[{"data":1,"prerenderedAt":35},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fplane-motion-rigid-body":3},{"topic":4,"trail":17,"children":30,"tasks":31,"alternates":32},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},220,"Plane Motion of a Rigid Body","en","theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fplane-motion-rigid-body","Plane motion of a rigid body: decomposition into translation of a reference point and rotation about that point.","This topic introduces plane motion of a rigid body and its representation as translation of a reference point combined with rotation of the body about that point.","\u003Cp>\u003Cstrong>Plane motion of a rigid body\u003C\u002Fstrong> is motion in which all points of the body move in planes parallel to a fixed plane. Its kinematics can be studied through the motion of a plane figure representing a section of the body parallel to the plane of motion.\u003C\u002Fp>\u003Ch2>Kinematic decomposition\u003C\u002Fh2>\u003Cp>At any instant, plane motion of a rigid body can be represented as translation with an arbitrarily selected reference point $A$ plus rotation about that point:\u003C\u002Fp>\u003Cp>$$\\vec r_B=\\vec r_A+\\vec r_{B\u002FA}.$$\u003C\u002Fp>\u003Cp>The position of the plane figure is specified by two coordinates of the reference point, such as $x_A(t)$ and $y_A(t)$, together with the orientation angle $\\varphi(t)$.\u003C\u002Fp>\u003Cp>The reference point can be selected arbitrarily. The translational component depends on that choice, but the angular velocity and angular acceleration of the plane figure at an instant do not.\u003C\u002Fp>\u003Ch2>Equations of motion of a plane figure\u003C\u002Fh2>\u003Cp>The three functions $x_A=x_A(t)$, $y_A=y_A(t)$, and $\\varphi=\\varphi(t)$ completely specify the figure position in the plane. The first two describe translation of the reference point and the third describes change of body orientation.\u003C\u002Fp>\u003Ch2>Translational and rotational components\u003C\u002Fh2>\u003Cp>Decomposing the motion into translation and rotation does not mean the body physically performs them one after another. It is a kinematic representation of one actual motion: the reference point moves while the figure simultaneously changes orientation.\u003C\u002Fp>\u003Ch2>Angular characteristics\u003C\u002Fh2>\u003Cp>Angular velocity is $\\omega=\\dot\\varphi$ and angular acceleration is $\\varepsilon=\\ddot\\varphi$. For plane motion, their vectors are perpendicular to the plane of motion.\u003C\u002Fp>\u003Ch2>Special cases\u003C\u002Fh2>\u003Cp>If $\\omega=0$ throughout an interval, the motion reduces to translation. If the reference point is fixed, the motion reduces to rotation about a fixed axis perpendicular to the plane.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A wheel rolling along a straight path undergoes plane motion: its center translates while the wheel simultaneously rotates. The motion of any point on the rim combines these two components.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>treating the reference point as a physically fixed point;\u003C\u002Fli>\u003Cli>assuming $\\omega$ depends on the selected reference point;\u003C\u002Fli>\u003Cli>describing plane-figure position only by one point’s coordinates and omitting $\\varphi$;\u003C\u002Fli>\u003Cli>confusing plane motion with pure translation.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},186,"kinematics-plane-motion-decomposition","definition",[18,22,26,29],{"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":6,"path":28},286,"theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion",{"id":5,"name":6,"path":8},[],[],{"en":33,"uk":34},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fplane-motion-rigid-body","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fploskoparalelnyi-rukh\u002Fploskoparalelnyi-rukh-tverdoho-tila",1787712536496]