13.1 Newton’s law of motion 1.Newton’s 2 nd law of motion (1) A particle subjected to an unbalanced force experiences an accelerationhaving the same direction as and a magnitude that is directly proportional to the force. =m m = mass of a particle =a quantitative measure of the resistance of the particle to a change in its velocity.

(2) The unbalanced force acting on the particle is proportional to the time rate of change of the particle’s linear momentum. (if m=constant) 2. Newton’s Law of Gravitational Attraction r m1m2

G = universal constant of gravitation = r = distance between centers of two particles Weight of a particle with mass m 1 = m =mg m 2 : mass of the earth r = distance between the earth center and the particle

g= = acceleration due to gravity =9.81 measured at a point on the surface of the earth at sea level and at a latitude of

p 13-2 The equations of motion 1.Equations of motion of a particle subjected to more than one force.

p Kinetic diagram of particle p. p ………...equation of motion Free body diagram of particle p.

D’A lembert principle inertia force vector Dynamic equilibrium diagram p ( 慣性力 ) 若 則此狀態為靜平衡 +

(1) Inertial frame path p p x y o 2. Inertial frame of reference (newtonian) A coordinate system is either fixed or translates in a given direction with a constant velocity. (2) Noninertial frame path p p o y x

z y x i Equation of motion of particle i. Dynamic equilibrium diagram of particle i. i 13-3 Equation of motion for a system of particle xyz: Inertial Coordinate System

resultant external force resultant internal force Equation of motion of a system of particles.

By definition of the center of mass for a system of particles. Position vector of the center of mass G. Total mass of all particles. Assume that no mass is entering or leaving the system.

Hence ： This equation justifies the application of the equation of motion to a body that is represented as a single particle.

Rectangular Coordinate system. z y x path Equation of motion of particle P. In rectangular components 13-4 Equations of motion ： Rectangular Coordinate

scalar eqns. Analysis procedure 1.Free Body Diagram. (1) Select the proper inertial coordinate system. (2) Draw the particle’s F.B.D. 2. Equation of motion (1) Apply the equations of motion in scalar form or vector form. or

(2) Friction force (3) Spring force 3. Equations of kinematics Apply for the solutions

Curve path of motion of a particle is known. =Tangential unit vector =Normal unit vector =Binormal unit vector = 13.5 Equation of Motion:Normal and Tangential Coordinates P n b t Curvepath

Or scalar form = = = 0 Equation of motion

1. Free body diagram Identify the unknowns in the problem. 2. Equation of motion Apply the equations of motion using normal and tangential coordinates. 3. Kinematics Formulate the tangential and normal components of acceleration. Analysis procedure

r z 13.6 Equation of Motion :Cylindrical coordinate Equation of motion in cylindrical coordinates

and Cylindrical or polar coordinates are suitable for a problem for which Data regarding the angular motion of the radial line r are given, or in Cases where the path can be conveniently expressed in terms of these coordinates.

Normal and Tangential force If the particle’s accelerated motion is not completely specified, then information regarding the directions or magnitudes of the forces acting on the particle must be known or computed. Now, consider the case in which the force P causes the particle to move along the path r=f(  ) as shown in the following figure. r=f(  ) :path of motion of particle P:External force on the particle F:Friction force along the tangent N:Normal force perpendicular to tangent of path

Direction of F & N dr :radial component rd  :transverse component ds:distance The directions of F and N can be specified relative to the radial coordinate r by computing the angle . Angle  is defined between the extended radial line and the tangent to the path. dr rd + positive direction of - negative direction of