tailieunhanh - On the relations between kinetic energy and linear momentum, angular momentum of the particle and of the rigid body

Using the definit~on for the partial derivative of a scalar in respect to the vector, this paper presents the relations between 'kinetic energy and linear momentum, angular momentum of the particle and of the rigid body. The obtained results are useful for the investigation of the dynamics of multibody systems. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 23, 2001, No 2 (110 - 115) ON THE RELATIONS BETWEEN KINETIC ENERGY AND LINEAR MOMENTUM, ANGULAR MOMENTUM OF THE PARTICLE AND OF THE RIGID BODY NGUYEN VAN KHANG Hanoi University of Technology ABSTRACT. Using the definit~on for the partial derivative of a scalar in respect to the vector, this paper presents the relations between 'kinetic energy and linear momentum , angular momentum of the particle and of the rigid body. The obtained results are useful for the investigation of the dynamics of multibody systems. 1. Introduction The linear and angular momentum of the particle are the basic dynamic quantities of Newton's mechanics. The kinetic energy of the particle is the basic dynamic quantity of Lagrange's mechanics [1 , 2]. In the present paper we use the definition for the partial derivative of a scalar a in respect to the vector x [4, 5] 8a 8X = oa . a 1T X = [ oa ' aa ' . . . ' 8a ] 0 Xn 8 X1 8X2 () in order to study the relation between the kinetic energy of the particle and the rigid body and their linear and angular momentum. 2. Relation between kinetic energy and linear momentum of the particle The expression for the linear momentum and the kinetic energy of the particle has the following form [1, 2] P=mv, 1 T = - mv 2 2 ' () where mis the mass and vis the velocity vector of the particle. In the reference system Oxyz (Fig. 1), the linear momentum and kinetic energy of the particle can be written as 110 . mz']T , P = [mx. , my, () z () Theorem 1. Partial derivative of kinetic energy of the particle in respect to its velocity vector is equal to the transposed vector of the linear momentum of the particle mv aT =PT av y () . Proof. According to definition () we find x · ~: = ar · a~T = [~~, ~: ' ~:J From the expression for the kinetic energy of the p article () it follows that 8T . ax = m x, 8T ay . =my , 8T . oi = mz , With it we .