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13.1 Newton’s law of motion 1.Newton’s 2nd law of motion (1) A particle subjected to an unbalanced force experiences an acceleration direction as F a F having the same and a magnitude that is directly proportional to the force. F =m a m = mass of a particle =a quantitative measure of the resistance of the particle to a change in its velocity. (2) The unbalanced force F acting on the particle is proportional to the time rate of change of the particle’s linear momentum. d F (mv ) dt dv dm m v dt dt ma (if m=constant) 2. Newton’s Law of Gravitational Attraction m1 m 2 FG 2 r r m1 m2 G = universal constant of gravitation = 66.73x10 12 m3 / kg s 2 r = distance between centers of two particles Weight of a particle with mass m1 = m mm 2 FG 2 r Gm 2 m( 2 ) r =mg m2 : mass of the earth r = distance between the earth center and the particle g= Gm 2 2 r = acceleration due to gravity m =9.81 s 2 measured at a point on the surface of the earth at sea level and at a latitude of 450 13-2 The equations of motion 1. Equations of motion of a particle subjected to more than one force. FR F ma a F1 F2 p Free body diagram of particle p. F1 p FR F F2 Kinetic diagram of particle p. ma p FR F ma ………...equation of motion D’A lembert principle FR ma 0 ma : inertia force vector Dynamic equilibrium diagram FR p ma (慣性力) + F 0 FR ma 0 FR ma 若 ma 0則此狀態為靜平衡 2. Inertial frame of reference (newtonian) A coordinate system is either fixed or translates in a given direction with a constant velocity. (1) Inertial frame ap y p v 0 path x o a p / o a p ao dvo a p ( ao 0) dt (2) Noninertial frame ap y o p a0 x path a p / o a p a0 13-3 Equation of motion for a system of particle Fi z fi ri i y xyz: Inertial Coordinate System x Equation of motion of particle i. Dynamic equilibrium diagram of particle i. Fi f i i mi ai Fi f i mi ai Fi resultant external force n f i resultant internal force f ij j 1 j i Equation of motion of a system of particles. ( F1 f1 ) ( F2 f 2 ) ...... ( Fi f i ) ...... m1a1 m2 a2 ...... mi ai ...... Fi fi mi ai fi 0 Fi mi ai By definition of the center of mass for a system of particles. mrG mi ri rG Position vector of the center of mass G. m mi Total mass of all particles. d2 d2 2 (mrG ) 2 ( mi ri ) dt dt Assume that no mass is entering or leaving the system. 2 d rG d ri m 2 mi 2 dt dt maG mi ai 2 Hence： F m a i G This equation justifies the application of the equation of motion to a body that is represented as a single particle. 13-4 Equations of motion：Rectangular Coordinate Fz z Fx x path Fy y Rectangular Coordinate system. Equation of motion of particle P. F ma In rectangular components Fxi Fy j Fz k maxi may j maz k F F F x max y may z maz 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. Fx max F ma or F F y may z maz (2) Friction force Ff k N (3) Spring force Fs ks 3. Equations of kinematics dv ds Apply a 、v 、ads vdv for the solutions dt dt 13.5 Equation of Motion:Normal and Tangential Coordinates Curve path of motion of a particle is known. Curve path b n un t ub ut P F ut =Tangential unit vector u n =Normal unit vector u b =Binormal unit vector = un ut F ma Equation of motion Ftut Fnun Fbub mat man Or scalar form Ft = mat F Fb u = m an = 0 at dv / dt an v2 Analysis procedure 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. v2 an a t dv / dt vdv / ds 2 2/3 1 dy / dx d 2 y / dx 2 13.6 Equation of Motion :Cylindrical coordinate F u z z z F u F u r r r Equation of motion in cylindrical coordinates F ma Fr u r Fu Fz u z ma r ma ma z F ma F ma F ma r z ar r r r z and a r 2 r az z 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 The directions of F and N can be specified relative to the radial coordinate r by computing the angle y. Angle y is defined between the extended radial line and the tangent to the path. dr :radial component rd :transverse component ds:distance rd dr rd r tan y dr dr d y + positive direction of - negative direction of