Kinematics of Particles

Transcription

1 nnouncements Recitation time is set to 8am eery Monday. Participation i credit will be gien to students t who uploads a good question or good answer to the Q& bulletin board. Suggestions? T s and I will be uploading questions/answers if necessary. During class, please try to ask in English. Upload your pictures. Download working model D, play around with it. 11-1

2 Introduction What are the two parts of Dynamics? - 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. 11 -

3 Tools and Mechanisms Wheel, Leer, Pulley, Hammer etc. nd moies

4 Why do we need dynamics? Why do we want to know kinematics? What are we really interested in? Let s understand what we are analyzing before we analyze! It soles problems. What kind of problems? Limited motion, limited force characteristics. Translate one motion to another. Translate one force to another. There is always a Desired motion, Desired force s motion and force that we can generate. Mechanisms enable us to generated the desired motion, force!!! Need to understand these mechanism s kinematics and kinetics! 11-4

5 Rectilinear Motion: Position, Velocity & cceleration 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., 3 = 6t t or in the form of a graph s. t. 11-5

6 Rectilinear Motion: Position, Velocity & cceleration Consider particle with motion gien by = 6t t = 3 d = 1t 3t dt a d d = = = 1 t dt dt 6 at t =, =, =, a = 1 m/s at t = s, = 16 m, = ma = 1 m/s, a = at t = 4 s, = ma = 3 m, =, a = -1 m/s at t = 6s, =, =-36 m/s, a = 4 m/s 11-6

7 Determination of the Motion of a Particle Recall, motion of a particle is known if position is known for all time t. Typically, conditions of motion are specified by the type of acceleration eperienced by the particle. Determination of elocity and position requires two successie integrations. i 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() 11-7

8 Determination of the Motion of a Particle cceleration gien as a function of time, a = f(t): cceleration gien as a function of position,, a = f(): ) 11-8

9 Determination of the Motion of a Particle cceleration gien as a function of elocity, a = f(): 11-9

10 Sample Problem 11. SOLUTION: all tossed with 1 m/s ertical elocity from window 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. 11-1

11 Sample Problem 11.3 SOLUTION: a = k rake mechanism used to reduce gun recoil consists of piston attached to barrel moing in fied cylinder filled with oil. s barrel recoils with initial elocity, piston moes and oil is forced through orifices in piston, causing piston and cylinder to decelerate at rate proportional to their elocity. Determine (t), (t), and ()

12 Uniform Rectilinear Motion For particle in uniform rectilinear motion, the acceleration is zero and the elocity is constant. d dt = = constant t d = dt = = t + t 11-1

13 Uniformly ccelerated Rectilinear Motion For particle in uniformly accelerated rectilinear motion, the acceleration of the particle is constant. d dt = a = constant d = a dt = + at t = at d dt = + at 1 = + t + at t ( ) 1 d = + at dt = t + at d d = a = constant d = a d = ( ) + a 1 ( ) = a( ) 11-13

14 Motion of Seeral Particles: Relatie Motion For particles moing along the same line, time should ldbe recorded dfrom the same starting instant and displacements should be measured from the same origin in the same direction. = + = = relatie position of with respect to = = + = relatie elocity of with respect to a = a a a = a + a = relatie acceleration of with respect to 11-14

15 Sample Problem 11.4 all thrown ertically from 1 m leel in eleator shaft with initial elocity of 18 m/s. t same instant, open-platform eleator passes 5 m leel moing upward at m/s. Determine (a) when and where ball hits eleator and (b) relatie elocity of ball and eleator at contact