{"id":9933,"date":"2021-01-25T22:19:24","date_gmt":"2021-01-26T03:19:24","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?p=9933"},"modified":"2024-10-05T18:06:19","modified_gmt":"2024-10-05T22:06:19","slug":"homework-1-e","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/chapter-1-discussion\/homework-1-e\/","title":{"rendered":"Homework H1.A.17"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-9934 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/01\/1E-292x300.png\" alt=\"\" width=\"292\" height=\"300\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/01\/1E-292x300.png 292w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/01\/1E.png 399w\" sizes=\"auto, (max-width: 292px) 100vw, 292px\" \/><\/p>\n<p>We encourage you to interact with your colleagues here on conversations about this homework problem.<\/p>\n<hr \/>\n<p>With the polar description, we write the velocity and acceleration vectors for a point in terms of the polar unit vectors <em><strong>e<\/strong><\/em><sub>r<\/sub> and <em><strong>e<\/strong><\/em><sub>\u03b8<\/sub>.\u00a0These two unit vectors are shown in <span style=\"color: #ff00ff\">magenta<\/span> in the animated GIF below.<\/p>\n<ul>\n<li><em><strong>e<\/strong><\/em><sub>r<\/sub>\u00a0points <span style=\"text-decoration: underline\"><em>outward<\/em><\/span> from the observer O toward point P<\/li>\n<li><em><strong>e<\/strong><\/em><sub>\u03b8<\/sub> is perpendicular to <em><strong>e<\/strong><\/em><sub>r<\/sub> and points in the positive <em>\u03b8<\/em> direction.<\/li>\n<\/ul>\n<p>The velocity (in <span style=\"color: #0000ff\">blue<\/span>) is seen to be tangent to the path, as expected, The \u00a0acceleration (in <span style=\"color: #ff0000\">red<\/span>) has both tangential and normal components: the normal component always points inward on the path, and the size and sign of the tangential component is directly tied to the rate of change of speed of P, also as expected.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2021\/01\/H1A_17a.gif\" width=\"566\" height=\"320\" \/><\/p>\n<p>In terms of problem solving, this problem is straight forward. Both <em>r<\/em> and <em>\u03b8<\/em> are given as functions of time. The components of the velocity and acceleration vectors are found directly from the differentiation of r and theta with respect to time. The chain rule of differentiation is NOT needed here.<\/p>\n<p>Some additional observations:<\/p>\n<ul>\n<li>Although r_ddot = constant, the r-component of acceleration is NOT constant.<\/li>\n<li>Although <em>\u03b8<\/em>_ddot = 0, the <em>\u03b8<\/em>-component of acceleration is NOT zero.<\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We encourage you to interact with your colleagues here on conversations about this homework problem. With the polar description, we write the velocity and acceleration vectors for a point in terms of the polar unit vectors er and e\u03b8.\u00a0These two unit vectors are shown in magenta in the animated GIF below. er\u00a0points outward from the &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/chapter-1-discussion\/homework-1-e\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Homework H1.A.17<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":3969,"featured_media":0,"parent":14839,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-9933","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/9933","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/users\/3969"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=9933"}],"version-history":[{"count":14,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/9933\/revisions"}],"predecessor-version":[{"id":16097,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/9933\/revisions\/16097"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/14839"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=9933"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}