[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fspatial-motion-of-a-rigid-body\u002Fspherical-motion-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},225,"Spherical Motion of a Rigid Body","en","theoretical-mechanics\u002Fkinematics\u002Fspatial-motion-of-a-rigid-body\u002Fspherical-motion-rigid-body","Spherical motion of a rigid body with one fixed point: angular velocity, instantaneous axis of rotation and point velocities.","This topic introduces spherical motion of a rigid body with one fixed point, including the instantaneous axis of rotation and angular-velocity vector.","\u003Cp>\u003Cstrong>Spherical motion of a rigid body\u003C\u002Fstrong> is three-dimensional motion in which one point of the body remains fixed. Every other point moves on a spherical surface centered at that fixed point.\u003C\u002Fp>\u003Ch2>Geometry of spherical motion\u003C\u002Fh2>\u003Cp>Let point $O$ be fixed. The distance $OP$ to any body point $P$ is constant, so the trajectory of $P$ lies on a sphere of radius $OP$. Unlike rotation about a fixed axis, the direction of the instantaneous rotation axis generally changes with time.\u003C\u002Fp>\u003Ch2>Instantaneous angular velocity\u003C\u002Fh2>\u003Cp>For a rigid body with a fixed point $O$, the velocity of any point $P$ at an instant is determined by the instantaneous angular velocity:\u003C\u002Fp>\u003Cp>$$\\vec v_P=\\vec\\omega\\times\\vec r_{P\u002FO}.$$\u003C\u002Fp>\u003Cp>Points on the instantaneous axis of rotation have zero velocity at that instant. For other points, velocity is perpendicular to the plane formed by $\\vec\\omega$ and $\\vec r_{P\u002FO}$.\u003C\u002Fp>\u003Cp>The vector $\\vec\\omega$ describes the instantaneous rotation of the body. The line through fixed point $O$ in the direction of $\\vec\\omega$ is the instantaneous axis of rotation.\u003C\u002Fp>\u003Ch2>Point velocities\u003C\u002Fh2>\u003Cp>The speed of point $P$ is $v_P=\\omega r_\\perp$, where $r_\\perp$ is the perpendicular distance from the point to the instantaneous axis. Points on the instantaneous axis therefore have zero velocity at that instant.\u003C\u002Fp>\u003Ch2>Changing instantaneous axis\u003C\u002Fh2>\u003Cp>The instantaneous axis is not generally one material line that remains fixed throughout the motion. Its position changes both within the body and in fixed space. This distinguishes general spherical motion from simple fixed-axis rotation.\u003C\u002Fp>\u003Ch2>Angular acceleration\u003C\u002Fh2>\u003Cp>Angular acceleration is $\\vec\\varepsilon=d\\vec\\omega\u002Fdt$ in the fixed reference frame. Because both magnitude and direction of $\\vec\\omega$ may change, $\\vec\\varepsilon$ and $\\vec\\omega$ are not generally parallel.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>If at an instant $\\omega=6$ rad\u002Fs and a point is at a perpendicular distance of 0.15 m from the instantaneous axis, its speed is $v=6\\cdot0.15=0.9$ m\u002Fs.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>assuming the instantaneous axis remains fixed throughout spherical motion;\u003C\u002Fli>\u003Cli>using the full distance $OP$ instead of perpendicular distance to the instantaneous axis in $v=\\omega r_\\perp$;\u003C\u002Fli>\u003Cli>assuming $\\vec\\varepsilon$ is always parallel to $\\vec\\omega$;\u003C\u002Fli>\u003Cli>confusing spherical rigid-body motion with the motion of a single particle on a sphere.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},196,"kinematics-spherical-motion-velocity","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},288,"Spatial Motion of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Fspatial-motion-of-a-rigid-body",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fspatial-motion-of-a-rigid-body\u002Fspherical-motion-rigid-body","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fprostorovyi-rukh-tverdoho-tila\u002Fsferychnyi-rukh-tverdoho-tila",1787712536901]