. Contributions. mptes Kendus the following example to illustratethat a motion which may be possible geometrically may be impossibledynamically. Three equal beads are attached to a weightless rod, oneat each end and the third at the mid point. The beadsand rod are constrained to remain on the surface of asmooth right circular cone. The problem is to deterniinethe motion for any given initial conditions when noexternal forces act on the particles. Call r, a, (p the polar coordinates of Pj (Fig- 1),P^P^ = 2a, and indicate derivatives with respect to tby dots.^) The kinetic energy of the three pa
. Contributions. mptes Kendus the following example to illustratethat a motion which may be possible geometrically may be impossibledynamically. Three equal beads are attached to a weightless rod, oneat each end and the third at the mid point. The beadsand rod are constrained to remain on the surface of asmooth right circular cone. The problem is to deterniinethe motion for any given initial conditions when noexternal forces act on the particles. Call r, a, (p the polar coordinates of Pj (Fig- 1),P^P^ = 2a, and indicate derivatives with respect to tby dots.^) The kinetic energy of the three particlesand also the angular momentum about OZ are con-stant, so we have 3P + (dr^ -r 2or) sin-a-^ sin-a^- = C, 3r-(/ =- along the radius vector r — r(f^ sin-(( = >.These equations can only be satislied by qr = 0, f = a constant if r -f then the centroid is not at the vertex of the cone, the rod moves. 1) Routh. Advanced Rigid Dynamics, sixtli edition, p. 206. 310 On Constrained Motion. Fig. 2. uniforralj along the same generator no matter what the initial con-ditions even though there is no geometrical reason why it should notmove from one generator to another. As stated by Routh loc. cit., ifthe motion is not initially along a given generator an impulsive coupleacts with the result that the rod moves along a generator. This result can also be explained as follows. The projections ofthe velocities of P^, P^, P^ along the tangents to the circular sectionsof the cone at Pj, Pg, P3 are in the same ratio _as the distances ofPj, Pg, P3 from the vertex of the cone. As the reaction of the coneis not normal to its axis, it follows that the ratios of these distancesOP^:OP^:OP^ will change. But the projections of the velocities ofPj, Pg, Pg along the tangents to the circular sections at the givenpoints cannot change as both the reaction of the cone and the stressin the rod are at right angle
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