[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fpoint-accelerations-plane-motion":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},222,"Accelerations of Points in Plane Motion","en","theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fpoint-accelerations-plane-motion","Acceleration relation for points of a plane rigid body: reference-point, tangential and normal components.","This topic explains how to determine accelerations of points in plane rigid-body motion from reference-point acceleration, angular velocity and angular acceleration.","\u003Cp>In \u003Cstrong>plane rigid-body motion\u003C\u002Fstrong>, accelerations of different points are related through the acceleration of a selected reference point, the angular velocity, and the angular acceleration of the body.\u003C\u002Fp>\u003Ch2>Acceleration relation\u003C\u002Fh2>\u003Cp>The acceleration of point $B$ of a plane rigid body relative to a selected reference point $A$ is:\u003C\u002Fp>\u003Cp>$$\\vec a_B=\\vec a_A+\\vec\\varepsilon\\times\\vec r_{B\u002FA}+\\vec\\omega\\times(\\vec\\omega\\times\\vec r_{B\u002FA}).$$\u003C\u002Fp>\u003Cp>The relative tangential component has magnitude $a^\\tau_{B\u002FA}=|\\varepsilon|AB$ and is perpendicular to $AB$. The normal component has magnitude $a^n_{B\u002FA}=\\omega^2AB$ and points from $B$ toward $A$.\u003C\u002Fp>\u003Cp>Unlike the velocity relation, the relative acceleration contains two components: a tangential component caused by changing angular velocity and a normal component caused by changing direction of relative velocity.\u003C\u002Fp>\u003Ch2>Tangential component\u003C\u002Fh2>\u003Cp>The vector $\\vec a^\\tau_{B\u002FA}=\\vec\\varepsilon\\times\\vec r_{B\u002FA}$ is perpendicular to $AB$. Its magnitude is $|\\varepsilon|AB$, and its direction follows from the sign of angular acceleration.\u003C\u002Fp>\u003Ch2>Normal component\u003C\u002Fh2>\u003Cp>The vector $\\vec a^n_{B\u002FA}=\\vec\\omega\\times(\\vec\\omega\\times\\vec r_{B\u002FA})$ points from $B$ toward reference point $A$ and has magnitude $\\omega^2AB$.\u003C\u002Fp>\u003Ch2>Choosing a reference point\u003C\u002Fh2>\u003Cp>A useful reference point is one whose acceleration is known or easily obtained from the constraints. The angular quantities $\\omega$ and $\\varepsilon$ do not depend on which reference point is selected.\u003C\u002Fp>\u003Ch2>Instantaneous center of acceleration\u003C\u002Fh2>\u003Cp>In some cases a point of the plane figure may have zero acceleration, but it must not be confused with the instantaneous center of zero velocity. Zero velocity at an instant does not imply zero acceleration.\u003C\u002Fp>\u003Ch2>Rolling without slipping\u003C\u002Fh2>\u003Cp>The contact point of a wheel rolling without slip on a fixed surface has zero instantaneous velocity, but generally nonzero acceleration. It therefore cannot be used as a fixed center for acceleration analysis in the same way that the IC is used for velocities.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For a link $AB=0.4$ m with $\\omega=5$ rad\u002Fs and $\\varepsilon=3$ rad\u002Fs², the relative acceleration components of $B$ with respect to $A$ have magnitudes $a^\\tau_{B\u002FA}=1.2$ m\u002Fs² and $a^n_{B\u002FA}=10$ m\u002Fs². They must be added vectorially to $\\vec a_A$.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>including only the tangential or only the normal component;\u003C\u002Fli>\u003Cli>directing the normal component from the reference point toward the other point;\u003C\u002Fli>\u003Cli>assuming the instantaneous center of zero velocity has zero acceleration;\u003C\u002Fli>\u003Cli>adding component magnitudes instead of vectors;\u003C\u002Fli>\u003Cli>using different $\\omega$ or $\\varepsilon$ values for different points of the same rigid body.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},190,"kinematics-plane-motion-acceleration-theorem","formula",[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},286,"Plane Motion of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fpoint-accelerations-plane-motion","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fploskoparalelnyi-rukh\u002Fpryskorennia-tochok-pry-ploskoparalelnomu-rusi",1787712536535]