[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Frotation-rigid-body-fixed-axis":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},217,"Rotation of a Rigid Body About a Fixed Axis","en","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Frotation-rigid-body-fixed-axis","Angular position, angular velocity and angular acceleration of a rigid body rotating about a fixed axis.","This topic covers the rotation law of a rigid body about a fixed axis, angular velocity, angular acceleration and the kinematic characteristics of points of the body.","\u003Cp>\u003Cstrong>Rotation of a rigid body about a fixed axis\u003C\u002Fstrong> is motion in which two points of the body, and therefore the line through them, remain fixed. This line is the axis of rotation. Every other point moves on a circle whose center lies on the axis.\u003C\u002Fp>\u003Ch2>Rotation law\u003C\u002Fh2>\u003Cp>Body orientation is specified by the angular position $\\varphi=\\varphi(t)$. A positive angular direction is selected in advance and determines the signs of angular velocity and angular acceleration.\u003C\u002Fp>\u003Ch2>Angular velocity and angular acceleration\u003C\u002Fh2>\u003Cp>If rotation about a fixed axis is described by the angular position $\\varphi=\\varphi(t)$, then:\u003C\u002Fp>\u003Cp>$$\\omega=\\frac{d\\varphi}{dt},\\qquad \\varepsilon=\\frac{d\\omega}{dt}=\\frac{d^2\\varphi}{dt^2}.$$\u003C\u002Fp>\u003Cp>The signs of $\\omega$ and $\\varepsilon$ depend on the selected positive angular direction. Equal signs mean the magnitude of angular velocity increases; opposite signs mean it decreases.\u003C\u002Fp>\u003Cp>In SI, angular position is measured in radians. Angular velocity is commonly expressed in rad\u002Fs and angular acceleration in rad\u002Fs²; the radian is dimensionless in SI, but retaining its symbol is useful for identifying angular quantities.\u003C\u002Fp>\u003Ch2>Vector description\u003C\u002Fh2>\u003Cp>The vector $\\vec\\omega$ lies along the rotation axis according to the right-hand rule. For a fixed axis, $\\vec\\varepsilon=d\\vec\\omega\u002Fdt$ also lies along that axis; its direction relative to $\\vec\\omega$ indicates whether the angular-speed magnitude is increasing or decreasing.\u003C\u002Fp>\u003Ch2>Uniform rotation\u003C\u002Fh2>\u003Cp>If $\\omega=const$, then $\\varepsilon=0$ and $\\varphi=\\varphi_0+\\omega t$. One complete revolution corresponds to an angular change of magnitude $2\\pi$ rad.\u003C\u002Fp>\u003Ch2>Constant angular acceleration\u003C\u002Fh2>\u003Cp>If $\\varepsilon=const$, then $\\omega=\\omega_0+\\varepsilon t$ and $\\varphi=\\varphi_0+\\omega_0t+\\varepsilon t^2\u002F2$. These relations are analogous to those for uniformly accelerated rectilinear motion.\u003C\u002Fp>\u003Ch2>Period and frequency\u003C\u002Fh2>\u003Cp>For uniform rotation, the period $T$ is the time for one revolution and the frequency $f=1\u002FT$ is the number of revolutions per unit time. Angular velocity is related by $\\omega=2\\pi\u002FT=2\\pi f$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A disk rotates uniformly at frequency $f=5$ Hz. Its angular velocity is $\\omega=2\\pi f=10\\pi\\approx31.4$ rad\u002Fs. In 2 s, the disk completes 10 revolutions.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing angular velocity with the linear velocity of a point on the body;\u003C\u002Fli>\u003Cli>using degrees in formulas intended for radians;\u003C\u002Fli>\u003Cli>ignoring the sign of $\\omega$ or $\\varepsilon$;\u003C\u002Fli>\u003Cli>assuming points on the rotation axis have nonzero linear velocity.\u003C\u002Fli>\u003C\u002Ful>",[13],{"id":14,"code":15,"type":16,"locale":7},179,"kinematics-angular-velocity-acceleration","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},285,"Basic Motions of a Rigid Body","theoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fkinematics\u002Fbasic-motions-of-a-rigid-body\u002Frotation-rigid-body-fixed-axis","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fkinematyka\u002Fnaiprostishi-rukhy-tverdoho-tila\u002Fobertannia-tverdoho-tila-navkolo-nerukhomoi-osi",1787712536387]