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Physics I Class 14 Introduction to Rotational Motion 14-1 Definitions Angular Position: Angular Displacement: (in radians) 0 Average or mean angular velocity isdefined as follows: 0 avg t t 0 t Instantaneous angular velocity or just “ angular velocity”: d lim t 0 t dt Wait a minute! How can an angle have a vector direction? 14-2 Direction of Angular Displacement and Angular Velocity •Use your right hand. •Curl your fingers in the direction of the rotation. •Out-stretched thumb points in the direction of the angular velocity. 14-3 Angular Acceleration Average angular acceleration is defined as follows: avg 0 t t0 t Instantaneous angular acceleration or just “angular acceleration”: d d 2 lim 2 t 0 t dt dt The easiest way to get the direction of the angular acceleration is to determine the direction of the angular velocity and then… If the object is speeding up, angular velocity and acceleration are in the same direction. If the object is slowing down, angular velocity and acceleration are in opposite directions. 14-4 Equations for Constant 1. 0 t t 0 2. 0 0 ( t t 0 ) 2 ( t t 0 ) 1 1 (0 )( t t 0 ) 3. 0 2 4. 0 ( t t 0 ) 2 ( t t 0 ) 1 5. 2 x v a 2 2 02 2 0 14-5 Relationships Among Linear and Angular Variables MUST express angles in radians. s r v r a tangential r a centripetal v 2 2 r 2 2 r r r The radial direction is defined to be + outward from the center. a radial a centripetal 14-6 Energy in Rotation Consider the kinetic energy in a rotating object. The center of mass of the object is not moving, but each particle (atom) in the object is moving at the same angular velocity (). K 12 m i v i 12 m i 2 ri 12 2 m i ri 2 2 2 The summation in the final expression occurs often when analyzing rotational motion. It is called the moment of inertia. 14-7 Moment of Inertia For a system of discrete “point” objects: I m i ri 2 For a solid object, use an integral where is the density: I r 2 dx dy dz We may ask you to calculate the moment of inertia for point objects, but we will give you a formula for a solid object or just give you its moment of inertia. I for a solid sphere: I for a spherical shell: I 52 M R 2 I 23 M R 2 14-8 Characteristics of the Moment of Inertia The moment of inertia of an object depends on Its mass. Its shape. The axis of rotation. NOT the angular velocity or acceleration. The moment of inertia is a measure of how difficult it is to get an object to start rotating or to slow down once started. For two or more objects rotating around a common axis, the total moment of inertia is the sum of each individual moment of inertia. I Ii 14-9 Introduction to Torque For linear motion, we have “F = m a”. For rotation, we have I The symbol “” is torque. We will define it more precisely next time. Torque and angular acceleration are always in the same direction in Physics 1. 14-10 Correspondence Between Linear and Rotational Motion x v a mI F K 12 I 2 I You will solve many rotation problems using exactly the same techniques you learned for linear motion problems. 14-11 Class #14 Take-Away Concepts 1. 2. Definitions of rotational quantities: , , . Centripetal and tangential acceleration. mi ri1 2 4. Rotational kinetic energy: K 2 I 5. Introduction to torque: I 3. Moment of inertia: I 6. Correspondence x mI v F 2 a 14-12 Activity #14 Introduction to Rotation Objective of the Activity: 1. 2. Think about rotation concepts. Try changing the moment of inertia of a simple object and see how that affects I . 14-13