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Dynamics in Two Dimensions: Solving Plane Motion Problems

Learn to solve motion problems in a plane with independent x and y accelerations using Newton's second law and kinematic equations. Includes examples on rocketing in the wind and projectile motion with and without air resistance.

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Dynamics in Two Dimensions: Solving Plane Motion Problems

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  1. Chapter 8 Lecture

  2. Chapter 8. Dynamics II: Motion in a Plane Chapter Goal: To learn how to solve problems about motion in a plane. Slide 8-2

  3. Chapter 8 Preview Slide 8-3

  4. Chapter 8 Preview Slide 8-4

  5. Chapter 8 Preview Slide 8-5

  6. Chapter 8 Preview Slide 8-6

  7. Chapter 8 Preview Slide 8-7

  8. Chapter 8 Preview Slide 8-8

  9. Chapter 8 Reading Quiz Slide 8-9

  10. Reading Question 8.1 When drag is included, the launch angle of a projectile which maximizes the range is • Greater than 45. • Equal to 45. • Less than 45. Slide 8-10

  11. Reading Question 8.1 When drag is included, the launch angle of a projectile which maximizes the range is • Greater than 45. • Equal to 45. • Less than 45. Slide 8-11

  12. Reading Question 8.2 Circular motion is best analyzed in a coordinate system with • x- and y-axes. • x-, y-, and z-axes. • x- and z-axes. • r-, t-, and z-axes. Slide 8-12

  13. Reading Question 8.2 Circular motion is best analyzed in a coordinate system with • x- and y-axes. • x-, y-, and z-axes. • x- and z-axes. • r-, t-, and z-axes. Slide 8-13

  14. Reading Question 8.3 This chapter studies • Uniform circular motion. • Nonuniform circular motion. • Orbital motion. • Both a and b. • All of a, b, and c. Slide 8-15

  15. Reading Question 8.3 This chapter studies • Uniform circular motion. • Nonuniform circular motion. • Orbital motion. • Both a and b. • All of a, b, and c. Slide 8-15

  16. Reading Question 8.4 For uniform circular motion, the net force • Points toward the center of the circle. • Points toward the outside of the circle. • Is tangent to the circle. • Is zero. Slide 8-16

  17. Reading Question 8.4 For uniform circular motion, the net force • Points toward the center of the circle. • Points toward the outside of the circle. • Is tangent to the circle. • Is zero. Slide 8-17

  18. Reading Question 8.5 The centrifugal force • Is a fictitious force. • Points toward the center of the circle. • Is provided by static friction. • All of the above. • B and C, but not A. Slide 8-18

  19. Reading Question 8.5 The centrifugal force • Is a fictitious force. • Points toward the center of the circle. • Is provided by static friction. • All of the above. • B and C, but not A. Slide 8-19

  20. Chapter 8 Content, Examples, and QuickCheck Questions Slide 8-20

  21. Dynamics in Two Dimensions • Suppose the x- and y-components of acceleration are independent of each other. • That is, ax does not depend on y or vy, and ay does not depend on x or vx. • Your problem-solving strategy is to: • Draw a pictorial representation: a motion diagram (if needed) and a free-body diagram. • Use Newton’s second law in component form: The force components (including proper signs) are found from the free-body diagram Slide 8-21

  22. Dynamics in Two Dimensions Dynamics in Two Dimensions • Solve for the acceleration. If the acceleration is constant, use the two-dimensional kinematic equations of Chapter 4 to find velocities and positions: Slide 8-22

  23. Example 8.1 Rocketing in the Wind Slide 8-23

  24. Example 8.1 Rocketing in the Wind Slide 8-24

  25. Example 8.1 Rocketing in the Wind Slide 8-25

  26. Example 8.1 Rocketing in the Wind Slide 8-26

  27. Example 8.1 Rocketing in the Wind Slide 8-27

  28. Example 8.1 Rocketing in the Wind Slide 8-28

  29. Example 8.1 Rocketing in the Wind Slide 8-29

  30. Projectile Motion: Review • In the absence of air resistance, a projectile moves under the influence of only gravity. • If we choose a coordinate system with a vertical y-axis, then • Consequently, from Newton’s second law, the acceleration is Slide 8-30

  31. Projectile Motion: Review • Consider a projectile with initial speed v0, and a launch angle of  above the horizontal. • In Chapter 4 we found that the distance it travels before it returns to the same elevation from which it was launched (the range) is: Trajectories of a projectile launched at different angles with a speed of 99 m/s. • The maximum range occurs for  45. • All of these results neglect the effect of air resistance. Slide 8-31

  32. Projectile Motion • For low-mass projectiles on earth, the effects of air resistance, or drag, are too large to ignore. • When drag is included, the angle for maximum range of a projectile depends both on its size and mass. • The optimum angle is roughly 35 for baseballs. • The flight of a golf ball is even more complex, because of the dimples and effects of spin. • Professional golfers achieve their maximum range at launch angles of barely 15! Slide 8-32

  33. Projectile Motion • The acceleration of a typical projectile subject to drag force from the air is: • The components of acceleration are not independent of each other. • These equations can only be solved numerically. • The figure shows the numerical solution for a 5-g plastic ball. Slide 8-33

  34. Uniform Circular Motion • When describing circular motion, it is convenient to define a moving rtz-coordinate system. • The origin moves along with a certain particle moving in a circular path. • The r-axis (radial) points from the particle toward the center of the circle. • The t-axis (tangential) is tangent to the circle, pointing in the ccw direction. • The z-axis is perpendicular to the plane of motion. Slide 8-34

  35. Uniform Circular Motion • A particle in uniform circular motion with angular velocity ω has velocity v =ωr, in the tangential direction. • The acceleration of uniform circular motion points to the center of the circle. • The rtz-components of and are: Slide 8-35

  36. QuickCheck 8.1 The diagram shows three points of a motion diagram. The particle changes direction with no change of speed. What is the acceleration at point 2? Slide 8-36

  37. QuickCheck 8.1 The diagram shows three points of a motion diagram. The particle changes direction with no change of speed. What is the acceleration at point 2? Acceleration of changing direction Slide 8-37

  38. QuickCheck 8.2 A toy car moves around a circular track at constant speed. It suddenly doubles its speed — a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of 1/4. 1/2. No change since the radius doesn’t change. 2. 4. Slide 8-38

  39. QuickCheck 8.2 A toy car moves around a circular track at constant speed. It suddenly doubles its speed—a change of a factor of 2. As a result, the centripetal acceleration changes by a factor of 1/4. 1/2. No change since the radius doesn’t change. 2. 4. Slide 8-39

  40. Example 8.2 The Ultracentrifuge Slide 8-40

  41. Example 8.2 The Ultracentrifuge Slide 8-41

  42. Dynamics of Uniform Circular Motion • An object in uniform circular motion is not traveling at a constant velocity in a straight line. • Consequently, the particle must have a net force acting on it • Without such a force, the object would move off in a straight line tangent to the circle. • The car would end up in the ditch! Highway and racetrack curves are banked to allow the normal force of the road to provide the centripetal acceleration of the turn. Slide 8-42

  43. Dynamics of Uniform Circular Motion • The figure shows a particle in uniform circular motion. • The net force must point in the radial direction, toward the center of the circle. • This centripetal force is not a new force; it must be provided by familiar forces. Slide 8-43

  44. QuickCheck 8.3 An ice hockey puck is tied by a string to a stake in the ice. The puck is then swung in a circle. What force or forces does the puck feel? A new force: the centripetal force. A new force: the centrifugal force. One or more of our familiar forces pushing outward. One or more of our familiar forces pulling inward. I have no clue. Slide 8-44

  45. QuickCheck 8.3 An ice hockey puck is tied by a string to a stake in the ice. The puck is then swung in a circle. What force or forces does the puck feel? A new force: the centripetal force. A new force: the centrifugal force. One or more of our familiar forces pushing outward. One or more of our familiar forces pulling inward. I have no clue. The rules about what is or is not a force haven’t changed. Force must be exerted at a point of contact (except for gravity). Force must have an identifiable agent doing the pushing or pulling. The net force must point in the direction of acceleration (Newton’s second law). Slide 8-45

  46. QuickCheck 8.4 • An ice hockey puck is tied by a string to a stake in the ice. The puck is then swung in a circle. What force is producing the centripetal acceleration of the puck? • Gravity • Air resistance • Friction • Normal force • Tension in the string Slide 8-46

  47. QuickCheck 8.4 • An ice hockey puck is tied by a string to a stake in the ice. The puck is then swung in a circle. What force is producing the centripetal acceleration of the puck? • Gravity • Air resistance • Friction • Normal force • Tension in the string Slide 8-47

  48. Example 8.4 Turning the Corner I Slide 8-48

  49. Example 8.4 Turning the Corner I • VISUALIZE • The second figure below shows the top view of a tire as it turns a corner. • The force that prevents the tire from sliding across a surface is static friction. • Static friction pushes sideways on the tire, perpendicular to the velocity, since the car is not speeding up or slowing down. • The free-body diagram, drawn from behind the car, shows the static friction pointing toward the center of the circle. Slide 8-49

  50. Example 8.4 Turning the Corner I Slide 8-50

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