Dynamics of a point mass

Pagina 1

1. DYNAMIC ANALYSIS OF A POINT MASS A point mass is a geometric (0-dimensional) point th at may be assigned a finite mass. Since a point has zero volume, the density of a point mass having a fin ite mass is infinite, so point masses do not exist in reality. However, it is often a useful simplificati on in real problems to consider bodies point masses, especially when the dimensions of the bodies are much less than the distances among them. Let us consider a point mass, m, that is supported on a frictionless surface (Figure 1.1), defined in a three dimensional space. Let us denote P the mass position. The considered point mass is subjected to a set of n1 external forces, Fj (with j = 1, 2, …, n1). These forces can be both driving and resistance forces. They are expressed by means of 3-D vectors the resultant of which is vector Fe (Figure 1.2). Let us assume that the z axis of the absolute Cartesian coordinate system coincides with the vertical direction and that the point mass is subjected to the gravitation field. Therefore, a further force, W = m g, where g is the gravity acceleration vector (- 9.81 m/s 2), is exerted on the point mass. In order to study the dynamic behaviour of the mass m it is necessary to isolate it, that is it is necessary to remove the supporting surface and show the corresponding reaction force, R vz, normal to the surface at point P. Let us denote by v the point mass velocity and t the unity vector that indicates the direction of v (Figure 1.3). Besides, let us denote by Fx-y the resultant of the vector forces Fex and Fey : this vector is contained in the plane γ (Figure 1.3) The absolute acceleration a of the point mass can be null or not null. In the latter case, the acceleration vector a can be decomposed into the two projections, an and at,…