Demonstration of path dependence of acceleration and time for ball rolling down a slope

Pri kotrljanju kuglice niz kosinu na nju djeluje sljedeće sile: sila teže koja privlači tijelo prema središtu Zemlje, reakcije podloge, sila trenja između podloge i kuglice te otpor zraka. U teorijskom dijelu rada pokazano je da se, pod određenim uvjetima, kuglica kotrlja jednoliko ubrzano bez klizanja. Pretpostavljena je kvadratna ovisnost, i o radijusu i o brzini centra mase, otpora zraka. Pertubativnim računom određena je analitička veza između gubitka mehaničke energije kuglice i njenog radijusa pri silazu i uzlazu kuglice po kosini. Za eksperimentalni dio rad konstruirane su, od pleksiglasa, dvije kosine s omjerom komponenti gravitacijskog ubrzanja u smjeru kosine 1:2. Na njima je potvrđena linearna ovisnost puta o ubrzanju. Strmija kosina ima dodatak koji služi za određivanje jedinice vremena. Pomoću dviju kuglica koje u istom vremenskom intervalu pređu puteve u omjeru 1:3 pokazana je kvadratna ovisnost puta o vremenu. Mjerenjem visine kuglice, na početku gibanja niz kosinu i nakon povratka na istu, pokazano je da je sila otpora zraka proporcionalna s kvadratom radijusa.When the ball is rolling down a slope, there are several forces acting on it: the weight that attracts the body toward the center of the Earth, the force of the surface reaction, the friction force between the surface of a slope and the ball, and the air resistance. In the theoretical part of the paper it has been shown that, under certain conditions, the ball is rolling with uniform acceleration without slipping. We assumed squared air resistance dependence of the radius and the speed of the center of mass. Pertubative calculations determined the analytical link between the loss of mechanical energy of ball and its radius while going down and up the slope. For the experimental part of the paper, two slopes with a ratio of components of gravity acceleration 1:2 in the direction of the slope are constructed using plexiglass. There we confirmed linear path dependence of the acceleration. The steep slope has a supplement that serves to determine the unit time. Using two balls that cross the paths in the 1: 3 ratio at the same time interval, the quadratic path dependence on time was shown. By measuring the height of the ball, at the beginning of the motion down the slope and after returning to the same, it was shown that the force of the air resistance is proportional to the radius square