Rotation of a Rigid Body Chapter Outline ENGINEERING PHYSICS I PHY 303K Motion of a Rigid Body Rotation about a Fixed Axis Chapter 12: Rotation of a Rigid Body Motion with Constant Angular Acceleration Motion with Time-Dependent Angular Acceleration Maxim Tsoi Kinetic Energy of Rotation; Moment of Inertia Physics Department, The University of Texas at Austin http://www.ph.utexas.edu/~tsoi/303K.htm 303K: Ch.12 303K: Ch.12 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Rigid object Angular position • It is convenient to represent position of P with its compact disc rotating about a fixed axis Analysis of a rotational motion can be greatly simplified by polar coordinates (r, ) through O assuming that an object in motion is rigid r distance from the origin (r=const) measured counterclockwise from a reference line •A rigid object is one that is nondeformable – that is, the relative • P moves through an arc of length s: locations of all particles of which the object is composed remain s constant s r r is measured in radians (rad) 180o= rad is angular position of the rigid object • We can associate the angle with the entire object as well as with an individual point 303K: Ch.12 303K: Ch.12 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Angular displacement and speed Right-hand rule • Angular displacement: • Angular velocity vector A particle on a rotating rigid object moves from • For rotation about a fixed axis, the f i A to B in time interval t only rotation that uniquely specifies the rotational motion is the direction • Average angular speed: along the axis of rotation f i • Right-hand rule: when the four fingers of the right hand are wrapped t f ti t in the direction of rotation, the • Instantaneous angular speed: extended right thumb points in the direction of d lim t0 t dt d units radians per second (rad/s) dt 303K: Ch.12 303K: Ch.12 1 Angular Position, Velocity, and Acceleration Rotational Motion with Constant Acceleration Angular acceleration Kinematic equations for constant angular acceleration • If the instantaneous angular speed of an • Using definitions of the angular acceleration and angular velocity: object changes from I to f in the time A particle on a rotating rigid object moves from interval t the object has an angular d f t A to B in time interval t d dt d dt t acceleration f i dt i 0 • Average angular acceleration: t t d f 1 2 d dt d dt i t dt f i i t 2 t dt i 0 0 f i t f ti t 2 2 f i t f i 2 f i • Instantaneous angular acceleration : f f 1 2 1 f i i t 2 t f i 2 i f t d i i lim v v a t v 2 v 2 2a x x t0 t dt xf xi x xf xi x f i 1 1 2 x f xi 2 v xi v xf t x f xi v xi t 2 axt units radians per second squared (rad/s2) 303K: Ch.12 303K: Ch.12 Example 12.1 Angular and Linear Quantities Rotational Kinetic Energy Relationships between angular and linear speed and acceleration Moment of inertia Rigid object rotates about a fixed axis every particle of the object Kinetic energy of an object is the energy associated with its motion through space moves in a circle whose center is the axis of rotation 2 2 m v m r • Kinetic energy of an • Relation of the linear ds d K i i i i v r r i 2 2 individual particle (tangential) speed to the dt dt angular velocity: • Total kinetic energy of K K 1 2 m r 2 R i 2 i i the rotating rigid object • Relation of the tangential dv d i i a r r acceleration to the angular t dt dt acceleration: I m r 2 • Moment of inertia i i measure of the resistance i 2 of an object to changes in • Centripetal acceleration in v 2 its rotational motion ac r terms of the angular speed: r 1 2 K R 2 I 2 2 2 4 • The total linear acceleration: a at ac at ac r 303K: Ch.12 303K: Ch.12 Example 12.2 Calculation of Moments of Inertia Calculation of Moments of Inertia Homogeneous rigid objects with different geometries Parallel-axis theorem Suppose the moment of inertia about an axis through the center of mass of an object is I . Moment of inertia: CM Than the moment of inertia about any axis parallel to and a distance D away from this axis is 2 2 I lim ri mi I I MD m 0 CM i i 2 r dm I r 2dm x 2 y2 dm 2 r dV x'x 2 y' y 2 dm CM CM x'2 y'2 dm •Density =m/V mass per unit volume 2x x'dm 2 y y'dm CM CM • Homogeneous object x 2 y2 dm =const CM CM 303K: Ch.12 Example 12.5-7 303K: Ch.12 Example 12.8 2 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object In general – very complex motion Condition for pure rolling motion Cylinder rolling on a straight path • As cylinder rotates through an angle , its center of mass moves a linear distance s=R ds d v R R CM dt dt • Trajectory of a particle on the rim (red) cycloid • Trajectory of the center of mass straight line • The magnitude of the linear acceleration of the center of mass If an object rolls without slipping on the surface (pure rolling motion) dvCM d aCM R R linear velocities of various points a simple relationship exists between its rotational and translational motions dt dt on and within the cylinder 303K: Ch.12 303K: Ch.12 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object Combination of translational and rotational motion Kinetic energy of rolling cylinder Mv2 I 2 K CM K CM 2 2 Applying the parallel-axis theorem I MR2 2 K CM I 2 2 K P 2 I 2 Mv 2 CM CM 2 2 303K: Ch.12 303K: Ch.12 Rolling Motion of a Rigid Object SUMMARY Using energy methods Rotation of a Rigid Body 2 2 ICM MvCM K d 2 2 • Instantaneous angular speed: 2 dt d 2 • Instantaneous angular acceleration: ICM vCM MvCM dt 2 R 2 • Kinematic equations for rotational motion: t t 1 t 2 2 2 1 t 1 ICM 2 f i f i i 2 f i 2 f i f i 2 i f 2 2 M vCM R • Relation between linear and angular s r v r at r quantities: Conservation of mechanical energy: • Moment of inertia of a rigid object: I r 2dm 1 2 1 • Rotational kinetic energy: K I K f U f K i U i 2gh 2 R 2 v CM 2 1 ICM MR 1 ICM 2 2 M vCM 0 0 Mgh R2 303K: Ch.12 303K: Ch.12 3.