By Thomas R. Kane
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Extra resources for Analytical Elements of Mechanics
1). Problem: Determine the magnitude and sense of FL, the L resolute of the force F shown in Fig. 6b. L FIG. 6b Solution: Let n be a unit vector parallel to line L and having the sense AB. 8n! 21b and FL = n Fn = - 7 . , FL has the sense BA (see Fig. 6b). 7 Given a line L and n vectors vt, i = 1, 2, . . , n, the resultant of the L resolutes of these vectors is equal to the L reso lute of the vectors' resultant. Proof: Let n be a unit vector parallel to L. , i = 1, . . 6) n Σ (n*v*n) t= l while the L resolute of the resultant of the vectors vt, i = 1, 2 , .
Problem: Find the angle (na, nb) between the unit vectors na and n& shown in Fig. 5. FIG. 5 Solution: Express na and n& in terms of the unit vectors m, n2, n3 shown in Fig. 1°, . 12): VL = nvn Notation (see Fig. 6a): FIG. 6a v L VL vp n Proof: pressed as a vector a line the L resolute of v the resolute of v perpendicular to L a unit vector parallel to L By definition, vL is parallel to n. 12) Substitute from Eq. (1) into Eq. 14 Hence, nv = VL Substitute into Eq. (1). Problem: Determine the magnitude and sense of FL, the L resolute of the force F shown in Fig.
N Σ < i=l From Fig. 4 If one assigns to the centroid of a set of points a strength equal to the sum of the strengths of the points of the set, then the first moment of the centroid with respect to any point is equal to the sum of the first moments of the points of the set with respect to this point. In this sense, the centroid can be regarded as repre senting the entire set of points. Proof (see Fig. 4 for notation): The sum of the first mo ments of the n points P t ·, i = 1, . . , n, with respect to 0 is equal Pi (Hi) / P Ρ*(Σ Νί) " FIG.