1 / 37

Chapter 10

Chapter 10. Rotational Motion and Torque. 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. 10.1. The location of point P can be described in polar coordinates (r , θ ).

dalton-morin
Download Presentation

Chapter 10

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related