1. Angular Kinematics Ch 4 Notes

2. What is a radian?

3. Angular Variables Angular Variables 1 Θ rad = 57.3 ͦ of twist 1 ω = 57.3 ͦ of twist in a second’s time = Θ rad / 1 s 1 α = change in angular speed equal to 57.3 of twist per second = Δω / 1 s The radian itself has no units S θ rad

4. Angular Kinematics Definitions Angular Velocity: “amount of twist per time” ώ = ΔΘ / Δt units: rad/s Angular Acceleration: “amount of change in twist per time” Angular Acceleration: “amount of change in twist per time” α = Δώ / Δt Units: rad/s 2

5. Linking Linear with Angular… v t = Δx/Δt = C/T = 2πr/T 1 rev = 2π radians = 6.28 radians Centripetal or radial acceleration a = a r = v 2 /r in linear units a c = a r = v 2 /r in linear units a c = a r = rώ 2 in angular units

6. Clockwise (CW) vs. Counterclockwise (CCW) CCW is +, CW is – throughout course

7. 1. Finding radial acceleration- linear units What is the radial acceleration of an object that spins with a linear speed of 4.00 m/s at a 0.80 m distance from the axis of rotation?

8. 2. Finding radial acceleration- angular units What is the radial acceleration of a motorcycle proceeding in a circular cage of diameter 14.0 m if it is moving at an angular velocity of 2.0 rad/s?

9. A closer look at acceleration… For objects that are moving, the acceleration can be broken down into components parallel to and perpendicular to motion. // to motion: tangential acceleration –Leads to more/less rpm, units m/s 2 Perp. to motion: radial acceleration or centripetal acceleration –Gives inward acceleration needed to make circular motion, units m/s 2

10. Galileo’s Formulas Work for Rotation!!!! Angular displacement: Θ(# radians spun) rad Angular velocity: ώ (radians spun per second) rad/s Angular acceleration: α (Δ in rps per sec) rad/s 2 V f = v i + aΔt ώ = ώ 0 + αΔt V f 2 = V i 2 + 2ax ώ 2 = ώ 0 2 + 2α(ΔΘ) x = v i t + ½at 2 Θ = ώ 0 t + 1/2αt 2

11. Rotation and Graphing Θ for x ώ for v α for a SLOPES TO GO UP AREAS UNDER CURVE TO GO DOWN

12. 3. What is angular acceleration? A piece of tape at the edge of the disk of radius 0.19 m slows from a speed of 2.00 m/s to a speed of 1.00 m/s in a 6.00 second time frame. What is the angular acceleration of the tape? What is the angular acceleration of the inside edge of the disk (r = 0.03 m)?

13. 4. What is the final angular velocity? An object being swung around with a radius of 0.50 m is subject to an angular acceleration of 0.25 rad/s 2 for a 3.00 second time frame. If it started with a speed of 8.00 m/s, what is its angular velocity after the three seconds have elapsed? What number of rotations will have occurred in this 3.0 sec time span?

14. 5. What is the angular acceleration? A disk is spun for a total of 62 rotations. It begins at an angular velocity of 1.8 rad/s and due to friction this value drops to 1.2 rad/s by the 62 nd rotation. What is this disk’s angular acceleration?

15. Moment of Inertia “Inertia of Spin” Think of it as “Rotational Mass” Measure of the unwillingness of a material to want to rotate Units: Kg*m 2 Shape affects moment If all mass on outside of orbit, I = mr 2 Others p. 298

16. Moment of inertia changes for the same material according to what spin axis you choose for it.

18. 5. Which has a greater Moment Which object requires more force to spin about its center of mass? –A hoop of diameter 15cm and 0.8kg –Or a disk of diameter 15cm and 0.8kg Justify your answer.

19. Torque Rotational Force τ = Torque [Nm] or [Foot-Pounds] Which F will cause the most spin?

20. Torque Torque depends on Force and Radius Only Forces perpendicular to Radius add torque

21. 6. Calculate the Torque Jimmy physics slams a door that has a length (measured from the hinges) of 0.75m. He applies a 20N force at an angle of 30degrees from the door.

22. Newton’s 2 nd Law Rotational Force Rotational Mass F net = m*a τ net = I*α

23. What has more kinetic energy, a knuckleball pitched at 70 mph, or a changeup pitched at 70 mph? Close but not identical!!!!!!

24. Rotational Kinetic Energy “Energy of Spin” as opposed to “Energy of Motion” E k Linear = ½mv 2 E k Rotational = ½ I ω 2 Units: [Joules,J] Objects that are rolling have both forms of E k !

25. Example: Total KE What is the total E k of a basketball (0.60 kg, diameter 30.0 cm) that is rolling at a speed of 2.00 m/s on the ground? (I spherical shell = 2/3 mr 2 )

26. Which object will reach the bottom first?

27. The disk cylinder has a lower moment due fact that mass closer to axis of rotation is easier to rotate Τ = F r T = I α