{"id":23,"date":"2020-08-28T05:36:52","date_gmt":"2020-08-28T05:36:52","guid":{"rendered":"http:\/\/spacican.com\/notes\/?p=23"},"modified":"2023-09-18T06:49:53","modified_gmt":"2023-09-18T06:49:53","slug":"engineering-mechanics-short-notes-gate","status":"publish","type":"post","link":"https:\/\/spacican.com\/notes\/engineering-mechanics-short-notes-gate\/","title":{"rendered":"Engineering mechanics (EM) short notes GATE"},"content":{"rendered":"\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<div class=\"wp-block-create-block-note sz-note\">\n<p><strong>Note:<\/strong> Incomplete<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-create-block-note sz-note\">\n<p><strong>Syllabus:<\/strong> Free-body diagrams and equilibrium; trusses and frames; virtual work; kinematics and dynamics of particles and of rigid bodies in plane motion; impulse and momentum (linear and angular) and energy formulations, collisions.<\/p>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Engineering Mechanics basics and important formulas<\/h2>\n\n\n\n<p>Engineering mechanics deals with the effect of forces on bodies. EM has three parts:<\/p>\n\n\n\n<div class=\"wp-block-create-block-negative-margin negative-margin-top\"><\/div>\n\n\n\n<ol>\n<li><strong>Statistics: <\/strong>Deals with forces acting on a body.\n<ul>\n<li>Example: Analysis of truss<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Kinematics: <\/strong>Studies only the motion of the bodies.\n<ul>\n<li>Example: Projectile motion<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Kinetics: <\/strong>Deals with the motion as well as different forces acting on the bodies.\n<ul>\n<li>Example: Collision<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Newton&#8217;s laws of motion:<\/h3>\n\n\n\n<ol>\n<li><strong>First law:<\/strong> Body remains in it&#8217;s state of rest or motion (inertia)<\/li>\n\n\n\n<li><strong>Second law:<\/strong> Force, \\(F\\) = <mark>Rate of change of linear momentum (\\(mv\\)) with respect to time<\/mark><br><span class=\"indenter-span\">or  <span class=\"border-curly-red\">\\(F = ma\\)<\/span>  <span class=\"hint-tooltip\">\\[ F = \\frac{ (mv_2 &#8211; mv_1) }{ t }\\] \\[\\ =\\ m\\frac{ (v_2 &#8211; v_1) }{ t }\\] \\[\\ =\\ ma \\]<\/span><\/span><\/li>\n\n\n\n<li><strong>Third law:<\/strong> To every action, there&#8217;s always an equal and opposite reaction.<\/li>\n<\/ol>\n\n\n\n<p class=\"accordion-button tertiary collapsed accordion-title\">Newton&#8217;s laws of rotation<\/p><div class=\"accordion-panel collapsed\">\n<ol>\n<li>Body remains in it&#8217;s state of rest or rotational motion<\/li>\n\n\n\n<li>Torque, \\(T\\) = Rate of change of angular momentum with respect to time<br><span class=\"indenter-span\">or \\[ T = \\frac{mv_2r &#8211; mv_1r}{t} = mr\\left(\\frac{v_2 &#8211; v_1}{t}\\right)\\ =\\ mra\\ =\\ mr^2\\alpha\\ \\ =\\ I\\alpha \\]<\/span><br><span class=\"indenter-span\">or <span class=\"border-curly-red\">\\(T = I\\alpha\\)<\/span><\/span><\/li>\n<\/ol>\n\n\n\n<p><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Mass, Force, Work done, Power, and Energy<\/h3>\n\n\n\n<ol>\n<li><strong>Mass:<\/strong><mark>Mass is the indication of quantity of material<\/mark>.\n<ul>\n<li><mark>Mass is a scalar quantity<\/mark><\/li>\n\n\n\n<li>usually denoted by a small letter \\( m \\)<\/li>\n\n\n\n<li>SI unit of mass is Kilogram, or <strong>Kg<\/strong><br><br><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Force: <\/strong><mark>An external agent or an intersection, that tends to change or maintain the motion of an object<\/mark>\n<ul>\n<li><mark>Force is a vector quantity<\/mark><\/li>\n\n\n\n<li>Usually denoted by capital letter \\( F \\)<\/li>\n\n\n\n<li><span class=\"border-curly-red\">\\( F=ma \\)<\/span><\/li>\n\n\n\n<li>SI unit of force is Newton, or <strong>N<\/strong><br><span class=\"indenter-span\">\\[ N=\\text{Kg}\\frac{m}{s^2} \\]<\/span><\/li>\n\n\n\n<li>Gravitational metric unit of force is Kilogram-force or <strong>Kgf<\/strong>\n<ul>\n<li>1 Kgf = 9.8 N<br><br><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Weight: <\/strong><mark>Weight of any body, is the force exerted by that body due to local gravity<\/mark>.\n<ul>\n<li>A body of mass \\(m\\) has weight, <span class=\"border-curly-red\">\\( w=mg \\) <\/span><span class=\"alignright\">( g is acceleration due to gravity )<\/span><\/li>\n\n\n\n<li><mark><strong>1 Kg<\/strong> mass experiences <strong>9.8 N<\/strong> or <strong>1 Kgf<\/strong> of force due to gravity<\/mark>.<br><br><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Work done:<\/strong><mark>Work is done when the an object is <u class=\"underline\">displaced<\/u> from it&#8217;s original position due to an applied force<\/mark>.<ul><li><mark>Work done is a scalar quantity<\/mark><\/li><li>Usually denoted by capital letter \\(W\\)<\/li><li><mark>Work done is the dot product of Force and displacement vector<\/mark>. Assuming a force F of constant magnitude, work done is calculated as:<ul><li><span class=\"border-curly-red\">\\( W=Fd\\cos\\theta \\)<\/span>    <span class=\"hint-tooltip\">\\(F\\) and \\(d\\) are the magnitude of Force and Displacement vectors, both force and displacement vectors are separated by an angle \\(\\theta\\)<\/span><\/li><li>When the angle between force and displacement vectors is zero (ie. body is moved only in the direction of applied force), then<br><span class=\"indenter-span\">\\( W=Fd \\)<\/span><\/li><\/ul><\/li><\/ul>\n<ul>\n<li><mark>Graphically, Work done is the area under the curve drawn by the force with respect to distance<\/mark>\n<ol>\n<li>When the applied force is constant with respect to the movement of the body, then<br><span class=\"indenter-span\">Work done, \\( W=Fx \\) <span class=\"hint-tooltip\">\\(x\\) is the distance moved by body in the direction of  applied force,<br>or in other words, It is the displacement of body in the direction of applied force<br>If \\(d\\) is the actual displacement of body then, \\( x=d\\cos\\theta \\)<\/span><\/span><\/li>\n\n\n\n<li>When the applied force is not constant (ie. when force is a function of \\(x\\)) ( example: spring force, force applied by a person to push a body ), then<br><span class=\"indenter-span\">Work done, \\[ W=\\int Fdx \\]<\/span><\/li>\n\n\n\n<li>When the force and \\(x\\) has a linear relation (ie. when \\( F\\propto x \\)) ( example: spring force, force applied by a UTM machine ), then<br><span class=\"indenter-span\">Work done, \\[ W=\\frac{1}{2}Fx \\]<\/span><\/li>\n<\/ol>\n<\/li>\n\n\n\n<li><mark>SI Unit of work done<\/mark> is <strong>Jule<\/strong>, or <strong>J <\/strong><br><span class=\"indenter-span\">\\[ J=Nm=Kg\\frac{m^2}{s^2} \\]<\/span><br><\/li>\n\n\n\n<li>#todo attach different workdone curves here<br><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Power :<\/strong><mark>Power is the amount of work done with respect to time<\/mark>\n<ul>\n<li><mark>Power is a scalar quantity<\/mark><\/li>\n\n\n\n<li>Usually denoted by capital letter \\(P\\)<\/li>\n\n\n\n<li><span class=\"border-curly-red\">\\[ P=\\frac{\\text{Work done}}{\\text{time}} \\]<\/span><\/li>\n\n\n\n<li>SI Unit of Power is <strong>Watt<\/strong>, or <strong>W<\/strong>\n<ul>\n<li>\\[ \\text{W}=\\frac{J}{s} \\]<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>&#8220;Watt hour&#8221;, or <strong>Wh<\/strong> is another unit of Power. It is equivalent to the amount of work done in one hour.<br><span class=\"indenter-span\">\\[ \\text{Wh}\\ =\\ \\frac{\\text{Jule}}{\\text{Hour}}\\ =\\ 3600\\ \\frac{J}{s}\\ =3600\\text{ Watt} \\]<\/span><br><br><br><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Energy:  <\/strong><mark>Energy of a system indicates the amount of work that can be done by the system<\/mark>.\n<ul>\n<li>More the energy \\( \\implies \\) More work can be done.<\/li>\n\n\n\n<li>Since it measures the amount of work done,\n<ul>\n<li>Energy is a scalar quantity.<\/li>\n\n\n\n<li>SI unit of energy is also Jule, or <strong>J<\/strong><\/li>\n\n\n\n<li>Other unit of energy includes &#8220;Watt hour&#8221;, or <strong>Wh<\/strong>. It is equivalent to the amount of work that can be done in one hour.<br><span class=\"indenter-span\">\\[ \\text{Wh}\\ =\\ \\frac{\\text{Jule}}{\\text{Hour}}\\ =\\ 3600\\ \\frac{J}{s}\\ =3600\\text{ Watt} \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>A system of mass m, moving with velocity \\(v\\) stores the kinetic energy,\n<ul>\n<li>\\[ K.E.\\ =\\frac{1}{2}mv^2 \\]<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>A system of mass \\(m\\) kept at height \\(h\\) above the ground, it can do work by falling. The energy stored by this system is potential energy,\n<ul>\n<li>\\( P.E.=mgh \\) <span class=\"alignright\"><\/span><span class=\"alignright\">( \\(g\\) is the acceleration due to gravity)<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">Moment of force (Torque)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-23.png\" alt=\"Moment of force F (torque) about point O\" class=\"wp-image-317\" style=\"width:379px;height:230px\" width=\"379\" height=\"230\"\/><figcaption class=\"wp-element-caption\">Moment of force F (torque) about point O<\/figcaption><\/figure><\/div>\n\n\n<p>Moment of a force \\(F\\) about a point O when the force is applied on a point P is given by:<\/p>\n\n\n\n<p><span class=\"border-curly-red\">\\( \\vec{M}=F\\ d\\sin\\theta\\ \\hat n \\)<\/span><\/p>\n\n\n\n<ul>\n<li>Moment of force (Torque) is a vector quantity.<\/li>\n\n\n\n<li>Magnitude of moment of force is:\n<ul>\n<li><span class=\"border-curly-red\">\\( M=Fd\\sin\\theta \\)<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>From the diagram, it is clear that:<br>Magnitude of <strong>moment of force = Force \\(\\times\\) perpendicular distance <\/strong>of point O from the line of action of force<\/li>\n\n\n\n<li>Moment of force is always taken with reference to a point (that may or may not lie on the line of action of force).<\/li>\n\n\n\n<li>Moment of force is also refereed to as <strong>Torque<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\" id=\"varignons_theorem\">Varignon&#8217;s theorem (Principle of moments)<\/h4>\n\n\n\n<p>Varignon&#8217;s theorem states that the Moment of Resultant of two forces is equal to the algebraic sum of moment of individual forces.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Law of moments<\/h4>\n\n\n\n<p>When a body is in equilibrium, the sum of all clockwise moments on the body is equal to the sum of all anti-clockwise moments.<br>\\( \\Rightarrow \\) For an equilibrium system, \\[ \\sum_{ }^{ }M=0 \\]<\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Couple<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-25.png\" alt=\"Couple\" class=\"wp-image-335\" style=\"width:266px;height:133px\" width=\"266\" height=\"133\"\/><figcaption class=\"wp-element-caption\">Couple of two forces of magnitude F, separated by distance l (arm of couple)<\/figcaption><\/figure><\/div>\n\n\n<p><mark>Two equal, parallel, and opposite forces, separated together by a finite distance form a <strong>couple<\/strong>.<\/mark> <\/p>\n\n\n\n<p>The distance between two opposite forces is called the <strong>Arm of Couple.<\/strong><\/p>\n\n\n\n<ul>\n<li>Moment of forces forming a couple is given as\n<ul>\n<li><span class=\"border-curly-red\">\\[ M=Fl \\]<\/span>  ( \\(l\\) is the arm of couple)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>The Moment of forces forming a couple <strong>remains same at every point<\/strong> that lies on the plane of couple.<\/li>\n\n\n\n<li>A couple can not be balanced by single force, It can be balanced only by another couple of opposite direction and equal moment of force.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading thin-col1\">Linear-Rotational Parallels<\/h3>\n\n\n\n<figure class=\"wp-block-table thin-col1\"><table><tbody><tr><td><strong>Sr.no.<\/strong><\/td><td class=\"has-text-align-left\" data-align=\"left\"><strong>Linear Motion<\/strong><\/td><td><strong>Rotational Motion<\/strong> ( radius = r )<\/td><\/tr><tr><td>1<\/td><td class=\"has-text-align-left\" data-align=\"left\">Displacement, \\(d\\)<\/td><td>Angular displacement, \\(\\theta\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\( \\theta=d\/r \\)<\/span><\/span><\/td><\/tr><tr><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">Velocity, \\(v\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ v=d\/t \\]<\/span><\/span><br><\/td><td>Angular velocity, \\( \\omega \\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ w=\\theta \/t \\]<\/span><br><span class=\"border-curly-black\">\\[ w=v\/r \\]<\/span><\/span><\/td><\/tr><tr><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">Acceleration, \\(a\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ a = v\/t \\]<\/span><\/span><br><\/td><td>Angular acceleration, \\(\\alpha\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[\\alpha = w\/t\\]<\/span><br><span class=\"border-curly-black\">\\[\\alpha = a\/r\\]<\/span><\/span><\/td><\/tr><tr><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">Mass, \\(m\\)<br><br><\/td><td>moment of Inertia, \\(I\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[I=mr^2\\]<\/span><\/span><\/td><\/tr><tr><td>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">Force, \\(F\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[F = ma\\]<\/span><\/span><\/td><td>Torque (moment of Force), \\(T\\) or \\( \\tau \\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[\\tau = I\\alpha\\]<\/span><br><span class=\"border-curly-black\">\\[\\tau = Fr\\]<\/span><\/span><\/td><\/tr><tr><td>6<\/td><td class=\"has-text-align-left\" data-align=\"left\">Work, \\(W\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ W=Fd \\]<\/span><\/span><\/td><td>Work, \\(W\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[W=\\tau\\theta\\]<\/span><\/span><\/td><\/tr><tr><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">Power, \\(P\\)<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[P=Fv\\]<\/span><\/span><\/td><td>Power, P<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[P=\\tau \\omega\\]<\/span><\/span><\/td><\/tr><tr><td>8<\/td><td class=\"has-text-align-left\" data-align=\"left\">Energy<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ K.E.=\\frac{1}{2}mv^2 \\]<\/span><\/span><\/td><td>Energy<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\[ K.E.=\\frac{1}{2}I\\omega^2 \\]<\/span><\/span><\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\">Linear-rotational parallel terms, This table can be used to convert equations of linear characteristics into the equations of rotational characteristics<\/figcaption><\/figure>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<hr class=\"wp-block-separator has-css-opacity\"\/>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Vectors<\/h2>\n\n\n\n<p>Read explanation for Vectors <a href=\"http:\/\/spacican.com\/notes\/vectors-a-complete-tutorial-for-beginners\/\">here<\/a>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Vector basics<\/h3>\n\n\n\n<ul>\n<li>If \\(\\vec{V}\\) or \\( \\textbf{V} \\) denotes a vector, then\n<ul>\n<li>It&#8217;s magnitude is denoted by \\( \\left|\\vec{V}\\right| \\) or \\( \\left|\\textbf{V}\\right| \\) or \\(V\\)<\/li>\n\n\n\n<li>A unit vector having the direction parallel to \\(\\vec{V}\\) is denoted by \\(\\hat{v}\\)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Unit vector parallel to a vector \\(\\vec{V}\\) is given by, <span class=\"border-curly-red\">\\[ \\hat{v}=\\vec{\\frac{V}{\\left|\\vec{V}\\right|}}=\\vec{\\frac{V}{V}} \\]<\/span><\/li>\n\n\n\n<li>Angle made by \\[ \\vec{V}=xi+yj \\] with X axis, <span class=\"border-curly-red\">\\[ \\theta=\\tan^{-1}\\left[\\frac{y}{x}\\right] \\]<\/span>\n<ul>\n<li>or, if Angle \\(\\theta\\) is known, components can be calculated as,<br><span class=\"indenter-span\"><span class=\"border-curly-red\">\\(x = V\\cos\\theta \\)<br>\\(y = V\\sin\\theta \\)<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Magnitude of vector \\[ \\vec{V}=xi+yj \\] is given by,  <span class=\"border-curly-red\">\\[ V=\\sqrt{x^2+y^2} \\]<\/span><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Parallelogram law of vectors<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-7.png\" alt=\"Parallelogram law of vectors\" class=\"wp-image-483\" style=\"width:438px;height:191px\" width=\"438\" height=\"191\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-7.png 908w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-7-768x336.png 768w\" sizes=\"(max-width: 438px) 100vw, 438px\" \/><figcaption class=\"wp-element-caption\">Parallelogram law of vectors<\/figcaption><\/figure><\/div>\n\n\n<p>If \\( \\vec{A}\\text{ and }\\vec{B} \\) are two vectors parallel-shifted to denote two sides of a parallelogram, <\/p>\n\n\n\n<p class=\"has-text-align-center\">then resultant, <span class=\"border-curly-red\">\\( \\vec{R}=\\vec{A}+\\vec{B} \\)<\/span><\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Triangle law of vectors<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-3.png\" alt=\"Triangle law\" class=\"wp-image-166\" style=\"width:519px;height:408px\" width=\"519\" height=\"408\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-3.png 981w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-3-768x604.png 768w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-3-300x236.png 300w\" sizes=\"(max-width: 519px) 100vw, 519px\" \/><figcaption class=\"wp-element-caption\">Triangle law<\/figcaption><\/figure><\/div>\n\n\n<p>For a system of 3 vectors as shown in figure,<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"border-curly-red\">\\[ \\begin{aligned} &amp;\\vec{R}=\\vec{A}+\\vec{B}\\\\ \\text{or}\\ \\ &amp; \\vec{A}=\\vec{R}-\\vec{B}\\\\ \\text{or}\\ \\ &amp;\\vec{B}=\\vec{R}-\\vec{A} \\end{aligned} \\] <\/span><\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Resultant and resolution of vectors<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-9.png\" alt=\"Resultant of two vectors when magnitude and angle between them is known\" class=\"wp-image-497\" style=\"width:316px;height:203px\" width=\"316\" height=\"203\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-9.png 782w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-9-768x494.png 768w\" sizes=\"(max-width: 316px) 100vw, 316px\" \/><figcaption class=\"wp-element-caption\">Two vectors A and B making angle alpha and beta with vector R <\/figcaption><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-10.png\" alt=\"Resultant of two vectors A and B perpendicular to each other\" class=\"wp-image-500\" style=\"width:302px;height:195px\" width=\"302\" height=\"195\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-10.png 782w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/09\/drawing-10-768x494.png 768w\" sizes=\"(max-width: 302px) 100vw, 302px\" \/><figcaption class=\"wp-element-caption\">Two orthogonal vector A and B making angle alpha and beta (90 &#8211; alpha) with vector R<\/figcaption><\/figure><\/div>\n\n\n<ul>\n<li>Resultant of two vectors \\(\\vec{A}\\) and \\(\\vec{B}\\) separated by angle \\(\\theta\\)\n<ul>\n<li>Magnitude, <span class=\"border-curly-red\">\\( R=\\sqrt{A^2+B^2+2AB\\cos\\theta} \\)<\/span><\/li>\n\n\n\n<li>Angle made from \\(\\vec{A}\\), <span class=\"border-curly-red\">\\[ \\alpha=\\tan^{-1}\\left[\\frac{B\\sin\\theta}{A+B\\cos\\theta}\\right] \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resultant of two <u class=\"underline\">orthogonal vectors<\/u> \\(\\vec{A}\\) and \\(\\vec{B}\\)\n<ul>\n<li>Magnitude, <span class=\"border-curly-red\">\\( R=\\sqrt{A^2+B^2} \\)<\/span><\/li>\n\n\n\n<li>Angle made from \\(\\vec{A}\\), <span class=\"border-curly-red\">\\[ \\alpha=\\tan^{-1}\\left[\\frac{B}{A}\\right] \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resolution of a vector \\(\\vec{R}\\) along two vectors \\(\\vec{A}\\) and \\(\\vec{B}\\) separated from \\(\\vec{R}\\) by angle \\(\\alpha\\) and \\(\\beta\\) respectively is given by:\n<ul>\n<li>Magnitude, <span class=\"border-curly-red\">\\[ A=R\\frac{\\sin\\beta}{\\sin\\left(\\alpha+\\beta\\right)} \\]<\/span><\/li>\n\n\n\n<li>Magnitude, <span class=\"border-curly-red\">\\[ B=R\\frac{\\sin\\alpha}{\\sin\\left(\\alpha+\\beta\\right)} \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resolution of a vector \\(\\vec{R}\\) along two orthogonal vectors \\(\\vec{A}\\) and \\(\\vec{B}\\) separated from \\(\\vec{R}\\) by angle \\(\\alpha\\) and \\[ 90^{\\circ}-\\alpha \\] receptively is given by:\n<ul>\n<li>Magnitude, <span class=\"border-curly-red\">\\( A=R\\cos\\alpha \\)<\/span><\/li>\n\n\n\n<li>Magnitude, <span class=\"border-curly-red\">\\( B=R\\sin\\alpha \\)<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Vector product<\/h3>\n\n\n\n<p>If \\(vec{A}\\) and \\(\\vec{B}\\) are two vectors, then<\/p>\n\n\n\n<ul>\n<li><strong>Dot product:<\/strong> It is the product of magnitude of \\(\\vec{A}\\) and magnitude of projection of \\(\\vec{B}\\) onto \\(\\vec{A}\\)<ul><li><span class=\"border-curly-red\">\\( \\vec{A}\\cdot\\vec{B}=AB\\cos\\theta \\)<\/span><\/li><\/ul>\n<ul>\n<li>Dot product of two vectors results in a <u class=\"underline\">scalar quantity<\/u><\/li>\n\n\n\n<li>Work done is an example of dot product of force and displacement vector.\n<ul>\n<li>\\( W=fd\\cos\\theta \\)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Cross product:<\/strong> Cross product of two vectors results in a <u class=\"underline\">vector quantity<\/u> whose magnitude is equal to the area of the parallelogram formed by two vectors, and direction is perpendicular to the plane containing the two vectors.\n<ul>\n<li><span class=\"border-curly-red\">\\( \\vec{A}\\times\\vec{B}=AB\\sin\\theta\\ \\ \\hat{n} \\)<\/span><\/li>\n\n\n\n<li>Torque is an example of cross product of force and position vector of point on which force is applied\n<ul>\n<li>\\( \\vec{T}=Fr\\sin\\theta\\ \\ \\hat{n} \\) <span class=\"hint-tooltip\">\\(r\\) is the position vector of point on which force was applied<br>\\(\\theta\\) is the angle between \\(F\\) and \\(r\\)<br>\\(\\hat{n}\\) is a unit vector perpendicular to both \\(F\\) and \\(r\\)<\/span><\/li>\n\n\n\n<li>or, \\( T=Fr\\sin\\theta\\ \\)<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>The direction of \\(\\hat{n}\\) is determined by using the Right hand rule.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-css-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Force system<\/h2>\n\n\n\n<p>Group of two or more forces acting on a single system\/object is called a force system ( or system of force ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Types of force system<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing.png\" alt=\"system of forces\" class=\"wp-image-58\" style=\"width:387px;height:323px\" width=\"387\" height=\"323\" title=\"system of forces\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing.png 1352w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-768x641.png 768w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-300x251.png 300w\" sizes=\"(max-width: 387px) 100vw, 387px\" \/><figcaption class=\"wp-element-caption\">System of (a) concurrent; (b) parallel like; (c) parallel unlike; and, (d) non-concurrent non-parallel forces<\/figcaption><\/figure><\/div>\n\n\n<ol start=\"1\">\n<li><strong>Co-planer forces:<\/strong> All forces lie in a single plane\n<ol>\n<li><strong>concurrent forces:<\/strong> All forces pass through a common point<\/li>\n\n\n\n<li><strong>parallel forces:<\/strong> All forces are parallel to each other\n<ol>\n<li><strong>like forces:<\/strong> Parallel forces acting in same direction<\/li>\n\n\n\n<li><strong>unlike forces:<\/strong> Parallel forces acting in opposite direction<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li>Non-cuncurrent non-parrallel forces<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li><strong>Non co-planer forces:<\/strong> System of non co-planer forces is system of two or more forces that lie in more than one plane.<br><span class=\"indenter-span\">Types of non co-planer forces are analogous to that of co-planer forces<\/span><\/li>\n<\/ol>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Resultant and resolution of Concurrent Forces<\/h3>\n\n\n\n<ul>\n<li>Resultant of two forces \\( F_1 \\) and \\(F_2\\) separated by angle \\(\\theta\\)\n<ul>\n<li>resultant, <span class=\"border-curly-red\">\\( R=\\sqrt{F_1^2+F_2^2+2F_1F_2\\cos\\theta} \\)<\/span><\/li>\n\n\n\n<li>angle of resultant \\(R\\) made with force \\(F_1\\), <span class=\"border-curly-red\">\\[ \\alpha=\\tan^{-1}\\left[\\frac{F_2\\sin\\theta}{F_1+F_2\\cos\\theta}\\right] \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resultant of two orthogonal forces \\(F_1\\) and \\(F_2\\)\n<ul>\n<li>resultant, <span class=\"border-curly-black\">\\( R=\\sqrt{F_1^2+F_2^2} \\)<\/span><\/li>\n\n\n\n<li>angle made with \\(F_1\\), <span class=\"border-curly-black\">\\[ \\alpha=\\tan^{-1}\\left[\\frac{F_2}{F_1}\\right] \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resolution of a force \\(F\\) into two forces \\(F_1\\) and \\(F_2\\) separated by angles \\(\\alpha\\) and \\(\\beta\\) respectively\n<ul>\n<li><span class=\"border-curly-red\">\\[ F_1=F\\frac{\\sin\\beta}{\\sin\\left(\\alpha+\\beta\\ \\right)} \\]<\/span><\/li>\n\n\n\n<li><span class=\"border-curly-red\">\\[ F_2=F\\frac{\\sin\\alpha}{\\sin\\left(\\alpha+\\beta\\ \\right)} \\]<\/span><\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Resolution of a force into two orthogonal forces \\(F_1\\) and \\(F_2\\) separated by angles \\(\\alpha\\) and \\(\\beta\\) ( \\( =\\alpha+90^{\\circ} \\) ) respectively\n<ul>\n<li><span class=\"border-curly-black\">\\( F_1\\ =\\ F\\cos\\alpha\\ =\\ F\\sin\\beta \\)<\/span><\/li>\n\n\n\n<li><span class=\"border-curly-black\">\\( F_2\\ =\\ F\\cos\\beta\\ =\\ F\\sin\\alpha \\)<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Theorem of Resolution (Theorem of resolved parts)<\/h3>\n\n\n\n<p>The algebraic sum of resolved parts of two vectors in any given direction equals to the resolved part of the resultant vector in the same direction.<\/p>\n\n\n\n<h3 class=\"wp-block-heading clear-both\">Principle of Transmissibility<\/h3>\n\n\n\n<p>States that, If the Force on a system is translated along its <u class=\"underline\">line of action<\/u> (keeping the direction unchanged), then the system remains unchanged &amp; unaffected.<\/p>\n\n\n\n<p>#todo diagram<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Resultant of Parallel forces<\/h3>\n\n\n\n<ul>\n<li>The resultant of two or more parallel forces is also parallel to those forces.<\/li>\n\n\n\n<li>The magnitude of Resultant of parallel forces is algebraic sum of magnitude of each force.<br>=&gt; <span class=\"border-curly-red\">\\[ R=\\sum F_i \\]<\/span><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-24.png\" alt=\"Resultant of parallel forces\" class=\"wp-image-331\" style=\"width:478px;height:255px\" width=\"478\" height=\"255\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-24.png 775w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-24-768x409.png 768w\" sizes=\"(max-width: 478px) 100vw, 478px\" \/><figcaption class=\"wp-element-caption\">Resultant of parallel forces<\/figcaption><\/figure><\/div>\n\n\n<ul>\n<li>The position of Resultant of parallel forces is obtained with the help of <a href=\"#varignons_theorem\">Varignon&#8217;s theorem<\/a> by equating Moment of Resultant force and algebraic sum of moment of each force.\n<ul>\n<li>\\[ Rx=\\sum_{i=1}^nF_ix_i \\]\n<ul>\n<li> <span class=\"border-curly-red\">\\[ x=\\frac{\\sum F_ix_i}{\\sum F_i} \\]<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p>Hence, For the system of parallel point forces as shown in above figure,<\/p>\n\n\n\n<p class=\"has-text-align-center\">resultant <span class=\"border-curly-black\">\\( R=F_1-F_2+F_3+F_4 \\)<\/span><\/p>\n\n\n\n<p>and, \\( Rx=F_1x_1-F_2x_2+F_3x_3+F_4x_4 \\) or,<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"border-curly-black\">\\[ x=\\frac{F_1x_1-F_2x_2+F_3x_3+F_4x_4}{R} \\]<\/span><\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading clear-both\">Polygon law of forces (or vectors)<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignleft size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-8.png\" alt=\"System of forces\" class=\"wp-image-187\" style=\"width:289px;height:205px\" width=\"289\" height=\"205\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-8.png 635w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-8-300x214.png 300w\" sizes=\"(max-width: 289px) 100vw, 289px\" \/><figcaption class=\"wp-element-caption\">System of forces<\/figcaption><\/figure><\/div>\n\n\n<p>Polygon law states that, <mark>If all the forces of a system can be represented by the sides of a polygon taken in same order with same angle\/direction, Then the resultant of those forces can be represented by the <u class=\"underline\">closing side<\/u> of that polygon taken in <u class=\"underline\">opposite order<\/u><\/mark>.<\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignleft size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-6.png\" alt=\"Polygon law\" class=\"wp-image-183\" style=\"width:291px;height:210px\" width=\"291\" height=\"210\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-6.png 638w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-6-300x216.png 300w\" sizes=\"(max-width: 291px) 100vw, 291px\" \/><figcaption class=\"wp-element-caption\">Polygon Law (Representing forces by sides of a polygon)<\/figcaption><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"alignleft size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-9.png\" alt=\"Polygon law\" class=\"wp-image-212\" style=\"width:295px;height:230px\" width=\"295\" height=\"230\"\/><figcaption class=\"wp-element-caption\">Polygon law (identifying resultant)<\/figcaption><\/figure><\/div>\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<p><mark>If the Polygon formed by the forces is a <strong>closed polygon<\/strong>, then the <strong>Resultant of the forces is Zero<\/strong><\/mark><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-11.png\" alt=\"system of balanced forces\" class=\"wp-image-225\" style=\"width:412px;height:193px\" width=\"412\" height=\"193\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-11.png 1269w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-11-768x359.png 768w\" sizes=\"(max-width: 412px) 100vw, 412px\" \/><figcaption class=\"wp-element-caption\">A system of balanced forces forms a closed polygon<\/figcaption><\/figure>\n\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Sine rule<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-12.png\" alt=\"sine rule for triangles\" class=\"wp-image-238\" style=\"width:303px;height:250px\" width=\"303\" height=\"250\"\/><figcaption class=\"wp-element-caption\">sine rule<\/figcaption><\/figure><\/div>\n\n\n<p>For any closed triangle with sides a, b, and c, having the angles \\( \\alpha,\\ \\beta,\\) and \\(\\gamma\\) opposite to the sides respectively,<br><br><span class=\"border-curly-red\">\\[ \\frac{a}{\\sin\\alpha}=\\frac{b}{\\sin\\beta}=\\frac{c}{\\sin\\gamma} \\]<\/span><\/p>\n\n\n\n<p>or \\[ \\frac{\\sin\\alpha}{a}=\\frac{\\sin\\beta}{b}=\\frac{\\sin\\gamma}{c} \\]<\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Lami&#8217;s theorem<\/h3>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-13.png\" alt=\"equilibrium system of concurrent forces (Lami's theorem)\" class=\"wp-image-244\" style=\"width:256px;height:211px\" width=\"256\" height=\"211\"\/><figcaption class=\"wp-element-caption\">Equilibrium system of concurrent forces (Lami&#8217;s theorem)<\/figcaption><\/figure><\/div>\n\n\n<p>Lami&#8217;s theorem states that, if the system of three concurrent forces is in equilibrium, then each force is proportional to the sin of angle made by other two forces<\/p>\n\n\n\n<p>For the equilibrium system shown in figure,<\/p>\n\n\n\n<p><span class=\"border-curly-red\">\\[ \\frac{F_1}{\\sin\\alpha}=\\frac{F_2}{\\sin\\beta}=\\frac{F_3}{\\sin\\gamma} \\]<\/span><\/p>\n\n\n\n<p>Lami&#8217;s theorem can be proved by applying polygon law and sin rule<\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<!--nextpage-->\n\n\n\n<h2 class=\"wp-block-heading\">Mechanical Equilibrium<\/h2>\n\n\n\n<p>A system is in mechanical equilibrium when the sum of all the forces and moment of forces acting upon the system is zero.<br> \\( \\Rightarrow \\) In an equilibrium system, \\[ \\sum_{ }^{ }F=\\sum_{ }^{ }M=0 \\]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Types of mechanical equilibrium<\/h3>\n\n\n\n<ul>\n<li><strong>Static Equilibrium:<\/strong> When a system is at rest and is in mechanically equilibrium, the system is said to be in static equilibrium.\n<ul>\n<li>A statically equilibrium system will always be at rest. It will start moving only when an external force is applied. By applying an external force, the system will no longer be in mechanical equilibrium.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Dynamic Equilibrium:<\/strong> If a system is in motion with a constant velocity and if the sum of all the forces acting upon it is zero, then the system is said to be in dynamic equilibrium.\n<ul>\n<li>A dynamically equilibrium system will keep moving with the constant velocity forever. The motion of  such system can be changed by applying an external force i.e. by braking the conditions of equilibrium.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Free body diagrams (FBD) <\/h2>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"alignright size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-15.png\" alt=\"free body diagram (mass on inclined plane)\" class=\"wp-image-250\" style=\"width:481px;height:251px\" width=\"481\" height=\"251\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-15.png 965w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-15-768x400.png 768w\" sizes=\"(max-width: 481px) 100vw, 481px\" \/><figcaption class=\"wp-element-caption\">Actual system (on left) with it&#8217;s Free Body Diagram (on right)<\/figcaption><\/figure><\/div>\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<p>A free body diagram of a system is the symbolic representation of system with the help of <u class=\"underline\">all the forces<\/u> (including reaction forces) acting on it. The various types of forces included (but not limited to) in FBD&#8217;s are Reaction force, Friction force, Gravitational force, and Applied force<\/p>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Points to remember when drawing Free body diagrams<\/h3>\n\n\n\n<ul>\n<li>Free body diagram must depict all the forces acting on the system.<\/li>\n\n\n\n<li>For normal contact surfaces (or at Hinged joints), Friction force along with normal force is also considered when drawing FBD&#8217;s. Both Friction force and Normal force add up to make the resultant reaction force that may or may not be normal to the contact surface.<\/li>\n\n\n\n<li>For smooth contact surfaces (or at roller supports), Friction force doesn&#8217;t exist, hence only Normal force is considered. The reaction force at smooth contact surfaces is equal to the normal force, always perpendicular to the surface at point of contact.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-16.png\" alt=\"Free body diagram of a rod supported on wall\" class=\"wp-image-268\" style=\"width:680px;height:308px\" width=\"680\" height=\"308\" srcset=\"https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-16.png 1106w, https:\/\/spacican.com\/notes\/wp-content\/uploads\/2020\/08\/drawing-16-768x349.png 768w\" sizes=\"(max-width: 680px) 100vw, 680px\" \/><figcaption class=\"wp-element-caption\">Free body diagram of a rod supported on wall with the help of rollers<\/figcaption><\/figure>\n\n\n\n<div class=\"wp-block-create-block-clear-both clear-both\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">More Free body diagram examples<\/h3>\n\n\n\n<p>#todo attach example images here.<\/p>\n\n\n\n<div class=\"wp-block-create-block-extra-margin-top extra-margin-top\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Trusses and frames<\/h2>\n\n\n\n<ul>\n<li>Truss is a group of beams (called members) assembled together to create a rigid structure.<\/li>\n<\/ul>\n\n\n\n<p>#todo diagram of a simple truss. mark members and joints with arrows<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Classification of truss<\/h3>\n\n\n\n<ul>\n<li><strong>Planar truss:<\/strong> All the members and joints of truss lie in a single plane.<\/li>\n\n\n\n<li><strong>Space truss:<\/strong> Members and Joints of such truss lie in 3 dimensions.<\/li>\n\n\n\n<li><strong>Perfect truss:<\/strong> A truss is said to be perfect when the number of members and joints of the truss has the following relation:<ul><li>\\( m=2j-3 \\)<\/li><li>Here \\(m\\) is number of members in the truss, and \\(j\\) is the number of joints in the truss.<\/li><\/ul>\n<ul>\n<li>#todo diagram of perfect truss<\/li>\n\n\n\n<li>A perfect truss is Rigid in nature, no deformation occurs when load is applied on such trusses.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Deficient truss:<\/strong> The number of members and joins of deficient truss hast the following relation:\n<ul>\n<li>\\( m&lt;2j-3 \\)<\/li>\n\n\n\n<li>#todo diagram of deficient truss<\/li>\n\n\n\n<li>Deficient trusses are not rigid in nature, any kind of load applied causes deformation in such truss.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Redundant truss:<\/strong> The number of members and joints of redundant truss has the following relation:\n<ul>\n<li>\\( m&gt;2j-3 \\)<\/li>\n\n\n\n<li>#todo diagram of redundant truss<\/li>\n\n\n\n<li>Redundant trusses are also rigid in nature.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Statistically determinant trusses:<\/strong><\/li>\n\n\n\n<li><strong>Statistically in-determinant trusses:<\/strong><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Read or print these complete Engineering mechanics short notes created for GATE revision with numerical problems.<\/p>\n","protected":false},"author":1,"featured_media":72,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[9,10,8,7],"_links":{"self":[{"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/posts\/23"}],"collection":[{"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/comments?post=23"}],"version-history":[{"count":546,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/posts\/23\/revisions"}],"predecessor-version":[{"id":1254,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/posts\/23\/revisions\/1254"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/media\/72"}],"wp:attachment":[{"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/categories?post=23"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spacican.com\/notes\/wp-json\/wp\/v2\/tags?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}