I 1

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ほだかふじがわ
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2 I 1

3 m = µm. cm = 1 4 km 3. 1 m = cm 4. 5 cm 3 = km = = = = π m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec kg g = kg 4. 7 m/min + 4 km/hr =.8 m/sec m 3. m/s 3. m/s 4. kg/m 3 5. m 3 /kg 6. kg m/s 7. kg m /s 8. kg m /s 9. kg m /s 3

4 cm =.5 1 m. 8 m/6 s = 1.3 m/s kg/ 4 3 π(1 cm)3 = 33 kg/ 4 3 π(.1 m)3 = kg/m d [m]. e At/m K [m] 3. t [s] 4. m [kg] 5. At/m A [kg/s] 6. r [m] 7. A = Cr A [kg/s], r [m] C [kg/m s] [kg], [m/s ] [N]=[kg m/s ]. G [kg /m ] [kg m/s ] G [m 3 /kg s ] 3. [m ] [Pa]=[kg/m s ] 4. [kg m/s ] [m]=[kg m /s ] [kg] [m/s] = [kg m /s ] 5. [kg/m s ] [m 3 ] = [kg m /s ] 6. nrt n [mol] T [K] R [J/mol K] = [kg m /s mol K] 7. 1 L 1 m 3 1/1 = [m 3 ] [L] V 1 1 3

5 1.7 1 kg=1 3 g, 1 m=1 cm 1 kg m /s = 1 3 (1 ) g cm /s = 1 7 g cm /s [L], [M], [T ] 1. h [L] g [LT ] h a g b [L a+b T b ] = [LT 1 ] a = 1/, b = 1/ gh h a g b [L a+b T b ] = [T ] a = 1/, b = 1/ h/g. [MLT ] k [MT ] P [T ] m [M] P = k a m b a = 1/, b = 1/ P = m/k 4

6 .1 1,, 5 3, 4, 6, (1,). (,1) 3. (cos(3 ), sin(3 )) = ( 3/, 1/) 4. 4 ( 3, ) 5. (6 cos(6 ), 6 sin(6 )) = (3, 3 3) 6. (3, 3 3) = 3(1, ) + 3 3(, 1) 3e x + 3 3e y.3 1. a = = 35. b = ( 1) = a b = ( 1) = 4 4. a b = (3 ( 1) 5, ( 1), 1 3 3) = ( 13, 16, 7).4 d = e x + e y d = 4 y 3 A d B 1 e y e x x 5

7 .5 1. A (, ) A B x y 45 B y = x 4 m B (, ) C B B x y 7 C (, 7). P A x B y P e y 3. C P 7 m, m (, 7) 4. C P (, 7) (, ) (, 7).6 x, y, v, A, ω (1) x =, y =, v = (3) A=1, ω=1 1-1 () A=1, ω=1-1 1 (4) A=1, ω= (x(t), y(t)) t 1. x = x +v t t = (x x )/v y(t) = y +t y = y +(x x ) /v. cos (ωt) + sin (ωt) = 1 x + y = A 3. x = A cos(ωt) = A(1 sin (ωt)) = A(1 (y/a) ) x = A y /A 4. x = A cos(ωt) = A(1 sin (ωt)) = A( cos (ωt) 1) sin (ωt) = (1 (x/a))/ cos (ωt) = (1 + (x/a))/ y = A sin(3ωt) = A sin(ωt)(3 cos (ωt) sin (ωt)) y = ±A(1 + x/a) (1 (x/a))/ 6

8 .7 1. x P = x P x O y P = y P y O. x x y y 1 e x = e x e y = e y 3. x P = x P cos θ y P sin θ y P = x P sin θ + y P cos θ 4. P O xy x P e x +y P e y O x y x P e x +y P e y x P e x + y P e y = x P e x + y P e y 5. e x = cos θe x + sin θe y e y = sin θe x + cos θe y 3 O x y (x P, ) x y θ θ θ θ θ θ.8 7

9 .9 [cm] 1. P (1,). Q (, ) 3. R (,) Q R 8

10 t [s] x [m] (x(1) x())/(1 ) = [m/s] 3. (x(.8) x())/(.8 ) =.4 [m/s] 4. (x(.6) x())/(.6 ) =.8 [m/s] 5. (x(.4) x())/(.4 ) = 1. [m/s] 6. (x(.) x())/(. ) = 1.6 [m/s] 7. t = t x( t) = t t 8. lim t ( t ) = x( t) x() t = t t t 9. t = m/s = t t = s v() = m/s t = 1 s x = m 3. t = v() = m/s t =.5 s t =.5 s x = 1 m t =.5 s v(.5) = m/s t =.5 s t = 1 s t =.5 s 1 m 4. t = s t =.5 s v() = m/s t =.5 s x =.5 m t =.5 s t =.5 s v(.5) = 1 m/s t =.5 s.5 + ( 1).5 =.75 m t =.5 s t =.75 s v(.5) = m/s t =.5 s =.75 m t =.75 s t = 1 s v(.75) = 1 m/s t = 1 s =.5 m x =.5 m 5. x =.5 6. t = x() = t x(t) 9

11 x(t) = x() + t=1 v(t) = + 1 t= (4t ) = m a v(t) = v, a(t) = b v(t) = v + gt, a(t) = g c v(t) = x γe γt + u, a(t) = x γ e γt d v(t) = γx e γt, a(t) = γ x e γt e v(t) = ωx sin(ωt), a(t) = ω x cos(ωt). x(t) a(t) t (d) e γt = (e) cos(ωt) = (d) a(t) = γ x(t) (e) a(t) = ω x(t) 3.4 x(t) y(t) t x y t x y 1. v = (1, 3), a = (, ). v = (1, t), a = (, 1) 3. v = ( 6 sin(3t), 6 cos(3t)), a = ( 18 cos(3t), 18 sin(3t)) 4. v = ( 6 sin(3t), 3 cos(3t)), a = ( 18 cos(3t), 9 sin(3t)) 5. v = ( 6 sin(3t), 6 cos(3t + 1)), a = ( 18 cos(3t), 18 sin(3t + 1)) u 1, k, t u 1 [LT 1 ] k [LT ] t [T ]. v = v = u 1 + k (t t ) t t = t 3. x y x x u 1 x /u 1 1

12 3 (1) 3 () 1 1 y y x x 3 (3) 3 (4) 1 1 y y x x 3 (5) 1 y x r e r 3. e r e x e y re r = r cos(ωt)e x + r sin(ωt)e y r e r = cos(ωt)e x + sin(ωt)e y 4. e ϕ = Ae x + Be y e r e ϕ = A cos(ωt) + B sin(ωt) = 11

13 e ϕ = 1 A + B = 1 e ϕ = sin(ωt)e x + cos(ωt)e y 5. e x e y de r = ω sin(ωt)e x + ω cos(ωt)e y = ωe ϕ 6. de ϕ = ω cos(ωt)e x ω sin(ωt)e y = ωe r x y L/v. (V t, ) 3. A: x y (vt, ) ((v + V )t, ) B: x y (, vt) (V t, vt) 4. T L/v A L + V T, L V T L B xy (V L/v, L) L (1 + V ) 1/ 5. ϵ = (V/v) L V m/s 7. δ v v δ = 15 m (3 14 m/s) (3 1 8 m/s) = m = 15 nm 1/4 1

14 /3. a =, b = 1, f(x) = x 3. N 1/3 =.333 a 1/8 =.15 b (3/15 =) 6/5 =.4 c (85/1 =) 57/ =.85 b a f(x)dx = lim f(x) x f(x) = x, a =, b = 1 [, 1] N N =, 5, 1 (b a)/n x C (1) 1 x + C () x 1 + C (3) log(x) + C (4) log(x + 3) + C (5) cos(x) + C (6) sin(x) + C (7) log cos(x) + C x (1) f(x) f(x) x f(x) x 1 C (1) 1 x + 3 () x (3) log(x) + (4) log(x + 3) log 4 + (5) cos(x) + cos(1) + (6) sin(x) sin(1) (7) log cos(x) + log cos(1) + 1. t x dx t t 3t dx = t 3t. t = 1 x = t = 1 x(1) = 13

15 3. 1 x(t) C x(t) = 3 t3 3 t + C t = 1 x(t = 1) = C = C = 5/6 t x(t) = 3 t3 3 t + 5 t = 4 [s] x(t = 4) = 39/ [m] x y(t) x(t) = x() + t v x (t) x(t) = x(t)e x + y(t)e y dx = v(t) (1) x y (1) x dx = v () x(t) v x(t) C x(t) = v t + C (3) x t = t x(t ) = x t = t x = v t + C (4) C x(t) C = x v t (5) x(t) = x + v (t t ) (6) 1. (x(t), y(t)) = (x + v (t t ), 1 ) g(t t ) + y (. (x(t), y(t)) = x, v ) γ (e γt e γt ) + y ( v 3. (x(t), y(t)) = ω (sin(ωt) sin(ωt )) + x, v ) ω (cos(ωt) cos(ωt )) + y 14

16 t = 1 x = [m] 3e x 3 [m/s] t = 1 x 1 [m/s ]. 3. r = x(t)e x d r = (4t 3)e x 4. x d x = 4t 3 t = 1 x(t = 1) =, dx (t = 1) = 3 d x = 4t 3 dx = t 3t + C C dx dx (t = 1) = 3 dx(t = 1) = 1 + C = 3 C = = t 3t t x x(t) = 3 t3 3 t t + D D x(t = 1) = x(t = 1) = 17 + D = D = 17/6 t 6 x(t) = 3 t3 3 t t dx/ x(t) t = 3 t = 3 7 m/s x = 4 3 [m] x(t) = 1 a t + v t + x. x(t) = 1 6 kt3 + v t + x 3. x(t) = β ( ω sin(ωt) + v + β ) t + x ω 4.8 a(t) = 1 4 t 1 t [s] a [m/s ] d x = 1 4 t 1 15

17 dx (t = ) = 4, x(t = ) = 5 x(t) = 1 48 t4 1 t 4t + 5 t =.5 x(.5) = 7.3 [m] x(t)e x + y(t)e y. d [x(t)e x + y(t)e y ] =.96e x +.7e y x(t) d x =.96, y(t) d y =.7 t = x(t = ) =, y(t = ) =, dx dy (t = ) =, (t = ) = 3. 4 x(t) =.48t, y(t) =.36t.48t e x +.36t e y 4. t = 1 = 6 x(6) m, y(6) m 5. x(t) y(t) t y = 3 4 x 3/ (x(t), y(t)) = (x + v x (t t ), 1 ) g(t t ) + v y (t t ) + y (. (x(t), y(t)) = x + v x (t t ), v ( ) ) v γ e γt + γ v (t t ) e γt + v y (t t ) + y 3. x(t) = v ω (sin(ωt) sin(ωt )) v (t t ) cos(ωt ) + v x (t t ) + x y(t) = v ω (cos(ωt) cos(ωt )) v (t t ) sin(ωt ) + v y (t t ) + y 16

18 f x = x f(x) f() = p p x f(x). ( ) f(x) log = 1 p ax f(x) = f(x) = pe ax / 5. f 5.3 (1) f(x) = f 1 af x () f(x) = [ (x f ) 1] 1/ (3) f(x) = log(x + e f ) 1. t a =.5v dv =.5v t = v(t = ) = dv. v t dv =.5 v v(t) dv v =.5 [ ] v(t) log =.5t log v(t) = e.5t 3. dx = e.5t x(t) x(t = ) = 4. t 3 x(t) C x(t) = 4e.5t + C x(t = ) = C = 4 x(t) = 4 4e.5t x t x 4 x = 4 t 17

19 5.4 v = u a /k 1 1 u du = 1 ( 1 1 u + 1 ) du = 1 [ ] 1 + u 1 + u log 1 u [ 1 t = log 1 + v(t) k/a ka 1 v(t) log 1 + v ] k/a k/a 1 v k/a v(t) 3 u = kv 1 cos u tan u du = du = log(sin(u)) sin u t = 1 ka log sin(kv) sin(kv ) v(t) kv(t) 1. v(t) = 1 [ a (a kv )e kt] k a γe t ka 1. v(t) = γ = 1 + v k/a k γe t ka v k/a 3. v(t) = 1 [ k sin 1 e ka(t t) sin(kv ) ] 3 kv(t) = π/ dv dx = a = v. v v dv = 1 d v v dv = av d v = v dv 1 d v = a(t)v(t) 18

20 t t t t d v = [ v(t) ] t t = v v 1 ( v(t) v ) = a(t)v(t) 3. t x(t) = dx dx dx = 1 dx = 1 v t x x a(t)v(t) = av 1 t x v dx = adx x a 4. a x x adx = a dx = a(x x ) x x 19

21 x ( [m/s ]) x ( [m]) 9 [1/s ]. d x = 9x 3. x(t) = Ce pt Cp e pt = 9Ce pt p = 9 p = ±3i 4. x(t) = C 1 e 3it x(t) = C e 3it C 1 C e ±3it = cos(3t) ± i sin(3t) x(t) = C 1 (cos(3t) + i sin(3t)) + C (cos(3t) i sin(3t)) = (C 1 + C ) cos(3it) + i(c 1 C ) sin(3it) A = C 1 + C, B = i(c 1 C ) x(t) = A cos(3t) + B sin(3t) 5. x() = A =, dx (t = ) = 3B = 3 A =, B = 1 6. t x(t) = cos(3t) + sin(3t) = 5 cos(3t α) α cos(α) = / 5, sin(α) = 1/ 5 x(t) = cos(3t α) = / 5 = cos(α) n t = πn 3 t = 3 α + πn 3 6. x 16x d x = 16x x(t) = Ce pt p p = ±4 C 1, C x(t) = C 1 e 4t + C e 4t x(t = ) = dx (t = ) = C 1 = C = 1 x(t) = e 4t + e 4t x(t) d x y(t) d y = p x = q y. x(t) x() = x dx (t = ) = y(t) y() = dy (t = ) = v 3. A, B, C, D x(t) = A cos(pt)+b sin(qt) y(t) = C cos(qt)+ D sin(qt) x(t) A = x, Bp = y(t) C =, Dq = v A = x, B =, C =, D = v /q

22 x(t) = x cos(pt) y(t) = (v /q) sin(qt) 4. (x(t), y(t)) = (3 cos(t), 3 sin(3t)).6 4 (ω = 1 ) 6.4 t 1. ωt t ω SI [s 1 ]. dx/ = v x dy/ = v y v x v y x(t) y(t) 3. dx/ = ωy dy/ = ωx d x d x = ω dy = ω x x(t) 4. x() = x dx/ = ωy t = dx/ dx/(t = ) = ωy 5. x(t) x(t) = A cos(ωt) + B sin(ωt) A B t = x(t = ) = A = x dx (t = ) = B = y A = x B = y x(t) = x cos(ωt) + y sin(ωt) y(t) = (1/ω)dx/ y(t) = x sin(ωt) + y cos(ωt) 6. x(t) = y(t) = x + y cos( ωt + α) x + y sin( ωt + α) α cos α = x / x + y sin α = y / x + y x + y 7. g > g = x g < x y 1

23 dv ω x. t f(t) f(t) df = 1 d f(t) v dv = 1 d v x dx = 1 d x 3. ( ) = t ( ) = ω t d v = 1 ( v(t) v() ) = 1 ( v(t) v ) d x = ω ( x(t) x() ) = ω x(t) v v = ω x v v = v ω x 4. t x = (v /ω) sin(u) x u sin 1 (ωx/v ) 1 v v ω (v /ω) sin (u) ω cos(u)du = 1 ω sin 1 (ωx/v ) t = 1 ( ) ωx ω sin 1 t x(t) = v v ω sin(ωt) du = 1 ( ) ωx ω sin 1 v

24 [1] 1, [] 5, [3] 1, [4] 3, [5], [6] 5, [7] 4, [8] 4, [9], [1], [11] 3, [1], [13], [14] 5, [15] 6, [16] 3, [17] 3, [18] 5, [19] 1, [], [1] [1] 3, [] 6, [3], [4] 1, [5] 5, [6], [7], [8] 8, [9] 6, [1] 1, [11], [1] 1, [13] 6, [14] 1, [15] 1, [16], [17] 4, [18] 7.4 [1] 5, [] 3, [3] 5, [4] [1] 4, [] 4, [3], [4] 3, [5] 1, [6] 5, [7] 7, [8] 3, [9] 3, [1] 6, [11] 9, [1] 3, [13] 3, [14] 6, [15] 3, [16] 3

25 II 4

26 8 1.. a A B B B b c g m mg 3. a b 8.1 a 1. x d x = a x a. x x t = x, v t = x(t = ) = x, dx/(t = ) = v 8. 5

27 v = dx e x a = d x e x. ( ) ( )=( ) x (a) m d x = (b) x md = mg (c) x md = kx m x(t) (a) (b) (c) t = x(t = ) = x, dx/(t = ) = (a) x(t) = Ax + B (b) x(t) = 1 gt + At + B (c) x(t) = A cos A B (a) x(t) = x (b) x(t) = 1 ( k/mt ) gt + x (c) x(t) = x cos ( k/mt ) ( k/mt ) + B sin 8.4 T A, T B T A sin α = T B sin β T A cos α + T B cos β = mg T A, T B T A = mg sin β sin(α + β), T mg sin α B = sin(α + β) 8.5 N f N = mg cos θ, f = mg sin θ 6

28 f = µn mg sin θ = µmg cos θ µ = mg sin θ mg cos θ = tan θ m/s m/s N N N N N 8.7 k 1 k 1. k 1 x k x (k 1 + k ) x x k 1 + k. k 1 x 1 k x x 1 + x F F = k x F F = k x k 1 x 1 = k x F = k x x 1 + x K K( x 1 + x ) = k x x 1 = (k /k 1 ) x K K = k 1 k /(k 1 + k ) 3. N K = k 1 + k + + k N 1 K = k 1 k k N µ mg. T cos α + T = µ mg T T = µ mg cos α + 1 7

29 3. T T cos α = µ mg cos α/( cos α+ 1) µ mg µ mg µ mg cos α cos α + 1 = 1 µ mg cos α + 1 µ g β β = v cos α + 1 τ τ = v β = v µ ( cos α + 1) g v vτ 1 βτ = µ ( cos α + 1) g 4. α α α T cos α = µ mg T cos α = cos α + 1 cos α 1 cos α 1 α 6 T 8.9 ( ) dy 1. l = x 1 + λ l dx λg l y (, λg l). x T x +T x + x = y T y +T y + T y λg l = 3. T x = x = T T x = T ( ) dy 4. y T y = λg x 1 + T y dx x = λg 1 + T x = T u 5. = 1 + t sinh 1 u sinh 1 sinh 6. x = T y = dt ( ) y dx = λg Ty 1 + Ty(x) T dt x y 1 + (Ty /T ) = λgdx ( Ty T ) ( ) λg u = T y /T 5 T y (x) = T sinh x T 7. x = y = y dy ( ) λg dx = sinh x T y(x) = T [ ( ) ] λg cosh x 1 + y λg T y = a cosh(x/a) 8

30 1. sinh x = ex e x. cosh x = ex + e x 3. tanh x = sinh x cosh x = ex e x e x + e x 4. cosh x sinh x = 1 d 5. cosh x = sinh x dx d 6. sinh x = cosh x dx 9

31 ( ). 1 N 3. m/s ( ) 4. 1 [N]/4 m/s = 3 kg [kg/m 3 ] 9. F/m F m, g F mg N z. z d z d z = 196 kg, N z 3. d z = 9.8 t = t = z = 3 m z() = 3 t = dz (t = ) = 4. z(t) = 4.9t z = = 4.9t + 3 t t =.78 s 6. 3 kg t =.78 s kg ( ) 3

32 9.4 x x z 196 N(= kg 9.8 m/s ) z t x x(t), z z(t) x d x, z d z d x = (x ), d z = 196 (z ) x(t) x(t = ) =, dx (t = ) = 15 z(t) z(t = ) = 3, dz (t = ) = x(t) = 15t, z(t) = 4.9t + 3 z = = 4.9t + 3 t =.78 s x x(t = 15) = = 1 m 9.5 x t = s t x(t) 1 d x = 4t x(t) t = s x(t = ) =, dx (t = ) = 3 x(t) = 1 15 t t 74 t = 1 s x t = s 1 s 15 (t = s ) x(1) = 156/15 = 83.7 m 9.6 t x F (t) F (t) = 5 4 t 5 t x(t) x 5 d x = 5 4 t 5 x(t = ) = 5 dx (t = ) = 4 x(t) x(t) = 1 48 t4 1 t 4t + 5 x(.5) = 5615/768 = 7.3 m 9.7 h/g gh 31

33 9.8 x = v = gt + v, x = 1 gt + v t + H x = t = v + v + gh g v = gt + v = v + gh x = v t h/g v h/g 9.1 (u x = (u + V cos θ)t, y = 1 gt + V sin θt V sin θ L = (u + V cos θ) g dl dθ = V g (( V sin θ) sin θ + (u + V cos θ) cos θ) = L sin θ + cos θ = 1 V (cos θ) + u cos θ V = cos θ cos θ = u + u + 8V 4V z mg mge z x z x m d x = z z md = mg 3. t = (x(), z()) = (, ( ) ) x v cos θ, z v sin θ t = dx dz (t = ), (t = ) = (v cos θ, v sin θ) x(t) = v t cos θ, z(t) = 1 gt + v t sin θ 3

34 4. t = x/v cos θ z z = gx v cos θ + x tan θ γ = tan θ z = g(1 + γ ) x + γx 1 + tan θ = 1/ cos θ 5. d x z = x d d = v γ g(1 + γ H z x ) v z H = γ g(1 + γ ) 6. 4 (x s, z s ) x = x s z = z s v v = gx s 1 + γ γ z s /x s θ v 7. 5 d H 6 v d = x γ s, H = γ z s /x s x s γ 4 γ z s /x s 8. x x p x p > x s x p = d/ γ < z s /x s γ > z s /x s γ > z s /x s H z c H < z c H x s γ < z c γ > z s /x s 4 γ z s /x s γ z s /x s > γ > z s /x s γ < z [ c z ] s Γ = z [ c z ] s x s z c x s z c 9. d γ γ = Γ d d min d min = Γ x s Γ z s /x s 1. H γ H γ γ min = z s /x s H min = z s d Hmin = x s (x s, z s ) γ γ x s 11. Γ = d min = 144 m d Hmin = 194 m v 33

35 13. γ = v = 41 m/s = 147 km/h x m d x = z z md = mg. t x(t) = v t z(t) = z 1 gt z(t) = t = z /g x x f = v z /g 3. H = R + x f R 4. H = z x f /R = z x f z (1 v /gr ) = v = gr

36 1 1.1 A B A B = AB cos(θ) A, B A, B θ A cos(θ) A B A B B dx F dx N.5 m = 1 J. F F = 4e x + e y x x = 1e x + 4e y F x = 4 J 3. F = xye x + 1 y e y ds = dxe x + dye y dw dw = F ds = xydx + 1 y dy (, 1) (3, 6) y = 5x 9 (3,6) (,1) [xydx + 1y ] dy = xy(x)dx + 1 y dy = x(5x 9)dx + 1 y dy = log = 3 J 3 J. 5 J 4 kg 5 m/s W = 5 y = W = 5 (x y)dx = 5 F x dx + (x )dx = [ x x ] 5 = F y dy

37 . x = 5 W = 5 5 F x dx + 6 F y dy 6 ( x + y)dy = 6 3. y = x x = y W = 6 F x dx + 6 F y dy 6 (x y)dx + 6 ( 5 + y)dy = [ 5y + y ] 6 = 1 ( x + y)dy = 6 xdx + 6 [ x ydy = F [J] F = x [m] N/m ] 6 = F = mg h W P = W t t = W P = mgh P = = 8.9s W dx = v. (ev B) dx = ev (v B) = eb (v v) = A (B C) = B (C A) = C (A B) mge y x m d x =, y y md = mg. x v x (t) = v x, y v y (t) = gt + v y t x(t) = v x t + x, y(t) = 1 gt + v y t + y x(t), y(t) t y = (g/v x)(x x ) + (v y /v x )(x x + y ) (x max, y max ) = (x + v x v y /g, y + v y/g) x t max = (x max x )/v x = v y /g 3. ds = dxe x + dye y = v x e x + v y e y = v x e x + (v y gt)e y 4. F = mge y F ds = mg(v y gt) 5. W = tmax F v = vy/g mg(v y gt) = 1 mv y 36

38 6. x y F = mge y y y max y = v y/g W = mg (y max y ) = 1 mv y ( 3. ) 1.8 dx 1. = rω sin(ωt)e x + r cos(ωt)e y. r rω 3. rω cos(ωt)e x rω sin(ωt)e y mrω cos(ωt)e x mrω sin(ωt)e y 4. t t + ds = v x e x + v y e y = rω sin(ωt)e x + rω cos(ωt)e y ( mrω cos(ωt)e x mrω sin(ωt)e y ) ( rω sin(ωt)ex + rω cos(ωt)e y ) =

39 (1) y () y cos(x) (3) y cos(xy) (4) xy cos(x y) (5) tan(x) (6) 1/y (7) 6x y exp[x y 3 ] (8) 11. U(x) du dx = F x F x U(x) 1. U(x) = mgx. U(x) = 1 kx 3. U(x) = A x x > +x x < x. du = 3x U(x) dx 3. U(x) = 3 x + U U U() = U = U(x) U(x) = 3 x 4. U(x = 4) = 4 J 5. x = J 5 kg 1 m/s 5/ J 5/ J x = 4 m 4 J K 5/ = K 4 K = 53/ J v mv / = 53/ m = 5 kg v = 53/5 m/s 6. x U(x) K x = x K K + U(x ) = K + U(x) K > K + U(x ) U(x) > U(x) x = U() = x < K + U(x ) > x = 4 m U(x ) = 4 J K > 4 J 5 kg 48/5 m/s 38

40 U = F dx = 1 kx 1 3 hx3 + C C =. U (x) = kx hx = x =, k h x = a = k/h U(a) = U ( ) k h = 1 k 3 6 h 3. x a U + k 3 U 1 6 h 4. 1 mv U(a) = 1 6 h k 3 /3mh k d x e x mge x x m d x = mg. x(t) = 1 gt + x v(t) = gt 3. du dx = mg 4. U(x) = mgx 5. x(t) v(t) E = 1 mv(t) + mgx(t) = 1 m( gt) + mg ( 1 ) gt + x = mgx 6. mg x mgx x(t) = t = x /g gt = gx (1/)m( gx ) = mgx Ave x A v. dv/ m dv = mg Av 39

41 3. dv/ = g (A/m)v v(t) 1 g + (A/m)v dv = t = (m/a) log [(g + (A/m)v(t))/g] v(t) v(t) = mg (1 e (A/m)t) A t v( ) = mg/a 4. v(t) v m dv = d ( ) 1 mv v = dx/ mgv(t) Av = d (mgx) Av t de = d ( ) 1 mv + mgx = Av < E E m/s. b = 6πηr m = ρ 4π 3 r3 v = ρgr 9η 1.m/s 11.8 m dv = bv log v = b m t + C log v = b m t+log v v = v e bt/m x x = m b v (e bt/m 1) t (e = x = m b v m d x = Av x, m d y = Av y F. v x v y x y v x (t) = v x e (A/m)t ( v y (t) = v y + F ) e (A/m)t F A A x(t) = mv x A 4 (1 e (A/m)t)

42 y(t) = m A ( v y + F ) ( 1 e (A/m)t) F A A t 3. A = 4. x(t) = v x t, y(t) = v y t + 1 F m t y A= A becomes large x r = x + y + z. x x/r GMmx r 3 e x GMmy r 3 e y GMmz r 3 e z 3. U(x, y, z) = GMm GMm = U(x, y, z) x, y, z r x + y + z 4. d (R cos(ωt), R sin(ωt)) = (Rω cos(ωt), Rω sin(ωt)) xy (R cos(ωt), R sin(ωt)) ( ) GMmR cos(ωt) GMmR sin(ωt) GM R 3, R 3 ω = R 3 5. ( Rω sin(ωt), Rω cos(ωt)) 1 mv = 1 mr ω = 1 GMm GMm R R 1 mv GMm R = GMm R 6. E = GMm/R E 1 = GMm/R 1 R > R

43 m d z = GM m (R + z). GM /R = 9.8 m/s 3. (64 km) (1 m) 4. t dz d z t = GM dz (R + z) dz d z = 1 ( ) d dz = 1 d v t = z v t z v(t) t t z t dz d z = 1 t d v = 1 v(t) 1 v GM dz z(t) (R + z) = GM (R + z) dz = GM R + z(t) GM R v(t) z(t) 1 v = GM R + z GM + 1 R v 5. z v v GM v = m/s = 11 km/s R 4

44 m /.5 = 4 N/m. 4/1 = 6.3 rad/s π/6.3 = 1. s 1/1. = 1. Hz = 6.8 N 6.8/9.8 =.69 kg 1. k, l, m, g l l = mg/k g 1/6 1/ x x 1x [N].5 kg.5 d x = 1x. d x = 4x x(t) A B x(t) = A cos ( 4t ) + B sin ( 4t ) t = x(t = ) =.1, dx (t = ) =.4 A =.1, B =.63 x(t) =.1 cos ( 4t ) +.63 sin ( 4t ) x(t) =.1 sin ( 4t α ) α cos(α) =.53, sin(α) =.85 ([rad] ) 3. x(t) dx dx t = 3 x(3) =.91 m, (t = 3) =.47 m/s 4. K(t) = 1.5 v x(t) K(t) = 1.5 (.1 4 cos ( 4t α )) ( =.7 cos 4t α ) 5. U(t) = 1 kx k = 1 N/m x(t) U(t) =.7 sin ( 4t α ) 6. t = K(t = ) =.7 cos (α) =. J, U(t = ) =.7 sin (α) =.5 J.7 J 7. K(t = 3) =.8 J, U(t = 3) =.4 J.7 J 8. sin θ + cos θ = 1 K(t) + U(t) =.7 J 43

45 F = kx x = C 1 sin ωt + C cos ωt C 1, C v v = dx = C 1ω cos ωt C ω sin ωt t = x =, v = v = C 1 sin ω + C cos ω C = v = C 1 ω cos ω C ω sin ω C 1 = v /ω x = v sin ωt ω. l = C 1 sin ωt + C cos ωt = C 1 ω cos ωt C ω sin ωt C 1, C C 1 = l sin ωt, C = l cos ωt x = l cos (ω(t t )) 3. v x v = dx = Aω cos(ωt) x v K = 1 mv = 1 ma ω cos (ωt), U = 1 kx = 1 ka sin (ωt) ω = k m k = mω U (sin θ) + (cos θ) = 1 K + U = 1 ma ω l k l = mg l = mg/k. z z = t z(t) mge z fe z z(t) < fe z = kz(t)e z z(t) > f = z f m d z = mg + f (t = ) z(t = ) = mg k h dz (t = ) = 44

46 z < d z = mg k k m z z(t) = mg + z(t) z(t) k z(t) d z = k m z z ω = k/m z z(t = ) = h, d z = z(t) z(t) = h cos(ωt) z(t) = mg h cos(ωt) z(t) < k h < mg k h > mg z < z > k m d z = mg z = t = t = mg k h cos(ωt ) z v v = dz ( mg ) (t = t ) = h sin(ωt ) = ω h z > k t = t v z(t) = 1 ( ( g(t t ) + v (t t ) v g = ω mg ) ) h g k z = l l < v/g mg(kl + mg) h < k < h < mg/k mg/k < h < mg(kl + mg)/k h v = dx = d [R cos ϕe x + R sin ϕe y ] = R dϕ sin ϕe x + R dϕ cos ϕe y. e x e y e r e ϕ e x r φ 3. a e x e y ( ) dϕ a = R e r + R d ϕ e ϕ 4. mg F g F g = φ mge x = mg cos ϕe r mg sin ϕe ϕ 5. T T e r T 6. m d x = F g + T F g T e r e ϕ e ϕ mr d ϕ = mg sin ϕ 7. ϕ sin ϕ ϕ 45

47 d ϕ = g R ϕ 8. ϕ ϕ ϕ = A cos(ωt) + B sin(ωt) ω = g/r A B dϕ = Aω sin(ωt) + Bω cos(ωt) A = ϕ, B = ϕ(t) = ϕ cos(ωt) Av k β, f f sin(βt) z m d x = Av kx + f sin(βt) m τ = m/a, ω = k/m, a = f/m. x sp (t) x hom (t) d x sp + 1 τ d x hom dx sp + 1 τ + ω x sp = a sin(βt) dx hom + ω x hom = x sp (t) + x hom (t) 3. A ϕ β A sin(βt + ϕ ) + Aβ τ (β ω )A (sin(βt) cos ϕ + cos(βt) sin ϕ ) + Aβ τ cos(βt + ϕ ) + ω A sin(βt + ϕ ) = a sin βt (cos(βt) cos ϕ sin(βt) sin ϕ ) = a sin(βt) t sin(βt) cos(βt) A tan ϕ (β ω )A cos ϕ Aβ τ sin ϕ = a (β ω )A sin ϕ + Aβ τ cos ϕ = A = a (ω β ) + β /τ tan ϕ = 46 β/τ β ω

48 tan(ϕ ) nπ 4. x hom (t) 5. x hom (t) x sp (t) + x hom (t) x sp (t) x sp (t) ω τ ρ M = ρ4πr /3, M = ρ4πx /3 M = M x3 R 3. F = GmM /x = GmMx/R 3 3. M 1 F = GmMx/R 3 = mgx/r 4. k = mg/r ω = k/m = g/r T/ = π/ω = π R/g = s r/r 1 B f(r) f(r) < f(r) >. U(r) x U(r) x F x = du dr r x = d( W ) dr r x = x x + y + z = 1 y z 3. x x r = f(r)x r x x + y + z = x r U ] x = du x [6 dr r = 8ϵ σ1 x 3σ6 r13 r 7 r y z r = 1/6 σ r < r r > r 4. 47

49 [1] 1, [], [3], [4], [5], [6], [7], [8] 4, [9], [1], [11] 1, [1], [13] 4, [14], [15] 4, [16], [17] 13.3 [1], [], [3], [4], [5], [6] 1//3/4/5/7/8, [7] 1, [8], [9] 1, [1] 6, [11], [1] 6, [13] 8, [14], [15] 1, [16] 1, [17] 13.4 [1] 4, [] 8, [3], [4] 8, [5], [6] 7, [7], [8] 1, [9], [1], [11] 5, [1] 4, [13], [14], [15], [16] 4, [17] 4, [18] 8, [19], [] [1] 9, [] 9, [3] 7, [4] 3, [5] 9, [6] 6, [7] 6, [8], [9] 3, [1], [11], [1] 3, [13], [14] 8, [15] [1] 3, [], [3] 1, [4] 1, [5] 5, [6] 1, [7], [8], [9] 1, [1], [11] 3 48

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pdf http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg