By Thomas R. Kane
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Additional resources for Analytical Elements of Mechanics. Dynamics
Dz2 + +. The values of v at z* and at z* + h, expressed in terms of their nt-, i = 1, 2, 3, components, are 3 3 1=1 Diff. of Vectors] SEC. 21^dz2\μ + . 1) 1 ! dz\ h2 A cftiJ + 2! dz2\ + ... Problem: In Fig. 1, R and R' represent two reference frames (coplanar rectangles), ni and n2 represent unit vectors fixed R FIG. 1 in R, and n/ and n2' are unit vectors fixed in R'. A vector function v is defined as v = z2 n/ where z is the angular displacement of R relative to R'. 01 rad in (a) R and (b) Rr.
2 The importance of rates of change of orientation as analytical tools is due to the fact that they furnish a means of replacing the process of differentiation with that of cross multiplication. 3 The expression for κ'ωζκ given in Sec. 1 appears to be unsymmetrical in a and b. This lack of symmetry is spurious. For, in view of Eq. (4), Sec. 1, R'wzR may be expressed in the form and this expression remains unaltered when the roles of a and b are interchanged. Problem: A rectangular plate P moves in such a way that the corners A and B remain on two nonintersecting lines while the FIG.
SS (ρ'2 + Ap' - p" + . 1, suppose that there exists a point 0 which is fixed in both R and R' and that p is the position vector of a point P relative to 0. Then P traces out a curve C in R and curve C in R'. Determine the cosine of the angle a between the tangents to C and C" at point P, for t = 0. Solution: Let r and τ' be vector tangents of C and C at P . 1, R 'dp\ = 3n2 + 4n! dt i = 0 SECS. 3 Diß. 1) depends on the choice of the scalar variable which governs the position of P on C: τ points in the direction in which P moves when the variable increases algebraically.
Analytical Elements of Mechanics. Dynamics by Thomas R. Kane