Mechanics of rigid body StaticsDynamics Equilibrium Galilei Newton Lagrange Euler KinematicsKinetics v=ds/dt a=dv/dt Σ F = 0 Σ F = m a mechanics of rigid body StaticsDynamics 12.1 Introduction

1.K inematics Analysis of the geometric aspects of motion. 2.P article A particle has a mass but negligible size and shape. 3.R ectilinear Kinematics Kinematics of objects moving along straight path and characterized by objects position, velocity and acceleration. 4.P osition (1) position vector r A vector used to specify the location of particle P at any instant from origin O Rectilinear Kinematics: Continuous motion

(2) position coordinate, S An algebraic scalar used to represent the position coordinate of particle P scalarvector 5.Displacement Change in position of a particle, vector (1) Displacement or from O to P. P’P r’ r ss r s s’ o

(2) Distance Total length of path traversed by the particle. A positive scalar. 6.Velocity (1) Average velocity (2) Instantaneous Velocity speed = magnitude of velocity = | v | Average speed = Total distance/elapsed time = (3) Speed

7.Acceleration (2) (Instantaneous) acceleration (1) Average acceleration 8.Relation involving a, s and v v=ds/dt, dt=ds/v a=dv/dt, dt=dv/a so, ds/v=dv/a vdv=ads

9.Constants acceleration a = a c

10.Analysis Procedure (2) Kinematic Equations A. Know the relationship between any two of the four variables a, v, a and t. B. Use the kinematic equations to determine the unknown varaibles (1)Coordinate System A. Establish a position coordinate s along the path. B. Specify the fixed origin and positive direction of the coordinate.

12.3 Rectilinear Kinematics : Erratic Motion ttt S Va

Givenmethod Kinematics egn Find S-t graph Measure slope V=ds/dt V-t graph Measure slope A=dv/dt a-t graph A-t graph Area integration v-t graph Area integration s-t graph A-s graph Area integration v-s graph Measure slope A=v(dv/ds) a-s graph

1. Curvilinear motion 1. Curvilinear motion The particle moves along a curved path. The particle moves along a curved path. Vector analysis will be used to formulate the particle’s position, velocity and acceleration. p 12-4 General Curvilinear Motion

s = change in position of particle form p to p’ o p r (t) s 2. Position 3. Displacement o p p’ s r r ’ r

= “tangent” to the curve at Pt.p = “tangent” to the path of motion 4. Velocity (1) average velocity 平均 (2) Instantaneous velocity 瞬時 (2) Instantaneous velocity 瞬時 o p p’ v r r ’ r (3) Speed

(2) Instantaneous acceleration which is not tangent to the curve of motion, but tangent to the hodograph. Hodograph 5. Acceleration (1) Average acceleration (1) Average acceleration = time rate of change of velocity vectors Hodogragh is a curve of the locus of points for the arrowhead of velocity vector.

xyz : fixed rectangular coordinate system 12-5 Curvilinear Motion : Rectangular components x y z θ r path p s

1. Position vector Here = magnitude of = unit vector = direction of

2. Velocity 0 00 tangent to the path

3. Acceleration

: initial velocity : Constant downward acceleration : velocity at any instant 12.6 Motion of a projectile y x v0v0 v0v0 a= - g j v (v 0 ) x (v 0 ) y

Position Vector (x,y components) = x + y initial position = x o + y o Velocity Vector = = x + y = Vx + Vy Acceleration Vector a = = + = -g = a x + a y V 0 = (V x )o + (V y )o (known)

1. Horizontal motion, a x =0 One independent eqn V x = (V x ) 0 + a x t = (V x ) 0 X = X 0 + (V x ) 0 t Same as 1 st Eq. X = X 0 + (V x ) 0 t

two independent eqns a y =-g constant 2. Vertical motion, a y =-g constant Can be derived from above two Eqs.

Path of motion of a particle is known. 1. Planar motion 12-7 Curvilinear Motion:Normal and Tangential components. o path o’ n ρ unun utut t p Here: t (tangent axis ): axis tangent to the curve at P and positive in the direction of increasing S; u t : unit vector n (normal axis ): axis perpendicular to t axis and directed from P toward to the center of curvature o’; u n : unit vector o’ = center of curvature  = radius of curvature p = origin of coordinate system tn s

(1) Path Function (known) (2) Velocity (3) Acceleration o’ ρ unun utut p ds dθdθ ut’ut’ unun du t p ut’ut’ utut dθdθ

a t : Change in magnitude of velocity a n : Change in direction of velocity If the path in y = f ( x )

1.Polar coordinates r p r Reference line r ： radial coordinate, ： transverse coordinate, (2) Position (3) Velocity 12-8 Curvilinear Motion ： Cylindrical Components (1) coordinates (r,  ) o 

rate of change of the length of the radial coordinate. angular velocity (rad/s)

(4) Acceleration angular acceleration

2. Cylindrical coordinates Position vector Velocity Acceleration z x y r 3D

12.9 Absolute Dependent Motion Analysis of Two Particles 1.Absolute Dependent Motion The motion of one particle depends on the corresponding motion of another particle when they are interconnected by inextensible cords which are wrapped around pulleys. A B A B

(1)position-coordinate equation A. Specify the location of particles using position coordinates having their origin located at a fixed point or datum line. B. Relate coordinates to the total length of card l T (2)Time Derivatives Take time derivatives of the position-coordinate equation to yield the required velocity and acceleration equations. 2.Analysis procedure

3. Example A B Datum (1)position-coordinate equation (2) Time Derivatives

12.10 Relative-Motion Analysis of Two Particles 1.Translating frames of reference A frame of reference whose axes do not rotate and are only permitted to translate relative to the fixed frame. xyz:fixed frame x’y’z’:translating frame moving with particle A r A 、 r B : absolute positions of particle A & B r B/A : relative position of B with respect to A x o y z x’ y’ z’ B A r B/A rBrB rArA

2.position vector B A O rArA rBrB r B/A rArA rBrB 3. velocity Vector V B/A : relative velocity observed from the translating frame. 4. acceleration vector a B/A :acceleration of B as seen by an observer located at A and translating with x’y’z’ frame.