{"id":3120,"date":"2021-07-16T15:05:11","date_gmt":"2021-07-16T09:35:11","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=3120"},"modified":"2024-09-30T08:11:05","modified_gmt":"2024-09-30T02:41:05","slug":"uniform-non-uniform-circular-motion","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/uniform-non-uniform-circular-motion\/","title":{"rendered":"Uniform &#038; Non-Uniform Circular Motion"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"3120\" class=\"elementor elementor-3120\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c5ddedc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c5ddedc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-b915d55\" data-id=\"b915d55\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5af2444 elementor-widget elementor-widget-text-editor\" data-id=\"5af2444\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"top\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Let&#8217;s explore circular motion! And we will begin with a brief understanding of <a href=\"#types_of_circular_motion\">types of circular motion<\/a> (uniform and non-uniform circular motion). Following which we will quickly dive into difference between circular motion (of a particle) and <a href=\"#rotational_motion\">rotational motion<\/a> (of a rigid body, where in different particles of the rigid body may be in circular motion moving at different speeds).<\/p><p>Then we will get into the <a href=\"#kinematics_of_circular_motion\">kinematics of the circular motion<\/a>, where we will talk about the centripetal and tangential accelerations (Note: in uniform circular motion we only have centripetal acceleration), and we will also touch upon ways to determine average acceleration or average velocity over a given duration of interest.<\/p><p>Afterwards, we will deep dive into uniform circular motion and we will study <a href=\"#horizontal_circular_motion\">horizontal circular motion<\/a>, <a href=\"#conical_pendulum\">conical pendulum<\/a> moving in a circle, <a href=\"#car_on_a_level_road\">circular motion of a car on a level road<\/a>, <a href=\"#car_on_a_banked_road\">circular motion of a car on a banked road<\/a> and circular motion at constant speed in vertical plane.<\/p><p>And then we will jump into <a href=\"#non-uniform_circular_motion\">non-uniform circular motion<\/a> and examine non-uniform circular motion in horizontal plane, accelerated circular motion of a car on a level road, and non-uniform motion in a vertical plane<\/p><p>Towards the end we will look at the <a href=\"#simple_harmonic_motion\">simple harmonic motion<\/a> of the $x$ and $y$ projections of the particle in uniform circular motion<\/p><p>With that, let&#8217;s start with types of circular motion for which you might want to carefully examine the video and carefully look at the involved forces and accelerations<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d04f58b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d04f58b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-063b3d3\" data-id=\"063b3d3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e764738 elementor-widget elementor-widget-text-editor\" data-id=\"e764738\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2fa2839 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2fa2839\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e0bbd12\" data-id=\"e0bbd12\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-38b381e elementor-widget elementor-widget-heading\" data-id=\"38b381e\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"types_of_circular_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Types of Circular Motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-476bbd6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"476bbd6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-aeca701\" data-id=\"aeca701\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0aea2fc elementor-widget elementor-widget-video\" data-id=\"0aea2fc\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575214426?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-89947e5 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"89947e5\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9db1ba3\" data-id=\"9db1ba3\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-23a7346 elementor-widget elementor-widget-text-editor\" data-id=\"23a7346\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>In uniform circular motion, the &#8216;speed&#8217; of the particle is constant. In other words, while the direction of the velocity is changing, it&#8217;s magnitude is constant. And this means that its angular velocity (number of radians covered per second) is also constant.\u00a0<\/p><p>So what does this mean for its acceleration?<\/p><p>Well as we will see later when discussing kinematics of circular motion, there is NO acceleration in the direction of the velocity but there is an acceleration perpendicular to it, which is, as we will shall see later, always pointed towards the center of the circle and hence the name centripetal acceleration<\/p><p>In non-uniform acceleration, the magnitude and direction of the velocity of the particle are changing with time, which means it has a component in the direction of the velocity (tangential acceleration) and an acceleration perpendicular to it&#8217;s direction (centripetal acceleration) which rotates the velocity vector.<\/p><p>Now before we get into the kinematics of circular motion, let&#8217;s understand the difference between two similar terms circular motion and rotational motion<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0e9aef9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0e9aef9\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-9ae77e7\" data-id=\"9ae77e7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4a6415b elementor-widget elementor-widget-text-editor\" data-id=\"4a6415b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4285c67 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4285c67\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5fb3c16\" data-id=\"5fb3c16\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9d0b410 elementor-widget elementor-widget-heading\" data-id=\"9d0b410\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"rotational_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Difference between Circular and Rotational Motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-07e39b8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"07e39b8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-32c9e95\" data-id=\"32c9e95\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f61b818 elementor-widget elementor-widget-video\" data-id=\"f61b818\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575254861?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7ca28d7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7ca28d7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3d4d9b6\" data-id=\"3d4d9b6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e8b4695 elementor-widget elementor-widget-text-editor\" data-id=\"e8b4695\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Circular motion typically refers to the motion of particle along a circle while rotational motion refers to rotation of a rigid body about some axis. Note that the speed of different particles of the rigid body is different.<\/p><p>Now let&#8217;s explore the kinematics of circular motion.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d37d3f1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d37d3f1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-74d00e6\" data-id=\"74d00e6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-be871a3 elementor-widget elementor-widget-heading\" data-id=\"be871a3\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"kinematics_of_circular_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Kinematics of circular motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-43e8378 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"43e8378\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-32a7fed\" data-id=\"32a7fed\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4387366 elementor-widget elementor-widget-heading\" data-id=\"4387366\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Centripetal acceleration in uniform circular motion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b239da3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b239da3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6304355\" data-id=\"6304355\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-61e74d6 elementor-widget elementor-widget-text-editor\" data-id=\"61e74d6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Let&#8217;s begin with particle in uniform circular motion and let&#8217;s determine the acceleration of this particle which is moving at constant speed.\u00a0<\/p><p>Note that we can do so either geometrically as highlighted in the video below (Note that the direction of the acceleration is towards the center) or we can determine acceleration algebraically by writing the position vector of particle in circular motion and then using $\\cfrac{d^2 \\overrightarrow{r}}{dt^2}$ to determine the acceleration vector $\\overrightarrow{a}$<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-939e34b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"939e34b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f76449b\" data-id=\"f76449b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-979963e elementor-widget elementor-widget-video\" data-id=\"979963e\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575382047?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-eecda91 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"eecda91\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6afa462\" data-id=\"6afa462\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9d6b683 elementor-widget elementor-widget-text-editor\" data-id=\"9d6b683\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Now let&#8217;s derive the expression for centripetal acceleration algebraically<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3017048 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3017048\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4450106\" data-id=\"4450106\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-09f034f elementor-widget elementor-widget-image\" data-id=\"09f034f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"768\" height=\"630\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation-768x630.png\" class=\"attachment-medium_large size-medium_large wp-image-3197\" alt=\"Uniform Circular Motion - Centripetal Acceleration - Algebraic Derivation\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation-768x630.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation-300x246.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation-1024x840.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation-1536x1260.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Uniform-Circular-Motion-Centripetal-Acceleration-Algebraic-Derivation.png 2031w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4704941 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4704941\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4221781\" data-id=\"4221781\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f7a2cbe elementor-widget elementor-widget-text-editor\" data-id=\"f7a2cbe\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>When the position vector of the particle is making an angle $\\theta$ (at some time $t$), it&#8217;s position vector $\\overrightarrow{r}$ can be written as:<\/p><p>$\\overrightarrow{r} = r(\\cos \\theta \\hat{i} + \\sin \\theta \\hat{j})$<\/p><p>or $\\overrightarrow{r} = r(\\cos \\omega t \\hat{i} + \\sin \\omega t \\hat{j})$, where angular velocity $\\omega =\\cfrac{v}{r}$<\/p><p>So, it&#8217;s acceleration at that position and time would be $\\cfrac{d^2 \\overrightarrow{r}}{dt^2}$<\/p><p>And $\\cfrac{d^2 \\overrightarrow{r}}{dt^2}=-\\omega^2 r (\\cos \\theta \\hat{i} + \\sin \\theta \\hat{j})$, which we can write as<\/p><p>$\\overrightarrow{a} =- \\cfrac{v^2}{r} (\\cos \\theta \\hat{i} + \\sin \\theta \\hat{j})$<\/p><p>Note that the magnitude of $\\overrightarrow{a}$ is $\\cfrac{v^2}{r}$ and it&#8217;s direction is $- \\hat{r}$ i.e. pointed towards the center and hence the name centripetal acceleration<\/p><p>Now let&#8217;s examine the accelerations in a non-uniform circular motion.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6a41d36 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6a41d36\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ee5411a\" data-id=\"ee5411a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3223e56 elementor-widget elementor-widget-heading\" data-id=\"3223e56\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Accelerations (tangential and centripetal) in non-uniform circular motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bfb7b95 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bfb7b95\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0f0a156\" data-id=\"0f0a156\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5ad9ebd elementor-widget elementor-widget-text-editor\" data-id=\"5ad9ebd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here in this animation a particle is undergoing circular motion in vertical plane with string tension $T$ and gravitational force $mg$ acting on it. Note that since there are no dissipative forces, total mechanical energy of the particle remains constant i.e. $U + K = $ constant at all times. And as you will expect, kinetic energy $K$ will be maximum at the bottom and potential energy $U$ will be maximum at the top.\u00a0<\/p><p>Also note that gravitational force, $mg \\hat{k}$, is constant at all times though tension $T$ is constantly changing in magnitude and direction. $T$ is maximum at the bottom $T = mg + m \\cfrac{v^2}{r}$ and it will be minimum at the top $T = m\\cfrac{v^2}{r} &#8211; mg$<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9472629 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9472629\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-81e30a6\" data-id=\"81e30a6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-05165d2 elementor-widget elementor-widget-video\" data-id=\"05165d2\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575399428?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-945bc82 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"945bc82\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1416070\" data-id=\"1416070\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b8fa4a7 elementor-widget elementor-widget-text-editor\" data-id=\"b8fa4a7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>So, as mentioned before, non uniform circular motion refers to scenarios where in the speed of the particle is also changing as it is moving along a circular path. In such cases, particle has tangential as well as centripetal acceleration. Note that magnitude of centripetal acceleration ($=\\cfrac{v^2}{r}$) is changing as speed $v$ is changing with time.<\/p><p>And once again, centripetal acceleration changes the direction of velocity vector while tangential acceleration changes the magnitude of velocity vector (i.e. speed $v$)<\/p><p>Now let&#8217;s deep dive into commonly encountered examples of uniform and non-uniform circular motion in text and we will try and understand a bit about the forces acting on the particle in circular motion\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f2f3363 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f2f3363\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-ee60e35\" data-id=\"ee60e35\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e6873df elementor-widget elementor-widget-text-editor\" data-id=\"e6873df\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-fc04c6d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"fc04c6d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-f43e711\" data-id=\"f43e711\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8b11130 elementor-widget elementor-widget-heading\" data-id=\"8b11130\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Uniform Circular Motion Examples<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c567924 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c567924\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-dcd4ffa\" data-id=\"dcd4ffa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-96b8151 elementor-widget elementor-widget-heading\" data-id=\"96b8151\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"horizontal_circular_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Uniform Circular Motion | Horizontal Motion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3fe17f1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3fe17f1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0e4521a\" data-id=\"0e4521a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3a6f181 elementor-widget elementor-widget-text-editor\" data-id=\"3a6f181\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Given that a mass, tied to one end of a spring of spring constant $k$, is rotating in a circle with constant speed $v$ or angular velocity $\\omega=\\cfrac{v}{r}$. How would you determine the radius of circle, given that the natural or free length of the spring is $l$?<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-29c754a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"29c754a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-26b0270\" data-id=\"26b0270\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a2e1e1e elementor-widget elementor-widget-image\" data-id=\"a2e1e1e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"768\" height=\"818\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Horizontal-Uniform-Circular-Motion-768x818.png\" class=\"attachment-medium_large size-medium_large wp-image-3219\" alt=\"Horizontal Uniform Circular Motion\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Horizontal-Uniform-Circular-Motion-768x818.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Horizontal-Uniform-Circular-Motion-282x300.png 282w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Horizontal-Uniform-Circular-Motion-961x1024.png 961w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Horizontal-Uniform-Circular-Motion.png 1327w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e082466 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e082466\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-912d750\" data-id=\"912d750\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f9f1f31 elementor-widget elementor-widget-text-editor\" data-id=\"f9f1f31\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We are interested in finding $l+x$ where $x$ is the amount by which spring is stretched.<\/p><p>So, let&#8217;s start with writing the force acting on the mass<\/p><p>$F = -kx$, with minus sign indicating that the force is a restoring force and as such, it will be pointed towards the center of the circle. Going forward we will drop the minus sign.<\/p><p>Now, given that the particle is moving around the circle at speed $v$, it will have a centripetal acceleration $a _r= \\cfrac{v^2}{l+x}$<\/p><p>Now, as per newton&#8217;s 2nd law, force acting on the particle $F$ and its acceleration $a_r$ must be related by the relation $F=ma_r$, So we have $kx = m \\cfrac{v^2}{l+x}$, solving which we will get $x$ and thus the radius of circle traced by the particle will be $l+x$.<\/p><p>With that let&#8217;s explore the motion of conical pendulum moving in a circle.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-cd6571a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"cd6571a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d5c75e6\" data-id=\"d5c75e6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ac386b6 elementor-widget elementor-widget-text-editor\" data-id=\"ac386b6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-63aec5c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"63aec5c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3ca9868\" data-id=\"3ca9868\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2d0c181 elementor-widget elementor-widget-heading\" data-id=\"2d0c181\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"conical_pendulum\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Uniform Circular Motion | Conical Pendulum moving in a circle<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-3d6deb4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3d6deb4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6333260\" data-id=\"6333260\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8dd0e9a elementor-widget elementor-widget-video\" data-id=\"8dd0e9a\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575414762?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-d18e747 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"d18e747\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-083b4aa\" data-id=\"083b4aa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-54178d1 elementor-widget elementor-widget-text-editor\" data-id=\"54178d1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Let&#8217;s say that the string makes an angle $\\theta$ with the vertical.<\/p><p>So, there are two external forces acting on the mass $m$, gravitational force $mg$ pointed vertically downwards and tension $T$ acting along the string as shown.<\/p><p>Now, there is no acceleration in the vertical direction, so net forces in the vertical or $y$ direction must be zero i.e. $F_{net,y} = T \\cos \\theta &#8211; mg = 0$ or $T \\cos \\theta = mg$<\/p><p>As for the horizontal direction, at any given point in time, there is a centripetal force in the amount $T \\sin \\theta$ pointed towards $O$ and since the mass is moving at speed $v$, it has a centripetal acceleration in the amount $\\cfrac{v^2}{r}$ (in the inertial frame of reference)<\/p><p>So as per newtons 2nd law $F=ma$, along the radius $T \\sin \\theta = m \\cfrac{v^2}{r}$<\/p><p>Solving the two equations we can determine the desired unknown.<\/p><p>Note that you may choose to write the equations in a frame of rotating at the same angular speed in which the observer in that reference frame will report that the particle is stationary and there is a centrifugal force in the amount $ma$ pointed radially outwards. (If you look closely you go from $F = ma$ in the inertial frame of reference to $F-ma = 0$ in the accelerated frame of reference i.e. you treat it as a force balance problem in the accelerated frame of reference)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9122838 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9122838\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-93e2b30\" data-id=\"93e2b30\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d5906ba elementor-widget elementor-widget-text-editor\" data-id=\"d5906ba\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e0075b1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e0075b1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4eec73f\" data-id=\"4eec73f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-543c19f elementor-widget elementor-widget-heading\" data-id=\"543c19f\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"car_on_a_level_road\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Uniform circular motion | circular motion of a car on a level road<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e4e787a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e4e787a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d33d0bf\" data-id=\"d33d0bf\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5d30151 elementor-widget elementor-widget-image\" data-id=\"5d30151\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"768\" height=\"590\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1-768x590.png\" class=\"attachment-medium_large size-medium_large wp-image-3227\" alt=\"Circular motion of a car on a level road\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1-768x590.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1-300x230.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1-1024x787.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1-1536x1180.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Circular-motion-of-a-car-on-a-level-road-rev1.png 1894w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-e81bfa3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"e81bfa3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-fc14e13\" data-id=\"fc14e13\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ce3055b elementor-widget elementor-widget-text-editor\" data-id=\"ce3055b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Given that the coefficient of friction $= \\mu$, let&#8217;s find the limiting speed of the car (beyond which the car will slip outwards)<\/p><p>Now the limiting frictional force, pointed towards the center of the circle, $=\\mu mg$ and centripetal acceleration $=\\cfrac{v^2}{r}$ (Note $v = \\omega r$<\/p><p>So, using the newton&#8217;s 2nd law $F = ma$, we get $f_s = m \\cfrac{v^2}{r}$ and in the limiting scenario, we will have $\\mu mg = m \\cfrac{v^2_{max}}{r}$ or $v_{max} = \\sqrt{\\mu g r}$<\/p><p>Now, let&#8217;s see how the scenario changes a bit when the car is on a banked road<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-69aec87 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"69aec87\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8a3a415\" data-id=\"8a3a415\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9a4712f elementor-widget elementor-widget-text-editor\" data-id=\"9a4712f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1e56e3f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1e56e3f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5fc99b2\" data-id=\"5fc99b2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3d4218f elementor-widget elementor-widget-heading\" data-id=\"3d4218f\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"car_on_a_banked_road\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Uniform Circular Motion | Car on a banked road<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2fececa elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2fececa\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8c5422e\" data-id=\"8c5422e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-346b814 elementor-widget elementor-widget-image\" data-id=\"346b814\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"653\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-768x653.png\" class=\"attachment-medium_large size-medium_large wp-image-3232\" alt=\"\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-768x653.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-300x255.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-1024x871.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-1536x1307.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Car-on-a-banked-road-rev1-2048x1742.png 2048w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2cdd90f elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2cdd90f\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-23ab75b\" data-id=\"23ab75b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-616a94d elementor-widget elementor-widget-text-editor\" data-id=\"616a94d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>There are 3 possible scenarios depending on the speed of the car, $v$.\u00a0<\/p><p>I) For certain speed $v_o$, there is no friction force between the car and the ground<\/p><p>II) For $v&gt;v_o$, frictional force would act down the incline as shown<\/p><p>III) For $v&lt;v_o$, frictional force would act up the incline as shown<\/p><p>So, let&#8217;s begin with establishing $v_o$ for which frictional force, $f_s = 0$<\/p><p>In this scenario, $N \\cos \\theta = mg$ and $N \\sin \\theta = m \\cfrac{v_o^2}{r}$. Dividing the two equations we get $v_o = \\sqrt{gr \\tan \\theta}$.<\/p><p>For $v&gt; v_o$, $N \\sin \\theta$ will not be able to provide the needed centripetal force and the car would have tendency to slip outwards (up the incline) and as such a frictional force would kick in. Now, the two equations in the vertical and horizontal direction would become<\/p><p>$N \\cos \\theta = mg + f_s \\sin \\theta$ and<\/p><p>$N \\sin \\theta + f_s \\cos \\theta = m \\cfrac{v^2}{r}$<\/p><p>Solving the two we can find the desired unknown. Note, to find $v_max$ at which car is about to slip, use $f_s = \\mu N$<\/p><p>We trust that you will be able to write equations for the 3rd scenario on your own.<\/p><p>With that, now let&#8217;s explore the motion of a daredevil&#8217;s bike tracing a vertical circular path at a constant speed<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-19fff9c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"19fff9c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-97e0b2d\" data-id=\"97e0b2d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-0a36090 elementor-widget elementor-widget-text-editor\" data-id=\"0a36090\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-39fb8ca elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"39fb8ca\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-de00a11\" data-id=\"de00a11\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e51f73 elementor-widget elementor-widget-heading\" data-id=\"7e51f73\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span style=\"font-size: 20px;font-style: normal;font-weight: 600\">Uniform Circular Motion | Vertical Circular Motion<\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1b7c2c3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1b7c2c3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e7154b0\" data-id=\"e7154b0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9b7950c elementor-widget elementor-widget-text-editor\" data-id=\"9b7950c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here let&#8217;s determine the normal force and frictional force on the bike at different points along the circular path<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ccb122d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ccb122d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-32344f0\" data-id=\"32344f0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3aa3198 elementor-widget elementor-widget-image\" data-id=\"3aa3198\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"2384\" height=\"2481\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed.png\" class=\"attachment-full size-full wp-image-3236\" alt=\"Vertical circular motion at constant speed\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed.png 2384w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed-288x300.png 288w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed-984x1024.png 984w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed-768x799.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed-1476x1536.png 1476w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Vertical-circular-motion-at-constant-speed-1968x2048.png 1968w\" sizes=\"(max-width: 2384px) 100vw, 2384px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-31dc0a7 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"31dc0a7\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7c3c96e\" data-id=\"7c3c96e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7e352c6 elementor-widget elementor-widget-text-editor\" data-id=\"7e352c6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>At A:\u00a0<span style=\"font-size: 15px;\">$N = mg + \\cfrac{mv^2}{r}$ and $f_s = 0$<\/span><\/p><p style=\"font-size: 15px;\">At B:\u00a0<span style=\"font-size: 15px;\">$N = \\cfrac{mv^2}{r}$ and $f_s = mg$<\/span><\/p><p style=\"font-size: 15px;\">At C:\u00a0<span style=\"font-size: 15px;\">$N = \\cfrac{mv^2}{r} &#8211; mg$ and $f_s = 0$<\/span><\/p><p style=\"font-size: 15px;\">At A:\u00a0<span style=\"font-size: 15px;\">$N = \\cfrac{mv^2}{r}$ and $f_s = mg$<\/span><\/p><p style=\"font-size: 15px;\"><span style=\"font-size: 15px;\">With that, we will now move to non uniform circular motion<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-12737bd elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"12737bd\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-0873a17\" data-id=\"0873a17\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dc3a818 elementor-widget elementor-widget-text-editor\" data-id=\"dc3a818\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-bcb621a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"bcb621a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-91c55d7\" data-id=\"91c55d7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c50c39d elementor-widget elementor-widget-heading\" data-id=\"c50c39d\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"non-uniform_circular_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Non uniform circular motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-9089f3d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"9089f3d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c90e90\" data-id=\"6c90e90\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-023a23f elementor-widget elementor-widget-text-editor\" data-id=\"023a23f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>In non-uniform circular motion, particle has a centripetal component as well as tangential component of the acceleration. The tangential component results in a change in its speed while the centripetal component of the accelerations changes direction of the velocity vector.<\/p>\n<p>So let&#8217;s go ahead and look at some examples of non uniform circular motion in horizontal and vertical planes.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-45df19c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"45df19c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-18e416e\" data-id=\"18e416e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4a5f2e2 elementor-widget elementor-widget-heading\" data-id=\"4a5f2e2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Non Uniform Circular Motion Examples<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2b7ce44 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2b7ce44\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5fef8a2\" data-id=\"5fef8a2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-e7710f2 elementor-widget elementor-widget-heading\" data-id=\"e7710f2\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Non uniform circular motion | Accelerated circular motion of a car on a level road<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b7e7297 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b7e7297\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c0daaf2\" data-id=\"c0daaf2\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2cf67eb elementor-widget elementor-widget-image\" data-id=\"2cf67eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img loading=\"lazy\" decoding=\"async\" width=\"768\" height=\"492\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-768x492.png\" class=\"attachment-medium_large size-medium_large wp-image-3249\" alt=\"Example of Non uniform Circular motion of a car on a level road\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-768x492.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-300x192.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-1024x655.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-1536x983.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/07\/Non-uniform-Circular-motion-of-a-car-on-a-level-road-2048x1311.png 2048w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-767b813 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"767b813\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d030f11\" data-id=\"d030f11\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9bdb747 elementor-widget elementor-widget-text-editor\" data-id=\"9bdb747\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here a component of the frictional force, $f_{s,t}$, is providing the tangential acceleration $a_t$ and another component\u00a0<span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$f_{s,r}$ is providing the centripetal acceleration<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">So, $f_{s,t} = ma_t$ and<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$f_{s,r} = \\cfrac{mv^2}{r}$<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">In the about to slip situation (i.e. when $f_s = \\mu mg$), we will have<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$\\sqrt{f_{s,t}^2 + f_{s,r}^2} = \\mu mg$<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5f19c24 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5f19c24\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1980b9a\" data-id=\"1980b9a\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-8221d78 elementor-widget elementor-widget-text-editor\" data-id=\"8221d78\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-68ddfd4 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"68ddfd4\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-4cff025\" data-id=\"4cff025\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fb2229b elementor-widget elementor-widget-heading\" data-id=\"fb2229b\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Non Uniform Circular Motion | Vertical Circular Motion<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-56463fc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"56463fc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c3d17ad\" data-id=\"c3d17ad\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-927aac0 elementor-widget elementor-widget-video\" data-id=\"927aac0\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575399428?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0c237ab elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0c237ab\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-20bef77\" data-id=\"20bef77\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a3ea0f1 elementor-widget elementor-widget-text-editor\" data-id=\"a3ea0f1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here is the particle is moving under the influence of gravitational force $mg$ and tension $T$. And since there are no dissipative forces involved here, the total mechanical energy ($=U+K$) will remain constant.<\/p><p>So, what is the minimum speed that the particle should have at the bottom, i.e. $v_b$ such that it completes the circle.<\/p><p>Well, to complete the circle, at the very top $\\cfrac{mv_t^2}{r} \\geq mg$ in other words speed at the top should be sufficient enough so that all of $mg$ is consumed in turning the direction of velocity at that point. Note that when $mg = \\cfrac{mv_t^2}{r}$, tension $T$ is zero at the very top.<\/p><p>or $v_{t,min} = \\sqrt{gr}$.<\/p><p>So, how much should $v_{b,min}$ be in order to have\u00a0<span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$v_{t,min} = \\sqrt{gr}$.<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">Well using conservation of mechanical energy, $U_b + K_b = U_t + K_t$, where assuming $U_b = 0$, $U_t$ would be $2mgr$, we will get $v_b =\u00a0<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">\\sqrt{5gr}<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$.<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">Hope that helped.<\/span><\/p><p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">Now let&#8217;s take a quick look at the motion of $x$ and $y$ projections of a particle in uniform circular motion<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5c901ba elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5c901ba\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-384a406\" data-id=\"384a406\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-80036fc elementor-widget elementor-widget-text-editor\" data-id=\"80036fc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"#top\"><span style=\"text-decoration: underline;\">Top<\/span><\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2297547 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2297547\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7d427b6\" data-id=\"7d427b6\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f929054 elementor-widget elementor-widget-heading\" data-id=\"f929054\" data-element_type=\"widget\" data-e-type=\"widget\" id=\"simple_harmonic_motion\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Simple Harmonic Motion and Uniform Circular Motion<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-866f780 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"866f780\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-31f51e9\" data-id=\"31f51e9\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cb0ad0f elementor-widget elementor-widget-video\" data-id=\"cb0ad0f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;video_type&quot;:&quot;vimeo&quot;,&quot;autoplay&quot;:&quot;yes&quot;,&quot;play_on_mobile&quot;:&quot;yes&quot;,&quot;mute&quot;:&quot;yes&quot;,&quot;loop&quot;:&quot;yes&quot;}\" data-widget_type=\"video.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-wrapper elementor-open-inline\">\n\t\t\t<iframe class=\"elementor-video-iframe\" allowfullscreen allow=\"autoplay\" title=\"vimeo Video Player\" src=\"https:\/\/player.vimeo.com\/video\/575671536?autoplay=1&amp;playsinline=1&amp;color&amp;autopause=0&amp;loop=1&amp;muted=1&amp;title=1&amp;portrait=1&amp;byline=1#t=\"><\/iframe>\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-b1cb929 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"b1cb929\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bdf4098\" data-id=\"bdf4098\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9fcb812 elementor-widget elementor-widget-text-editor\" data-id=\"9fcb812\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>As discussed earlier in the section on kinematics of circular motion, acceleration vector of the particle at time $t$, is given by<\/p>\n<p>$\\overrightarrow{a} = -\\cfrac{v^2}{r} (\\cos \\omega t \\hat{i} + \\sin \\omega t \\hat{j})$<\/p>\n<p>or $a_x = -\\cfrac{v^2}{r} \\cos \\omega t$ and $a_y = -\\cfrac{v^2}{r} \\sin \\omega t$.<\/p>\n<p>In other words the $x$ projection is moving with an acceleration $a_x = -\\cfrac{v^2}{r} \\cos \\omega t$ (simple harmonic motion) and $y$ projection is moving with an acceleration $a_y = -\\cfrac{v^2}{r} \\sin \\omega t$ (also a simple harmonic motion).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-083a707 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"083a707\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-a095e32\" data-id=\"a095e32\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9965f26 elementor-widget elementor-widget-text-editor\" data-id=\"9965f26\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>And that concludes the discussion on circular motion. Hope it helped. Do drop a comment if you have any questions on this topic or have any suggestions for us.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Let&#8217;s explore circular motion and understand types of circular motion (uniform and non-uniform circular motions) and kinematics of circular 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