DYNAMICS. Kinematics of Particles Engineering Dynamics Lecture Note VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER

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1 27 The McGraw-Hill Companies, Inc. All rights resered. Eighth E CHAPTER 11 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdinand P. Beer E. Russell Johnston, Jr. Kinematics of Particles Lecture Notes: J. Walt Oler Teas Tech Uniersity Engineering Dynamics Lecture Note

2 Contents Introduction Rectilinear Motion: Position, Velocity & Acceleration Determination of the Motion of a Particle Uniform Rectilinear-Motion Uniformly Accelerated Rectilinear- Motion Motion of Seeral Particles: Relatie Motion Motion of Seeral Particles: Dependent Motion Graphical Solution of Rectilinear-Motion Problems Curilinear Motion: Position, Velocity & Acceleration Deriaties of Vector Functions Rectangular Components of Velocity and Acceleration Motion Relatie to a Frame in Translation Tangential and Normal Components Radial and Transerse Components 27 The McGraw-Hill Companies, Inc. All rights resered. 11-2

3 Introduction Dynamics includes: - Kinematics: study of the geometry of motion. Kinematics is used to relate displacement, elocity, acceleration, and time without reference to the cause of motion. - Kinetics: study of the relations eisting between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by gien forces or to determine the forces required to produce a gien motion. Rectilinear motion: position, elocity, and acceleration of a particle as it moes along a straight line. Curilinear motion: position, elocity, and acceleration of a particle as it moes along a cured line in two or three dimensions. 27 The McGraw-Hill Companies, Inc. All rights resered. 11-3

4 Rectilinear Motion: Position, Velocity & Acceleration Particle moing along a straight line is said to be in rectilinear motion. Position coordinate of a particle is defined by positie or negatie distance of particle from a fied origin on the line. The motion of a particle is known if the position coordinate for particle is known for eery alue of time t. Motion of the particle may be epressed in the form of a function, e.g., 2 3 6t t or in the form of a graph s. t. 27 The McGraw-Hill Companies, Inc. All rights resered. 11-4

5 Rectilinear Motion: Position, Velocity & Acceleration 27 The McGraw-Hill Companies, Inc. All rights resered. 11-5

6 Rectilinear Motion: Position, Velocity & Acceleration 27 The McGraw-Hill Companies, Inc. All rights resered. 11-6

7 Rectilinear Motion: Position, Velocity & Acceleration Consider particle with motion gien by 2 6t t 3 at t =, at t = 2 s, at t = 4 s, at t = 6 s, 27 The McGraw-Hill Companies, Inc. All rights resered. 11-7

8 Determination of the Motion of a Particle Recall, motion of a particle is known if position is known for all time t. Typically, cons of motion are specified by the type of acceleration eperienced by the particle. Determination of elocity and position requires two successie integrations. Three classes of motion may be defined for: - acceleration gien as a function of time, a = f(t) - acceleration gien as a function of position, a = f() - acceleration gien as a function of elocity, a = f() 27 The McGraw-Hill Companies, Inc. All rights resered. 11-8

9 Determination of the Motion of a Particle Acceleration gien as a function of time, a = f(t): t t d a f t d f t d f t d t d t t d Acceleration gien as a function of position, a = f(): d d or f d d d a d f or a t t f t t t t d d f d f 2 2 t t d 27 The McGraw-Hill Companies, Inc. All rights resered. 11-9

10 Determination of the Motion of a Particle Acceleration gien as a function of elocity, a = f(): d t d d t a d f a f t f t d f d f d d f t t d d f t t d f 27 The McGraw-Hill Companies, Inc. All rights resered. 11-1

11 Sample Problem 11.2 Ball tossed with 1 m/s ertical elocity from window 2 m aboe ground. Determine: elocity and eleation aboe ground at time t, highest eleation reached by ball and corresponding time, and time when ball will hit the ground and corresponding elocity. 27 The McGraw-Hill Companies, Inc. All rights resered

12 Sample Problem The McGraw-Hill Companies, Inc. All rights resered

13 Uniform Rectilinear Motion For particle in uniform rectilinear motion, the acceleration is zero and the elocity is constant. d d constant t t t 27 The McGraw-Hill Companies, Inc. All rights resered

14 Uniformly Accelerated Rectilinear Motion For particle in uniformly accelerated rectilinear motion, the acceleration of the particle is constant. d a at constant d a t at d at t 1 2 at 2 d t at t 1 2 at 2 d d 2 a 2 2a constant d a d a 27 The McGraw-Hill Companies, Inc. All rights resered

15 Eample: Kinematics relations 27 The McGraw-Hill Companies, Inc. All rights resered

16 Eercise 27 The McGraw-Hill Companies, Inc. All rights resered

17 Homework 27 The McGraw-Hill Companies, Inc. All rights resered

18 Motion of Seeral Particles: Relatie Motion For particles moing along the same line, time should be recorded from the same starting instant and displacements should be measured from the same origin in the same direction. 27 The McGraw-Hill Companies, Inc. All rights resered

19 Sample Problem 11.4 Ball thrown ertically from 12 m leel in eleator shaft with initial elocity of 18 m/s. At same instant, open-platform eleator passes 5 m leel moing upward at 2 m/s. Determine (a) when and where ball hits eleator and (b) relatie elocity of ball and eleator at contact. 27 The McGraw-Hill Companies, Inc. All rights resered

20 Sample Problem The McGraw-Hill Companies, Inc. All rights resered. 11-2

21 Sample Problem The McGraw-Hill Companies, Inc. All rights resered

22 Motion of Seeral Particles: Dependent Motion 27 The McGraw-Hill Companies, Inc. All rights resered

23 Sample Problem 11.5 Pulley D is attached to a collar which is pulled down at 3 cm/s. At t =, collar A starts moing down from K with constant acceleration and zero initial elocity. Knowing that elocity of collar A is 12 cm/s as it passes L, determine the change in eleation, elocity, and acceleration of block B when block A is at L. 27 The McGraw-Hill Companies, Inc. All rights resered

24 Sample Problem The McGraw-Hill Companies, Inc. All rights resered

25 Sample Problem The McGraw-Hill Companies, Inc. All rights resered

26 Eercise 27 The McGraw-Hill Companies, Inc. All rights resered

27 27 The McGraw-Hill Companies, Inc. All rights resered

28 27 The McGraw-Hill Companies, Inc. All rights resered

29 Homework 27 The McGraw-Hill Companies, Inc. All rights resered