[{"data":1,"prerenderedAt":37},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Fpoint-velocities-accelerations-fixed-axis-rotation":3},{"topic":4,"trail":18,"children":32,"tasks":33,"alternates":34},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},218,"Velocities and Accelerations of Points in Fixed-Axis Rotation","en","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Fpoint-velocities-accelerations-fixed-axis-rotation","Point Velocities and Accelerations in Fixed-Axis Rotation","Linear velocity, tangential, normal and total acceleration of rigid-body points during rotation about a fixed axis.","This topic relates rigid-body angular velocity and angular acceleration to the linear velocity, tangential acceleration, normal acceleration and total acceleration of its points.","\u003Cp>During \u003Cstrong>rotation of a rigid body about a fixed axis\u003C\u002Fstrong>, all points share the same angular velocity $\\omega$ and angular acceleration $\\varepsilon$, but their linear velocities and accelerations depend on perpendicular distance from the axis.\u003C\u002Fp>\u003Ch2>Linear velocity\u003C\u002Fh2>\u003Cp>For a point of a rigid body at perpendicular distance $r$ from the fixed rotation axis, the linear speed is:\u003C\u002Fp>\u003Cp>$$v=\\omega r.$$\u003C\u002Fp>\u003Cp>The velocity vector is tangent to the circle traced by the point and perpendicular to the radius $r$.\u003C\u002Fp>\u003Cp>The farther a point is from the axis, the greater its speed for the same $\\omega$. Points located directly on the axis have $r=0$ and remain fixed.\u003C\u002Fp>\u003Ch2>Vector velocity relation\u003C\u002Fh2>\u003Cp>For a point whose position from the axis is represented by $\\vec r$, velocity can be written as $\\vec v=\\vec\\omega\\times\\vec r$. The cross product automatically gives the tangential direction according to the right-hand rule.\u003C\u002Fp>\u003Ch2>Tangential acceleration\u003C\u002Fh2>\u003Cp>A change in speed produces the tangential component $a_\\tau=\\varepsilon r$. It is tangent to the circular path, with direction determined by the sign of angular acceleration.\u003C\u002Fp>\u003Ch2>Normal acceleration\u003C\u002Fh2>\u003Cp>The change in velocity direction produces the normal component $a_n=\\omega^2r=v^2\u002Fr$, directed from the point toward the rotation axis.\u003C\u002Fp>\u003Ch2>Total acceleration\u003C\u002Fh2>\u003Cp>The tangential and normal components are perpendicular, so $a=\\sqrt{a_\\tau^2+a_n^2}=r\\sqrt{\\varepsilon^2+\\omega^4}$. In vector form, $\\vec a=\\vec\\varepsilon\\times\\vec r+\\vec\\omega\\times(\\vec\\omega\\times\\vec r)$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A point on a disk is $r=0.20$ m from the axis. With $\\omega=10$ rad\u002Fs and $\\varepsilon=4$ rad\u002Fs², $v=2$ m\u002Fs, $a_\\tau=0.8$ m\u002Fs², $a_n=20$ m\u002Fs², and $a\\approx20.02$ m\u002Fs².\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>using distance from the body center instead of perpendicular distance from the axis;\u003C\u002Fli>\u003Cli>confusing $a_\\tau=\\varepsilon r$ with $a_n=\\omega^2r$;\u003C\u002Fli>\u003Cli>assuming point acceleration is zero when $\\omega$ is constant;\u003C\u002Fli>\u003Cli>adding tangential and normal accelerations algebraically instead of vectorially.\u003C\u002Fli>\u003C\u002Ful>",[14],{"id":15,"code":16,"type":17,"locale":7},182,"kinematics-fixed-axis-point-velocity","formula",[19,23,27,31],{"id":20,"name":21,"path":22},81,"Theoretical Mechanics","theoretical-mechanics",{"id":24,"name":25,"path":26},137,"Kinematics","theoretical-mechanics\u002Fkinematics",{"id":28,"name":29,"path":30},285,"Basic Motions of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body",{"id":5,"name":6,"path":8},[],[],{"en":35,"uk":36},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Fpoint-velocities-accelerations-fixed-axis-rotation","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fnaiprostishi-rukhy-tverdoho-tila\u002Fshvydkosti-pryskorennia-tochok-pry-obertanni",1787712536400]