{"id":2988,"date":"2018-06-03T02:21:22","date_gmt":"2018-06-02T18:21:22","guid":{"rendered":"https:\/\/gurumuda.net\/physics\/?p=2988"},"modified":"2023-08-06T14:21:36","modified_gmt":"2023-08-06T14:21:36","slug":"wheels-connected-by-belts-problems-and-solutions","status":"publish","type":"post","link":"https:\/\/gurumuda.net\/physics\/wheels-connected-by-belts-problems-and-solutions.htm","title":{"rendered":"Wheels connected by belts – problems and solutions","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

Wheels connected by belts – problems and solutions<\/p>\n

1. Three wheels are connected as show<\/span>n in the figure <\/span>below.<\/span><\/span><\/span><\/p>\n

If RA<\/sub> = 10 cm, RB <\/sub>= 4 cm, and RC<\/sub> = 40 cm, then the <\/span>ratio <\/span>of the angular velocity<\/a> of wheel A and wheel C is …<\/span><\/span><\/span><\/p>\n

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

Radius of wheel <\/span><\/span>A (r<\/span><\/span>A<\/span><\/sub>) = 10 cm<\/span><\/span><\/p>\n

Radius of wheel <\/span><\/span>B (r<\/span><\/span>B<\/span><\/sub>) = 4 cm<\/span><\/span><\/p>\n

Radius of wheel <\/span><\/span>C (r<\/span><\/span>C<\/span><\/sub>) = 40 cm<\/span><\/span><\/p>\n

Wanted:<\/u><\/span><\/span> the ratio of the angular velocity of wheel A and wheel C<\/span><\/span><\/span><\/p>\n

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

The angular velocity of wheel A and C <\/b><\/span><\/span><\/p>\n

The circumference of wheel A is much larger than the circumference of wheel C. When the C wheel has been circularly rotated one circle (360o<\/sup>), during the same time interval the wheel A not yet rotates one circle (360o<\/sup>). Thus, the angular speed of the wheel A is not equal to the angular speed of the wheel C.<\/span><\/span><\/p>\n

However, wheel A and wheel C are interconnected through ropes, so that in the same time interval, the distance<\/a> traveled by the edge of the wheel A is equal to the distance traveled by the edge of the wheel C. Thus the linear speed of the edge of the wheel C (vC<\/sub>) equal to the linear speed<\/a> of the edge of the wheel A (vA<\/sub>).<\/span><\/span><\/p>\n

vA <\/sub>= vC <\/sub><\/span><\/span><\/p>\n

rA<\/sub> \u03c9A<\/sub> = rC <\/sub>\u03c9C<\/sub> <\/span><\/span><\/p>\n

10 \u03c9A<\/sub> = 40 \u03c9C<\/sub> <\/span><\/span><\/p>\n

\u03c9A<\/sub> \/ \u03c9C <\/sub>= 40 \/ 10 <\/span><\/span><\/p>\n

\u03c9A <\/sub>\/ \u03c9C<\/sub> = 4 \/ 1<\/span><\/span><\/p>\n

<\/p>\n

2. Wheels B and C have the same axis of rotation and wheel A is tangent to wheel B. If radius <\/span>of wheel <\/span>A = radius <\/span>of wheel <\/span>C = 30 cm, the radius <\/span>of wheel <\/span>B = 60 cm, <\/span>then determine the ratio of the <\/span>linear speed between wheel A, B, and <\/span>C.<\/span><\/span><\/span><\/p>\n

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

Radius of wheel A (rA<\/sub>) = 30 cm = 0.3 meters\"Wheels<\/span><\/span><\/p>\n

Radius of wheel <\/span><\/span>B (r<\/span><\/span>B<\/span><\/sub>) = 60 cm = 0.6 meter<\/span><\/span>s<\/span><\/span><\/p>\n

Radius of wheel <\/span><\/span>C (r<\/span><\/span>C<\/span><\/sub>) = 30 cm = 0.3 meter<\/span><\/span>s<\/span><\/span><\/p>\n

Wanted :<\/u><\/span><\/span> ratio of the linear speed between wheel A, B , and C.<\/span><\/span><\/span><\/p>\n

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

The linear speed of the edge of the whee<\/b>l A <\/b>:<\/b><\/span><\/span><\/p>\n

Wheel A and wheel B are interconnected as shown in figure below, therefore the angular velocity of the wheel A is not equal to the angular velocity of the wheel B. This is because the circumference of wheel B is larger than wheel A. During the same time interval, when wheel A around one circle (360o<\/sup>)<\/a>, wheel B not yet around one circle (360o<\/sup>). However, during the same time interval, the distance traveled by the edge of wheel A is equal to the distance traveled by the edge of wheel B. Thus the linear velocity of the edge of the wheel A (vA<\/sub>) is equal to the linear velocity of the edge of the wheel B (vB<\/sub>).<\/span><\/span><\/p>\n

The linear speed of the edge of wheel A :<\/span><\/span><\/p>\n

v<\/span><\/span>A<\/span><\/sub> = r<\/span><\/span>A<\/span><\/sub> \u03c9<\/span><\/span>A<\/span><\/sub> = 0.3 \u03c9<\/span><\/span>A<\/span><\/span><\/sub><\/p>\n

T<\/strong>he linear speed of the edge of the whee<\/b>l B <\/b>:<\/b><\/span><\/span><\/p>\n

Wheel B and wheel B stick together, therefore, wheel B and wheel C rotate together. When wheel B around one circle (360o<\/sup>) than during the same time interval, wheel C also around one circle (360o<\/sup>). Since it rotates together, then the angular speed of wheel B (\u03c9B<\/sub>) is equal to the angular speed of wheel C (\u03c9C<\/sub>) = \u03c9. But the linear speed of wheel B (vB) is not equal to the linear speed of wheel C (vC<\/sub>)<\/span><\/span><\/p>\n

The linear speed of the edge of wheel B :<\/span><\/span><\/p>\n

v<\/span><\/span>B<\/span><\/sub> = r<\/span><\/span>B<\/span><\/sub> \u03c9<\/span><\/span>B <\/span><\/sub>= <\/span><\/span>0.6 \u03c9<\/span><\/span>B<\/span><\/sub> = <\/span><\/span>0.6 \u03c9<\/span><\/span><\/p>\n

The linear speed of the edge of wheel C :<\/span><\/span><\/p>\n

v<\/span><\/span>C<\/span><\/sub> = r<\/span><\/span>C<\/span><\/sub> \u03c9<\/span><\/span>C <\/span><\/sub>= <\/span><\/span>0.3 \u03c9<\/span><\/span>C<\/span> <\/sub>= <\/span><\/span>0.3 \u03c9<\/span><\/span><\/p>\n

The linear speed of the edge of wheel A <\/b><\/span><\/span>(v<\/b><\/span><\/span>A<\/b><\/span><\/sub>) <\/b><\/span><\/span>same as the linear speed of the edge of wh<\/b><\/span><\/span>e<\/b><\/span><\/span>el B <\/b><\/span><\/span>(v<\/b><\/span><\/span>B<\/b><\/span><\/sub>)<\/b><\/span><\/span><\/p>\n

v<\/span><\/span>A<\/span><\/sub> = v<\/span><\/span>B<\/span><\/sub><\/p>\n

0.3 \u03c9<\/span><\/span>A<\/span><\/sub> = <\/span><\/span>0.6 \u03c9<\/span><\/span><\/p>\n

\u03c9<\/span><\/span>A<\/span><\/sub> = 0.6 \u03c9 \/ 0.3<\/span><\/span><\/p>\n

\u03c9<\/span><\/span>A<\/span><\/sub> = 2 \u03c9<\/span><\/span><\/p>\n

The linear speed of the edge of wheel A <\/b><\/span><\/span>(v<\/b><\/span><\/span>A<\/b><\/span><\/sub>) :<\/b><\/span><\/span><\/p>\n

v<\/span><\/span>A<\/span><\/sub> = 0.3 \u03c9<\/span><\/span>A <\/span><\/sub>= 0.3 (<\/span><\/span>2 \u03c9) = 0.6 \u03c9 <\/span><\/span><\/p>\n

The ratio of the linear speed between wheel A, B, and C.<\/b><\/span><\/span><\/span><\/p>\n

vA: vB: v<\/span><\/span>C<\/span><\/sub><\/p>\n

0.6 \u03c9 : 0.6 \u03c9 : 0.3 \u03c9<\/span><\/span><\/p>\n

0.6 : 0.6 : 0.3<\/span><\/span><\/p>\n

6: 6 : 3<\/span><\/span><\/p>\n

2: 2: 1<\/span><\/span><\/p>\n

<\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"excerpt":{"rendered":"

Wheels connected by belts – problems and solutions 1. Three wheels are connected as shown in the figure below. If RA = 10 cm, RB = 4 cm, and RC = 40 cm, then the ratio of the angular velocity of wheel A and wheel C is … Known : Radius of wheel A (rA) … Read more<\/a><\/p>\n","protected":false,"gt_translate_keys":[{"key":"rendered","format":"html"}]},"author":1,"featured_media":0,"comment_status":"closed","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":"Wheels connected by belts - 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-2988","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\/2988","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=2988"}],"version-history":[{"count":1,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2988\/revisions"}],"predecessor-version":[{"id":8531,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/2988\/revisions\/8531"}],"wp:attachment":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/media?parent=2988"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/categories?post=2988"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/tags?post=2988"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}