Dr. Jie Zou PHY 1151G Department of Physics1 Chapter 10 Rotational Kinematics and Energy

Dr. Jie Zou PHY 1151G Department of Physics2 Outline Angular Position Angular Velocity Angular Acceleration Kinematics Equations for Rotations with Constant Acceleration Examples

Dr. Jie Zou PHY 1151G Department of Physics3 Angular position Definition of angular position,  :  : an angle measured from reference line. The reference line defines  = 0. Sign convention for angular position:  > 0 for counterclockwise rotation from reference line;  < 0 for clockwise rotation from reference line. Units to measure angles: SI units: radians (rad); other units: degrees (º) and revolutions (rev). 1 rev = 360º = 2π rad, 1 rad = 57.3º.

Dr. Jie Zou PHY 1151G Department of Physics4 Angular velocity Angular displacement  =  f -  i. Average angular velocity:  av =  /  t. SI units: radians per second (rad/s). Sign convention for angular velocity:  > 0 for counterclockwise rotation.  < 0 for clockwise rotation. Angular speed: The speed of rotation or the magnitude of the angular velocity.

Dr. Jie Zou PHY 1151G Department of Physics5 Period of Rotation Definition of period: The time to complete one revolution, T, is referred to as the period. T = 2  / . Here  is the angular speed in rad/s. SI units for T: second (s).

Dr. Jie Zou PHY 1151G Department of Physics6 Exercise 10-1 and 10-2 (a) An old record player rotates clockwise at 33 rpm (revolutions per minute). What is its angular velocity in rad/s? (b) If a CD rotates at 22.0 rad/s, what is its angular speed in rpm? (c) Find the period of a record that is rotating at 45 rpm.

Dr. Jie Zou PHY 1151G Department of Physics7 Angular acceleration Average angular acceleration  av =  /  t = (  f -  i )/  t SI units: radians per second per second (rad/s 2 ). Determination of the sign of the angular acceleration: If  and  have the same sign, the speed of rotation (angular speed) is increasing. If  and  have opposite signs, the speed of rotation (angular speed) is decreasing.

Dr. Jie Zou PHY 1151G Department of Physics8 Exercise 10-3 As the wind dies, a windmill that was rotating at 2.1 rad/s begins to slow down with a constant angular acceleration of 0.45 rad/s 2. How long does it take for the windmill to come to a complete stop?

Dr. Jie Zou PHY 1151G Department of Physics9 Linear QuantityAngular Quantity xvaxva Linear-to-angular analogies Linear Equation (a = constant) Angular equation (  = constant) v = v 0 +  t  =  0 +  t x = x 0 + v 0 t + at 2 /2  =  0 +  0 t +  t 2 /2 v 2 = v a(x - x 0 )  2 =   (  -  0 ) Kinematics Equations for Rotational Motions

Dr. Jie Zou PHY 1151G Department of Physics10 Example 10-1 To throw a curve ball, a pitcher gives the ball an initial angular speed of 36.0 rad/s. When the catcher gloves the ball s later, its angular speed has decreased (due to air resistance) to 34.2 rad/s. (a) What is the ball’s angular acceleration, assuming it to be constant? (b) How many revolutions does the ball make before being caught?

Dr. Jie Zou PHY 1151G Department of Physics11 Active Example 10-1: A pulley rotating in the counterclockwise direction is attached to a mass suspended from a string. The mass causes the pulley’s angular velocity to decrease with a constant angular acceleration  = rad/s 2. (a) If the pulley’s initial angular velocity is  0 = 5.40 rad/s, how long does it take for the pulley to come to rest? (b) Through what angle does the pulley turn this time?

Dr. Jie Zou PHY 1151G Department of Physics12 Homework See online homework assignment on