{"id":7760,"date":"2020-06-09T16:53:18","date_gmt":"2020-06-09T20:53:18","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me274\/?page_id=7760"},"modified":"2024-10-05T18:09:21","modified_gmt":"2024-10-05T22:09:21","slug":"polar-description","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/polar-description\/","title":{"rendered":"Polar description"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-7763\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-300x114.jpg\" alt=\"\" width=\"537\" height=\"204\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-300x114.jpg 300w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-1024x388.jpg 1024w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-768x291.jpg 768w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-1536x583.jpg 1536w, https:\/\/www.purdue.edu\/freeform\/me274\/wp-content\/uploads\/sites\/15\/2020\/05\/figure07-2048x777.jpg 2048w\" sizes=\"auto, (max-width: 537px) 100vw, 537px\" \/><\/p>\n<p>Particle P travels on an elliptical path, as shown above. The equation for an ellipse in polar form is provided above, with the origin of the polar coordinates at point O.<\/p>\n<p>Recall that the radial unit vector <em><strong>e<\/strong><\/em><sub>r\u00a0<\/sub>points OUTWARD from O to P, and the transverse unit vector <em><strong>e<\/strong><sub>theta<\/sub><\/em>\u00a0is perpendicular to <em><strong>e<\/strong><\/em><sub>r<\/sub> and pointing in the positive <em>theta<\/em> direction. This is shown in the figure above.<\/p>\n<p>The animation below shows the results of a calculation of the velocity and acceleration of P using the above polar description of the path corresponding to a constant <em>theta_dot<\/em>. Try this out on your own.<\/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\/2020\/05\/section_01A_02.gif\" width=\"621\" height=\"313\" \/><\/p>\n<p>As you watch the results above, focus on the orientation of the polar unit vectors with respect to the velocity and acceleration vectors. From this:<\/p>\n<ul>\n<li>Can you tell when <em>r_dot<\/em> is positive? Just envision the projection of \u00a0the velocity vector onto <em><strong>e<\/strong><\/em><sub>r<\/sub>.<\/li>\n<li>Can you tell when <em>r_dotdot<\/em> is positive? This is more difficult &#8211; why?<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Particle P travels on an elliptical path, as shown above. The equation for an ellipse in polar form is provided above, with the origin of the polar coordinates at point O. Recall that the radial unit vector er\u00a0points OUTWARD from O to P, and the transverse unit vector etheta\u00a0is perpendicular to er and pointing in &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me274\/course-material\/animations\/polar-description\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Polar description<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":14,"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-7760","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/7760","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\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/comments?post=7760"}],"version-history":[{"count":4,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/7760\/revisions"}],"predecessor-version":[{"id":7767,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/7760\/revisions\/7767"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/pages\/14"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me274\/wp-json\/wp\/v2\/media?parent=7760"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}