twisting effect of a force <b>F<\/b> applied to a rotating object which is at position <b>r<\/b> from its axis of rotation.<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d78f7463314c4945dd5a3a81a7b138240d5866b6\" alt=\"{\\boldsymbol {\\tau }}=\\mathbf {r} \\times \\mathbf {F} ,\" \/><\/span><\/dd>\n<\/dl>\n<p>A net torque acting upon an object will produce an angular acceleration of the object according to<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/10b7bf2030bd73d3f0748336701432cc33e3f2a5\" alt=\"{\\boldsymbol {\\tau }}=I{\\boldsymbol {\\alpha }},\" \/><\/span><\/dd>\n<\/dl>\n<p>just as <b>F<\/b> = <i>m<\/i><b>a<\/b> in linear dynamics.<\/p>\n<p>The work done by a torque acting on an object equals the magnitude of the torque times the angle through which the torque is applied<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/777e9a864e7a24973765abe75398add8755138ed\" alt=\"W=\\tau \\theta .\\!\" \/><\/span><\/dd>\n<\/dl>\n<p>The power of a torque is equal to the work done by the torque per unit time<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6f300cd9b28a97feb1b471dc93f084357a5f5c4d\" alt=\"P=\\tau \\omega .\\!\" \/><\/span><\/dd>\n<\/dl>\n<p><span style=\"text-decoration: underline;\"><strong>Angular momentum<\/strong><\/span><\/p>\n<p>The angular momentum <b>L<\/b> is a measure of the difficulty of bringing a rotating object to rest. It is given by<\/p>\n<dl>\n<dd><strong>L = mv\u00d7r<\/strong><\/dd>\n<\/dl>\n<p>Angular momentum is related to angular velocity by<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/afb18c91ec4284e8e4142e3b7fea1b3cb54baf40\" alt=\"{\\mathbf {L}}=I{\\boldsymbol {\\omega }},\" \/><\/span><\/dd>\n<dd><span class=\"mwe-math-element\">just as <b>p<\/b> = <i>m<\/i><b>v<\/b> in linear dynamics.<\/span><\/dd>\n<\/dl>\n<p>The greater the angular momentum of the spinning object such as a top, the greater its tendency to continue to spin.<\/p>\n<p>The Angular Momentum of a rotating body is proportional to its mass and to how rapidly it is turning. In addition the angular momentum depends on how the mass is distributed relative to the axis of rotation: the further away the mass is located from the axis of rotation, the greater the angular momentum . A flat disk such as a record turntable has less angular momentum than a hollow cylinder of the same mass and velocity of rotation.<\/p>\n<p>Like linear momentum, angular momentum is vector quantity, and its conservation implies that the direction of the spin axis tends to remain unchanged. For this reason the spinning top remains upright whereas a stationary one falls over immediately.<\/p>\n<p>Torque and angular momentum are related according to<\/p>\n<dl>\n<dd><span class=\"mwe-math-element\"><img decoding=\"async\" class=\"mwe-math-fallback-image-inline\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/122a95cd7f28328ee8ce7861958e6a72e9ff5bb1\" alt=\"{\\boldsymbol {\\tau }}={\\frac {d{\\mathbf {L}}}{dt}},\" \/><\/span><\/dd>\n<\/dl>\n<p>just as <b>F<\/b> = <i>d<\/i><b>p<\/b>\/<i>dt<\/i> in linear dynamics<\/p>\n<p><span style=\"text-decoration: underline;\"><strong>Law of conservation of angular momentum<\/strong><\/span><\/p>\n<p>If there is no external torque acting on a system the angular momentum remains unchanged.<\/p>\n<h3><strong>L=I\u03c9=constant<\/strong><\/h3>\n<p><span style=\"text-decoration: underline;\"><strong>Rotational kinetic energy<\/strong><\/span><\/p>\n<p>When an object rotates about an axis every particle in that body moves in a circle then rotational kinetic energy is the sum of total kinetic energy of particle.<\/p>\n<h3><strong>Rotational kinetic energy =\u00a0\u00bdI\u03c9\u00b2<\/strong><\/h3>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A body is said to perform a pure rotational motion if every particle in the body moves in a circular path such that the centers of all those circles lie on a single straight line called as the axis of rotation. Angular acceleration When an object rotates its angular velocity changes with time.Angular acceleration is [&hellip;]<\/p>\n","protected":false},"author":842,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[16,1948],"tags":[],"class_list":["post-6250","post","type-post","status-publish","format-standard","hentry","category-physics","category-unit-02-en"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Rotational motion - Learning &amp; Education Portal<\/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:\/\/astan.lk\/al_virtualclassroom\/rotational-motion\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Rotational motion - Learning &amp; Education Portal\" \/>\n<meta property=\"og:description\" content=\"A body is said to perform a pure rotational motion if every particle in the body moves in a circular path such that the centers of all those circles lie on a single straight line called as the axis of rotation. 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