{"id":2819,"date":"2017-08-26T05:48:25","date_gmt":"2017-08-26T05:48:25","guid":{"rendered":"http:\/\/physicscatalyst.com\/graduation\/?p=2819"},"modified":"2022-04-15T07:25:23","modified_gmt":"2022-04-15T07:25:23","slug":"position-of-a-particle","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/graduation\/position-of-a-particle\/","title":{"rendered":"Position of a particle (Mechanics)"},"content":{"rendered":"<pre>This article \" Position of a particle \" is for B.Sc. students. We have also defined inertial and non inertial frame of reference in this article.<\/pre>\n<h2><span lang=\"EN-US\">Position of a particle<\/span><\/h2>\n<h3><span lang=\"EN-US\">(a) By Position Vector:-<\/span><\/h3>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">The line<\/span><span lang=\"EN-US\"> joining the point P and the origin \u2018O\u2019 of the frame of reference that is $\\vec{PO}$ is known as position vector and is also represented by $\\vec{r}$.<\/span><\/p>\n<p><a href=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/position-vector.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2821\" src=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/position-vector-300x247.png\" alt=\"Position of a particle\" width=\"300\" height=\"247\" srcset=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/position-vector-300x247.png 300w, https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/position-vector.png 347w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<h3><span lang=\"EN-US\">(b) By coordinates:-<\/span><\/h3>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">The position<\/span><span lang=\"EN-US\"> of a point P can also be represented in the coordinate system in the form of mutually perpendicular x, y, and z axis.<\/span><\/p>\n<p><a href=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/by-coordinate.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2820\" src=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/by-coordinate-300x262.png\" alt=\"Position of a particle\" width=\"300\" height=\"262\" srcset=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/by-coordinate-300x262.png 300w, https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/by-coordinate.png 374w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<h2><span lang=\"EN-US\">Position of a moving particle<\/span><\/h2>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">If any point changes its position from $P_1}$ to $P_2$ with respect to time then position vector of point will be $\\vec{r}$ at time $t$ and its displacement vector would be given by<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{P_1P_2}=(\\vec{r_2}-\\vec{r_1})$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Where $P_1$ and $P_2$ is the position of moving particle at time $t_1$ and $t_2$ and&nbsp; $\\vec{r_1}$ and $\\vec{r_2}$ are position vectors at time $t_1$ and $t_2$ .<\/span><\/p>\n<p><a href=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/moving-particle.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2822\" src=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/moving-particle-300x261.png\" alt=\"Position of a particle\" width=\"300\" height=\"261\" srcset=\"https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/moving-particle-300x261.png 300w, https:\/\/physicscatalyst.com\/graduation\/wp-content\/uploads\/2017\/08\/moving-particle.png 467w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Now time interval is <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\Delta t={{t}_{2}}-{{t}_{1}}$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Displacement of the particle is given by <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\Delta \\vec r = \\overrightarrow {{r_2}}&nbsp; &#8211; \\overrightarrow {{r_1}} $<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Rate of change of position or velocity is given by <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{v}=\\frac{\\vec{r_2}-\\vec{r_1}}{t_2 \u2013 t_1}=\\frac{\\Delta \\vec{r}}{\\Delta t}$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">If time interval is very very less i.e. $\\Delta t\\to 0$ then the change in position with respect to time is called instantaneous velocity.<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Or, <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{v}=\\underset{\\Delta t\\to 0}{\\mathop{\\lim }}\\,\\frac{\\Delta \\vec{r}}{\\Delta t}$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">By the definition of differentiation <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\underset{\\Delta t\\to 0}{\\mathop{\\lim }}\\,\\frac{\\Delta \\vec{r}}{\\Delta t}=\\frac{d\\vec{r}}{dt}$ <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">If the velocity of particle is changing with respect to time then this is called rate of change of velocity or <b>acceleration<\/b>. Let $\\vec{v_1}$ and $\\vec{v_2}$ be the velocities of particle at time $t_1$ and $t_2}$ respectively then,<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Change in velocity = $\\vec{v_2}-\\vec{v_1}=\\Delta \\vec{v}$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Change in time = $t_2 \u2013 t_1 = \\Delta t$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Rate of change of velocity would be<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{a}=\\frac{\\overrightarrow{{{v}_{2}}}-\\overrightarrow{{{v}_{1}}}}{{{t}_{2}}-{{t}_{1}}}=\\frac{\\Delta \\vec{v}}{\\Delta t}$ &nbsp;&nbsp;<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">If time interval is very very less i.e. $\\Delta t\\to 0$ then the change in velocity with respect to time is called instantaneous acceleration i.e.,<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{a}=\\underset{\\Delta t\\to 0}{\\mathop{\\lim }}\\,\\frac{\\Delta \\vec{v}}{\\Delta t}=\\frac{d}{dt}\\left( \\frac{d\\vec{r}}{dt} \\right)=\\frac{{{d}^{2}}\\vec{r}}{d{{t}^{2}}}$<\/span><\/p>\n<h2><span lang=\"EN-US\">Inertial and Non-Inertial frame of reference<\/span><\/h2>\n<p class=\"MsoNormal\"><span lang=\"EN-US\"><a href=\"https:\/\/physicscatalyst.com\/graduation\/frame-reference\/\" target=\"_blank\" rel=\"noopener noreferrer\">Frames of reference<\/a> are of two types<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">(1) Inertial frame of reference<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">(2) Non Inertial frame of reference<\/span><\/p>\n<h3><span lang=\"EN-US\">1. Inertial frame of reference:-<\/span><\/h3>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">If no external force is acting on the particle then the frame of reference with respect to which the particle seems to be moving at constant velocity or with no acceleration is known as inertial frame of reference. Since no external force is acting on the particle we have<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{F}=m\\frac{d^2 \\vec{r}}{dt^2}=0$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Therefore, acceleration<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$\\vec{a}=\\frac{d^2 \\vec{r}}{dt^2}=0$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">or, velocity <\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">$v=constant$<\/span><\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">Clearly in absence of external force the particle will move with a constant velocity in the frame of reference. Therefore for inertial frame of reference Newton\u2019s first and second laws of motion are always valid. By analysis it is found that such frames are either at rest or move with constant velocity. Hence for an observer at rest or moving with constant velocity frame of reference are always inertial frames of reference.<\/span><\/p>\n<h3><span lang=\"EN-US\">2. Non-inertial frame of reference:-<\/span><\/h3>\n<p>&nbsp;<\/p>\n<p class=\"MsoNormal\"><span lang=\"EN-US\">In non-inertial frame of reference Newton\u2019s first and second laws of motion does not holds true. In non-inertial frame of reference observer or frame of reference accelerates and particle or object observed relative to them seems to be accelerated in the absence of any externally applied force.<\/span><\/p>\n<p>In this article we have covered how to define&nbsp;Position of a particle. For further reference you can visit the links given below<\/p>\n<div class=\"callout large primary\">\n<h4>Recommended Books and texts<\/h4>\n<p><strong>Books<\/strong><br \/>\n<a href=\"http:\/\/amzn.to\/2p8XLzq\" target=\"_blank\" rel=\"noopener noreferrer\">Mechanics: For Students of B.Sc (Pass and Hons.): D.S. Mathur<\/a><br \/>\n<a href=\"http:\/\/amzn.to\/2nTpMLi\" target=\"_blank\" rel=\"noopener noreferrer\">An Introduction to Mechanics (SIE) by David Kleppner, Robert Kolenkow<\/a><br \/>\n<a href=\"http:\/\/amzn.to\/2p8XWe4\" target=\"_blank\" rel=\"noopener noreferrer\">Physics for Degree Students B.Sc. First Year<\/a><br \/>\n<strong>Texts<\/strong><br \/>\n1st Law and Newtonian space and time.<br \/>\n<a href=\"https:\/\/en.wikipedia.org\/wiki\/Absolute_time_and_space\" target=\"_blank\" rel=\"noopener noreferrer\">Absolute time and space<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This article &#8221; Position of a particle &#8221; is for B.Sc. students. We have also defined inertial and non inertial frame of reference in this article. Position of a particle (a) By Position Vector:- The line joining the point P and the origin \u2018O\u2019 of the frame of reference that is $\\vec{PO}$ is known as [&hellip;]<\/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":[1],"tags":[],"class_list":["post-2819","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Position of a particle (Mechanics) - Learn about education and B.Sc. 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