{"id":10042,"date":"2015-05-19T10:23:30","date_gmt":"2015-05-19T14:23:30","guid":{"rendered":"http:\/\/scienceblogs.com\/principles\/?p=10042"},"modified":"2015-05-19T10:23:30","modified_gmt":"2015-05-19T14:23:30","slug":"toy-roller-coasters-and-the-energy-principle","status":"publish","type":"post","link":"http:\/\/chadorzel.com\/principles\/2015\/05\/19\/toy-roller-coasters-and-the-energy-principle\/","title":{"rendered":"Toy Roller Coasters and the Energy Principle"},"content":{"rendered":"<p>One of the points I make repeatedly in teaching introductory mechanics (as I&#8217;m doing this term) is that absolutely every problem students encounter can, in principle, be solved using just Newton&#8217;s Laws or, in the terminology used by Matter and Interactions, the Momentum Principle. You don&#8217;t strictly <em>need<\/em> any of the other stuff we talk about, like energy or angular momentum.<\/p>\n<p>Of course, just because you <em>can<\/em> solve any problem using the Momentum Principle doesn&#8217;t mean that you <em>want<\/em> to solve those problems that way. As an example of a problem that&#8217;s really annoying to solve with just the Momentum Principle, I generally break out a toy looping roller coaster that we have in the department&#8217;s collection of demo gear. And, sinc eI have a slow-motion camera now, I shot some video of it:<\/p>\n<p><iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/prPCxKTAY9U\" frameborder=\"0\" allowfullscreen><\/iframe><\/p>\n<p>That&#8217;s three clips spliced together, partly because the camera&#8217;s format doesn&#8217;t play nice with Tracker Video (sigh&#8230;), but mostly because it makes a useful point, namely that if you don&#8217;t start the car high enough up the track, the car won&#8217;t make it all the way around the loop.<\/p>\n<p>It&#8217;s easy enough to calculate the speed the car needs to have at the top to make it around using just the Momentum Principle, but finding the position on the hill where it will reach the necessary speed at the top of the loop is a harder matter. You can get a good estimate of the start position very quickly if you think about the problem using energy methods, though&#8211; at the top of the loop, the cart has to have a minimum speed, and it&#8217;s above the ground by some amount, so you can work out the total kinetic plus potential energy it has as it makes the loop. all of that had to come from the gravitational potential it has at the top of the hill, which lets you get the start height. And, indeed, that&#8217;s how I figured out where to have my student hold the cart for the second and third clips in the video&#8230;<\/p>\n<p>Since I have this, of course, I can crank the clip into <a href=\"https:\/\/www.cabrillo.edu\/~dbrown\/tracker\/\">Tracker Video Analysis<\/a> (once the idiot format problem is fixed, sigh&#8230;) and measure the position of the car as a function of time. Which gets me a reconstruction of the track that looks like this:<\/p>\n<figure id=\"attachment_10043\" aria-describedby=\"caption-attachment-10043\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/scienceblogs.com\/principles\/files\/2015\/05\/sm_roller_coaster_reconstructed.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/scienceblogs.com\/principles\/files\/2015\/05\/sm_roller_coaster_reconstructed.jpg\" alt=\"Reconstruction of the roller coaster track from positions measured in Tracker Video.\" width=\"600\" height=\"469\" class=\"size-full wp-image-10043\" srcset=\"http:\/\/chadorzel.com\/principles\/wp-content\/uploads\/2015\/05\/sm_roller_coaster_reconstructed.jpg 600w, http:\/\/chadorzel.com\/principles\/wp-content\/uploads\/2015\/05\/sm_roller_coaster_reconstructed-300x235.jpg 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-10043\" class=\"wp-caption-text\">Reconstruction of the roller coaster track from positions measured in Tracker Video.<\/figcaption><\/figure>\n<p>(Yeah, it&#8217;s a little oblong, but I was selecting these positions using a trackpoint on a laptop, so I wouldn&#8217;t read too much into that&#8230;) Those positions let you work out the velocity of the car as it goes, which can be combined to find the speed:<\/p>\n<figure id=\"attachment_10044\" aria-describedby=\"caption-attachment-10044\" style=\"width: 600px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/scienceblogs.com\/principles\/files\/2015\/05\/sm_roller_coaster_speed.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/scienceblogs.com\/principles\/files\/2015\/05\/sm_roller_coaster_speed.jpg\" alt=\"Speed of the car going around the loop, as a function of time. Smoothed by taking a running average of five points.\" width=\"600\" height=\"464\" class=\"size-full wp-image-10044\" srcset=\"http:\/\/chadorzel.com\/principles\/wp-content\/uploads\/2015\/05\/sm_roller_coaster_speed.jpg 600w, http:\/\/chadorzel.com\/principles\/wp-content\/uploads\/2015\/05\/sm_roller_coaster_speed-300x232.jpg 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/a><figcaption id=\"caption-attachment-10044\" class=\"wp-caption-text\">Speed of the car going around the loop, as a function of time. Smoothed by taking a running average of five points.<\/figcaption><\/figure>\n<p>This shows basically what you expect: the speed increases to a maximum at the bottom of the hill, drops as the cart climbs the loop, reaching a minimum at the top, then goes back up on the way down. The reconstruction of the loop shows a radius of about 15cm, which implies a minimum speed to complete the loop of around 1.2 m\/s. The minimum seen in the speed graph is a bit over 2 m\/s, a nice safety margin.<\/p>\n<p>Since <a href=\"http:\/\/www.wired.com\/author\/rhettallain\/\">Rhett<\/a> is already done for the semester (I have two more weeks of class, grumble mutter grump), I guess it falls on me to assign you some physics homework, so:<\/p>\n<p>&#8212; The graphs above are for the first of the video clips. Do your own video analysis of the second and third clips, and find the minimum height for the release.<\/p>\n<p>&#8212; Use the difference between the peaks in the speed graph to estimate the force of friction and air resistance acting on the car as it rolls along the track.<\/p>\n<p>&#8212; Use the reconstruction of the roller coaster loop to write a VPython simulation of the car moving on the track using only the Momentum Principle.<\/p>\n<p>Send your homework to Rhett for grading, since he has lots of free time these days.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>One of the points I make repeatedly in teaching introductory mechanics (as I&#8217;m doing this term) is that absolutely every problem students encounter can, in principle, be solved using just Newton&#8217;s Laws or, in the terminology used by Matter and Interactions, the Momentum Principle. You don&#8217;t strictly need any of the other stuff we talk&hellip; <a class=\"more-link\" href=\"http:\/\/chadorzel.com\/principles\/2015\/05\/19\/toy-roller-coasters-and-the-energy-principle\/\">Continue reading <span class=\"screen-reader-text\">Toy Roller Coasters and the Energy Principle<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":10045,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[134,13,19,7,11,136],"tags":[],"class_list":["post-10042","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-course_reports","category-education","category-experiment","category-physics","category-science","category-video","entry"],"_links":{"self":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/10042","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/comments?post=10042"}],"version-history":[{"count":0,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/10042\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media\/10045"}],"wp:attachment":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media?parent=10042"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/categories?post=10042"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/tags?post=10042"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}