{"id":593,"date":"2018-02-12T17:30:41","date_gmt":"2018-02-12T09:30:41","guid":{"rendered":"https:\/\/gurumuda.net\/physics\/?p=593"},"modified":"2018-02-12T17:30:41","modified_gmt":"2018-02-12T09:30:41","slug":"area-expansion-problems-and-solutions","status":"publish","type":"post","link":"https:\/\/gurumuda.net\/physics\/area-expansion-problems-and-solutions.htm","title":{"rendered":"Area expansion \u2013 problems and solutions","gt_translate_keys":[{"key":"rendered","format":"text"}]},"content":{"rendered":"

1. At 20 <\/span><\/span>o<\/span><\/span><\/sup>C, the length of a sheet of steel is 50 cm and the width is 30 cm. If the coefficient of linear expansion<\/a> for steel is 10<\/span><\/span>-5<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup>, determine the change in area and the final area at 60 <\/span><\/span>o<\/span><\/span><\/sup>C.<\/span><\/span><\/p>\n

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

The initial temperature<\/a> (T<\/span><\/span>1<\/span><\/sub>) = 20<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

The final temperature (T<\/span><\/span>2<\/span><\/sub>) = 60<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

The change in temperature (<\/span><\/span>\u0394<\/span><\/span>T) = 60<\/span><\/span>o<\/span><\/span><\/sup>C \u2013 20<\/span><\/span>o<\/span><\/span><\/sup>C = 40<\/span><\/span>o<\/span><\/span><\/sup>C <\/span><\/span><\/p>\n

The initial area (A<\/span><\/span>1<\/span><\/sub>) = length x width = 50 cm x 30 cm = 1500 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The coefficient of linear expansion for steel (<\/span><\/span>\u03b1) = <\/span><\/span>10<\/span><\/span>-5<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

The coefficient of area expansion<\/a> for steel (<\/span><\/span>\u03b2) = 2\u03b1 = <\/span><\/span>2 x 10<\/span><\/span>-5<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

Wanted :<\/u><\/span><\/span> The change in area (<\/span><\/span>\u0394A<\/span><\/span>)<\/span><\/span><\/p>\n

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

The change in area (<\/span><\/span>\u0394A<\/span><\/span>) :<\/span><\/span><\/p>\n

\u0394A = \u03b2 <\/span><\/span>A<\/span><\/span>1<\/sub> <\/span><\/span>\u0394T<\/span><\/span><\/p>\n

\u0394A = (<\/span><\/span>2 x 10<\/span><\/span>-5<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup>)(1500 cm<\/span><\/span>2<\/span><\/span><\/sup>)(40<\/span><\/span>o<\/span><\/span><\/sup>C) <\/span><\/span><\/p>\n

\u0394A = (80 <\/span><\/span>x 10<\/span><\/span>-5<\/span><\/span><\/sup>)(1500 cm<\/span><\/span>2<\/span><\/span><\/sup>) <\/span><\/span><\/p>\n

\u0394A = 1<\/span><\/span>20,000 x 10<\/span><\/span>-5 <\/span><\/span><\/sup>cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

\u0394A = 1.<\/span><\/span>2 x 10<\/span><\/span>5<\/span><\/span><\/sup> x 10<\/span><\/span>-5 <\/span><\/span><\/sup>cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

\u0394A = 1.<\/span><\/span>2 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The final area (<\/span><\/span>A<\/span><\/span>2<\/span><\/sub>) :<\/span><\/span><\/p>\n

A<\/span><\/span>2 <\/span><\/sub>= A<\/span><\/span>1 <\/span><\/sub>+ <\/span><\/span>\u0394A<\/span><\/span><\/p>\n

A<\/span><\/span>2 <\/span><\/sub>= 1500 cm<\/span><\/span>2<\/span><\/span><\/sup> + <\/span><\/span>1.<\/span><\/span>2 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

A<\/span><\/span>2<\/sub> <\/span><\/span>= 1501<\/span><\/span>.<\/span><\/span>2 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

<\/p>\n

2. At 30 <\/span><\/span>o<\/span><\/span><\/sup>C, the area of a sheet of aluminum is 40 cm<\/span><\/span>2 <\/span><\/span><\/sup>and the coefficient of linear expansion is 24 x 10<\/span><\/span>-6<\/span><\/span><\/sup> \/<\/span><\/span>o<\/span><\/span><\/sup>C. Determine the final temperature if the final area is 40.2 cm<\/span><\/span>2<\/span><\/span><\/sup>. <\/span><\/span><\/p>\n

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

The initial temperature (T<\/span><\/span>1<\/span><\/sub>) = 30<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

The coefficient of linear expansion (<\/span><\/span>\u03b1) = <\/span><\/span>24 x 10<\/span><\/span>-6<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

The coefficient of area expansion (<\/span><\/span>\u03b2) = 2a = <\/span><\/span>2 x 24 x 10<\/span><\/span>-6<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup> = 48 x <\/span><\/span>10<\/span><\/span>-6<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

The initial area (A<\/span><\/span>1<\/span><\/sub>) = 40 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The final area (A<\/span><\/span>2<\/span><\/sub>) = 40.2 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

The change in area (<\/span><\/span>\u0394A) = <\/span><\/span>40.2 cm<\/span><\/span>2 <\/span><\/span><\/sup>\u2013 40 cm<\/span><\/span>2<\/span><\/span><\/sup> = 0.2 cm<\/span><\/span>2<\/span><\/span><\/sup><\/p>\n

Wanted : <\/u><\/span><\/span> Determine the final temperature (T<\/span><\/span>2<\/span><\/sub>)<\/span><\/span><\/p>\n

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

Formula of the change in area (<\/span><\/span>\u0394A) <\/span><\/span>:<\/span><\/span><\/p>\n

\u0394A = \u03b2 <\/span><\/span>A<\/span><\/span>1 <\/span><\/sub>\u0394T <\/span><\/span><\/p>\n

The final temperature (T<\/span><\/span>2<\/span><\/sub>) :<\/span><\/span><\/p>\n

\u0394A <\/span><\/span>=\u00a0<\/span><\/span>\u03b2 <\/span><\/span>A<\/span><\/span>1 <\/span><\/sub>(T<\/span><\/span>2<\/span><\/sub> – T<\/span><\/span>1<\/span><\/sub>)<\/span><\/span><\/p>\n

0.2 cm<\/span><\/span>2<\/span><\/span><\/sup> = (48 x <\/span><\/span>10<\/span><\/span>-6<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup>)(40 cm<\/span><\/span>2<\/span><\/span><\/sup>)<\/span><\/span>(T<\/span><\/span>2<\/span><\/sub> – <\/span><\/span>30<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span>)<\/span><\/span><\/p>\n

0.2<\/span><\/span> = (1920 x <\/span><\/span>10<\/span><\/span>-6<\/span><\/span><\/sup>)<\/span><\/span>(T<\/span><\/span>2<\/span><\/sub> – <\/span><\/span>30<\/span><\/span>)<\/span><\/span><\/p>\n

0.2<\/span><\/span> = (1.920 x <\/span><\/span>10<\/span><\/span>-3<\/span><\/span><\/sup>)<\/span><\/span>(T<\/span><\/span>2<\/span><\/sub> – 30)<\/span><\/span><\/p>\n

0.2<\/span><\/span> = (2 x <\/span><\/span>10<\/span><\/span>-3<\/span><\/span><\/sup>)<\/span><\/span>(T<\/span><\/span>2<\/span><\/sub> \u2013 30)<\/span><\/span><\/p>\n

0.2 \/ (2<\/span><\/span> x 10<\/span><\/span>-3<\/span><\/span><\/sup>) = <\/span><\/span>T<\/span><\/span>2<\/span><\/sub> \u2013 30<\/span><\/span><\/p>\n

0.1<\/span><\/span> x 10<\/span><\/span>3<\/span><\/span><\/sup> = <\/span><\/span>T<\/span><\/span>2<\/span><\/sub> \u2013 30<\/span><\/span><\/p>\n

1 x 10<\/span><\/span>2<\/span><\/span><\/sup> = T<\/span><\/span>2<\/span><\/sub> \u2013 30<\/span><\/span><\/p>\n

100 = T<\/span><\/span>2 <\/span><\/sub>\u2013 30<\/span><\/span><\/p>\n

100 + 30 = T<\/span><\/span>2<\/span><\/sub><\/p>\n

T<\/span><\/span>2<\/span><\/sub> = 130<\/span><\/span><\/p>\n

The final temperature = 130<\/span><\/span>o<\/span><\/span><\/sup>C <\/span><\/span><\/p>\n

<\/p>\n

3. The radius of a ring at 20 <\/span><\/span>o<\/span><\/span><\/sup>C is 20 cm. If the final radius at 100 <\/span><\/span>o<\/span><\/span><\/sup>C is 20.5 cm, determine the coefficient of area expansion and the coefficient of linear expansion…<\/span><\/span><\/p>\n

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

The initial temperature (T<\/span><\/span>1<\/span><\/sub>) = 30<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

The final temperature (T<\/span><\/span>2<\/span><\/sub>) = 100<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

The change in temperature (<\/span><\/span>\u0394T) = 100<\/span><\/span>o<\/span><\/span><\/sup>C \u2013 30<\/span><\/span>o<\/span><\/span><\/sup>C = 70<\/span><\/span>o<\/span><\/span><\/sup>C <\/span><\/span><\/p>\n

The initial radius (r<\/span><\/span>1<\/span><\/sub>) = 20 cm<\/span><\/span><\/p>\n

The final radius (r<\/span><\/span>2<\/span><\/sub>) = 20.5 cm<\/span><\/span><\/p>\n

Wanted :<\/u><\/span><\/span> The coefficient of area expansion (<\/span><\/span>\u03b2)<\/span><\/span><\/p>\n

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

The initial area (A<\/span><\/span>1<\/span><\/sub>) =\u00a0<\/span><\/span>\u03c0 r<\/span><\/span>1<\/span><\/sub>2<\/span><\/span><\/sup> = (3.14)(20 cm)<\/span><\/span>2<\/span><\/span><\/sup> = (3.14)(400 cm<\/span><\/span>2<\/span><\/span><\/sup>) = 1256 cm<\/span><\/span>2 <\/span><\/span><\/sup><\/p>\n

The final area (A<\/span><\/span>2<\/span><\/sub>) =\u00a0<\/span><\/span>\u03c0 r<\/span><\/span>2<\/span><\/sub>2<\/span><\/span><\/sup> = (3.14)(20.5 cm)<\/span><\/span>2<\/span><\/span><\/sup> = (3.14)(420.25 cm<\/span><\/span>2<\/span><\/span><\/sup>) = 1319.585 cm<\/span><\/span>2 <\/span><\/span><\/sup><\/p>\n

The change in area (<\/span><\/span>\u0394A) = <\/span><\/span>1319.585 cm<\/span><\/span>2 <\/span><\/span><\/sup>–<\/span><\/span> 1256 cm<\/span><\/span>2<\/span><\/span><\/sup> = 63.585 cm<\/span><\/span>2 <\/span><\/span><\/sup><\/p>\n

Formula of the change in area (<\/span><\/span>\u0394A) <\/span><\/span>:<\/span><\/span><\/p>\n

\u0394A = \u03b2 <\/span><\/span>A<\/span><\/span>1 <\/span><\/sub>\u0394T <\/span><\/span><\/p>\n

The coefficient of area expansion :<\/span><\/span><\/p>\n

\u0394A = \u03b2 <\/span><\/span>A<\/span><\/span>1 <\/span><\/sub>\u0394T <\/span><\/span><\/p>\n

63.585 cm<\/span><\/span>2 <\/span><\/span><\/sup>= <\/span><\/span>b (1<\/span><\/span>256 cm<\/span><\/span>2<\/span><\/span><\/sup>)(70 <\/span><\/span>o<\/span><\/span><\/sup>C)<\/span><\/span><\/p>\n

63.585 = <\/span><\/span>b (<\/span><\/span>87,920 <\/span><\/span>o<\/span><\/span><\/sup>C)<\/span><\/span><\/p>\n

\u03b2 = <\/span><\/span>63.585 \/ 87,920 <\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

\u03b2 = 0.00072 <\/span><\/span>\/<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

\u03b2 = 7.2 <\/span><\/span>x 1<\/span><\/span>0<\/span><\/span>-4<\/span><\/span><\/sup> \/<\/span><\/span>o<\/span><\/span><\/sup>C<\/span><\/span><\/p>\n

\u03b2 = 7.2 <\/span><\/span>x 1<\/span><\/span>0<\/span><\/span>-4<\/span><\/span><\/sup> o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

The coefficient of linear expansion (<\/span><\/span>\u03b1) :<\/span><\/span><\/p>\n

\u03b2 = 2 \u03b1
\n<\/span><\/span><\/p>\n

\u03b1 = \u03b2 \/ 2<\/span><\/span><\/p>\n

\u03b1 = (7.2 <\/span><\/span>x 1<\/span><\/span>0<\/span><\/span>-4<\/span><\/span><\/sup>) \/ 2<\/span><\/span><\/p>\n

\u03b1 = 3.6 <\/span><\/span>x 1<\/span><\/span>0<\/span><\/span>-4 <\/span><\/span><\/sup>o<\/span><\/span><\/sup>C<\/span><\/span>-1<\/span><\/span><\/sup><\/p>\n

[wpdm_package id=’698′]<\/p>\n

    \n
  1. Converting temperature scales<\/a><\/li>\n
  2. Linear expansion<\/a><\/li>\n
  3. Area expansion<\/a><\/li>\n
  4. Volume expansion<\/a><\/li>\n
  5. Heat<\/a><\/li>\n
  6. Mechanical equivalent of heat<\/a><\/li>\n
  7. Specific heat and heat capacity<\/a><\/li>\n
  8. Latent heat, heat of fusion, heat of vaporization<\/a><\/li>\n
  9. Energy conservation for heat transfer<\/a><\/li>\n<\/ol>\n

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

    1. At 20 oC, the length of a sheet of steel is 50 cm and the width is 30 cm. If the coefficient of linear expansion for steel is 10-5 oC-1, determine the change in area and the final area at 60 oC. Known : The initial temperature (T1) = 20oC The final temperature (T2) … 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":"Area expansion \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-593","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\/593","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=593"}],"version-history":[{"count":0,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/posts\/593\/revisions"}],"wp:attachment":[{"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/media?parent=593"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/categories?post=593"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gurumuda.net\/physics\/wp-json\/wp\/v2\/tags?post=593"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}