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Dynamics II Chapter 6

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  1. Dynamics IIChapter 6

  2. Chapter Objectives • Discuss dynamics in terms of: • Work • Energy • Power • Momentum • Conservation laws • Oscillations

  3. Work and Energy • Consider a force F. • Sometimes, the object moves in the same direction as F • That means a parallel displacement vector d exists • Such a displacement requires Energy

  4. A word about Energy • Uses of energy: • Lifting a weight to a given height • Charging a battery • Boiling water • Heating an object • Types of energy: • Mechanical energy • Electrical energy • Heat

  5. Work Done by Constant Force cont’d • Work done in a straight line is: • F = force • d = distance traveled • Work expressed as N·m. • 1 Nm=1 J oule (J) W=Fd

  6. Work Done by Constant Force cont’d • Work done at an angle θ is: W = Fd cosθ

  7. Work Done by Variable Force • Example: throwing a javelin • Force is zero, then rises to a maximum, then decreases back to zero as javelin is released • F varies with x • Consider Figure 6-4.

  8. How does one calculateFd?

  9. Divide distance into small intervals

  10. Work Done by Variable Force cont’d • Variable force can be divided into intervals • Interval 1 has force value F1 and so on. • Work done on interval 1 is: • Δx = (xf-xi)/10 • Width of each interval on axis W=F1Δx

  11. Work Done by Variable Force cont’d Error decreases as number of intervals increases… Integral gets the size of each interval down to the infinitesimal

  12. One more trick in numerical intergration • Consider the force as a trapezoid, not a rectangle

  13. Energy • Energy can be viewed as the capacity to do work • When we deal with “mechanical work”, we’re talking about • Kinetic Energy (KE) • Gravitational Potential Energy (PE) • Strain Potential Energy

  14. Kinetic Energy • Energy possessed because of motion • Kinetic energy is expressed as: • Given as SI unit Joules (J)

  15. Work and Energy • Consider the work required to accelerate and object • Distance moved: • v=at, so t=v/a, so: • F=ma, so a=F/m, so: • Work = Fd =

  16. Kinetic Energy cont’d • Example: • Calculate the kinetic energy of a sprinter of mass 70 kg moving at 10 m/s. • KE = 0.5(70 kg)(10 m/s)2 • = 3500 J

  17. Gravitational Potential Energy • Energy possessed by a system due to position of a body • Also called potential energy PE = mgh

  18. Gravitational Potential Energy cont’d • Example: • A pole vaulter of body mass 70 kg succeeds in clearing a bar that is 6 m above the ground. Calculate his potential energy at the top of the vault.

  19. Gravitational Potential Energy cont’d • Example: h = 6m g = 9.81m/s2 PE = (70 kg)(9.81 m/s2)(6 m) = 4120 J

  20. Conservation of Mechanical Energy • Kinetic energy • Gravitational potential energy • Strain potential energy

  21. Conservation of Mechanical Energy cont’d • Total energy in a system does not change

  22. We’re lying to you again… • A real system inevitably involves friction • Presence of friction means energy is converted to heat or sound • Thermal energy: energy created when heat is generated

  23. Power • The rate of doing work • The rate of energy transfer • Given as SI units Watts (W) or J/s

  24. Impulse–Momentum Relationship • Impulsive forces: • Forces that occur over very short time intervals • Usually milliseconds • Bodies deform • Example: bat striking ball • Change of momentum of ball = Impulse • Consider Figure 6-8.

  25. Newton’s 2nd Law: Realizing Force is variable and using smallest intervals possible: Change in momentum = Impulse

  26. Collisions in One Dimension • Consider the following collisions: • Hockey puck with stick • Golf ball and club • Ball and bat in baseball • Foot and ball in soccer

  27. Equal and opposite reaction

  28. Collisions in One Dimension cont’d • Force between colliding partners exists for very short time • Impulse on objects is the same but with opposite signs • Change in momentum is the same but in the opposite direction

  29. Collisions in One Dimension cont’d • m1 and m2 = masses of objects 1 and 2 • v1f and v1i = final and initial velocities of object 1 • v2f and v2i = final and initial velocities of object 2

  30. Gets us back to conservation of linear momentum

  31. Elastic and Inelastic Collisions • Elastic collisions: • Kinetic energy is conserved • Consider a ball dropped on a floor from height h0 that bounces all the way back up

  32. Perfectly Elastic Impact Velocity of system is conserved Drop a Superball (close to perfect) Perfectly Plastic Impact Permanent deformation of at least one body Bodies do not separate Drop a lump of clay Extremes

  33. In Between • Coefficient of Restitution 0 1 Inelastic Perfectly Plastic Perfectly Elastic

  34. Elastic and Inelastic Collisions cont’d • Inelastic collisions: • Total kinetic energy is decreased but some other energy is increased • 0 < e < 1

  35. Coefficient of Restitution • Depends on BOTH bodies: • Basketball and gym floor • Baseball and bat • Tennis ball and racquet

  36. Velocities before impact Velocities after impact

  37. V2 V1 V3 V4

  38. One Simplification • If one body is stationary (e.g. a floor), the equation becomes simpler • Example: ball dropped to floor from height hd will bounce to a height of hb

  39. Bat-Ball Games and Spin • Speed of ball after impact is increased by: • Increasing the mass of the bat • Decreasing the mass of the ball • Increasing the initial velocity of the bat or ball • Increasing the angle of incidence • Increasing the value of e

  40. Backspin increases frictional effects when bouncing

  41. Topspin decreases frictional effects when bouncing

  42. Change in range due to angle of impact

  43. Oscillations • Hooke’s law: • Extension x of a spring is proportional to applied force F • k is spring constant/stiffness • given as N/m F=-kx

  44. Oscillations cont’d • Extension of spring is: