By Donald T. Greenwood
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Let us use the principle of virtual work. The applied forces acting on the system are the spring force F and those due to gravity. 277) Since δx1 and δx2 are independent virtual displacements, their coefﬁcients must each be zero. 282) Notice that, at the equilibrium conﬁguration, the angle θ and force F do not depend on the spring stiffness k. So let us consider the case in which k is inﬁnite; that is, replace the spring by a massless rigid rod of length L. Then δx1 and δx2 are not longer independent.
214) that is, the order of partial differentiation is immaterial. In terms of a ji s we have ∂a ji ∂a jk = ∂qk ∂qi (i, k = 1, . . , n) ∂a jt ∂a ji = ∂qi ∂t (i = 1, . . 215) ˙ t) is These are the exactness conditions for integrability. 212) and if it is either (1) exact as it stands or (2) can be made exact through multiplication by an integrating factor of the form M j (q, t). 212). 215) are satisﬁed, it follows that ∂fj ∂ q˙ i d dt − ∂fj =0 ∂qi (i = 1, . . 219) is satisﬁed for all qs we see that the exactness conditions must apply.
189) Thus, in their ﬁnal motion, the blocks each move with a velocity 12 v0 and a separation somewhat larger than the original value. 9 Two stacked blocks, each of mass m, slide relative to each other as they move along a horizontal ﬂoor, as shown in Fig. 18. Suppose that the initial conditions are x A (0) = 0, x˙ A (0) = v0 , x B (0) = 0, x˙ B (0) = 0. 1 between block B and the ﬂoor. 2mg. 18. 1g. 195) The ﬁnal displacement of block B is equal to its average velocity multiplied by the total time.
Advanced Dynamics by Donald T. Greenwood