ࡱ> +RT'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQ3 Djbjb^^ 8h<h< 4l&&&&&&&$VVVVPDV ((() + + + + + + , `W u&"W &&((a&(&() J&&&&) z &&  VV  66  Expansion of the Einstein Notation: s'kl = aki alj sij where, for example, aki is the cosine of the angle between the k-axis in the primed system and the i-axis in the unprimed system.  i = 1, j = 1 i = 1, j = 2 i = 1, j = 3 i = 2, j = 1 i = 2, j = 2 i = 2, j = 3 i = 3, j = 1 i = 3, j = 2 i = 3, j = 3 s'11 = a11a11s11 + a11a12s12 + a11a13s13 + a12a11s21 + a12a12s22 + a12a13s23 + a13a11s31 + a13a12s32 + a13a13s33 k = 1, l = 1 s'12 = a11a21s11 + a11a22s12 + a11a23s13 + a12a21s21 + a12a22s22 + a12a23s23 + a13a21s31 + a13a22s32 + a13a23s33 k = 1, l = 2 s'13 = a11a31s11 + a11a32s12 + a11a33s13 + a12a31s21 + a12a32s22 + a12a33s23 + a13a31s31 + a13a32s32 + a13a33s33 k = 1, l = 3 s'21 = a21a11s11 + a21a12s12 + a21a13s13 + a22a11s21 + a22a12s22 + a22a13s23 + a23a11s31 + a23a12s32 + a23a13s33 k = 2, l = 1 s'22 = a21a21s11 + a21a22s12 + a21a23s13 + a22a21s21 + a22a22s22 + a22a23s23 + a23a21s31 + a23a22s32 + a23a23s33 k = 2, l = 2 s'23 = a21a31s11 + a21a32s12 + a21a33s13 + a22a31s21 + a22a32s22 + a22a33s23 + a23a31s31 + a23a32s32 + a23a33s33 k = 2, l = 3 s'31 = a31a11s11 + a31a12s12 + a31a13s13 + a32a11s21 + a32a12s22 + a32a13s23 + a33a11s31 + a33a12s32 + a33a13s33 k = 3, l = 1 s'32 = a31a21s11 + a31a22s12 + a31a23s13 + a32a21s21 + a32a22s22 + a32a23s23 + a33a21s31 + a33a22s32 + a33a23s33 k = 3, l = 2 s'33 = a31a31s11 + a31a32s12 + a31a33s13 + a32a31s21 + a32a32s22 + a32a33s23 + a33a31s31 + a33a32s32 + a33a33s33 k = 3, l = 3 HP1 The sketch below shows a person (who shall remain unnamed) hanging from a chandelier during small gathering in celebration of the resumption of the academic year. The person s mass is 90 kg, plus the mass of recently consumed beverages, for a total of 120 kg.  (a) What is the magnitude of the force exerted on the ceiling? (b) Express this force in terms of a vector using a coordinate system keyed to the ground. (c) Express the same force in terms of a vector keyed to the ceiling (i.e., has two of its axes in the plane of the ceiling). (d) If the rod holding the chandelier up is 1/2 inch in diameter, what is the stress on the rod? HP2 A wire is 1/16 inch in diameter and carries a load of fifty pounds. What is the force in Newtons that the wire experiences from this loading? What is the axial stress in the wire in MPa ? HP3 A 1 cm diameter cylinder is anchored at the top and uniformy loaded at the bottom by a total mass of 30 kg, as in the sketch below:  Using the logic presented in the potato analogy for stress at a point, show that no stresses act on planes cut parallel to the axis of the cylinder. (Neglect any end-effects, air pressure, and the weight of the cylinder itself.) MSM 355. 2002 D. S. 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