1. FE Exam: Dynamics review D. A. Lyn School of Civil Engineering 21 February 2012

2. Preliminaries Units (relevant quantities: g, displacement, velocity, acceleration, energy, momentum, etc.) Notation (dot, vector) Vectors (components and directions/signs, addition (graphical), dot and cross products, vector polygons) Coordinate systems (Cartesian and curvilinear, fixed and moving or relative, unit vectors) Statics (free body diagram)

3. Classification of dynamics and problems Kinematics: description of motion without reference to forces –Particle (no rotation about itself) and rigid-body –Coordinate systems (Cartesian, curvilinear, rotation) –Constraints on motion Kinetics: inclusion of forces (mass, or momentum or energy) –Types of forces: conservative (gravitational, spring, elastic collisions) and non-conservative (friction, inelastic collisions) –Newton’s 2 nd law: linear and angular momentum Use of free body diagram to deal with external forces –Particles and rigid body (system of particles) –Impulse (time involved) and momentum still working with vectors –Work (distances involved) and energy (velocities involved) working with scalars (usually easier)

4. Particle kinematics General relations between displacement ( r ), velocity ( u ), and acceleration ( a ) Given a formula for (or graph of) r as function of t, take derivatives to find u and a –Given a formula for u or a as function of t, integrate to find r or u  Special case: constant acceleration,

5. Sample problems The position of a particle moving horizontally is described by, with s in m and t in s. At t = 2 s, what is its acceleration? Soln: Take derivatives of s with respect to t, and evaluate at t =2s ( ) so a ( t =2s) = 4 m/s 2.derivatives Projectile problem: A projectile is launched with an initial speed of v 0 =100 ft/s at  =30 ° to the horizontal, what is the horizontal distance, L, covered by the projectile when it lands again? Soln: constant acceleration (only gravitationalconstant acceleration acceleration involved) problem, so apply formulae in two directions wish to find L = x end -x 0, for y end -y 0 =0, so we solve L=v 0x t end and 0=v 0y t end -g(t end 2 /2) for t end and L; t end =3.1 s and L=269.2 ft

6. Kinetics of a particle Linear momentum, L = mu (appearance of mass, i.e., inertia) Newton’s 2 nd law: Forces determined from free body diagram (as in statics) –Types of forces: gravitational, frictional, external Angular momentum (about a point O), Newton’s 2 nd law: Impulse (used in impact and collision problems), – momentum conservation: – mini-problem: A golf ball of mass 50-g is hit with a club. If the initial velocity of the ball is 20 m/s, what is the impulse imparted to the ball? If the contact duration was 0.05 s, what was the average force on the ball?

7. Problem: kinetics of a particle (truck) A truck of weight W = 4000 lbf moves down a  =10° incline at an initial speed of u 0 = 20 ft/s. A constant braking force of F brk =1200 lbf is experienced by the truck from a time, t = 0. What is the distance covered by the truck before it stops from the time that the braking force is applied? kinematics problem: kinematic Notes: forces involved – kinetics problem, rectilinear (straight-line) motion: determine net force on truck in direction of motion, apply Newton’s 2 nd law to evaluate distance covered From free body diagram, sum of forces in direction of motion,

8. Curvilinear coordinates and motion Plane motion (motion on a surface, i.e., in only two dimensions) –Tangential ( t ) and normal ( n ) coordinates where  is the radius of curvature of particle path –Radial ( r ) and transverse (  ) or polar coordinates –Special case: pure circular motion at an angular frequency,  t  (  is the angular acceleration)

9. Particle kinetics problem Find the tension, T, in the string and the angular acceleration, , if at the position shown the a sphere of mass, m= 10 kg, has a tangential velocity of v 0 =4 m/s. Choose a polar coordinate system, perform free body analysis to determine sum of forces, and set equal to ma.coordinate

10. Energy and work Work of a force, F, resulting in a change in position from state 1 to state 2: –Constant force in rectilinear motion, F x  x 2 -x 1 ) –Gravitational force, -W  y 2 -y 1 ), y>0 upwards –Spring force, -k(x 2 2 -x 1 2 )/2, (x 2 <x 1, returning to undeformed state) Kinetic energy, Relation between work and kinetic energy: for conservative forces (such as gravitational and spring forces, but not frictional forces), a potential energy function, V, can be defined such that –Gravitational force: V = Wy, spring force, V=kx 2 /2 For conservative forces, an equation for conservation of energy can be expressed as or

11. A problem solved using energy principles A 2-kg block (A) rests on a frictionless plane inclined at an angle  =30 °. It is attached by an inextensible cable to a 3-kg block (B) and to a fixed support. Assume pulleys are frictionless and weightless. If initially both blocks are stationary, how far will the 2-kg block travel before its speed is 4 m/s? Motion constraints: s B =s A /2 (and  y A =-2  y B sin  ), and v B =v A /2 Frictionless system  conservative gravitational forces only, only distances and speeds explicitly involved  apply energy equationenergy equation

12. Constrained motion, reference frames, relative motion Constrained-motion problems – choice of reference frames: relative motion (in a plane) Choice of reference frames – motion relative to a point A in a moving reference frame –For plane motion, note direction of components, e.g.,  r B/A is perpendicular to r B/A, etc. –For points on the same rigid body,

13. Problem: Kinematics of rigid body example The end A of rod AB of length L = 0.6 m moves at velocity V A = 2 m/s and acceleration, a A = 0.2 m/s 2, both to the left, at the instant shown, when  = 60 °. What is the velocity, V B, and acceleration, a B, of end B at the same instant? Pure kinematics problem:

14. Kinetics of a system of particles (or rigid body) For a system of particles (or a rigid body), analysis is performed in terms of the mass center, G, located at radial vector, r G, and total mass m Equations of motions: where a G is the acceleration of the mass center, and H G is the angular momentum about the mass center − For a system with no external forces or moment acting, then linear momentum, L, and angular momentum, H, is conserved, i.e., remains constant For a system of particles (or a rigid body), and where the mass moment of inertia I is defined by (Standard formulae for I = mk 2, where k is the radius of gyration, for standard bodies are listed in tables; be careful about which axis I is defined, whether centroidal axis or not, remember parallel axis theorem)

16. Problem: two-particle system A particle A of mass m and and a particle B, of mass 2 m are connected by rigid massless rod of length R. If mass B is suddenly given a vertical velocity v perpendicular to the connecting rod, determine the location of the mass center, the velocity of the mass center, the angular momentum, and the angular velocity of the system soon after the motion begins.mass center

17. Problem: rigid-body kinetics What is the angular acceleration, , of the 60-kg (cylindrical) pulley of radius R = 0.2 m and the tension in the cable if a 30-kg block is attached to the end of the cable? Analysis of block − Kinematic constraint ( a block = R  ) Analysis of pulleypulley

18. Dynamics Outline and Problem - Solutions as Provided by Kaplan

19. Copyright Kaplan AEC Education, 2008 Dynamics Outline Overview DYNAMICS, p. 205 KINEMATICS OF A PARTICLE, p. 206 Relating Distance, Velocity and the Tangential Component of Acceleration Constant Tangential Acceleration Rectilinear Motion Rectangular Cartesian Coordinates Circular Cylindrical Coordinates Circular Path

20. Copyright Kaplan AEC Education, 2008 Dynamics Outline Overview Continued RIGID BODY KINEMATICS, p. 203 The Constraint of Rigidity The Angular Velocity Vector Instantaneous Center of Zero Velocity Accelerations in Rigid Bodies

21. Copyright Kaplan AEC Education, 2008 Dynamics Outline Overview Continued NEWTON’S LAWS OF MOTION, p. 210 Applications to a Particle Systems of Particles Linear Momentum and Center of Mass Impulse and Momentum Moments of Force and Momentum

22. Copyright Kaplan AEC Education, 2008 Dynamics Outline Overview Continued WORK AND KINETIC ENERGY, p. 219 A Single Particle Work of a Constant Force Distance-Dependent Central Force

23. Copyright Kaplan AEC Education, 2008 Dynamics Outline Overview Continued KINETICS OF RIGID BODIES, p. 225 Moment Relationships for Planar Motion Work and Kinetic Energy

24. Copyright Kaplan AEC Education, 2008 Kinematics of Particles—1D Motion

25. Copyright Kaplan AEC Education, 2008 Solution

26. Copyright Kaplan AEC Education, 2008 Kinematics of Particles—1D Motion

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29. Copyright Kaplan AEC Education, 2008 2D Motion—Rectangular Cartesian Coordinates

30. Copyright Kaplan AEC Education, 2008 Solution

31. Copyright Kaplan AEC Education, 2008 2D Motion—Plane Polar Coordinates

32. Copyright Kaplan AEC Education, 2008 Solution

33. Copyright Kaplan AEC Education, 2008 Instantaneous Center of Zero Velocity

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35. Copyright Kaplan AEC Education, 2008 Evaluation of Accelerations

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39. Copyright Kaplan AEC Education, 2008 Newton’s 2 nd Law

40. Copyright Kaplan AEC Education, 2008 Solution

41. Copyright Kaplan AEC Education, 2008 Solution (continued)

42. Copyright Kaplan AEC Education, 2008 Solution (continued)

43. Copyright Kaplan AEC Education, 2008 Newton’s 2 nd Law

44. Copyright Kaplan AEC Education, 2008 Solution

45. Copyright Kaplan AEC Education, 2008 Work & Kinetic Energy

46. Copyright Kaplan AEC Education, 2008 Solution

47. Copyright Kaplan AEC Education, 2008 Moments of Force & Momentum

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49. Copyright Kaplan AEC Education, 2008 Solution (continued)

50. Copyright Kaplan AEC Education, 2008 Solution (continued)

51. Copyright Kaplan AEC Education, 2008 Work & Kinetic Energy

52. Copyright Kaplan AEC Education, 2008 Solution

53. Copyright Kaplan AEC Education, 2008 Solution

54. Copyright Kaplan AEC Education, 2008 Work & Kinetic Energy

55. Copyright Kaplan AEC Education, 2008 Solution

56. Copyright Kaplan AEC Education, 2008 Solution (continued)

57. Copyright Kaplan AEC Education, 2008 Work & Kinetic Energy

58. Copyright Kaplan AEC Education, 2008 Solution

59. Copyright Kaplan AEC Education, 2008 Solution (continued)