[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fpoint-velocities-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},221,"Velocities of Points in Plane Motion","en","theoretical-mechanics\u002Fkinematics\u002Frigid-body-plane-motion\u002Fpoint-velocities-plane-motion","Relative velocity relation for a plane rigid body, instantaneous center of zero velocity and determination of point velocities.","This topic covers point velocities in plane rigid-body motion using a reference point, angular velocity and the instantaneous center of zero velocity.","\u003Cp>In \u003Cstrong>plane rigid-body motion\u003C\u002Fstrong>, different points generally have different velocities. Their velocities are related by the rigid-body relative-velocity equation.\u003C\u002Fp>\u003Ch2>Velocity relation\u003C\u002Fh2>\u003Cp>The velocity of any point $B$ of a plane rigid body equals the vector sum of the velocity of a selected reference point $A$ and the velocity of $B$ due to rotation about $A$:\u003C\u002Fp>\u003Cp>$$\\vec v_B=\\vec v_A+\\vec\\omega\\times\\vec r_{B\u002FA}.$$\u003C\u002Fp>\u003Cp>The relative component is perpendicular to segment $AB$, with magnitude $v_{B\u002FA}=\\omega\\,AB$.\u003C\u002Fp>\u003Cp>Choosing a point $A$ with known velocity as the reference point allows the velocity of any other point $B$ to be determined. The vector $\\vec\\omega\\times\\vec r_{B\u002FA}$ is always perpendicular to $AB$.\u003C\u002Fp>\u003Ch2>Velocity projections\u003C\u002Fh2>\u003Cp>Because the relative velocity $\\vec v_{B\u002FA}$ is perpendicular to $AB$, the velocity components of two points of a rigid body projected onto the line joining them are equal. This property often determines unknown components without a complete vector construction.\u003C\u002Fp>\u003Ch2>Instantaneous center of zero velocity\u003C\u002Fh2>\u003Cp>The \u003Cstrong>instantaneous center of zero velocity (IC)\u003C\u002Fstrong> is a point in the plane whose velocity is zero at the instant considered. If a finite IC $P$ exists, the velocity field of the figure at that instant is equivalent to instantaneous rotation about $P$.\u003C\u002Fp>\u003Ch2>Locating the IC\u003C\u002Fh2>\u003Cp>If the velocity directions of two points are known, draw through each point a line perpendicular to its velocity. Their intersection is the IC when the lines meet at a finite point. For pure translation, the IC is regarded as lying at infinity.\u003C\u002Fp>\u003Ch2>Velocity from the IC\u003C\u002Fh2>\u003Cp>For a point $A$ and known IC $P$, $v_A=|\\omega|PA$. Thus $v_A\u002Fv_B=PA\u002FPB$. Velocity directions are perpendicular to $PA$ and $PB$ and must correspond to one consistent sense of instantaneous rotation.\u003C\u002Fp>\u003Ch2>Rolling without slipping\u003C\u002Fh2>\u003Cp>For a wheel rolling without slip on a fixed surface, the contact point has zero instantaneous velocity and is the IC. Therefore the wheel-center speed satisfies $v_C=\\omega R$ in magnitude.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>If the IC of a plane link is at $P$, $PA=0.2$ m, $PB=0.5$ m, and $v_A=1$ m\u002Fs, then $|\\omega|=1\u002F0.2=5$ rad\u002Fs and $v_B=5\\cdot0.5=2.5$ m\u002Fs.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>treating the IC as one material point fixed for a finite time interval;\u003C\u002Fli>\u003Cli>locating the IC along velocity directions instead of along perpendiculars to them;\u003C\u002Fli>\u003Cli>using $v_A\u002Fv_B=PA\u002FPB$ without a common IC;\u003C\u002Fli>\u003Cli>assuming zero contact-point velocity when rolling includes slip.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},187,"kinematics-plane-motion-velocity-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-velocities-plane-motion","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fploskoparalelnyi-rukh\u002Fshvydkosti-tochok-pry-ploskoparalelnomu-rusi",1787712536509]