1. Unit 8: Circular Motion

2. Section A: Angular Units Corresponding Textbook Sections: –10.1 PA Assessment Anchors: –S11.C.3.1

3. Angular Position Defined as the angle, , that a line from the axle to a spot on the wheel makes with a reference line Unit: Radian (rad) [dimensionless]

4. Sign convention for angular position: If  > 0, counterclockwise rotation If  < 0, clockwise rotation

5. Converting between degrees and radians 1 revolution = 360  = 2  rad 1 rad = 57.3  Convert the same way you would between any other units.

6. Section B: Angular / Linear Relationships Corresponding Textbook Sections: –10.3 PA Assessment Anchors: –S11.C.3.1

7. Arc Length The arc length is the distance from a reference line to a spot of interest on a circle. Equation: s = r 

8. Angular Velocity Symbol:  Units: s -1 or 1/s

9. Sign Convention for  If  > 0Counterclockwise rotation If  < 0Clockwise rotation

10. Practice Problem #1 An old phonograph rotates clockwise at 33⅓ rpm. What is the angular velocity in rad/s?

11. Practice Problem #2 If a CD rotates at 22 rad/s, what is its angular speed in rpm?

12. Period The period is the time it takes to complete one revolution. Units: seconds (s)

13. Practice Problem #3 Find the period of a record that is rotating at 45 rpm.

14. Angular Acceleration The change in angular speed of a rotating object per unit of time. Units: rad/s 2

15. Practice Problem #4 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?

16. Section C: Angular Kinematics Corresponding Textbook Sections: –10.2 PA Assessment Anchors: –S11.C.3.1

17. Relationship between angular and linear quantities Linear QuantityAngular Quantity x  vω aα Based on these relationships, we can rewrite the kinematics equations from 1-D and 2-D Kinematics

18. Angular Kinematics Equations

19. So, basically… These are just variations of equations we already know how to use. They work the same way as the linear equations. We’ll use the same setup as before: Data table, equation, picture, etc…

20. Practice Problem #1 To throw a curveball, a pitcher gives the ball an initial angular speed of 36 rad/s. When the catcher gloves the ball 0.595 s later, its angular speed has decreased to 34.2 rad/s. What is the ball’s angular acceleration?

21. Practice Problem #2 Based on the last problem, how many revolutions does the ball make before being caught?

22. Practice Problem #2 Refer to Example 10-2 on page 280

23. Section D: Torque Corresponding Textbook Sections: –11.1, 11.2 PA Assessment Anchors: –S11.C.3.1

24. What is Torque? Torque is the rotational equivalent of force It depends on: –Force applied –Distance from the force to the axis of rotation

25. More on Torque… Equation: Units: Nm Greek Letter “tau” Axis of Rotation (where it turns)

26. Practice Problem #1 If the minimum required torque to open a door is 3.1 Nm, what force must be applied if: –r = 0.94 m –r = 0.35 m

27. Section E: Moment of Inertia Corresponding Textbook Sections: –10.5 PA Assessment Anchors: –S11.C.3.1

28. What is “Moment of Inertia”? The “rotational mass” of an object –Rotational mass depends on actual mass, radius, and distribution of mass Useful for determining rotational KE: Moment of inertia

30. Practice Problem #1 What is the moment of inertia of a hollow sphere with mass of 40 kg and radius of 3 m?

31. Practice Problem #2 A grindstone with radius of 0.61 m is being used to sharpen an axe. If the linear speed of the stone relative to the ax is 1.5 m/s, and the stones rotational KE is 13 J, what is its moment of inertia?