By Brian H. Chirgwin, Charles Plumpton
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Extra info for A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics
2 a cost a. CHAPTER II SETS OF FORCES: EQUILIBRIUM 2:1 Introduction We assume that the reader is familiar with the definitions and concepts of elementary mechanics, in particular with force. (See Vol. ) In Vol. III we confined our attention to forces whose lines of action were parallel to one plane and to the motions of bodies parallel to this plane, the typical rigid body being a lamina. Following on our considerations of the motion of a rigid body in three dimensions we now discuss forces in three dimensions whose lines of action, in general, do not intersect.
52) that the velocity of P is the sum of the separate velocities it would have if the body were given each of the angular velocities alone. There are two special cases we consider in more detail. 1. *. v = e x [ay. = we e. 53) where (col. + co2) rG = co, rA . This definition shows that 0 is the centroid of masses co, at 0 and co, at A. (In Fig. ) Hence the two parallel angular velocities are equivalent to a velocity equal to their sum about an axis through the centroid. Clearly, this result can be generalised to give the sum of any number of parallel angular velocities.
We do not need to know the action at 0 and so we take moments for the rod OA, about an axis through 0 perpendicular to the plane AOG. R . AG — W AG — Wk0 = O. ( 1) If we consider the equilibrium of the whole structure we see that the only external forces are the three weights W, and the three reactions R. Hence, vertical resolution 52 A COURSE OF MATHEMATICS for the whole system gives (2) R= W. •. W AG = Wk0, a sine = kO. (3) This equation has a root 0 = 0, which we reject as a solution of this problem.
A Course of Mathematics for Engineers and Scientists. Volume 6: Advanced Theoretical Mechanics by Brian H. Chirgwin, Charles Plumpton