[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftranslation-rigid-body":3},{"topic":4,"trail":17,"children":31,"tasks":32,"alternates":33},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},216,"Translation of a Rigid Body","en","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftranslation-rigid-body","Rigid-body translation: trajectories, velocities and accelerations of points and the fundamental properties of translational motion.","This topic explains rigid-body translation and shows that at any instant all points of a translating rigid body have identical velocity and acceleration vectors.","\u003Cp>\u003Cstrong>Translation of a rigid body\u003C\u002Fstrong> is motion in which every line fixed in the body remains parallel to its initial orientation. The body therefore does not rotate relative to the selected reference frame.\u003C\u002Fp>\u003Ch2>Fundamental property\u003C\u002Fh2>\u003Cp>During translation, every line segment fixed in the rigid body remains parallel to itself. For any two points $A$ and $B$ of the body:\u003C\u002Fp>\u003Cp>$$\\vec v_A=\\vec v_B,\\qquad \\vec a_A=\\vec a_B.$$\u003C\u002Fp>\u003Cp>The trajectories of different points have the same shape and differ by a constant vector offset.\u003C\u002Fp>\u003Cp>Consequently, the kinematics of a translating rigid body can be determined by studying any one of its points. The velocity and acceleration found for that point at an instant apply to every other point at the same instant.\u003C\u002Fp>\u003Ch2>Point trajectories\u003C\u002Fh2>\u003Cp>Let $\\vec r_B=\\vec r_A+\\vec r_{AB}$, where $\\vec r_{AB}$ is constant because the body is rigid and its orientation does not change. The trajectory of point $B$ is therefore the trajectory of point $A$ shifted by the constant vector $\\vec r_{AB}$.\u003C\u002Fp>\u003Ch2>Velocities\u003C\u002Fh2>\u003Cp>Differentiating the position relation gives $\\vec v_B=\\vec v_A$ because $d\\vec r_{AB}\u002Fdt=0$. All points have identical velocity vectors at each instant even though their trajectories occupy different locations in space.\u003C\u002Fp>\u003Ch2>Accelerations\u003C\u002Fh2>\u003Cp>Differentiating again gives $\\vec a_B=\\vec a_A$. Thus, for velocity and acceleration analysis, rigid-body translation is kinematically equivalent to the motion of a single particle.\u003C\u002Fp>\u003Ch2>Translation need not be rectilinear\u003C\u002Fh2>\u003Cp>The term “translation” describes unchanged body orientation, not the shape of its trajectory. For example, with a suitable suspension, a Ferris-wheel cabin can undergo curvilinear translation while remaining vertically oriented.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>If one point of a translating platform has velocity $\\vec v=(2\\vec i+3\\vec j)$ m\u002Fs at an instant, every other point has the same velocity vector at that instant. Their acceleration vectors are likewise identical.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>assuming translation must be rectilinear;\u003C\u002Fli>\u003Cli>confusing translation with motion at zero acceleration;\u003C\u002Fli>\u003Cli>assigning different angular velocities to different points of a translating body;\u003C\u002Fli>\u003Cli>assuming identical trajectory shapes must occupy the same geometric curve.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},178,"kinematics-rigid-body-translation","definition",[18,22,26,30],{"id":19,"name":20,"path":21},81,"Theoretical Mechanics","theoretical-mechanics",{"id":23,"name":24,"path":25},137,"Kinematics","theoretical-mechanics\u002Fkinematics",{"id":27,"name":28,"path":29},285,"Basic Motions of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Ftranslation-rigid-body","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fnaiprostishi-rukhy-tverdoho-tila\u002Fpostupalnyi-rukh-tverdoho-tila",1787712536361]