Angular Kinematics

1. Angular Kinematics Angular motion occurs when all points on an object move in circular paths about the same fixed axis. Chapter 6 in the text Dr. Sasho MacKenzie - HK 376

2. LINEAR ANGULAR Vectors Displacement Velocity Acceleration Scalars Distance Speed KINEMATICS Previous Class Dr. Sasho MacKenzie - HK 376

3. What is an angle? • An angle is formed by the intersection of two lines. • The symbol for angle is  (Theta). • Angles can be measured in degrees or radians (rads). 1 rad = 180/Pi = 57.3 Dr. Sasho MacKenzie - HK 376

4. Arc length Angular displacement (rads)  radius Angular Displacement Angular displacement is the change in angular position experienced by a rotating line. A vector quantity. Dr. Sasho MacKenzie - HK 376

5. Direction of an angular vector • Not like linear vectors. Angular vectors are perpendicular to the plane of motion. • Must use right hand rule • Curl fingers of rt. hand in the direction of rotation. • The direction of your extended thumb is the direction of the angular displacement vector. • A counterclockwise finger curl means the thumb is pointing in the positive direction Dr. Sasho MacKenzie - HK 376

6. Link between Linear Distance and Angular Displacement • Radius is the link between linear and angular kinematics • If the angulardisplacement is measured in radians, then the lineardistance (arc length) is equal to the angular displacement times the radius. • L = *r Dr. Sasho MacKenzie - HK 376

7. Angular Velocity • The rate of change of angular displacement. • Average angular velocity equals angular displacement divided time. • The symbol is  (omega). • Angular velocity is a vector found using the rt. hand rule. Dr. Sasho MacKenzie - HK 376

8. Angular Distance and Angular Speed • Angular distance and angular speed define a magnitude of rotation but no direction as they are scalar quantities. Dr. Sasho MacKenzie - HK 376

9. Link between Linear and Angular Velocity • The link is radius. • All points on a golf club undergo the same angular displacement and therefore the same angular velocity. • But they trace out different arc lengths based on their radius, therefore their linear velocities must be different. Dr. Sasho MacKenzie - HK 376

10. axis of rotation The instantaneous linear velocity (VT) is equal to the instantaneous angular velocity times the radius. VT , the instantaneous speed, is at a tangent to the clubhead path. Points on Golf Club The clubhead moves a longer distance (arc length) in the same time. Therefore, it must have a higher linear velocity. VT The longer the club, the faster the linear velocity of the head. Dr. Sasho MacKenzie - HK 376

11. Fixed Reference Frame axis of rotation R y T R T x New (and convenient) Reference Frame Moving Reference Frame R: radial T: tangential Dr. Sasho MacKenzie - HK 376

12. Angular Acceleration • The rate of change of angular velocity. • Average angular acceleration equals the change in angular velocity divided by time. • The symbol is , (alpha) • Angular acceleration is a vector found using the rt. hand rule. Dr. Sasho MacKenzie - HK 376

13. Angular Acceleration • Angular acceleration occurs when something spins faster and faster or slower and slower, or when the object’s axis of spin changes direction. Dr. Sasho MacKenzie - HK 376

14. N W E S Track Example Usain Bolt runs the curve of this 200 m in 11 s. Assume he ran on the inside line of lane 1, which makes a semicircle (r = 36.5 m) for the first part of the race. His speed after the curve was 11.5 m/s. Circle Circumference = 2r; Circle Diameter = 2r; r is radius • What distance was run on the curve? • What was his angular displacement ? • What was his average angular velocity? • What was his average angular acceleration? Start 36.5 m Finish Dr. Sasho MacKenzie - HK 376

15. Figure Skater Examples • While spinning clockwise in the air, a figure skater completes 400 degrees of rotation. • What was the skater’s angular distance covered in radians? • What was the skater’s angular displacement covered in radians? • If a figure skater has an initial angular velocity of 12 radians/second and undergoes an angular acceleration of 10 radians/second/second for 0.5 seconds, what is the skater’s final angular velocity? Dr. Sasho MacKenzie - HK 376

16. Linear Acceleration and Rotation When an object has angular motion, it is often easier to express linear acceleration relative to a reference frame that moves with the object. So, instead of describing acceleration in the fixed X and Y directions, we consider… • Centripetal (radial) acceleration • Calculated using angular velocity • Tangential acceleration • Calculated using angular acceleration Dr. Sasho MacKenzie - HK 376

17. Centripetal Acceleration • The component of linear acceleration directed towards the axis of rotation (center of the circle). • Associated with the change in direction of an object moving in a circle. Changes the direction of the velocity vector (arrow). Dr. Sasho MacKenzie - HK 376

18. Caused by radial acceleration Vf Vi  Vi r Vf Resultant Vector Centripetal Acceleration and VT (VT) (VT) aR V Vf = Vi + V Dr. Sasho MacKenzie - HK 376

19. Tangential Acceleration • The component of linear acceleration tangent to the circular path (perpendicular to the radius). • Associated with the speeding up of an object moving in a circle. Increases the length of the tangential velocity arrow. • Equal to the angular acceleration times the radius. Dr. Sasho MacKenzie - HK 376

20. Constant Angular Motion  = 360 º (6.28 rad) t = 3 s r = 2 m • = 0, therefore  is constant at = 0, therefore Vt is constant For this example, instantaneous values are the same as average values! Dr. Sasho MacKenzie - HK 376

21. Accelerated Angular Motion A  = 360 º (6.28 rad) t = 3 s r = 2 m D B C True or False • Between A and B,  is > 0? • Between D and A,  is > 0? • Omega () is always  0? TFT Instantaneous values are different than average values! Dr. Sasho MacKenzie - HK 376

22. Visual Comparison Which has/have the… • greatest ? • greatest Vt? • greatest aR? • smallest magnitude of at at the 9 O’clock position? B A C C A and C B C Dr. Sasho MacKenzie - HK 376

23. Angular Kinematics Summary Angular Angular Displacement Linear Angular Velocity Centripetal Acceleration Angular Acceleration Tangential Acceleration Dr. Sasho MacKenzie - HK 376

24. N W E S Track Example #2 At 4 s into his 200 m race, Bolt is running with a speed of 7 m/s. At 9 s, his speed is at 10 m/s. Assume he is running on the inside line of lane 1, which makes a semicircle (r = 36.5 m). • What’s his radial acceleration at 4 s? • What’s his radial acceleration at 9 s? • What’s his angular velocity at 4 s? • What’s his angular velocity at 9 s? • What’s his average angular acceleration between 4 s and 9 s? • What’s his average tangential acceleration between 4 s and 9 s? t = 4s Start 36.5 m t = 9 s Finish Dr. Sasho MacKenzie - HK 376

25. Example Problem • The cyclists shown on the next page are rounding a turn at the bottom of a hill. The path they follow in doing this is a gentle curve that becomes progressively sharper as they near the corner. The radius of the path followed by one of these riders is 20 m at one point in the initial gentle part of the turn, and then decreases to a minimum value of 17 m, 1.5 s later. Her tangential velocity at these two instants are 12 m/s and 11.5 m/s respectively. What is her radial acceleration at the two points? What is her average tangential acceleration between the two points? Dr. Sasho MacKenzie - HK 376

26. Biking On a Curve Dr. Sasho MacKenzie - HK 376