ch8 - Rotational Motion Chapter Outline GENERAL PHYSICS PHY...

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Unformatted text preview: Rotational Motion Chapter Outline GENERAL PHYSICS PHY 302K Angular Quantities Constant Angular Acceleration Chapter 8: Rotational Motion Rolling Motion Torque Maxim Tsoi Rotational Dynamics: Torque and Rotational Inertia Rotational Kinetic Energy Angular Momentum and Its Conservation Physics Department, The University of Texas at Austin http://www.ph.utexas.edu/~tsoi/ 302K: Ch.8 302K: Ch.8 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Rigid object Angular position • It is convenient to represent position of P with its Analysis of a rotational motion can be greatly simplified by assuming that an object in motion is rigid polar coordinates (r, ) compact disc rotating about a fixed axis through O 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 locations of all particles of which the object is composed remain • P moves through an arc of length s: constant s r s 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 302K: Ch.8 302K: Ch.8 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Angular displacement and speed • Angular displacement: f i • Average angular speed: f i t f ti t • Instantaneous angular speed: t 0 t A particle on a rotating rigid object moves from A to B in time interval t Right-hand rule • Angular velocity vector • For rotation about a fixed axis, the only rotation that uniquely specifies the rotational motion is the direction along the axis of rotation • Right-hand rule: when the four fingers of the right hand are wrapped in the direction of rotation, the extended right thumb points in the direction of lim units radians per second (rad/s) 302K: Ch.8 d dt 302K: Ch.8 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 object changes from I to f in the time interval t the object has an angular acceleration A particle on a rotating rigid object moves from A to B in time interval t • Using definitions of the average angular acceleration and angular velocity: • Average angular acceleration: f i t f ti t t t f i i t 1 t 2 2 2 i2 2 f i f f i t • Instantaneous angular acceleration : f i i t t 1 2 lim t 0 t f i t f i 1 i f t 2 2 2 2 v xf v xi 2a x x f x i v xf v xi a x t x f xi units radians per second squared (rad/s2) 302K: Ch.8 1 2 v xi v xf t x f x i v xi t 1 a x t 2 2 Example 1 302K: Ch.8 Angular and Linear Quantities Rotational Kinetic Energy Relationships between angular and linear speed and acceleration Rigid object rotates about a fixed axis every particle of the object Moment of inertia 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 • Relation of the linear v (tangential) speed to the angular velocity: • Relation of the tangential acceleration to the angular acceleration: mi v i2 m i ri 2 2 2 s r r t t Ki K R K i 1 2 mi ri2 2 v r at r t t i • Kinetic energy of an individual particle • Total kinetic energy of the rotating rigid object i I mi ri2 • Moment of inertia i • Centripetal acceleration in ac terms of the angular speed: • The total linear acceleration: 2 v r 2 r measure of the resistance of an object to changes in its rotational motion K R 1 I 2 2 a at ac at2 ac2 r 2 4 302K: Ch.8 Example 2 302K: Ch.8 Calculation of Moments of Inertia Rolling Motion of a Rigid Object Homogeneous rigid objects with different geometries In general – very complex motion Moment of inertia: I lim mi 0 Cylinder rolling on a straight path r m 2 i i i • Density =m/V mass per unit volume • Homogeneous object • Trajectory of a particle on the rim (red) cycloid =const • Trajectory of the center of mass straight line If an object rolls without slipping on the surface (pure rolling motion) a simple relationship exists between its rotational and translational motions 302K: Ch.8 Example 5 302K: Ch.8 2 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object Condition for pure rolling motion Combination of translational and rotational motion • As cylinder rotates through an angle , its center of mass moves a linear distance s=R vCM s R R t t • The magnitude of the linear acceleration of the center of mass aCM vCM R R t t linear velocities of various points on and within the cylinder 302K: Ch.8 302K: Ch.8 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object Kinetic energy of rolling cylinder Using energy methods K I 2 K CM 2 2 MvCM K 2 I CM Mv 2 2 2 2 CM 2 2 I v MvCM CM CM 2 R 2 I CM 2 1 2 M vCM 2 R Conservation of mechanical energy: I 2 K P 2 K f U f Ki Ui 2 1 I CM M vCM 0 0 Mgh 2 2 R I P I CM MR 2 302K: Ch.8 2 gh vCM 2 1 I CM MR 1 2 302K: Ch.8 Relationship between and Torque Torque and moment arm Rotational analog of Newton’s second law • When a force is exerted on a rigid Ft ma t object pivoted about an axis, the object tends to rotate about that axis mr r mr I 2 • The tendency of a force to rotate an object about some axis is measured by a vector quantity called torque • For a rigid object: vector point to the line of action of F • If two or more forces tend to produce rotating in a circle of radius r under the influence of a tangential force Ft and a radial force Fr Torque acting on a particle is proportional to its angular acceleration! rF sin Fd d – moment arm - distance from pivot • Consider a particle of mass m Ft r ma t r 1 Ft ma t r Ft rat m r 2 m 2 r m I 2 F1d1 F2 d 2 rotation of a rigid object net torque 302K: Ch.8 302K: Ch.8 Example 7-8 3 Angular Momentum Conservation of Angular Momentum Angular momentum of a rotating rigid object 3rd conservation law • Consider a rigid object rotating about a fixed axis • The total angular momentum of a system is constant in both magnitude and direction if the external torque acting on the system is zero I I I I t I t 0 t 0 angular momentum of the whole object: Ltot const L I • Change of the object’s angular I momentum is equal to the net external torque L 0 t or Li L f • L=I=const if mass of the rotating system undergoes redistribution in some way L t system’s moment of inertia I changes angular speed changes • Example spinning figure skater 302K: Ch.8 302K: Ch.8 SUMMARY Rotational Motion Angular acceleration: t t • Kinematic equations for rotational motion: • Angular speed: f i t f i i t 1 t 2 2 2 i2 2 f i f • Relation between linear and angular quantities: s r • Moment of inertia of a rigid object: f i 1 i f t 2 v r a t r I m i ri2 i • Rotational kinetic energy: • Torque associated with force F: K R 1 I 2 2 rF sin • Newton’s 2nd law for rotational motion: I • Angular momentum of a rigid object rotating about a fixed axis (z): Lz I • If net external torque acting on a system is zero total angular momentum of the system is conserved Li L f 302K: Ch.8 4 ...
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