{"id":4812,"date":"2022-09-01T11:49:59","date_gmt":"2022-09-01T09:49:59","guid":{"rendered":"http:\/\/www.les.unina.it\/?page_id=4812"},"modified":"2024-01-08T18:39:48","modified_gmt":"2024-01-08T17:39:48","slug":"motion-of-pendulum","status":"publish","type":"page","link":"https:\/\/www.les.unina.it\/index.php\/risorse-didattiche-2\/motion-of-pendulum\/","title":{"rendered":"Oscillating Systems"},"content":{"rendered":"\n<h1 class=\"wp-block-heading has-text-align-center\">Oscillating Systems<\/h1>\n\n\n\n<p class=\"has-text-align-left has-black-color has-text-color wp-block-paragraph\">Besides planetary motion, another gravitation induced dynamics, extremely relevant for its importance for the foundation and history of concept, measure and mathematical model of time, is the <strong>motion of a pendulum<\/strong>. As an aside, it is worth mentioning that textbooks are sometimes reporting confusing tautologies about the \u201coperational\u201d definition of time interval and of time measurement units. From one side it is stressed the necessity to find (not define) a \u201cperiodic\u201d phenomenon, then the small oscillations of pendulum are found experimentally isochronous using a chronometer. In this way, the important observation of the universal proportionality between periods of different systems oscillating around an equilibrium position is substituted by a tautological definition. We will show that the pendulum motion can be thoroughly examined via the discretisation procedure described for the Gravitational Systems. In particular, the process can be implemented in a spreadsheet and allows to analyse the oscillatory motion of a pendulum, for generic initial conditions, in presence of viscous friction and forcing terms (e.g. the one meant to model the escapement of the pendulum clock).<\/p>\n\n\n\n<p class=\"has-black-color has-text-color wp-block-paragraph\">An example of a non-linear oscillating system is the Van der Pol oscillator. It was introduced in 1927 by electrical engineer Balthazar Van der Pol to describe oscillations in a triode, and is one of the first examples of self-sustained oscillations. The model predicts transitions between regimes in which energy grows and decays. The system has a limit cycle.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Oscillatore libero<\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vRprEOrz1D7jOVu7AVmJs_9baa2b0P2z0lFIDnzsInqraLJN46U5JHcWxclP7PehA\/pubhtml?gid=875774189&amp;single=true&amp;widget=true&amp;headers=false\"><\/iframe><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" seamless=\"\" frameborder=\"0\" scrolling=\"no\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vRprEOrz1D7jOVu7AVmJs_9baa2b0P2z0lFIDnzsInqraLJN46U5JHcWxclP7PehA\/pubchart?oid=404782302&amp;format=interactive\"><\/iframe><\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1TNqMNEy8_45TP6y1ho8wMNy9PCBsSFvR\/edit?usp=sharing&amp;ouid=106351140137531752533&amp;rtpof=true&amp;sd=true\" target=\"_blank\" rel=\"noreferrer noopener\">Go to Spreadsheet<\/a><\/p>\n\n\n\n<div style=\"height:75px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Oscillatore Smorzato<\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vSzbq6U3JvsnnkvR-TEhz1IhiYc712CWbnLzjG7Set08RcRVaOiS4CmfCSXV0Tmrw\/pubhtml?gid=1828382812&amp;single=true&amp;widget=true&amp;headers=false\"><\/iframe><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" seamless=\"\" frameborder=\"0\" scrolling=\"no\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vSzbq6U3JvsnnkvR-TEhz1IhiYc712CWbnLzjG7Set08RcRVaOiS4CmfCSXV0Tmrw\/pubchart?oid=75906126&amp;format=interactive\"><\/iframe><\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1TNqMNEy8_45TP6y1ho8wMNy9PCBsSFvR\/edit?usp=sharing&amp;ouid=106351140137531752533&amp;rtpof=true&amp;sd=true\" target=\"_blank\" rel=\"noreferrer noopener\">Go to Spreadsheet<\/a><\/p>\n\n\n\n<div style=\"height:75px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Oscillatore Smorzato Forzato<\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vTb5Kbe3-6VQ0rI_pcxYlu_z5CS8YpQmd9qs3phJavGYfn9qeIja8Hwz1Ztj6wHUg\/pubhtml?gid=498724168&amp;single=true&amp;widget=true&amp;headers=false\"><\/iframe><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" seamless=\"\" frameborder=\"0\" scrolling=\"no\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vTb5Kbe3-6VQ0rI_pcxYlu_z5CS8YpQmd9qs3phJavGYfn9qeIja8Hwz1Ztj6wHUg\/pubchart?oid=360437488&amp;format=interactive\"><\/iframe><\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1hg6pQq_TJErIfaN8BsGqOMcbiqwA8Aj8\/edit?usp=sharing&amp;ouid=106351140137531752533&amp;rtpof=true&amp;sd=true\" target=\"_blank\" rel=\"noreferrer noopener\">Go to Spreadsheet<\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading has-black-color has-text-color has-medium-font-size\">Oscillatore di van Der Pol<\/h2>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-8f761849 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vQYQuRA9jOg_LVczvaVIXlUuFt-CzCckgJkUvIH_d0MRPpZ8KFGQybCw_HuDLZG3iojbEhnFbFKmqO3\/pubhtml?widget=true&amp;headers=false\"><\/iframe><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><p><iframe loading=\"lazy\" width=\"459\" height=\"288\" seamless frameborder=\"0\" scrolling=\"no\" src=\"https:\/\/docs.google.com\/spreadsheets\/d\/e\/2PACX-1vQYQuRA9jOg_LVczvaVIXlUuFt-CzCckgJkUvIH_d0MRPpZ8KFGQybCw_HuDLZG3iojbEhnFbFKmqO3\/pubchart?oid=1932593334&amp;format=interactive\"><\/iframe><\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/docs.google.com\/spreadsheets\/d\/1U-QGv8gP_KROdaKvTUtLZZExQIIIA55Up9i1H4Fq24M\/edit#gid=0\" target=\"_blank\" rel=\"noreferrer noopener\">Go to Spreadsheet<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Oscillating Systems Besides planetary motion, another gravitation induced dynamics, extremely relevant for its importance for the foundation and history of concept, measure and mathematical model of time, is the motion &#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":4419,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-4812","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/pages\/4812","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/comments?post=4812"}],"version-history":[{"count":13,"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/pages\/4812\/revisions"}],"predecessor-version":[{"id":7513,"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/pages\/4812\/revisions\/7513"}],"up":[{"embeddable":true,"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/pages\/4419"}],"wp:attachment":[{"href":"https:\/\/www.les.unina.it\/index.php\/wp-json\/wp\/v2\/media?parent=4812"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}