Rotational Motion 2 Coming around again to a theater near you

Angular Velocity and Acceleration Look at a rigid body that rotates about a fixed axis in analyzing rotational motion. By fixed axis it means an axis that is at rest in some inertial frame of reference and does not change direction relative to that frame. y Diagram shows a rigid body rotating about a stationary axis passing through point O and perpendicular to the plane of the diagram. Line OP is fixed in the body and rotates with it. The angle 2 serves as a coordinate for rotational motion p p 2 2 x O Usually measure the angle 2 in radians One radian (1 rad) is the angle subtended at the center of a circle by an arc with a length equal to the radius of the circle Angle 2 is subtended by an arc of length s on a circle of radius r; 2 (in radians) is equal to s divided by r. s or 1 rad r

Angular Velocity Rotational motion of a rigid body can be described in terms of the rate change of 2. Average angular velocity = (omega) Because the body is rigid, all lines in it rotate through the same angle in the same time. At any instant, every part of a rigid body has the same angular velocity y p p 22 21 x O

Angular Acceleration When the angular velocity of a rigid body changes, it has an angular acceleration. If T1 and T2 are the instantaneous angular velocities at times t1 and t2, define the average angular acceleration Vav (alpha) as… When the angular acceleration is constant, equations for angular velocity and angular position can be derived. Let To be the angular velocity at time zero for a rigid body, and let T be its angular velocity at any later time t. The angular acceleration V is constant and equal to the average value for any interval.

Angular Acceleration If at the initial time t = 0 the body is at angular position 2o and has angular velocity To, then its angular position 2 at any later time t is the sum of three terms : its initial angular position 2o, plus the rotational Tot it would have if the angular velocity were constant, plus an additional rotation 2Vt2 caused by the changing angular velocity.

Velocity and Acceleration Relations When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. The speed of a particle is directly proportional to the body’s angular velocity; the faster the body rotates, the greater the speed of each particle time derivation

Tangential component of acceleration atan Velocity and Acceleration Relations Tangential component of acceleration atan

Radial component of acceleration arad Velocity and Acceleration Relations Radial component of acceleration arad The vector sum of the radial and tangential components of acceleration of a particle in a rotating body is linear acceleration…..a When using these equations the angular quantities must be expressed in radians

Example Problem A discuss thrower turns with an angular acceleration V=50 rad/s2, moving the discus in a circle of radius 0.80 m. Find the radial and tangential components of acceleration of the discus and the magnitude of its acceleration at the instant when the angular velocity is 10.0 rad/s Solution

Comparison of linear and angular motion with constant acceleration a = constant = constant These results are valid only when the angular acceleration is constant.