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3 [ ] [ ] [ ] 1687 (Prinkipia) (The three laws of motion) (1) ( ) (2) ( ) (3) () Newtonian mechanics [ ] * * [ ] [ ] [ ] Issac Newton 1642/ , [ ] A B A (B A B A B A

4 A., I., ( ),, 1939 R. P., ( ),, 1968 D. L., J. R., (),, 1996 (1) nishii@sci.yamaguchi-u.ac.jp (2) (3)

5 i ii ii ( ) * () * * * () * () Free-Body Diagrams

6 * * * * *

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8 vi

9 ( ), [ ] 1 1/10 ( 3.3cm) [ ] (inch) (feet) (SI [ ] ) ( ) 1m ( ) SI ( ) [m], [kg], [s] [m][kg][s] MKS SI 2006 [ ] 1/3 m 1 (wikipedia(ja) ) [ ] 1 inch= 2.54 cm 1 feet=12 inch 1/12 inch (1/12 uncia ) 1 inch [ ] Le Système International d Unités The International System of Units. [s], [m], [kg], [A], [K], [mol], [cd] Cs 9,192,631,

10 kg 2.2 [ ] Albert Einstein, , 1933, 299, ± km/s [ ] [ ] [ ] Galileo Galilei, , [ ] (1752 ) [ ] [ ] vector vectum vector() x (position) x(t) [ ] P O x x x(t + h) = x(t) + x 1.1 P (distance) (scalar) [ ], (Origin), [ ] scale 1849

11 1.2 3 t P x(t) h x(t + h) displacement x = x(t + h) x(t) ( 1.1) (velocity) P h x(t) x(t + h) x = x(t + h) x(t), v v = x h x(t + h) x(t) = h (1.1) 2 x h v(t) v(t) = lim h 0 x(t + h) x(t) h (1.2) (differentiation) x dx dt, ẋ x [ ] (speed) (velocity) v (1.1) (1.2) MKS ẋ [ ] (Gottfried Wilhelm Leibniz, , ) d dt ẋ [ ] = [ ] = [m/s] (1.3) [ ] [ 1.1] km/hr 8.23m 23.77m ( ) ( % ) [ ]

12 4 1 % l l = (8.23 m) 2 + (23.77 m) 2 (8 m) 2 + (24 m) 2 = m m l = m 249.4km/hr m = m/( s) = /m / 3 s / 3 /m = 0.36 s (1.4) [ ] x 1 (1 + x) n 1 + nx 1 + x 1 + x/ / = 25.4 [ ] 2% (1) m km/h m/s (2) (3) 36 s

13 1.2 5 [ 1.2] A,B x A (t), x B (t) 2 A, B t x A (t) = x B (t) = 3 km (1) h = 10 A x A (t + h) = 9 km a t t + h b t t + h (km/h m/s ) (2) B t h = 10 x B (t h) = 13 km a t h t b t h t (km/h ) (3) B t t + h x = 10 km t + h B [ ] (1) a t t + h A x x = x A (t + h) x A (t) = 9 km 3 km = 6 km ( ) ( ) b v v = x h = 6 km 10 min 6 km = hr 6 km 60 = 10 hr = 36 km/hr ( ) = m s = 10 m/s ( )

14 6 1 (2) a t h t B x x = x B (t) x B (t h) = 3 km 13 km = 10 km ( ) ( ) 10 km b v x(t) x(t h) v = h 10 km = 10 min 10 km = 10 min 60 min/hr 10 km 60 = 10 hr = 60 km/hr (3) t t + h B x x = x(t + h) x(t) t + h x(t + h) x(t + h) = x(t) + x = 3 km 10 km = 7 km. B t + h 7 km [ 1.3] [ ] 50 position [m] 100 O time [s] v(t) 0 t < 10 s v = 100 m 0 m 10 s 0 s = 10 m/s.

15 t < 20 s v = 100 m 100 m 20 s 10 s 20 t < 30 s v = 50 m 100 m 30 s 20 s = 0 m/s. = 5 m/s. 10 m/s (0 t < 10s) v(t) = 0 m/s (10 t < 20s) 5 m/s (20 t < 30s) velocity [m/s] 10 O time [s] ( ) 1.2 ) 10t [m] (0 t < 10s) x(t) = 100 [m] (10 t < 20s) 5t [m/s] (20 t < 30s) ( ) t ( ( )) [ ] [ ] ( ) t [] (acceleration) t t + h h a a = v(t + h) v(t) h t (1.2) a(t) = lim h 0 v(t + h) v(t) h (1.3) a v(t) x a = dv dt = v a = d2 x dt 2 = ẍ ẍ (1.2) (1.3) MKS [] = [ ] [ ] = [m/s] = [m/s 2 ] (1.4) [s]

16 8 1 [ 1.4] t = velocity [m/s] 4 O time [s] (1) 0 (2) (3) 11 (4) [ ] (1) 0 t = 4 6 s (2) t = 6 9 s (3) a 0 t < 4 s 4 s t < 6 s 6 s t < 9 s 9 s t < 12 s a = a = a = a = 8 m/s 0 m/s 4 s 0 s 8 m/s 8 m/s 6 s 4 s 4 m/s 8 m/s 9 s 6 s 0 m/s ( 4) m/s 11 s 9 s = 2 m/s 2 = 0 m/s 2 = 4 m/s 2 = 2 m/s 2 2 m/s 2 (0 s t < 4 s) 0 m/s 2 (4 s t < 6 s) a(t) = 4 m/s 2 (6 s t < 9 s) 2 m/s 2 (9 s t < 11 s) ( ) (1.5) ) 2t m/s (0 s t < 4 s) 8 m/s (4 s t < 6 s) v(t) = 4(t 8) m/s (6 t < 8 s) 2(t 11) m/s (8 t < 11 s) (1.6)

17 1.3 9 t ( (1.5)) (4) t = 6 8 s t = 9 11 s t = 6 8 s t = 9 11 s [ 1.5] x t x = at 2 (a: ) [ ] (1) ẋ ẍ ẋ = dx dt = 2at ẍ = dẋ dt = 2a x(t) ẋ(t) ẍ(t) d dt d dt 位置 速度 加速度 x dt ẋ dt ẍ 1.3,, (integration) ( 1.3) km v(t) v(t)

18 10 1 (indefinite integral) v(t)dt (1.7) x(t) v(t) (ẋ(t) = v(t)) v(t)dt = x(t) + C (C ) (1.8) v v 0 v = v 0 C ( t = 0 ) O x x 1 x 0 O t x = v 0 t + x 0 x = v 0 t + x 1 t v(t) t 0 t f ( ) t = t f t 0 n x ( 1.5) 1.4 v( ) x( ), (t = 0 ) x v(t 0 ) t + v(t 1 ) t + v(t 2 ) t + + v(t n 1 ) t n 1 = v(t k ) t (1.9) k=0 t 1 = t 0 + t, t 2 = t t,, t f = t n = t 0 + n t t n x x = lim n k=0 n 1 v(t k ) t. lim n 1 n k=0 tf, n v(t) t dt t 0 O v(t) v(t) t t 0 t 1 t 2 t k t k+1 t n = t f t 1.5 t x = tf t 0 v(t)dt (1.10) v(t) (definite integral) () v(t)dt

19 v(t) X(t), dx(t) dt ( (1.8) ) = v(t) x(t) X(t) = x(t) + C (1.11) t 0 t x x = X(t) X(t 0 ) = {x(t) + C} {x(t 0 ) + C} (1.10) x = t = x(t) x(t 0 ) (1.12) t 0 v(t)dt = x(t) t t 0 = x(t) x(t 0 ) (1.13) (1.13) x(t) = x + x(t 0 ) = t t 0 v(t)dt + x(t 0 ) x(t) x(t 0 ) a(t) v(t) = t t 0 a(t)dt + v(t 0 ), v(t) a(t) t 0 t x v x = v = t t 0 v(t)dt = x(t) x(t 0 ) t t 0 a(t)dt = v(t) v(t 0 ) x(t) = v(t) = t t 0 v(t)dt + x(t 0 ) t t 0 a(t)dt + v(t 0 )

20 12 1 [ 1.6] (1) 1 v(t) (t = 0, 1, 2,, 10 [s]) t = 1 s v(t) t 10 (2) t (3) v t [ ] (1) t=0 v(t) t = (v(0) + v(1) + v(2) + + v(9) + v(10)) t = ( ( 4) + ( 2)) m/s 1 s = 34 m ( ) (1.14) t (2) 1.7 (2 + 8) s 8 m/s 3 s ( 4) m/s = 34 m ( ) (3) (1.6) velocity [m/s] 8 4 O time [s] velocity [m/s] 8 4 O time [s] x = 4 0 2tdt dt ( 4(t 8))dt + = t t 6 4 2(t 8) (t 11) = 34 m ( ) (t 11)dt 1.4 x(t) ẍ(t) (

21 1.4 13

22 14 1

23 ( ) ( ) (Newton s first law of motion: Law of inertia) [1] (force) [ ] () [ ] (Aristotle,BC ) 1 [ ] [ ] Nicolaus Copernicus, , or

24 16 2 [ ] wiki/galileo_galilei (Wikipedia(ja) ) [ ] (Christian Huygens, , ) (Constantijn Huygens) 2.3 [ ] [2]p.31 (Vincenzo Galilei) 1/16 (Wikipedia(ja) ) 19 (1583 ) [ ] [ ] (1656 ) [ ] *1 (2.3 ) (2.2 ) *1 (p.35) 1586

25 2.1 ( ) * [ ] () ( ) [ ] [ ] A B [ ] ( ) ( ) [ ] [ ] [ ] 2.5 [ ]

26 18 2 [ ] ( 7.5 ) [ ] [ 2.1] 2.2 () (Newton s second law of motion : Law of acceleration) [ ] (Leonhard Euler, ) Mechanica (equation of motion) [ ] ma = F (2.1) a F m ( ) (mass) (inertia mass) 1 kg (2.1) m a = F m (2.2) MKS (2.1) [ ] = [] = [kg m/s 2 ] (2.3) [N]( ) [N] = [kg m/s 2 ] (2.4) [N] a F = ma (2.5)

27 2.2 () 19 (inverse dynamics) ( ) (2.1) ( ) (forward dynamics) m [ ] (material point) x m O x 2.1 mẍ = 0 (2.6) [ ] m [ ] ( ) ẍ = 0 ẋ = 0dt = C 1 (C 1 : ) (2.7) [ ] ẍ = 0 ẋ(0) = v 0 C 1 = v 0 ẋ = v 0 (2.8) ẋ v 0 (2.8) (v 0 0) (v 0 = 0) (2.7) x = v 0 t + C 2 (C 1, C 2 : ) (2.9) x(0) = x 0 x = v 0 t + x 0 (2.10) [ 2.2] (1) (t = 0 ) 0 v 0 ( 0) t = 0 (2)

28 F mẍ = F m O m F 2.2 x ẍ = F m F ẋ = m dt = F m t + C 1 (C 1 ) (2.11) ( ) F x = m t + C 1 dt = 1 F 2 m t2 + C 1 t + C 2 (C 2 ) (2.12) 1 2 (2.11),(2.12) F = 0 (2.7)(2.9) n [ ] x(t) t = 0 x(0) = 0, ẋ(0) = 0 [ 2.3] F m t = 0 [ ] 2.2.4

29 2.2 () 21 [ 2.1] 1.4 (1) (2) (3) 800kg [ ] (1) (2) (3) F F = 800 [kg] ( 4) [m/s 2 ] = 3200 [N] (2.13) 3200 N [ 2.2] m t = 0 0 t t 1 [ ] F t 1 < t t 2 F t 2 [ ] (1) 0 t 1 (2) t 2 t 1 (3) t 1 t 2 (4) (5) 0 (6) 0 (7)

30 22 2 [ ] [ ] 0 t t 1 x ( 2.3) [ ] (i) 0 t t 1 O O m (a) (b) F F m 2.3 (a) 0 t t 1 (b) t 1 t t 2 x x mẍ = F (2.14) [ ] 1 F/m m [ ] ẍ = F m F ẍdt = m dt (2.15) ẋ = F t + C (C : ) (2.16) m ẋ(0) = C ẋ(0) = 0 C = 0 ẋ = F m t (2.17) x = 1 F 2 m t2 + C (C : ) (2.18) (t = 0) x(0) = C x(0) = 0 C = 0 (ii) t 1 t t 2 x = 1 F 2 m t2 (2.19) mẍ = F (2.20)

31 2.2 () 23 [ ] ẋ = F m t + C (2.22) m ẍ = F m (2.21) (2.21) (2.23) C [ ] ẍdt = F m dt ẋ = F m (t t 1) + C (C : ) (2.23) t 1 ẋ(t 1 ) = C (2.17) ẋ(t 1 ) = F m t 1 C = F m t 1 [ ] [ ] (2.24) ẋ = F m (t 2t 1) ẋ = F m (t t 1) + F m t 1 = F m {(t t 1) t 1 } (2.24) x = F { } 1 m 2 (t t 1) 2 t 1 (t t 1 ) + C (C : ) x(t 1 ) = C [ ] (2.19) x(t 1 ) = 1 2 C = 1 F 2 m t2 1 x = F { } 1 m 2 (t t 1) 2 t 1 (t t 1 ) + 1 F 2 m t2 1 = F { (t t1 ) 2 2t 1 (t t 1 ) t 2 2m 1} (1) 0 a 0 t t (2.17) t > t 1 0 (2.24) ẋ = 0 F m O F m v F m t 1 O F m t 1 F m t2 1 (2.25) t t [ ] (2.24) ẋ = F m (t 2t 1) x = F m ( 1 2 t2 2t 1 t) + C 0 = F m {(t t 1) t 1 } t = 2t 1 ( ) 0 2 F m t 1 x F m t2 1 1 F 2 m t2 1 O t t 1 2t 1 3t 1 (2 + 2)t 1 2.4

32 24 2 (2) (2.25) x(t) = 0 t T = t t 1 T T = t t 1 0 = F { (t t1 ) 2 2t 1 (t t 1 ) t 2 2m 1} T 2 2t 1 T t 2 1 = 0 (2.26) T = (1 ± 2)t 1 (2.27) t = T + t 1 t > t 1 t 2 [ ] [ ] t = (2 2)t1 t 2 = (2 + 2)t 1 ( ) (3) (2.15)(2.21), (2.17)(2.24), (2.19)(2.25) 2.4 = (1 ± 2)t 1 + t 1 = (2 ± 2)t 1 (2.28) t = 0 t = t 1 t = 2t 1 F F t = (2 + F 2)t 1 t = 3t 1 F O 2m t2 F 1 m t (4) t 1 < t < 2t 1 ( 2.5) (5) t 1 < t < 2t 1 x t = 2t 1 ( 2.5) v = (6) v(t) = t 0 a(t)dt + v(0) (2.29) v(0) = 0 v(t) = = t 0 t0 1 a(t)dt t a(t)dt + a(t)dt t 1 (2.30) a F m O F m v F m t 1 O F m t 1 2 F m t 1 x F m t2 1 1 F 2 m t2 1 O S D t t 1 2t 1 3t 1 (2 + 2)t t t

33 2.2 () 25 a > 0 t 1 t v = 0 0 ( 2.6) (7) x(t) = x(0) = 0 t 0 x(t) = v(t)dt + x(0) (2.31) t 0 v(t)dt (2.32) y = v(t) t t = 0 0 x = 0 t 2 S D ( 2.6) * (2.1) [1] [1] d (mv) = F (2.33) dt mv (linear momentum momentum) (2.1) (2.1) [ ] [ ] 5.1

34 26 2 [ 2.4]? * (2.1) F (2.1) [ ] [3], p.14 [ ] m/s 2 (gravitational acceleration) g g 9.8 m/s 2

35 (universal gravitation) m F 2.7 mÿ = F F y 2.7 F ÿ = g F = mg (2.34) F m m F = mg [ ] (, weight) 1kg 1[kgw]() [ ] 1 kg [ ] [ ] 1 kgw 1 kg m/s 2 1kgw=1kg 9.8m/s 2 = 9.8N 10 N 1 kg [ ] 2 [ ] 2.1 (1) 1kg ()? (2) (3) 1 [N] ( [ ] [ ] (aether, ether) (p.16) (wikipedia(ja) ) [ ] [ ], 3 (1) [ ] (Johannes Kepler, , ) [ ] Tycho Brahe, ,

36 28 2 (2) (3) Moon [ ] [1]p.1 [ ] Earth ( 2.8) [ ] ( [4]p.681) ( p.696) [1] 1 [ ] G = [m 3 /kg s 2 ] [ ] mg F 2.8 ( ) ( ) ( ) ( 2.9) F = G Mm r 2 (2.35) G [ ] (gravitational constant of universe) r [ ] M m [ ] 5.4 () 2.6 (gravitational mass) () r F F m M (mass)

37 (equivalence principle) [ ] ( 2.10) F = G Mm ˆr (2.36) r2 [ ] ˆr [ ] F ˆr F r [ ] m M 2.10 m M [ ] ˆr r ˆr = r r [ ] [ 2.5] 1kg 1mm?? [ ] ( ) () (2.34) 2 (2.35) Earth mg = G Mm R 2 R (2.34) (2.35) ( 2.11) mg = G Mm R 2 (2.37) 2.11 R R

38 30 2 M m R g = G M R 2 (2.38) [ ] (2.37) m, (2.38) [ ] (Henry Cavendish , ) 1874 http: // [ ] p.102[5] m [ ] m F R G (2.38) M r [ ] Wikimedia commons [ ] G = [m 3 /kg s 2 ] G = [m 3 /kg s 2 ] [ ] Eratosthenes, BC275 -BC195 [ ] km 50 [ 2.3]

39 [ ] L L = [km/day] 50 [day] [km] R R = L 2π [km] R = [km] [ ] 2 [ 2.4] (1) g G, R, M (2) g 9.8 [m/s 2 ] G = [m 3 /kg s 2 ] [ ] (1) m mg, G GmM/R 2 [ ] [km], [km] mg = G mm R 2 g = G M R 2 ( ) (2.39) (2) (2.39) M = g G R2 M = 9.8 [m/s 2 ] [m 3 /kg s 2 ] ( [m]) 2 = [kg] R = [km] M = [kg]

40 32 2 M = [kg] ( ) [ 2.5] y y = 0 m y 0 m v 0 t = 0, g (1) (2) (3) y 0 = 20 m v 0 = 0 m/s (g 10 m/s 2 ) (4) y 0 = 20 m 1 s v 0 t = 0 [ ] ÿ = g [ ] (1) [ ] mÿ = mg (2.40) y y0 O mg (2) (2.40) m ÿdt = ( g)dt ẏ = gt + C 1 (2.41) ẏdt = ( gt) + C 1 dt y = g 2 t2 + C 1 t + C 2 (2.42) (2.41)(2.42) t = 0 ẏ(0) = C 1, y(0) = C 2 (2.43) ẏ 0 (0) = v 0, y(0) = y 0 C 1 = v 0, C 2 = y 0 (2.42) y = 1 2 gt2 + v 0 t + y 0 ( ) (2.44)

41 (3) y 0 = 20 m, v 0 = 0 m/s (2.44) y(t) = 0 m t t 2 s 0 = 1 2 gt m t 2 = 2 20 m g = 2 20 m 9.8 m/s 2 ( ) (4) (2.44) v 0 4 s (2.45) v 0 = y y gt2 t y 0 = 20 m, t = 1 s y(1) = 0 m v 0 (2.46) v 0 = 0 m 20 m m/s2 (1 s) 2 1 s 15 m/s ( ) (2.47) v 0 < 0 t = 0 ( ) [ 2.6] m θ v 0 x y g [ ] (1) (2) () t (3) (y = 0) (4) (5)?? (6) 100km m? (1) ( 2.13) { mẍ = 0 ( ) (2.48) mÿ = mg y v 0 sin θ θ v 0 cos θ v 0 mg 2.13 x

42 34 2 (2) m { ẋ = C x ẏ = gt + C y ( C x, C y ) (2.49) { x = Cx t + C x2 y = g 2 t2 + C y t + C y2 ( C x2, C y2 ) (2.50) (2.49)(2.50) t = 0 { { ẋ(0) = C x x(0) = C x2 (2.51) ẏ(0) = C y y(0) = C y2 { ẋ(0) = v 0 cos θ ẏ(0) = v 0 sin θ { x(0) = 0 y(0) = 0 (2.52) (2.51)(2.52) { C x = v 0 cos θ { C x2 = 0 = v 0 sin θ C y2 = 0 C y (2.53) (2.49)(2.50) { ẋ = v 0 cos θ ẏ = gt + sin θ x = v 0 cos θ t y = 1 2 gt2 + v 0 sin θ t (2.54) ( ) (2.55) (2.54) g = 0 (2.54) (2.55) = + ( 2.14) (2.55) t (y = ax 2 + bx + c ) (3) (2.55) 2 y = 0 t y 1 2 gt2 x 0 = g 2 t2 + v 0 sin θ t (2.56) t = 0, 2v 0 g sin θ (2.57) 2.14 t 1 2 gt2

43 t = t = 2v 0 g sin θ ( ) (2.58) m g [ ] (4) (2.58) (2.55) 1 [ ] x = v 0 cos θ 2v 0 g sin θ = v2 0 sin 2θ ( ) (2.59) g m (5) ( (2.59)) sin 2θ 1 θ = π/4 ( ) (2.60) x = v2 0 g ( ) (2.61) (6) (2.61) v = 100km/h x = (100 km/h)2 9.8 m/s 2 78 m ( ) [ ] ( 2.15) [ ] (Simon Stevin, , ) wiki/simon_stevin

44 36 2 mg sin θ θ θ mg F θ F cos θ (a) (b) 2.16 (a) (b) θ F ( 2.16) [ 2.7] M m M g [ ] (1) M m (2) 2 (1) 2 M m (1) M m F M, F m F M = Mg sin 30 = 1 2 Mg ( ) 3 F m = mg sin 60 = 2 mg (2) F M = F m FM M mg m m Fm 2.17 ( ) ( ) 3 2 Mg = 1 2 mg M m = 3 (2.62) M m 3/1

45 [ 2.6] 2.16 A 2.18 AB CA AB CA B C 2.18 ( ) [ ] 1687 F 1 F 2 F 3 F 2 F 1 + F 2 F F 1 + F 2 + F 3 = 0 [ ] Pierre Varignon, , ( 2.19) F 1 F 2 F m F i (i = 1,..., N) N ma = F i (2.63) [ 2.7] F 1 F 2 F 3 F 1 F 2 F 3 i= * F 1 F 2 F 1 F 2 F 1, F 2 F

46 38 2 ( 2.19) F 1 F 2 F 3 F 1 F (a) (F 1 = F 2 ) F 1 ϕ F 1 F 2 ( 2.21(b)) F 3 F 3 F 3 F 2 F 2 F 1 F 2 F F 1, F 2 F 1, F 2 F 3 = F 1 cos ϕ + F 2 cos(θ ϕ) (2.64) θ F 1 F 2 F 1 = F 1, F 2 = F 2, F 3 = F 3 F 3 ϕ d F 3 dϕ = F 1 sin ϕ + F 2 sin(θ ϕ) = 0 (2.65) ϕ F 1 F 2 F 3 F 1 = F 2 F 1 sin ϕ = F 2 sin(θ ϕ) (2.66) ( 2.21(a)) (2.65) F 2 F 2 F 3 θ ϕ F 1 F 1 F 3 F 2 F 3 θ ϕ F 1 (a) (b) 2.21 (a) F 1,F 2 F 1 (2) F 1,F 2 F () *

47 2.5 () * 39 A x s(t) x O x A O s(t) x 2.22 x(t) x(t) a, b x = x + s(t) (2.67) s(t) = at + b (2.68) (2.67) x = x + at + b (2.69) (galilean transformation) [ 2.8] A x x A F A mẍ = F (2.70) m x = F (2.71) [ ] x x (2.69) (2.70) (2.71) ẍ = x m x = F 0 ( (2.69) a = 0 ) (galilean invariance)

48 40 2 ( ) [ 2.9] a s(t) = a 2 t2 (2.72) m x = F ma (2.73) [ ] (2.67) (2.72) ẍ = x + a (2.74) (2.70) m( x + a) = F

49 2.6 () 41 m x = F ma (2.75) (!) [ 2.8] 2.6 () (Newton s Third law of motion : Law of reciprocal actions) ) [1] 2.3 (Law of reciprocal actions) [ ] S F i 0 F i = 0 i S [ ] [ ] 5 [ ]

50 42 2 N y Object mg m x y x m Ground N mg (a) (b) 2.23 (a) (b) Free-Body Diagrams 2.7 Free-Body Diagrams [ ] Free-Body Diagrams [ ] Free-Body Diagrams [ ] (1) (2) (3) [ ] [ ] [ ] (4) [ ] 5 (5) [ ] N mg Free-Body Diagrams (1) ( ) (2) + -

51 2.7 Free-Body Diagrams 43 mg y x N N mg mg m y Object x Ground mg mg mg m (a) (b) 2.24 (a)free-body Diagrams N (b) [ 2.10] m 2.23(a) g (1) Free-Body Diagrams (2) (3) [ ] (1) 2.23(b) Free-Body Diagrams mg [ ] (normal force) [ ] N (ground reaction force) [ ] (3.3 ) [ ] normal normal force N Free-Body Diagrams 2.24(a) (2) 2.23(a) x, y

52 44 2 [ ] mẍ = N + mg x = (x, y), N = (N, 0), g = (0, g) + [ ] { mẍ = 0 mÿ = N mg ÿ = 0 (2.76) m 0 = N mg N = mg N = mg > 0 ( ) mg Free-Body Diagrams mg 2.24(b)

53 T T T T mg ceil string ball (a) (b) 3.1 (a) (b) Free- Body Diagrams. (tension, tensile strength) [ ] [ ] T T T T T T T T () ( 3.3) [ ] tension T [ ] 3.2

54 46 3 Free-Body Diagrams,?? [ 3.1] 3.4 A, B A F A, B m A, m B A, B T g (1) A B Free-Body Diagrams x xa xb O 3.4 F A B (2) x A, B x A, x B A, B x (3) A, B a T [ ] ( ) A B Free-Body Diagrams 3.5 ( ) 3.5 A, B { m A ẍ A = F m A g T (3.1) m B ẍ B = T m B g F T ( ) (3.1) ẍ A = ẍ B = a { m A a m B a = F m A g T = T m B g T A mag B mbg 3.5 A B Free- Body Diagrams F a = g m A + m m B B T = F m A + m B F = 0 A, B 0 m A = 0 m B = 0 0 F

55 [ 3.2] θ A A O x B A, B m A, m B y B θ A O g (1) A, B ( ) Free-Body Diagrams (2) A x B y A, B (3) ẍ = ÿ ẍ (4) A, B [ ] ( ) T, A N Free- Body Diagrams A T T T T N mag sinθ mbg mbg N mag θ θ mag B slope A 3.6 A, B ( ) Free-Body Diagrams ( ) Free-Body Diagrams A, B { m A ẍ m B ÿ = T m A g sin θ = m B g T (3.2) (θ = 0) A θ = 0 A

56 48 3 ( ) (3.2) ẍ = ÿ ẍ = m B m A sin θ g A + m B T = m Am B (1 + sin θ)g m A + m B (3.3) m A = 0 m B ( g) 0 m B = 0 m A ( g sin θ) 0 ( ) A, B 0 (3.3) ẍ = 0 0 = m B m A sin θ g m A + m B m B g = m A g sin θ A B θ = 90 m A = m B θ = 0 m B = [ ] (Robert Fooke, , ) ( org/wiki/ ) [ ] [ ] x k kx k [ ] [ ] [ ]

57 [ 3.3] 3.7 ( k) m x O x k O 3.7 m x x g (1) F, () N Free-Body Diagrams (2) x < 0 x > 0 (3) x(0) = a, ẋ(0) = 0 [ ] (1) N N Free-Body kx kx kx Diagrams 3.8 kx mg x < 0 N kx mg kx x (2) mẍ = kx ( ) (3.4) (x > 0) ( kx < 0) (x < 0) ( kx > 0) (3) (3.4) x(t) = A cos(ωt + ϕ) (3.5), A(> 0) ω [ ] ϕ t = 0 [ ] SI [rad/s]

58 50 3 (3.4) (3.5) d dt x = d A cos(ωt + ϕ) dt = ωa sin(ωt + ϕ) d 2 dt 2 x = d ( ωa sin(ωt + ϕ)) dt = ω 2 A cos(ωt + ϕ) (3.4) = ω 2 x (3.6) ω 2 mx = kx k ω = m ω, (3.5) t = 0 k x(t) = A cos( t + ϕ) (3.7) m x(0) = A cos ϕ ẋ(0) = Aω sin ϕ, x(0) = a, ẋ(0) = 0 a = A cos ϕ (3.8) 0 = Aω sin ϕ (3.9) (3.8) A 0, cos ϕ > 0 (3.9) sin ϕ = 0 ϕ = 0 sin ϕ = 0 ϕ = π cos ϕ > 0 ) (3.8) A = a x x = a cos k m t (3.10) [ ] T ω T [s] ω [rad/s] T = 2π/ω a T = 2π/ω = 2π m [ ] k (k ) m

59 [ 3.4] k x m ( ) O k [ ] (1) (2) (3) (4) A t = 0 t (1) 3.9 (2) mẍ = kx + mg = k(x mg ) ( ) (3.11) k (3) 0 (3.11) 0 [ ] x 0 k(x 0 mg k ) = 0 x 0 = mg k ( ) x kx mg m 3.9 (4) A t = 0 x = A + x 0 (3.4) x(t) = A cos ωt + x 0 (3.12), ω ẋ = ωa sin ωt ẍ = ω 2 A cos ωt = ω 2 (x x 0 ) (3.11) ω 2 m = k k ω = m [ ] 0

60 52 3 ω, (3.12) k x(t) = A cos m t + x 0 k = A cos m t + mg k ( ) t = v [ ] (Leonardo da Vinci, , ) [ ] Wikipedia(ja) History of science: Friction, abc/history.htm [ ] [ ] Guillaume Amontons, , [ ] (Charles Augustin de Coulomb, , ) [ ] Wikipedia(ja) (frictional force) 200 [ ] [ ] (1) (2) ( ) [ ] 200 [ ] 100 [ ] (static friction) (kinetic friction) [ ] (1) (2) F N N mg F mg v Free-Body Diagrams F N ( 3.10,3.11) (1) (2) (3) (

61 ) µ 0 N µ 0 N (4) (5) ( ) µ N µn (6) (µ 0 N) (µn) Free- Body Diagrams [ 3.5] m ( 3.12) µ 0, µ g y x 3.12 (1) F F f N a Free-Body Diagrams b x y c F f F (2) F 1 F 1 (3) F f [ ] (1)(a) Free-Body Diagrams 3.13 (1)(b) { mẍ = F F f mÿ = N mg (3.13) Ff N F mg Ff F (1)(c) ẍ = ÿ = 0 { 0 = F F f 0 = N mg (3.14) N mg 3.13 Free- Body Diagrams

62 54 3 { F f N = F = mg (3.15) (F f = F ) (2) µ 0 N F f µ 0 N (3.16) (3.15) F µ 0 mg (3.17) F F > µ 0 mg (3.18) F 1 = µ 0 mg (3) F f µn (3.15) F f = µn = µmg ( ) (3.19) [ 3.6] θ m ( 3.14) µ 0, µ g (1) Free-Body Diagrams m θ 3.14 (2) (3) m, g θ (4) θ θ (5) (6)

63 [ ] (1) N, F 3.15 (2) x y { mẍ mÿ = mg sin θ F = N mg cos θ (3.20) (3) ẍ = 0 y F x mg cos θ N N mg sin θ θ mg F θ mg F = mg sin θ (3) (3.21) 3.15 Free-Body Diagrams (4) µ 0 N F µ 0 N (3.22) N ÿ = 0 (3.20) (3.22) (3.21) N = mg cos θ (3.23) mg sin θ µ 0 mg cos θ tan θ µ 0 θ = tan 1 µ 0 (5) 3.15 µn = µmg cos θ (6) (3.20) mẍ = mg sin θ µmg cos θ ẍ = g(sin θ µ cos θ)

64 (viscosity) v σ > 0 σv σv σ [σ] = [F ] [v] = [N] [m/s] = [Ns/m] = [kg/s] O y 3.16 σv mg Free-Body Diagrams [ 3.7] v( ) m 3.16 σ) y (1) 3.16 (v ) (2) (3) m σ (4) [ ] (1) v = ẏ > 0( ) v = ẏ < 0 σv (2) mÿ = σv + mg ( ) (3.24) σ = 0

65 (3) ẏ = v (3.24) m v = σv + mg (3.25) v = σ mg (v m σ ) V = v mg σ (3.26) V = σ m V (3.27) V V = σ m V V dt = σ m dt log V = σ m t + C V = ±e σ m t+c (C ) ±e C C V = Ce σ m t (3.28) (3.26) v = Ce σ m t + mg σ ( ) (3.29) v v 0 = mg σ v 3.17 m σ v 0 v mg σ v = Ce m σ t + mg σ C C = 0 v = mg σ (4) 3.18 v = ẏ v 3.18 v O y t 3.18 σv mg

66 58 3 mÿ = σv mg (3.30) ÿ = σ mg (v + m σ ) (3.31) V = v + mg σ (3.32) (3.27) V (3.28) (3.32) v = Ce σ m t mg σ ( ) (3.33) v v 0 = mg σ

67 59 4 [6] [ ] [ ] 4.1 [ ] ( René Descartes, , ) ( ) [ ] F, v ( v) 2 = F, v = F 4.1 m F O m F x mẍ = F (4.1) F t 1 t 2 = t 1 + t v = ẋ t dt F F dt (4.1) dt m vdt = F dt

68 60 4 dt F dt F F dt t 1 t 2 t 1 t 2 = t 1 + t ( 4.2) t2 t 1 m vdt = mv 2 mv 1 = t2 t 1 t2 F dt t 1 F dt. (4.2) v 1 = v(t 1 ), v 2 = v(t 2 ) (F dt) = t 1 + t F O F F dt t 1 t t + dt t 2 dt 4.2 ( ) mv ( ) (F dt) ( ) (impulse) mv (momentum) (mẍ = F ) (F = 0) mv F (4.2) mv 2 mv 1 = F (t 2 t 1 ) F t = F t (4.3) (4.3) MKS t [] = [ ] = [kg m/s] [ ] = [ ] = [kg m/s] [ 4.1] (1) (2) (3) [ ] t

69 F F ( 4.3) F t M m x M, x m v M = ẋ M, v m = ẋ m 4.3 { Mẍ M = F mẍ m = F (4.4) (t = 0 (t = t) t 0 t 0 Mẍ M dt mẍ m dt = O F x M x m x 4.3 Free-Body Diagrams = t 0 t 0 F dt F dt F (4.5) { M v M ( t) M v M (0) m v m ( t) m v m (0) = F t = F t (4.6) ((1) ) ((2) ) (M > m) ((3) ) F ( 4.1) t 1 t 2 x(t 1 ) x(t 2 ) = x(t 1 ) + x v = ẋ F x dx F F dx (4.1) dx m vdx = F dx

70 62 4 dx F F F dx dt F F dx F F dx x 1 = x(t 1 ) x 2 = x(t 2 ) t 1 t 2 = t 1 + t ( O x 1 x x + dxx 2 dx 4.4 x 4.4) x2 x2 mẍdx = F dx (4.7) x 1 x 1 dx v = ẋ dx dt dx = ẋdt = vdt x2 t2 x 1 mẍdx = t 1 m vvdt = 1 2 mv2 t 2 t 1 = 1 2 mv mv2 1. (4.8) t2 [ ] I = vvdt t 1 t2 I = vvdt t 1 = vv t 2 t1 = v 2 t 2 t1 I I = 1 2 v2 t 2 t 1 t2 t 1 v vdt v 1 = v(t 1 ) = ẋ(t 1 ) v 2 = v(t 2 ) = ẋ(t 2 ) d dt v2 = 2 vv [ ] (4.7) 1 2 mv2 2 1 x2 2 mv2 1 = F dx (4.9) x mv2 ( ) (F dx) ( ) 1 2 mv2 (kinetic energy) (F dx) (work) (mẍ = F ) [ ] 1 2 mv2 = (Gustave Gaspard Coriolis , ) (work) 1829 Calcul de l Effet des Machines (Calculation of the Effect of Machines) F [ ] (F = 0) 1 2 mv2 (t) (F 0) 0 0 [ ] [ ] 0

71 F (4.9) 1 2 mv mv2 1 = F (x 2 x 1 ) 1 2 mv mv2 1 = F x (4.10) F x (4.10) MKS [ ] = [ 2 ] = [kg m 2 /s 2 ] [ ] = [ ] = [kg m 2 /s 2 ] [J](, Joule [ ] ) [ 4.1] F F [ ] 5.1 [ 4.2] 4.1 F m (1) t 2 = t 1 + t [s] x 2 v 2 x 1, v 1 (2) t 1 t 2 (1). (3) F t. (4) (2) (3) F [6]

72 64 4 [ 4.2] 1kg g = 9.8m/s (1) 1 (2) 1 (3) 10 m [ ] (1) t[s] v(t) mv(1) mv(0) = mg 1[s] 1 0 v(1) = v(0) g 1[s] = 9.8[m/s] (4.11) 1 9.8m/s( 9.8m/s) (2) t x(t), 1 2 mv2 (1) 1 2 mv2 (0) = mg {x(1) x(0)} = mgx(1) (4.11) x(1s) x(1) = 1 { v 2 (1) v 2 (0) } 2g 1 = 2 9.8[m/s 2 ] (9.8[m/s])2 = 4.9m (4.12) [m] (3) 10m v mv mv2 (0) = mg ( 10[m] 0[m]) mg 10[m] = 98 [N] v 10 v 2 10 = 2g 10[m] = 196[m 2 /s 2 ] v 10 = 196[m 2 /s 2 ] = (7 2) 2 [m/s] 14[m/s] ( )

73 [ 4.3] m t = 0 θ F 1 F 2 t x l (1) (2) t F 1, F 2 (3) t (4) [ ] (1) ( ) x y N?? { mẍ = F 1 cos θ F 2 mÿ = N + F 1 sin θ mg 0 t t mẍdt = 0 t mÿdt = 0 { mẋ( t) mẋ(0) mẏ( t) mẏ(0) t 0 t 0 (4.13) y F 2 y (F 1 cos θ F 2 )dt x F 2 x (N + F 1 sin θ mg)dt = (F 1 cos θ F 2 ) t = (N + F sin θ mg) t N mg F 1 θ F 1 sin θ F θ F 1 cos θ (4.14) y = ẏ(t) = 0 (4.14) 0 (F 1 cos θ F 2 ) t (2) x (4.13) l 0 mẍdx = = l 0 l 0 (F 1 cos θ F 2 )dx F 1 cos θdx + l 0 ( F 2 )dx

74 66 4 F 1 F mẋ2 ( t) 1 2 mẋ2 (0) = F 1 cos θ l + ( F 2 l) (4.15) (4.8) F 1 F 2 F 1 cos θ l F 2 l F F (3) (4.15) mg 0 F 0 (4) (4.15) F 1 cos θ l + ( F 2 l) 4.3 * m x, F (t) mẍ = F (t) (4.16)

75 4.3 * t 1 t 2 F (4.16) t2 t2 mẍdt = t 1 mv(t 2 ) mv(t 1 ) = t 1 t2 F (t)dt t 1 F (t)dt (4.17) v = ẋ mv 1 F mv(t 2 ) mv(t 1 ) = F t (4.18) t = t 2 t 1 (4.3) x 1 x 2 x 2 x 2 x 1 mẍ dx = x 1 F (t) dx (4.19) x 2 x 1 mẍ dx = t2 t 1 mẍ dx dt dt = 1 2 mv mv2 1 (4.20) v 1 = v(t 1 ), v 2 = v(t 2 ) (4.19) 1 2 mv2 2 1 x 2 2 mv2 1 = F (t) dx (4.21) x 1 F (t) W

76 68 4 F (4.17) (4.21) F 1 2 mv2 (t 2 ) 1 2 mv2 (t 1 ) = F x (4.22) x = x 2 x 1 F θ F cos θ x ( ) 4.6 (4.22) F x = F x cos θ (4.23) θ F x F cos θ F ( 4.6) (4.21) dx dt dx = v dt W W = = x(t 2) x(t 1 ) t2 F (t) dx t 1 F (t) vdt (4.24) [ 4.3] (4.20) x = (x, y, z), dx = (dx, dy, dz) 4.4 ( ) 4.4.1

77 y ( ) () 4.7 m y O 4.7 mg mÿ = mg (4.25) y = y 0, y = h U(h), y = h y = y 0 ( y = h y = y 0 ), U(h) = y0 h ( mg)dy = mg(h y 0 ) (4.26) (y 0 = 0) U(h) = 0 h ( mg)dy = mgh (4.27) [ 4.4: ] U(y) U(y) = mgy [ ] y 1 = y(t 1 ) y 2 = y(t 2 ) (4.25) 1 2 mv mv2 1 = y2 y 1 ( mg)dy v 1 = ẏ(t 1 ), v 2 = ẏ(t 2 ), (4.26) U(h) 1 2 mv mv2 1 = = y0 y 1 ( mg)dy + y0 y 1 ( mg)dy = U(y 1 ) U(y 2 ) y2 y 0 y0 y 2 ( mg)dy ( mg)dy

78 mv2 1 + U(y 1 ) = 1 2 mv2 2 + U(y 2 ) (4.28) [ ] (y = y 1 y = y 2 ) [ ] (mechanical energy) [ 4.5] ( (4.25)) (1) t = 0, t (2) t ( ) [ 4.4] m h x m l θ h θ O (1) l (2) h h h (3) l [ ] (1) 4.8 mẍ = mg sin θ (4.29)

79 x = l x = 0 0 l ( mg sin θ)dx = mg sin θ x 0 l = mgl sin θ ( ) [ ] mg sin θ l W x N mg sin θ θ mg 4.8 W = mg sin θ l = mgl sin θ ( ) (4.30) (2) l cos θ = h (4.30) mgl sin θ = mgh (4.31) ( (4.30) θ = π/2 ) (3) (4.29) x = l x = 0 () 1 2 mv2 l 1 2 mv2 0 = mgh v 0 v 0 = 0 v l l v = 2gh ( ) (4.32) 1 2 mv2 0 + mgh = 1 2 mv2 l + mg 0 ( ) v l

80 72 4 y y = h y = 0 y = h y 1 y 0 ( 4.9) y 1 y 0 h mg [ ] 4.5 y1 y0 y0 ( mg)dy + ( mg)dy = ( mg)dy h y 1 h 4.9 y y 0, 2 3 (4.4.4 ) (conservative force) [ ] [ ] [ ] 4.6 [ ] 4.5 [ ] 4.7 (nonconservative force) * m x 1 x 2 W 12 g x 2 W 12 = mg dx (4.32) x 1 dx θ x 2 W 12 = mg dx cos θ x 1 x 2 y 1 y 2 dy dy = dx cos θ (4.33) y ( 4.10) x 1 θ dx dy = dx cos(π θ) = dx cos θ mg W 12 = y2 mg( dy) = mg(y 1 y 2 ) y 1 (4.34) 4.10

81 U(y) x 0 y 0 W 12 x 2 W 12 = mg dx x 1 x 0 = x 1 x 0 mg dx x 2 mg dx = U(y 1 ) U(y 2 ) (4.35) y U(y) = mgy 4.5 F (x) F (x) U U(x) = x 0 x F (x) dx = x F (x) dx x 0 (4.36) x [ 4.6] k m k O m x x (1) x m (2) x (elastic potential energy) (3) l (4) x 1 x 3 x 2 (x 1 < x 2 < x 3 )

82 [ 4.7] 2.3 M A r m B r = r = r = θ B A (1) B A r 4.11 r + a r O (2) B a A r + a A A (3) r = 0 [ ] (Joseph-Louis Lagrange, ) (George Green, , ) (wikipedia(ja) ) (wikipedia(ja) ) * 4.5 (4.36) U F (x) = U(x) (4.37) F U(x) F (potential) [ ] F U(x) U(x) F U(x) + C (C: ) C (potential energy) [ 4.8] x = (x, y, z), F = (f x, f y, f z ), dx = (dx, dy, dz) (4.36) (x, y, z) (4.37) 4.1 F (1) U(y) = mgy (y ) (2) U(r) = 1 r (r = (x, y, z) )

83 [ 4.5] 4.4 x F m l F (1) h O θ (2) l F (3) l [ ] (1) 4.12 x mẍ = mg sin θ F (4.38) (2) (4.38) x = l x = 0 ( 4.4 ) 1 2 mv2 l 1 2 mv2 0 = mgh + F l. (4.39) ( ) ( ) mv2 l + mg 0 2 mv2 0 + mgh = F l. (4.40) F mg N θ F (3) v 0 = 0 (4.39) ( v = 2 gh + F l ) m ( ) (4.40)

84 * 3 m F C F x 1 = x(t 1 ) x 2 = x(t 2 ) x 1 x 2 mẍ = F C + F (4.41) x 2 x 2 x 2 x 1 mẍ dx = x 1 F C dx + x 1 F dx (4.42) F C U { } 1 1 x 2 2 mv2 2 + U(x 2 ) 2 mv2 1 + U(x 1 ) = F dx (4.43) x 1 v 1 = ẋ(t 1 ), v 2 = ẋ(t 2 ) U(x) F F 4.7 [ ] 5.2 [ ], ( )

85 [ 4.6] m v 0 l µ g [ ] (1) F, N (2) F m, µ (3) F (4) l v 0 (5) µ = km, (1) x, y mẍ = F (4.44) mÿ = N mg (4.45) (2) ÿ = 0 (4.45) N = mg (4.46) F = µn = µmg (4.47) (3) (4.44) x = 0 x = l l 0 mẍdx = l 0 ( F )dx 1 2 m mv2 0 = F l 1 2 mv2 0 = F l (4.48) l mv2 0 F l (4) (4.47) (4.48) l = v2 0 2µg (4.49) l m

86 78 4 (5) v = 72km/h, µ = 0.5, g = 9.8m/s 2 l (4.49) l = v2 0 2µg (72km/h) 2 = m/s 2 = ( m/( s)) 2 9.8m/s 2 40m ( ), 1,, (20m ) [ 4.9] m x 1 x 3 x 2 (x 1 < x 2 < x 3 ) x 1 x 2 x 3 x µ g

87 A B A? B [ ] m A, m B A, B v pre A m A v pre A, vpre B + m Bv pre B vpost = m Av post A A, vpost B + m Bv post B (5.1) A,B [ ] (p.59) (p.16) m A v A (t) + m B v B (t) = C (C:) (5.2)

88 80 5 [ 5.1] 5.1 m A, m B A, B A, B v A (t), v B (t) t A B F (t) (1) (2) (5.2)(5.1) [ ] (1) A, B { m A ẍ A (t) m B ẍ B (t) = F (t) = F (t) (5.3) F (t) 0 F = 0 (2) 2 (5.3) m A ẍ A (t) + m B ẍ B (t) = 0 d dt (m Av A (t) + m B v B (t)) = 0 (5.4) (5.2) m A v A (t) + m B v B (t) = C (C : ) (5.5) (5.1) 2 (5.3) 2 [ 5.2] M ( M) v ( m m) x xm xm O (1) V M, m, v (2) [ ]?

89 (1) M 0 + mv = (M + m)v (5.6) V = m v ( ) (5.7) M + m (1) V (2) (m M) (3) (m M) v (2) 1 2 (M + m)v mv2 = 1 ( m (M + m) 2 = 1 mm 2 M + m ) mv2 M + m v2 (5.8) (inelastic collision) (elastic collision) [ 5.3] 2 A, B B v A A,B? () A,B m A, m B [ ] A,B v A, v B. m A v = m A v A + m B v B (5.9) 1 2 m Av 2 = 1 2 m Av 2 A m Bv 2 B. (5.10)

90 82 5 (5.9) v A = v m B m A v B (5.10) v A (5.9) v B 2 { v A = v v A = m A m B v v B = 0, m A + m B v B = 2m A v m A + m B ( ) m A = m B { v A = 0 v B = v A B A [ 5.4] m 2 B v A A 30 A,B [ ] A x A y y A,B v A, v B 1 2 mv2 = 1 2 mv2 A mv2 B, (5.11), v A = v A, v B = v B y v v A v B θ x v 2 = v 2 A + v 2 B (5.12). mv = mv A + mv B v = v A + v B (5.13)

91 v = (v, 0) 2 v A v B v (5.2) v A θ v A v v A = (v A cos 30, v A sin 30 ) = ( 3 2 v A, 1 2 v A) (5.14) 2 [ 1] (5.13) v B 5.2 v B = v v A (5.15) (5.14) v B = (v 3 2 v A, 1 2 v A) (5.16) (5.12) v A 2 v A = 0, 3 2 v (5.17) (5.14), (5.16) v A, v B 2 { v A = (0, 0) v B = (v, 0), v A = v B = ( ) v, 4 v ( ) v, 4 v ( ) [ 2] (5.13) v B v B = (v v A ) (v v A ) (5.12) v 2 = v 2 A + v 2 B + 2v A v B (5.18) v A v B = 0 (5.19) 2 v A v B [ ] ( 5.2) [ ]va v B (5.12) v B v A ( (5.14)) v B = ( 1 2 v B, 3 2 v B) (5.20) (5.14) (5.13) v A = 3v B (5.21)

92 84 5 v A > 0 ( (5.14) ) v B > 0 (5.14),(5.20) (5.12) v B = v/2 (5.20) v B v A = ( v, 4 v) v B = ( 1 ( ) 3 4 v, 4 v) N N m 1,m 2,...,m N v 1,v 2,v 3,...,v N N m 1 v 1 + m 2 v 2 + m N v N = C (C: ) 5.1 (1) (2) 5.2 [ ] [ ] 0, 5.3 (Photo from http: //en.wikipedia.org/wiki/image: Galileo_Thermometer_closeup.jpg)

93 [ ] [ ] 1714 [ ] 1738 ( [ ] ) * [ ] [ ] ( 5.4) [ ] 5.4 (Photo from Joule%27s_Apparatus_%28Harper%27s_Scan% 29.png) [ ] [ ] (Daniel Fahrenheit, ) [ ] Daniel Bernouli, , [ ] (Robert Boyle, ), 1662 [ ] (Japmes Prescott Joule, , ) ( ) (p.30) (p.74) [ ] (0 [ ] Julius Robert von Mayer, *1 (James Clerk Maxwell, ) (p.30)

94 86 5 [ ] Hermann Ludwig Ferdinand von Helmholtz, [ ] (Johannes Diderik van der Waals, , ) 1873 [ ] 1847 [ ] 1910 [ ] [ ] Robert Brown , [ ] Theodor Svedberg, , 1926 [ ] (Brownian motion) (1827) [ ] (1905) 5.5. (Zweistein 2006, Wikimedia Commons ) 1905 [ ] [ 5.1]? 1 2 m v 2 T [ ] ( ) 0.012kg 12 1 ([mol]) SI 1 2 m v 2 = 3 2 kt k = [J/K] 1 mol [ ] N A *2 *2 1mol 0.012kg 12 ( Avogadro, Lorenzo Romano Amedeo Carlo, , ) 1811

95 N Am v 2 = 3 2 RT R = kn A = [JK 1 mol 1 ] 27 C 300 K [ ] N A 14 g A B A B FA A f 5.6 F A, F B ( 5.6) A B x A, x B A B f { m A x A m B x B = F A + f = F B f f FB B (5.22) [ ] [K] 0 ( 0.01C ) (William Thomson, ) m A ẍ A + m B ẍ B = F A + F B (5.23) A B (m A + m B )Ẍ = F A + F B (5.24) A B X x A, x B? (5.23) (5.24) X = m Ax A + m B x B m A + m B (5.25) X A B (center of mass) (center of gravity) X M m A + m B (5.24) MẌ = F A + F B (5.26) F A = F B = 0, MẌ = 0.

96 88 5 MẊ = C (C ) [ 5.5] 5.2 m [ ] (1) x M, x m X (2) M, m, v (1) X (5.25) x M xm m xm O X = Mx M + mx m M + m (5.27) (2) (5.27) Ẋ = Mẋ M + mẋ m M + m (5.28) ẋ m = v Ẋ = m M + m v (5.29) N x 1, x 2,, x N X = m N 1x 1 + m 2 x m N x N i=1 = m ix i m 1 + m m N M M = m 1 + m m N = N m i (5.30) i=1

97 F 1, F 2, N MẌ = F 1 + F F N = F i (5.31) [ ] (rigid body) [ ] 6 i=1 [ ] [ ] [ 5.2] (5.31)

98 90 5

99 y 1 1 θ 1 (radian [ ] [rad] ) 1 O 1 x (1 1[deg] ) (SI ) [ ] (radian) radius [ 6.1] θ θ [deg] θ [rad] ( 1 y P r P r θ O x O OP = r OP θ xy x 6.2 (polar coordinate system) θ r r ( 1 θ [rad]= 2π 360 θ

100 92 6 (radius) r r (r, θ) (x, y) { x = r cos θ y = r sin θ θ θ 2 θ [ 6.1] m r ω y (x, y) t = 0 x xy xy r O r ωt x r = (x, y) = (r cos ωt, r sin ωt) (6.1) (1) ṙ (2) r a = r a ω b v (3) F F a ω b v (4) F (5) r ṙ

101 [ ] (1) (6.1) v v = ṙ = ( rω sin ωt, rω cos ωt) (6.2) y v F r v = v = rω O x r v = (6.3) ( 6.3) (2) (6.2) r = v = ( rω 2 cos ωt, rω 2 sin ωt) = ω 2 r (6.4) r = rω 2 (6.5) (6.2) v ( ω ) 2 r = r (6.6) r (3) (6.4) m m r = mω 2 r (6.7) F ( 6.3) F = mrω 2 = mv2 r (6.8) (centripetal force) (4) F ( ) (6.4) (6.3) F v = mω 2 r ṙ = 0 (6.9)

102 94 6 F v (5) r r = (r cos θ(t), sin θ(t)) (6.10) ṙ = ( r θ sin θ, θ cos θ) (6.11) r ṙ = (r cos θ, sin θ) ( r θ sin θ, θ cos θ) = 0 [ 6.2] l l (1) (2) v 0 (3) JR 360km km m JR km 600km [ ] (1) ( 6.4) (2) v N mg N mg F F = mv2 l/2 = 2mv2 l (6.12) 6.4

103 F N [ ] F = N + mg (6.13) (6.12) (6.13) [ ] N = 2mv2 l mg (6.14) N N 0 v v lg 2 v = 1 2 mv2 0 = mgl mv2 v 0 = 2gl + v 2 (6.15) lg 2 v 0 = 2gl + lg 2 5 = lg ( ) (6.16) 2 l (3) (6.16) l l = 2 v0 2 5 g (6.17) v 0 = 360 km/h = 100 m/s, g = 9.8 m/s 2 l = 2 (100 m/s) m/s 2 400m v 0 = 600 km/h l = 400 ( ) m 1111 m 360

104 [ 6.3] l m θ g O θ l (1) T m [ ] (2) x y (3) θ << 1 sin θ θ θ (4) (1) 6.5 T (2) { mẍ mÿ = T sin θ = mg T cos θ (6.18) (3) (x, y) = (l sin θ, l cos θ) { ẋ = l θ cos θ ẏ = l θ sin θ { ẍ = l( θ cos θ θ 2 sin θ) ÿ = l( θ sin θ θ 2 cos θ) (6.19) (6.20) θ T T sin θ 6.5 T cos θ mg (6.18) { ml( θ cos θ θ 2 sin θ) ml( θ sin θ θ 2 cos θ) = T sin θ = mg T cos θ ( cos θ) ( sin θ) ml θ = mg sin θ (6.21) θ 1 sin θ θ ml θ = mgθ θ = g l θ (6.22)

105 θ = A sin(ωt + ϕ), (A, ω : ) (6.23) θ = ω 2 A sin(ωt + ϕ) = ω 2 θ ω 2 θ = g l θ g ω = l (6.23) ( ) g θ = A sin l t + ϕ ( ) (6.24) A, A ϕ l T = 2π/ω = 2π g (θ 1) (p.16) (4) 6.5 T mg cos θ (6.21) ( sin θ) + ( cos θ) ml θ 2 = T mg cos θ (6.25) 6.2 ml θ 2 (6.24) ( ) 2 ml θ g g 2 = ml A l cos( l t + ϕ) ( ) g = mga 2 cos 2 l t + ϕ ( ) [ 6.2] (1) (2) 0 (3) 6.4

106 98 6 [ 6.3] M, R, G ( 2 (1) (2) [ ] 6.5 [ 6.4] R M m r, ω F F = m(r r)ω 2 = Mrω 2 (6.26) r, m M, R, ω F = G Mm R 2 (6.27) [ ] 27.3 (2π/ω) G M R 2.3 ( 2 M = kg, R = m, G = m 3 /s 2 kg v 1 v 1 GM/R = 7.90 km/s, v 2 v 2 2GM/R = 11.2 km/s

107 [ ] [ ] [ 6.5] 89 40,000km km 38.4 km [ ] (Hipparchus, BC190- BC125 ) () [ ] [ ] [ 6.6] [ ] (p.16) 10 [ ] (p.48) [ ],,1962,p [ 6.7] kg 6.6 (p.27) 3 1

108 100 6 [ 6.4] m F v v v v r F mr v = 0 (6.28) r r r = r [ ] F v v t r O mv (t + t) mv (t) = 0 (6.29) mrv (t + t) mrv (t) = 0 (6.30) mrv (angular momentum) t t 0 mrv (t + t) mrv (t) t mrv (t) l(t) = 0 (6.31) d dt (mrv ) = 0 (6.32) dl dt = 0 (6.33) v t 6.6 r + r r = v t t r = v t S r F Planet S = 1 2 rv t (6.34) Sun 6.6 ( )

109 t 0 ds dt = 1 2 rv (6.35) (6.35) d 2 S dt 2 = 0 (6.36) d 2 S dt 2 = 1 d 2 dt (rv ) (6.37) ( (6.32)) 0 (6.36)

110 102 6

111 [ 7.1] r m F r ϕ θ F t r θ O θ F r ϕ 7.1 (1) F W F = F, r, θ (2) t K θ(t), θ(t + t) (3) K = W mr 2 θ = F r sin ϕ (7.1)

112 104 7 [ ] (1) r(t) r r = r θ O F F sin ϕ ( 7.2) F W W = (F sin ϕ) (r θ) = F r θ sin ϕ (1) (2) F F sin ϕ r θ 7.2 ϕ (3) K = W K = 1 2 m(r θ(t + t)) m(r θ(t)) 2 (2) t 1 2 m(r θ(t + t)) m(r θ(t)) 2 = F r θ sin ϕ 1 ( θ(t + t)) 2 ( θ(t)) 2 2 mr2 = F r θ t t sin ϕ t t 0 (7.1) 1 d 2 mr2 dt ( θ) 2 = F r dθ dt sin ϕ 1 2 mr2 2 θ θ = F r θ sin ϕ mr 2 θ = F r sin ϕ (7.1) mr 2 (moment of inertia) F r sin ϕ (torque) (moment of force) 7.2 r F sin ϕ (7.1) I = mr 2, τ = F r sin ϕ I θ = τ (7.2)

113 r m θ, g ( 1 (1) (2) (3) θ m (7.1) t t t+ t mr 2 θ(t + t) mr 2 θ(t) = F r sin ϕdt (7.3) mr 2 θ (angular momentum) τ = F r sin ϕ t mr 2 θ(t + t) mr 2 θ(t) = F r sin ϕ t (7.4) mr 2 θ(t) l(t) l(t + t) l(t) = τ t (7.5) 0 (7.5) t t 0 l(t + t) l(t) lim = τ t 0 t l(t) = τ (7.6) 7.2 ( 1 (1) ml 2 (2) mgl sin θ (3) ml 2 θ = mgl sin θ

114 * (7.1) r F τ = r F τ F τ F τ ( ) [ 7.1] 7.1 p = mṙ ṗ = F (7.7) r r ṗ = r F (7.8) (7.1) [ ] l = r ṗ l = τ (7.9) l = r p [ 7.2] l = r p mr 2 θ m i N ( 7.3) P i (i = 1,..., N) m i, O r i O θ O OP i ϕ i r i O 7.3 (7.2) τ i, (i = 1,..., N τ ) N m i ri 2 d 2 dt 2 (θ + ϕ N τ i) = τ i (7.10) i i ϕ i θ

115 ϕ i N m i ri 2 θ N τ = τ i (7.11) i i I = N m i ri 2 (7.12) i I I θ = τ (7.13) τ [ 7.3] M I, I COM r COM I = I COM + MrCOM 2 (7.14) 7.3 l m M ( 2 (1) (2) [ 7.4] T T = 1 2 I θ 2 (7.15) ( 2 (1)I = m( l 2 )2 + M( l 2 )2 (2){m( l 2 )2 + M( l 2 )2 } θ = Mg sin θ l l + mg sin θ 2 2

116 108 7 l, m ( 7.4(a)) ρ ρ = m (7.16) l x x + dx ρdx I ( 7.5) O (a) (b) 7.4 x x + dx I = x 2 ρdx (7.17) x [0, l] I t 7.5 ρdx I t = l (7.16) ρl = m 0 x 2 ρdx = 1 3 l3 ρ = 1 3 ml2 (7.18) I c ( 7.4(b)) I c = l 2 l 2 x 2 ρdx = 1 12 ml2 (I t > I c ) I = ρ(x 2 + y 2 )ds (7.19) S ds S ρ (1) (2)

117 (Photo by E.Muybridge, 1887) ( 7.7) F (x,y) = + (x,y) y x θ F φ θ F φ 7.7 [ 7.2] 7.7 l F F ϕ I [ ] { mẍ = F mÿ = 0 (7.20)

118 110 7 I θ = l F sin ϕ (7.21) ( ) = mrω 2 = m v2 r ( ) (centrifugal force) m r ω v [ 7.5] 280km (1)? (2) (3) [ ] [ ] ( ) ( ) [ ]

119 ? (Coriolis effect, ) [ ] [ 7.6] (1) m ( ) r, r r t ω ω a (2) ( ) ( ) t 0 2mω dr dt [ ] p.62 [ 7.7] [ 7.8] ω ω t (1)? (2) 2mωv

120 第 7 章 剛体の運動 112 [解説] コリオリの力はこれまでの設 問から類推できるように 回転座標系 上で運動する物体に対して働くみか けの力であって その向きは運動の方 向と直交しており 大きさは 2mωv となる 地球上ではこのコリオリの 力が気象や大洋の流れに影響を及ぼ している その代表例の一つが台風 など強い低気圧による渦であり そ の渦の向きは北半球と南半球では逆 になる (図 7.8) 図 7.8 アイスランドに発生した低気圧 (Photo by NASA org/wiki/coriolis_force)

121 113 [1] S.. ( )., [2] D. L., J. R.. ()., [3] A., I.. ( )., [4]. 3., [5] R. P., R. B., M. L.. ( )., [6].., 2001.

122 114

123 115 Symbols [J] [kgw] [mol] [N] [rad] A acceleration aether Amontons, Guillaume Aristotle Avogadro, Lorenzo Romano Amedeo Carlo B Bernouli, Daniel Boyle, Roboert Brahe, Tycho Brown, Robert Brownian motion C Cavendish, Henry , 85 center of gravity of mass centrifugal force centripetal force collision elastic inelastic conservative force Copernicus, Nicolaus Coriolis effect Coriolis,Gustave Gaspard D da Vinci, Leonardo de Coulomb, Charles Augustin Descartes, René differentiation displacement distance dynamics forward inverse E Einstein, Albert , 86 elastic collision elastic potential energy equation of motion equivalence principle Eratosthenes ether F Fahrenheit, Daniel feet Fooke, Robert force forward dynamics Free-Body Diagrams frictional force G galilean invariance transformation Galilei, Galileo , 15 Galilei, Vincenzo gravitational acceleration gravitational constant of universe Green, George , 85 ground reaction force H Helmholtz, Hermann von Hipparchus Huygens, Christian Huygens, Constantijn I impulse inch inelastic collision integral definite indefinite integration inverse dynamics J Joule Joule, Japmes Prescott , 85 K Kepler, Johannes kinetic energy kinetic friction L Lagrange, Joseph-Louis Law of acceleration of inertia of reciprocal actions Leibniz, Gottfried Wilhelm M mass , 28 gravitational inertia material point Maxwell,James Clerk Mayer, Julius Robert von mechanical energy MKS moment of force of inertia momentum , 60 angular , 105 linear N Newton s first law of motion s second law of motion s Third law of motion Newton, Issac i Newtonian mechanics i nonconservative force normal force P polar coordinate system position potential energy Prinkipia i, 26 R radian radius rigid body S scalar SI speed static friction Stevin, Simon Svedberg, Theodor T tensile strength tension Thomson, William torque U universal gravitation V van der Waals, Johannes Diderik Varignon, Pierre

124 116 velocity viscosity W weight work , , , , , 105, , , , , , , , , , , , , , 17, , , , , , , i, 3, i , , 26, 27, , , i, , , 27, i , ,