{"id":1288,"date":"2018-03-01T12:42:47","date_gmt":"2018-03-01T04:42:47","guid":{"rendered":"https:\/\/gurumuda.net\/physics\/?p=1288"},"modified":"2018-03-01T12:42:47","modified_gmt":"2018-03-01T04:42:47","slug":"rotational-kinetic-energy-problems-and-solutions","status":"publish","type":"post","link":"https:\/\/gurumuda.net\/physics\/rotational-kinetic-energy-problems-and-solutions.htm","title":{"rendered":"Rotational kinetic energy \u2013 problems and solutions","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

1. <\/span><\/span>An object has the moment of inertia<\/a> of 1 kg <\/span><\/span>m<\/span><\/span>2 <\/span><\/span><\/sup>rotates at a constant angular speed of <\/span><\/span>2 rad\/s. <\/span><\/span>What is the rotational kinetic energy<\/a> of the object?<\/span><\/span><\/p>\n

Known :<\/span><\/span><\/u><\/p>\n

The moment of inertia <\/span><\/span>(I) = 1 kg <\/span><\/span>m<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The angular velocity <\/a><\/span><\/span>(<\/span><\/span>\u03c9) = <\/span><\/span>2 rad\/s<\/span><\/span><\/p>\n

Wanted:<\/u><\/span><\/span> The rotational kinetic energy <\/span><\/span>(K<\/span><\/span>E<\/span><\/span>)<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/span><\/p>\n

The formula of the rotational kinetic energy :<\/span><\/span><\/p>\n

K<\/span><\/span>E <\/span><\/span>= 1\/2 I <\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

K<\/i><\/span><\/span>E <\/i><\/span><\/span>= <\/i><\/span><\/span>the rotational kinetic energy <\/i><\/span><\/span>(kg m<\/i><\/span><\/span>2<\/i><\/span><\/span><\/sup>\/s<\/i><\/span><\/span>2<\/i><\/span><\/span><\/sup>), I = <\/i><\/span><\/span>the moment of inertia <\/i><\/span><\/span>(kg m<\/i><\/span><\/span>2<\/i><\/span><\/span><\/sup>), <\/i><\/span><\/span>\u03c9 <\/i><\/span><\/span>= <\/i><\/span><\/span>the angular velocity <\/i><\/span><\/span>(rad\/s)<\/i><\/span><\/span><\/p>\n

The rotational kinetic energy :<\/span><\/span><\/p>\n

KE = 1\/2 I <\/span><\/span>\u03c9<\/span><\/span>2<\/span><\/span><\/sup> = 1\/2 (1)(2)<\/span><\/span>2 <\/span><\/span><\/sup>= 1\/2 (1)(4)<\/span><\/span> = 2 Joule <\/span><\/span><\/p>\n

<\/p>\n

2. <\/span><\/span>A 20-kg <\/span><\/span>cylinder pulley <\/span><\/span>with a radius of 0.2 m rotates at a constant angular speed of 4 rad\/s. What is the rotational kinetic energy of the pulley? <\/span><\/span><\/p>\n

\"RotationalKnown :<\/u><\/span><\/span><\/p>\n

Mass<\/span><\/span><\/a> of cylinder pulley <\/span><\/span>(m) <\/span><\/span>= 20 k<\/span><\/span>g<\/span><\/span><\/p>\n

The radius of cylinder <\/span><\/span>(r) = 0.2 m<\/span><\/span><\/p>\n

The angular speed <\/span><\/span>(\u03c9) = 4 ra<\/span><\/span>d\/s<\/span><\/span><\/p>\n

Wanted :<\/u><\/span><\/span> What is the rotational kinetic energy<\/span><\/span><\/p>\n

Solution ;<\/u><\/span><\/span><\/p>\n

Formula of the moment inertia of cylinder :<\/span><\/span><\/p>\n

I <\/span><\/span>= 1\/2 m r<\/span><\/span>2 <\/span><\/span><\/sup><\/p>\n

I = <\/i><\/span><\/span>the moment of inertia (<\/i><\/span><\/span>kg m<\/i><\/span><\/span>2<\/i><\/span><\/span><\/sup>), m = mass (kg), <\/i><\/span><\/span>r = <\/i><\/span><\/span>radius<\/i><\/span><\/span> (meter)<\/i><\/span><\/span><\/p>\n

The moment of inertia of cylinder pulley : <\/span><\/span><\/p>\n

I = 1\/2 (20)(0.2)<\/span><\/span>2<\/span><\/span><\/sup> = (10)(0.04) = 0.4 kg m<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The rotational kinetic energy of the pulley : <\/span><\/span><\/p>\n

KE <\/span><\/span>= 1\/2 I \u03c9<\/span><\/span>2<\/span><\/span><\/sup> = 1\/2 (0.4)(4)<\/span><\/span>2 <\/span><\/span><\/sup>= (0.2)(16)<\/span><\/span> = 3.2 Joule <\/span><\/span><\/p>\n

<\/p>\n

3. <\/span><\/span>A-<\/span><\/span>10 kg <\/span><\/span>ball with radius of <\/span><\/span>0.1 m <\/span><\/span>rotates at a constant of <\/span><\/span>10 rad\/s. <\/span><\/span>What is the kinetic energy of the ball.<\/span><\/span><\/p>\n

Known :
\n<\/u><\/span><\/span><\/p>\n

Mass <\/span><\/span>of ball<\/span><\/span> (m) <\/span><\/span>= 10 <\/span><\/span>kg<\/span><\/span><\/p>\n

Radius of ball<\/span><\/span> (r) = 0.1 m<\/span><\/span><\/p>\n

Angular velocity <\/span><\/span>(\u03c9) = 10 r<\/span><\/span>ad\/s<\/span><\/span><\/p>\n

Wanted :<\/u><\/span><\/span> The rotational kinetic energy<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/span><\/p>\n

Formula of the moment of inertia :<\/span><\/span><\/p>\n

I <\/span><\/span>= (2\/5) m r<\/span><\/span>2 <\/span><\/span><\/sup><\/p>\n

I = <\/i><\/span><\/span>moment of inertia<\/i><\/span><\/span> (kg m<\/i><\/span><\/span>2<\/i><\/span><\/span><\/sup>), m = mass (kg), <\/i><\/span><\/span>r = <\/i><\/span><\/span>radius <\/i><\/span><\/span>(m)<\/i><\/span><\/span><\/p>\n

Moment of inertia of the ball :<\/span><\/span><\/p>\n

I = (2\/5)(10)(0.1)<\/span><\/span>2<\/span><\/span><\/sup> = (4)(0.01) = 0.04 kg m<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The rotational kinetic energy of the ball :<\/span><\/span><\/p>\n

K<\/span><\/span>E<\/span><\/span> = 1\/2 I \u03c9<\/span><\/span>2<\/span><\/span><\/sup> = 1\/2 (0.04)(10)<\/span><\/span>2 <\/span><\/span><\/sup>= (0.02)(100)<\/span><\/span> = 2 Joule <\/span><\/span><\/p>\n

<\/p>\n

4. <\/span><\/span>A <\/span><\/span>0.5-<\/span><\/span>kg particle rotates at a constant angular speed of <\/span><\/span>2 rad\/s. <\/span><\/span>What is the rotational kinetic energy of the particle if the radius of circle is <\/span><\/span>10 cm.<\/span><\/span><\/p>\n

Known :<\/span><\/span><\/u><\/p>\n

Mass <\/span><\/span>of particle <\/span><\/span>(m) <\/span><\/span>= 0.5 k<\/span><\/span>g<\/span><\/span><\/p>\n

The radius of ball <\/span><\/span>(r) = 10 cm = 10\/100 = 0.1 m<\/span><\/span><\/p>\n

The angular speed <\/span><\/span>(\u03c9) = 2 rad\/<\/span><\/span>s<\/span><\/span><\/p>\n

Wanted :<\/u><\/span><\/span> The rotational kinetic energy<\/span><\/span><\/p>\n

Solution :<\/u><\/span><\/span><\/p>\n

Moment of inertia for particle :<\/span><\/span><\/p>\n

I <\/span><\/span>= m r<\/span><\/span>2 <\/span><\/span><\/sup>= (0.5)(0.1)<\/span><\/span>2<\/span><\/span><\/sup> = (0.5)(0.01) = 0.005 kg m<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The rotational kinetic energy :<\/span><\/span><\/p>\n

K<\/span><\/span>E<\/span><\/span> = 1\/2 I \u03c9<\/span><\/span>2<\/span><\/span><\/sup> = 1\/2 (0.005)(2)<\/span><\/span>2 <\/span><\/span><\/sup>= 1\/2 (0.005)(4)<\/span><\/span> = (0.005)(2)<\/span><\/span> = 0.01 Joule <\/span><\/span><\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

1. An object has the moment of inertia of 1 kg m2 rotates at a constant angular speed of 2 rad\/s. What is the rotational kinetic energy of the object? Known : The moment of inertia (I) = 1 kg m2 The angular velocity (\u03c9) = 2 rad\/s Wanted: The rotational kinetic energy (KE) Solution … Read more<\/a><\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_seopress_titles_title":"","_seopress_titles_desc":"","_seopress_robots_index":"","_seopress_robots_follow":"","_seopress_robots_imageindex":"","_seopress_robots_snippet":"","_seopress_robots_primary_cat":"","_seopress_robots_breadcrumbs":"","_seopress_robots_freeze_modified_date":"","_seopress_robots_custom_modified_date":"","_seopress_robots_canonical":"","_seopress_social_fb_title":"","_seopress_social_fb_desc":"","_seopress_social_fb_img":"","_seopress_social_fb_img_attachment_id":0,"_seopress_social_fb_img_width":0,"_seopress_social_fb_img_height":0,"_seopress_social_twitter_title":"","_seopress_social_twitter_desc":"","_seopress_social_twitter_img":"","_seopress_social_twitter_img_attachment_id":0,"_seopress_social_twitter_img_width":0,"_seopress_social_twitter_img_height":0,"_seopress_redirections_value":"","_seopress_redirections_enabled":"","_seopress_redirections_enabled_regex":"","_seopress_redirections_logged_status":"","_seopress_redirections_param":"","_seopress_redirections_type":0,"_seopress_analysis_target_kw":"Rotational kinetic energy \u2013 problems and solutions","_seopress_news_disabled":"","_seopress_video_disabled":"","_seopress_video":[],"_seopress_pro_schemas_manual":[],"_seopress_pro_rich_snippets_disable_all":"","_seopress_pro_rich_snippets_disable":[],"_seopress_pro_schemas":[],"footnotes":""},"categories":[3],"tags":[],"class_list":["post-1288","post","type-post","status-publish","format-standard","hentry","category-solved-problems-in-basic-physics"],"gt_translate_keys":[{"key":"link","format":"url"}],"_links":{"self":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/1288","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/comments?post=1288"}],"version-history":[{"count":0,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/1288\/revisions"}],"wp:attachment":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/media?parent=1288"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/categories?post=1288"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/tags?post=1288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}