Chapter 10

1. Chapter 10 Rotational Motion and Torque

2. 10.1- Angular Position, Velocity and Acceleration • For a rigid rotating object a point P will rotate in a circle of radius r away from the axis of rotation.

3. 10.1 • The location of point P can be described in polar coordinates (r , θ). • The circular distance traveled is called the arc length according to • When θ is measured in radians (1 radian is the angle swept by an arc length equal to the radius).

4. 10.1 • Angles measured in radians, degrees, revolutions 2πrad = 360o = 1 rev • Angular displacement- the change in angular position

5. 10.1 • Angular Velocity- the rate of change in angular displacement • For constant rotations or averages • For Angular Position as function of time • Measured in rad/s or rev/s

6. 10.1 • Quick Quizzes p 294 • Angular Acceleration- the rate of change of angular velocity or

7. 10.1 • Angular Velocity/Acceleration Vector Directions- Right Hand Rule • Generally CCW is positive, CW is negative • Acceleration direction Points the same direction as ω, if ω is increasing, antiparallel if ω decreases

8. 10.1 Quick Quiz p. 296

9. 10.2 Rotational Kinematics • Tracking the increasing and decreasing rotation can be done with the same relationships as increasing and decreasing linear motion. • Remember Δx  Δθ v  ω a  α

10. 10.2 • Quick Quiz p. 297 • Example 10.1

11. 10.3 Angular and Linear Quantities • When an object rotates on any axis, every particle in that object travels in a circle of constant radius (distance from axis) • The motion of each point can be described linearly about the circular path • Tangential Velocity-

12. 10.3 • Tangential Acceleration- • We also know there is a centripetal acceleration

13. 10.3 • The resultant acceleration- • Quick Quizzes p. 298 • Examples 10.2

14. 10.4 • Rotational Kinetic Energy- the kinetic energy of a single particle in a rotating object is… • The Total Kinetic Energy would be the sum of all Ki • Which can be rewritten...

15. 10.4 • This is a new term we will call Moment of Inertia • Moment of Inertia has Dimensions ML2 and units kg.m2 (~ rotational counterpart to mass) • Rotational Kinetic Energy-

16. 10.4 • Quick Quiz p. 301 • Examples 10.3, 10.4

17. 10.5 Calculating Moments of Inertia • We can evaluate the moment of inertia of an extended object by adding up the M.o.I. for an infinite number of small particles.

18. 10.5 • Its generally easier to calculate based on the volume of elements rather than mass so using for small elements…. • We have… • If ρ is constant, the integral can be completed based on the geometric shape of the object.

19. 10.5 • Volumetric Density- ρ (mass per unit volume) • Surface Mass Density- σ (mass per unit area) • (Sheet of uniform thickness (t) σ = ρt) • Linear Mass Density- λ (mass per unit length) • (Rod of uniform cross sectional area (A) λ = M/L = ρA) See Board Diagrams

20. 10.5 • Example 10.5-10.7 • Common M.o.I. for high symmetry shapes (p. 304) • Parallel Axis Theorem

21. 10.5 • Example 10.8

22. 10.6 Torque • Torque- the tendency of a force to cause rotation about an axis • Where r is the distance from the axis of rotation and Fsinφ is the perpendicular component of the force • Where F is the force and d is the “moment arm.”

23. 10.6 • Moment Arm- (lever arm) the perpendicular distance from the axis of rotation to the “line of action”

24. 10.6 • Torque is a vector has dimensions ML2T-2 which are the same as work, units will also be N.m • Even though they have the same dimensions and units, they are two very different concepts. • Work is a scalar product of two vectors • Torque is a vector product of two vectors

25. 10.6 • The direction of the torque vector follows the right hand rule for rotation, and CCW torques will be considered positive, CW torques negative. • Quick Quizzes p. 307 • Example 10.9

26. 10.7 Torque and Angular Acceleration • Consider a tangential and radial force on a particle. • The Ft causes a tangential acceleration.

27. 10.7 • We can also look at the torque caused by the tangential force. • And since…

28. 10.7 • Newton’s 2nd Law (Rotational Analog) • Quick Quiz p. 309 • Review Examples 10.10-10.13

29. 10.8 Work, Power, and Energy • Rotational Analogs for Work, Power, and Energy • Work • Energy • Work-KE Theorem • Power

30. 10.8 • Quick Quiz p. 314 • Examples 10.14, 10.15

31. 10.9 Rolling Motion • For an object rolling in a straight line path the translational motion of its center of mass can be related to its angular displacment, velocity and acceleration. • Condition for Pure Rolling Motion- no slipping • If there is no slip, then every point on the outside of the wheel contacts the ground and following relationships hold.

32. 10.9

33. 10.9

34. 10.9 • Linear distance traveled (translational displacment)- • CofM Velocity (trans. vel.)- • CofMAccel. (trans. accel.)-

35. 10.9

36. 10.9 • Total Kinetic Energy for a rolling object Ktot = Kr+ Kcm (using just translational speed) (using just angular speed)

37. 10.9 • Friction must be present to give the torque causing rotation, but does not cause a loss of energy because the point of contact does not slide on the surface. • With zero friction the object would slide, not roll. Quick Quizzes p. 319 Example 10.16