{"id":221,"date":"2016-07-28T06:32:22","date_gmt":"2016-07-28T06:32:22","guid":{"rendered":"http:\/\/physicscatalyst.com\/graduation\/?p=221"},"modified":"2020-02-25T06:20:47","modified_gmt":"2020-02-25T06:20:47","slug":"constraints-in-physics-classical-mechanics-with-examples","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/graduation\/constraints-in-physics-classical-mechanics-with-examples\/","title":{"rendered":"constraints in physics (classical mechanics) with examples"},"content":{"rendered":"<div class=\"tcy\">In this article learn about Constraints in physics used in classical mechanics. Constraints limit the motion of the system. We can use constraints to find&nbsp;the <a href=\"https:\/\/physicscatalyst.com\/graduation\/degrees-of-freedom-mechanics\" target=\"_blank\" rel=\"noopener noreferrer\">degrees of freedom<\/a> that that must be calculated before solving dynamical problems.<\/div>\n<div><\/div>\n<h2>Constraints<\/h2>\n<ul>\n<li>A constrained motion is a motion which cannot proceed arbitrarily in any manner.<\/li>\n<li>Particle motion can be restricted to occur (1) along with some specified path (2) on the surface (plane or curved) arbitrarily oriented in space.<\/li>\n<li>Imposing constraints on a mechanical system is done to simplify the mathematical description of the system.<\/li>\n<li>Constraints expressed in the form of equation $f(x_1,y_1,z_1,&#8230;&#8230;,x_n,y_n,z_n :t)=0$ are called&nbsp;<span style=\"text-decoration: underline;\"><strong> holonomic constraints<\/strong><\/span>.<\/li>\n<li>Constraints not expressed in this fashion are called <span style=\"text-decoration: underline;\"><strong>non-holonomic constraints<\/strong><\/span>.<\/li>\n<li><strong>Scleronomic&nbsp;<\/strong><b>constraints<\/b>&nbsp;are independent of time.<\/li>\n<li>Constraints containing time explicitly are called &nbsp;<strong>rehonomic<\/strong>.<\/li>\n<\/ul>\n<p>Therefore a constraint is either<\/p>\n<blockquote><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Scleronomic where constraints relations does not depend on time or rheonomic where constraints relations depends explicitly on time<\/p><\/blockquote>\n<p>or<\/p>\n<blockquote><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Holonomic where constraints relations can be made independent of velocity or non-holonomic where these relations are irreducible functions of velocity<\/p><\/blockquote>\n<p>Constraints types of some physical systems are given below in the table<\/p>\n<p><a href=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2016\/07\/constraints.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-224 size-full\" src=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2016\/07\/constraints.png\" alt=\"constraints\" width=\"445\" height=\"387\" srcset=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2016\/07\/constraints.png 445w, https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2016\/07\/constraints-300x261.png 300w\" sizes=\"auto, (max-width: 445px) 100vw, 445px\" \/><\/a><\/p>\n<h3>More about constraints<\/h3>\n<p>Sometimes motion of a particle or system of particles is restricted by one or more conditions. The limitations on the motion of the system are called <strong>constraints<\/strong>. The number of coordinates needed to specify the dynamical system becomes smaller when constraints are present in the system. Hence the degree of freedom of a dynamical system is defined as the minimum number of independent coordinates required to simplify the system completely along with the constraints. Thus if k is the number of constraints and N is the number of particles in the system possessing motion in three dimensions then the number of degrees of freedom are given by<br \/>\nn=3N-k&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1)<br \/>\nthus the above system has n degrees of freedom.<br \/>\nConstraints may be classified in many ways. If the condition of constraints can be expressed as equations connecting the coordinates of the particles and possibly the time having the form<br \/>\nf(r<sub>1<\/sub>,r<sub>2<\/sub>,&#8230;&#8230;t)=0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2)<br \/>\nthen constraints are said to be holonomic and the simplest example of holonomic constraints is a rigid body. In case of rigid body motion, the distance between any two particles of the body remains fixed and do not change with the tie. If r<sub>i<\/sub>&nbsp;and r<sub>j<\/sub>&nbsp;are the position vectors of the i&#8217;th and j&#8217;th particles then , the distance between the is given by<br \/>\n|r<sub>i<\/sub>-r<sub>j<\/sub>|=c<sub>ij<\/sub>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3)<br \/>\nThe constraints which are not expressible in the form of equation 2 are called non-holonomic for example, the motion of a particle placed on the surface of a sphere of radius a will be described as<br \/>\n|r|\u2265a or, r-a\u22650<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This article is about <em><span style=\"text-decoration: underline;\"><strong>Degree of freedom and constraints<\/strong><\/span><\/em> in classical mechanics. You can use this article for studying B.Sc, JAM , GATE and CSIR-NET Physics. Definition: &#8211; The minimum number of independent variables or co-ordinates required for specifying the position of a dynamical system consisting of one or more particle is called <b>Degree of freedom<\/b>.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[18],"tags":[],"class_list":["post-221","post","type-post","status-publish","format-standard","hentry","category-classical-mechanics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>constraints in physics (classical mechanics) with examples<\/title>\n<meta name=\"description\" content=\"In this article learn about Constraints in physics used in classicsal mechanics. 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