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noted by Kazuma MATSUDA 2016 5 29 1. (force) (mechancs) (statcs) (dynamcs) 1.1 1 1 O F OA O (pont of applcaton) (lne of acton) (vector) F F F F 1.2 ( ) 1 (kg-wt) (Internatonal System of Unt, SI) kg SI 1

kgf SI N SI 1 kg 1 m/s 2 1 N 1 N = 1 [kg] 1 [m/s 2 ] = 1 kg m/s 2 g = 9.80665 m/s 2 1 kgf = 9.81 kg m/s 2 = 9.81 N 1 N = 0.102 kgf 1.3 1. 1 2(a) O F 1, F 2 OC R R F 1 F 2 (resultant force) 1 2(b) OAC (force trangle) R 1 3 F 1 F 2 α R 2 = F 1 2 + F 2 2 2F 1 F 2 cos(180 α) = F 1 2 + F 2 2 + 2F 1 F 2 cos α R = F 1 2 + F 2 2 + 2F 1 F 2 cos α (1.1) R F 1 θ F 2 sn θ = θ R sn(180 α) = R sn α sn θ = F 2 R sn α (1.2) 2

2. (component of force) F 1 5 1 6 F F x, F y F x θ F x = F cos θ, F y = F sn θ (1.3) F F F = F x 2 + F y 2 (1.4) tan θ = F y F x θ = tan 1 F y F x (1.5) 3. 1 7(a) O F 1, F 2,, F N (b) F 1 F 2 OA2 F 3 F 1, F 2, F 3 OA3 F 4 R ( OAN ) OA 1 A 2 A N (force polygon) 1 8 R x, y x, y R x = F 1 cos θ 1 + F 2 cos θ 2 + + F N cos θ N R y = F 1 sn θ 1 + F 2 sn θ 2 + + F N sn θ N (1.6) 3

R = F cos θ 2 + F sn θ 2 (1.7) tan θ = F sn θ F cos θ θ = tan 1 F sn θ (1.8) F cos θ 1.4 1 7 A N O x, y (1.6) R x = R y = F cos θ = 0 F sn θ = 0 (1.9) 1 11(a) (b) 180 α, 180 β, 180 γ 4

F 1 sn(180 α) = F 2 sn(180 β) = F 3 sn(180 γ) F 1 sn α = F 2 sn β = F 3 sn γ (1.10) (Lam s theorem) 1.5 1. 1 14 F d (moment of force) M = Fd (1.11) d (arm) N m 2. F 1, F 2 O 1 15 O OA x O-xy 5

F 1, F 2 x θ 1, θ 2 O d 1, d 2 OA a d 1, d 2 d 1 = a sn θ 1, d 2 = a sn θ 2 M 1 = F 1 d 1 = F 1 a sn θ 1 M 1 = F 2 d 2 = F 2 a sn θ 2 (1.6) M 1 + M 2 = (F 1 sn θ 1 + F 2 sn θ 2 )a = Ra sn θ R F 1 F 2 R θ x a sn θ = d R O M 1 + M 2 = M (1.12) (Vargnon) 1 16 xy P(x, y) F O F x, F y M = F y x F x y (1.13) F y F x 3. (couple) 1 18 O 6

M = F OA F OB = F BA = Fd (1.14) d (arm of couple) M ( 1 19) ( 1 20) 4. 1 21(a) A F (b) F F, F A F A F O F Fd A (c) F Fd F d Fd 7

1.6 1. 1 23 A, B F 1 F 2 F 1 F 2 O 2. A, B F 1, F 2 1 24 A, B F, F R 1, R 2 O O R 1, R 2 R F 1, F 2 R F 1 F 2 R = F 1 + F 2 (1.15) R AB C AC OC = F F 1, BC OC = F F 2 AC BC = F F 1 F 2 = (1.16) F F 1 F 2 C AB F 1, F 2 1 25 AB C 1 26 O F 1, F 2 R d 1, d 2 d (1.16) 8

d 1 d d d 2 = F 2 F 1 F 1 (d 1 d) = F 2 (d d 2 ) F 1 d 1 + F 2 d 2 = F 1 d + F 2 d = (F 1 + F 2 )d = Rd ( (1.15)) (1.18) F 1 d 1 + F 2 d 2 F 1 F 2 O Rd R 3. ( ) 1 27(a) F 1, F 2, F 3, F 4 (b) P 0 P 1 P 2 P 3 P 4 P 5 P 0 P 4 (force dagram) R O F 1 Q 1 OP 1 Q 1 Q 2 F 2 Q 2 Q 2 OP 2 Q 2 Q 3 F 3 Q 3 Q 4 Q 1 Q 2 OP 0 OP 4 Q 1 Q 0. Q 4 Q 0 Q 0 R 1 27(b) F 1, F 2, F 3, F 4 OP1 (F 1 ) P1 O OP2 (F 2 ) P2 O OP3 (F 3 ) P3 O OP4 (F 4 ) R R R P0 O OP4 P0 O, OP4 Q 0 Q 1 Q 0 Q 4 Q 0 Q 0, Q 1, Q 2, Q 3, Q 4 (funcular dagram) (strng) 9

1 28(b) (a) F 1 Q 1 (a) OP 1, OP 2, OP 3, OP 4 (b) R Q 0 4. ( ) O-xy F P (x, y ) x θ F O x, y F cos θ, F sn θ M = (F sn θ )x (F cos θ )y R = F cos θ tan θ = F cos θ 2 + F sn θ x θ F sn θ F sn θ 2 (1.19) θ = tan 1 (1.20) F cos θ M = Rr = F (x sn θ y cos θ ) (1.21) (1.21) r 5. 10

F cos θ = 0, F sn θ = 0 F (x sn θ y cos θ ) = 0 (1.22) 1.7 1. A, B A B (Newton ) A B (reactor force) 2. 1 32 (a) R (b) R (c) R M 11

1.8 (member) (framework) (truss) (jont) (tenson member) (compresson member) (Rahmen) 1. (nternal force) 2. 12

1 1 1 38 100 kg m = 100 kg 20 T 1 30 T 2 Lam mg sn(20 + 30 ) = T 1 sn(180 30 ) = T 2 sn(180 20 ) 100 [kg] 9.81 [m/s2 ] sn 50 = T 1 [N] sn 150 = T 2 [N] sn 160 T 1 = 100 9.81 = 640 [N] sn 150 sn 50 sn 160 T 2 = 100 9.81 sn 50 = 437.9 440 [N] Ans. T 1 = 640 N, T 2 = 440 N 1 2 50 kg 30 m = 50 kg N F Lam mg sn(30 + 90 ) = N sn 90 = 50 [kg] 9.81 [m/s2 ] sn 120 = N [N] sn 90 = F sn(180 30 ) F [N] sn 150 13

sn 90 N = 50 9.81 sn 120 = 566 570 [N] sn 150 F = 50 9.81 sn 120 = 283 280 [N] Ans. 280 N 1 3 1 39 150 N y x A A(x, y) x = 0.45 [m] cos 30 y = 0.2 [m] + 0.45 [m] sn 30 x 60 30 = 30 F = 150 N F y = 150 sn 30 F x = 150 cos 30 M M = F y x F x y = 150 sn 30 0.45 cos 30 ( 150 cos 30 ) (0.2 + 0.45 sn 30 ) = 29.2 + 55.2 = 84.4 84 [N m] Ans. 84 N 1 4 1 40 l F P F M 14

M = Fl P θ P M M = P cos θ b 2 2 + P sn θ b 2 3 2 θ = 90 M = P 0 b + P 1 = P b 3 b 3 M = M Fl = P b 3 P = 3Fl b Ans. P = 3Fl b 1 5 1 41 A, B 90 N P 300 N AB P O AB C O B x m C 90 [N] (x + 0.5)[m] + 300 [N] x [m] 90 [N] (0.5 x) [m] = 0 90x 45 + 300x 45 + 90x = 0 300x = 90 x = 0.3 Ans. O B 30 cm 15

1 6 1.42 m l 30 θ N 1 N 2 g N 1 mg + N 2 cos 30 = 0 (1) (2) F N 2 sn 30 = 0 (2) N 2 = F sn 30 (1) cos 30 N 1 mg + F sn 30 = 0 1 N 1 mg + F tan 30 = 0 N 1 mg + F 3 = 0 (3) mg cos θ l 2 + F sn θ l N 1 cos θ l = 0 (4) (3) mg 2 l + F sn θ cos θ l N 1l = 0 mg 2 + F tan θ N 1 = 0 mg 2 + F tan θ mg + F 3 = 0 F ( tan θ + 3 ) mg 2 = 0 N 1 = mg 2 F ( tan θ + 3 ) = mg 2 mg F = 2 ( tan θ + 3 ) + F tan θ (4) 16

Ans. F = mg 2 ( tan θ + 3 ) 1 7 1 43 m (a) T T 1, T 2, T 3 T 1 + T 2 mg = 0 T 1 = T 2 T 1 = T 2 = mg 2 2T 2 = mg (1) r T 1 r + rt 2 = 0 T 1 + T 2 = 0 T 1 = T 2 T 2 T 3 = 0 T 2 = T 3 (2) T 3 = T (3) (1) 2T 2 mg = 0 2T 3 mg = 0 ( (2)) 2T mg = 0 ( (3)) T = mg 2 17

(b) T 1,, T 6 T 2 + T 3 + T 6 mg = 0 (1) T 2 = T 6 (2) T 1 = T (3) T 4 = T 5 (4) T 1 T 2 = 0 T 2 = T 1 (5) T 4 + T 3 = 0 T 3 = T 4 (6) T 5 + T 6 = 0 T 6 = T 5 (7) (1) 2T 2 + T 3 mg = 0 ( (2)) 2T 2 + T 6 mg = 0 ( (6), (4) T 3 = T 6 ) 2T 2 + T 2 mg = 0 3T 2 mg = 0 3T mg =0 ( (5), (3)) T = mg 3 Ans. (a) : T = mg mg, (b) : T = 2 3 1 8 1 44 30 1 33 18

R T R θ R sn θ 800 [N] 1200 [N] + T sn 30 = 0 R sn θ 2000 + 1 2 T = 0 (1) R cos θ T cos 30 = 0 3 R cos θ 2 T = 0 (2) (1) (2) θ 800 [N] 0.25 [m] 1200 [N] (0.25 + 0.35) [m] + T sn 30 [N] (0.25 + 0.35 + 0.4) [m] = 0 200 720 + 1 2 T 1 = 0 1 2 T = 920 T = 1840 [N] R sn θ = 2000 1 T = 2000 920 2 = 1080 1100 [N] 3 3 R cos θ = 2 T = 2 1840 = 1593 1600 [N] R = R 2 sn 2 θ + R 2 cos 2 θ = 1080 2 + 1593 2 = 1924 1900 [N] tan θ = R sn θ R cos θ = 1080 1593 1 1080 θ = tan 1593 = 34 1 11 R sn θ = R A, T > R B 19

Ans. T = 1.8 kn, R = 1.9 kn 1 9 1 45 500 N A, B R A, R B 500 + R A + R B F = 0 (1) A 500 [N] 0.3 [m] + R B [N] 0.35 [m] F [N] 0.32[m] = 0 150 + 0.35R B 0.55F = 0 0.35R B = 150 + 0.55F R B = 428 + 1.57F (2) B 500 [N] (0.3 + 0.35) [m] R A [N] 0.35 [m] F [N] 0.2[m] = 0 500 0.65 0.35R A 0.2F = 0 0.35R A = 325 0.2F R A = 928 0.27F (3) F B A F R B (3) (1) 500 + 928 0.57F + 0 F = 0 1.57F = 428 F = 272.6 273 [N] R A = 0 (2) (1) 500 + 0 428 + 1.57F F = 0 0.57F = 928 F = 1628 [N] 20

Ans. F = 273 N 1328 N 1 10 1 46 Ans. 21

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