2. Angular Motion

3. Objectives: Define and apply concepts of angular displacement, velocity, and acceleration.Define and apply concepts of angular displacement, velocity, and acceleration. Draw analogies relating rotational-motion parameters ( , ,  ) to linear (x, v, a) and solve rotational problems.Draw analogies relating rotational-motion parameters ( , ,  ) to linear (x, v, a) and solve rotational problems. Write and apply relationships between linear and angular parameters.Write and apply relationships between linear and angular parameters.

4. Rotational Displacement,  Consider a disk that rotates from A to B: A B  Angular displacement  : Measured in revolutions, degrees, or radians. 1 rev = 360 0 = 2  rad The best measure for rotation of rigid bodies is the radian.

5. Definition of the Radian One radian is the angle  subtended at the center of a circle by an arc length s equal to the radius R of the circle. 1 rad = = 57.3 0 RRRR s

6. Example 1: A rope is wrapped many times around a drum of radius 50 cm. How many revolutions of the drum are required to raise a bucket to a height of 20 m? h = 20 m R  = 40 rad Now, 1 rev = 2  rad  = 6.37 rev

7. Example 2: A bicycle tire has a radius of 25 cm. If the wheel makes 400 rev, how far will the bike have traveled?  = 2513 rad s =  R = 2513 rad (0.25 m) s = 628 m

8. Angular Velocity Angular velocity,  is the rate of change in angular displacement. (radians per second.)  fAngular frequency f (rev/s).  f  Angular frequency f (rev/s). Angular velocity can also be given as the frequency of revolution, f (rev/s or rpm):  Angular velocity in rad/s.  Angular velocity in rad/s. tttt

9. Example 3: A rope is wrapped many times around a drum of radius 20 cm. What is the angular velocity of the drum if it lifts the bucket to 10 m in 5 s? h = 10 m R  = 10.0 rad/s  tt 50 rad 5 s  = 50 rad

10. Example 3: In the previous example, what is the frequency of revolution for the drum? Recall that  = 10.0 rad/s. h = 10 m R f  = 95.5 rpm Or, since 60 s = 1 min:

11. Angular Acceleration Angular acceleration is the rate of change in angular velocity. (Radians per sec per sec.) The angular acceleration can also be found from the change in frequency, as follows:

12. Example 4: The block is lifted from rest until the angular velocity of the drum is 16 rad/s after a time of 4 s. What is the average angular acceleration? h = 20 m R  = 4.00 rad/s 2 0

13. Angular and Linear Speed From the definition of angular displacement: s =  R Linear vs. angular displacement v =  R Linear speed = angular speed x radius

14. Angular and Linear Acceleration: From the velocity relationship we have: v =  R Linear vs. angular velocity a =  R Linear accel. = angular accel. x radius

15. Examples: R 1 = 20 cm R 2 = 40 cm R1R1 R2R2 A B   = 0;  f = 20 rad/s t = 4 s What is final linear speed at points A and B? Consider flat rotating disk: v Af = 4 m/s v Af =  Af R 1 = (20 rad/s)(0.2 m); v Af = 4 m/s v Bf = 8 m/s v Af =  Bf R 1 = (20 rad/s)(0.4 m); v Bf = 8 m/s

16. Acceleration Example R 1 = 20 cm R 2 = 40 cm What is the average angular and linear acceleration at B? R1R1 R2R2 A B   = 0;  f = 20 rad/s t = 4 s Consider flat rotating disk:  = 5.00 rad/s 2 a =  R = (5 rad/s 2 )(0.4 m) a  = 2.00 m/s 2

17. A Comparison: Linear vs. Angular

18. Linear Example: A car traveling initially at 20 m/s comes to a stop in a distance of 100 m. What was the acceleration? 100 m v o = 20 m/s v f = 0 m/s Select Equation: a = = 0 - v o 2 2s -(20 m/s) 2 2(100 m) a = -2.00 m/s 2

19. Angular analogy: Angular analogy: A disk (R = 50 cm), rotating at 600 rev/min comes to a stop after making 50 rev. What is the acceleration? Select Equation:  = = 0 -  o 2 2  -(62.8 rad/s) 2 2(314 rad)  = -6.29 m/s 2 R  o = 600 rpm  f = 0 rpm  = 50 rev 50 rev = 314 rad

20. Problem Solving Strategy:  Draw and label sketch of problem.  Indicate + direction of rotation.  List givens and state what is to be found. Given: ____, _____, _____ ( ,  ,  f, ,t) Find: ____, _____  Select equation containing one and not the other of the unknown quantities, and solve for the unknown.

21. Example 5: A drum is rotating clockwise initially at 100 rpm and undergoes a constant counterclockwise acceleration of 3 rad/s 2 for 2 s. What is the angular displacement?  = -14.9 rad Given: Given:  o = -100 rpm; t = 2 s  = +3 rad/s 2  = -20.9 rad + 6 rad Net displacement is clockwise (-) R 

22. Summary of Formulas for Rotation