Fundamentals of Physics

1. Fundamentals of Physics Chapter 10 Rotation • Translation & Rotation • Rotational Variables • Angular Position • Angular Displacement • Angular Velocity • Angular Acceleration • Angular Vectors??? • Rotation & Constant Angular Acceleration • Relating Linear & Angular Variables • Kinetic Energy of Rotation • Calculating the Rotational Inertia • Torque • Newton’s 2nd Law for Rotation • Work & Rotational Kinetic Energy Review & Summary Questions Exercises & Problems Physics 2111 Fundamentals of Physics

2. Motions • Motions Translation - motion along a straight line Rotation - motion about an axis • Rigid Body - rotates without any change in shape • Rotates with all its parts locked together • Axis of Rotation (Rotation Axis) • Every point of the body moves in a circle centered on the axis Physics 2111 Fundamentals of Physics

3. Rotational Variables Reference Line - fixed in the body, rotating with the body, perpendicular to the rotation axis. Top View: 1 revolution = 360O = 2 p radians 1 radian = 57.3O = 0.159 revolution Units:q is in radians s and r in length units Physics 2111 Fundamentals of Physics

4. Rotational Variables All points in the disk move through the same angular displacement, dq, but the distance that a typical point Pi moves, dsi, depends on its distance from the center of rotation. Units:q is in radians 1 revolution = 360O = 2 p radians 1 radian = 57.3O = 0.159 revolution Physics 2111 Fundamentals of Physics

5. Rotational Variables Angular Displacement x Angular Acceleration Angular Velocity v a a la Chapter 2 Physics 2111 Fundamentals of Physics

6. Shoot a long arrow through a narrow rotating gap • r = .30 m, 8 gaps • = 2.5 rev/s 20 cm arrow minimum speed ? 8 gaps 20 cm Independent of distance from center! Physics 2111 Fundamentals of Physics

7. Angular Quantities as Vectors? Right Hand Rule - curl your right hand about the axis of rotation with your fingers pointing in the direction of rotation; your extended thumb points in the direction of the angular velocity vector. The angular velocity vector defines the axis of rotation, not a direction in which something moves. Physics 2111 Fundamentals of Physics

8. A Right Hand Screw advances in the direction of its angular velocity. Right-Hand Rule Right Hand Rule - curl your right hand about the axis of rotation with your fingers pointing in the direction of rotation; your extended thumb points in the direction of the angular velocity vector. The length of the angular velocity vector indicates the rate of rotation (radians per second) Angular Acceleration is similarly defined. Physics 2111 Fundamentals of Physics

9. Angular Vectors? Angular displacementscannot be treated as vectors. • Vectors can be added in any order; but this is not true for angular displacements. “Rotation operations do not commute.” (more on this in the next chapter) Physics 2111 Fundamentals of Physics

10. Rotation with Constant Angular Acceleration Physics 2111 Fundamentals of Physics

11. A wheel rotating with constant acceleration Starting from rest: w0 = 0 Constant acceleration: a = 2.00 rad/s2 90 radians in a particular 3 second time interval How long turning before start of interval? Angular velocity at start ~ 14 revolutions Physics 2111 Fundamentals of Physics

12. Relating Angular & Linear Variables Velocity: Acceleration: Tangential: Units: (m/s) (rad/s) (m) Radial: Physics 2111 Fundamentals of Physics

13. Angular Acceleration & Angular Velocity are vectors: Tangential: Centripetal: If the angular velocity is increasing (spinning faster & faster), the angular acceleration is in the same direction as the angular velocity. If the angular velocity is decreasing, the angular acceleration is . . . Uniform Circular Motion: Physics 2111 Fundamentals of Physics

14. Measure the speed of light 500 slots r = 5.0 cm L = 500 m c = 3 x 105 km/s w, vt Physics 2111 Fundamentals of Physics

15. 2 wheels connected by a belt ra = 10 cm rc = 25 cm aa = 1.6 rad/s2 How long for c to reach 100 rev/min? Every point of the belt is moving at the same velocity Physics 2111 Fundamentals of Physics

16. Kinetic Energy of Rotation Consider a rotating body as a set of particles with different speeds. Then the kinetic energy of the body is: Or: Rotational K.E. (kg m2) “Moment of Inertia” aka “Rotational Inertia” Physics 2111 Fundamentals of Physics

17. Moment of Inertia Uniform rod of length L and mass M about a perpendicular axis through one end: Physics 2111 Fundamentals of Physics

18. Rotational Inertia: Uniform disk of radius R and mass M about a perpendicular axis through its center: Uniform hoop of radius R and mass M about a perpendicular axis through its center: Physics 2111 Fundamentals of Physics

19. Some Moments of Inertia Physics 2111 Fundamentals of Physics

20. Calculating Moments of Inertia Parallel Axis Theorem: Physics 2111 Fundamentals of Physics

21. Calculating Moments of Inertia Parallel Axis Theorem: Physics 2111 Fundamentals of Physics

22. Rank rotational inertia about these axes: Imax 1 2 4 3 Imin Parallel Axis Theorem: Physics 2111 Fundamentals of Physics

23. Rotational Inertia = ? Table 11-2: Parallel Axis Theorem: Physics 2111 Fundamentals of Physics

24. Torque - a force causes an object to rotate Physics 2111 Fundamentals of Physics

25. Vector (Cross) Product Magnitude: Direction: “the right-hand rule” cx = ay bz - by az cy = az bx - bz ax cz = ax by - bx ay “Anti-commutative”: http://physics.syr.edu/courses/java-suite/crosspro.html Physics 2111 Fundamentals of Physics

26. The Vector Cross Product Physics 2111 Fundamentals of Physics

27. Newton’s 2nd Law for Rotation Consider a point rotating about an axis: Physics 2111 Fundamentals of Physics

28. rod is not to turn Physics 2111 Fundamentals of Physics

29. Work & Rotational Kinetic Energy Work – Kinetic Energy Theorem: Consider a point rotating about an axis: Power in rotating an object about a fixed axis: Physics 2111 Fundamentals of Physics

30. Corresponding Relationships Physics 2111 Fundamentals of Physics

31. T T T Massless cord wrapped around a massive wheel M = 2 kg R = 0.2 m I = 0.05 kg m2 M = 2 kg P = 3 N a, a = ? P = 3 N Physics 2111 Fundamentals of Physics

32. The box descends after being released R = 0.20 m I = 0.4 kg m2 KEB = 6 J KEW = ? h = ? Box Kbox = 6 J Physics 2111 Fundamentals of Physics