. Peraire 16.07 Dynamics Ve Lecture D19-2D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture D7 for particle dynamics and extend it to 2D rigid body dynamics. Kinetic Energy for a 2D Rigid Body We start by recalling the kinetic energy expression for a system of particles derived in lecture D17, y G G T v2+ There n is the total number of particles, mi denotes the mass of particle i, and ri is the position vector of particle i with respect to the center of mass, G. Also, m=2ia mi is the total mass of the system, and UG is the velocity of the center of mass. The above expression states that the kinetic energy of a system of particles equals the kinetic energy of a particle of mass m moving with the velocity of the center of mass plus the kinetic energy due to the motion of the particles relative to the center of mass, G. For a 2D rigid body, the velocity of all particles relative to the center of mass is a pure rotation. Thus, we can write r1=w×r remore 42=∑2m(xn)间x)=∑2mn22 here we have used the fact that w and r are perpendicular. The term 2i, mir/2 is ea lized as he moment of inertia, IG, about the center of mass, G. Therefore, for a 2D rigid body, the kinetic energy simplyJ. Peraire 16.07 Dynamics Fall 2004 Version 1.1 Lecture D19 - 2D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture D7 for particle dynamics, and extend it to 2D rigid body dynamics. Kinetic Energy for a 2D Rigid Body We start by recalling the kinetic energy expression for a system of particles derived in lecture D17, T = 1 2 mv2 G + Xn i=1 1 2 mir˙ ′ i 2 , where n is the total number of particles, mi denotes the mass of particle i, and r ′ i is the position vector of particle i with respect to the center of mass, G. Also, m = Pn i=1 mi is the total mass of the system, and vG is the velocity of the center of mass. The above expression states that the kinetic energy of a system of particles equals the kinetic energy of a particle of mass m moving with the velocity of the center of mass, plus the kinetic energy due to the motion of the particles relative to the center of mass, G. For a 2D rigid body, the velocity of all particles relative to the center of mass is a pure rotation. Thus, we can write r˙ ′ i = ω × r ′ i . Therefore, we have Xn i=1 1 2 mir˙ ′ i 2 = Xn i=1 1 2 mi(ω × r ′ i ) · (ω × r ′ i ) = Xn i=1 1 2 mir ′ i 2ω 2 , where we have used the fact that ω and r ′ i are perpendicular. The term Pn i=1 mir ′ i 2 is easily recognized as the moment of inertia, IG, about the center of mass, G. Therefore, for a 2D rigid body, the kinetic energy is simply, T = 1 2 mv2 G + 1 2 IGω 2 . (1) 1