By Thomas R. Kane
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Additional resources for Analytical Elements of Mechanics. Dynamics
This follows from the definition of arc-length. Thus lim%l = l (A) Now R C l Hence ~ P l Substitute: R " (iIi)l! Rçp£, 2\ ds2 + + and Thus h P R dp ds R 2 dp ds2 ¥-[(*£)' + * - - + lim hj = h-+0 A r/^vr. L \ ds \Rdp\ = ! B) dp\ ■μ/2 -|i/2 (B) Diff. of Vectors] SECS. 1) ds/ Hence τ = R 33 dp\ ds\ és. ds While this expression for r appears to be simpler than the one given in Sec. 1, it is frequently less convenient, because the functional dependence of p on s is often more complicated than p's dependence on some other variable.
1) ds/ Hence τ = R 33 dp\ ds\ és. ds While this expression for r appears to be simpler than the one given in Sec. 1, it is frequently less convenient, because the functional dependence of p on s is often more complicated than p's dependence on some other variable. 4 The plane passing through P and normal to τ is called the normal plane of C at P. 1 If a curve C is fixed in a reference frame R (see Fig. 1a) and B is the binormal to C at a point P of C, a unit vector ß parallel to B is called a vector binormal of C at P and is given by ß = ρ' X Ρ' IP' X Ρ'Ί where p is the position vector of P relative to a point 0 fixed in R and primes denote differentiation with respect to z in R, z being any scalar variable such that the position of P on C depends on z.
3 Diff. of Vectors] 25 where 0 is the angular displacement of a line d fixed in D, relative to a line c fixed in C, 0 being regarded as positive when generated -C FIG. 3a by a clockwise (as seen by the reader) rotation of d (or D) relative to c (or C). A and B are points fixed in D. Draw a sketch showing the time-derivative in C at t* of the position vector p of B relative to A. Solution: Let n be a unit vector parallel to line AB. 5) dt t* dt\ Then p Let k be a unit vector perpendicular to the plane of the disc D, choosing the sense of k such that D rotates clockwise during a k rotation of D relative to C (see Fig.