m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d

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1 m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m v α = + e αk m e αk m t+c t+c t+c v = α + K exp ( αk m t + C K exp ( αk m t + C U( +T( =E(

2 . a. U: ( F (x kv F (x = mg F (x = kx F (x = k x n+ x F (x = G mm x ˆ U = F (x x. b. U x = F (x T = mv.. a. m v m x = F (x x = F (x x ˆ ˆ mvv = F (x x ˆ mv = F (x x = const

3 . b. mv + U (x = const mv v + U = 0 mv v + U x x = 0 m v = U x m x = kx v = Aω cos (ωt + θ 0 ω = k m x = A sin (ωt + θ 0 U = T = U + T = ka a. U (x = mgx 3. b. U (x = kx 3

4 3.3 c. ˆ U (x = F (x x E = U + T T = E U 3.4. F (x = G Mm x ˆ x ( U (x = G Mm x x = G Mm x E = U + T > 0 mv = G Mm x v = G M x GM x = g v = gx = = 0 3 = km/s 4 4. ( U + T = E (t ex. E : E (t E (t : 4

5 4. E = mv + U (x = const m v = E U v = m x (E U m = x + f E = T + U E 0 E = ( mv + U (x = mv v + U = mv v + U x x ( = v m v + U x = E = vf E (t E (t = ˆ t t vf 4.. E (x = mv + U (x x = f (t mv = E (x U (x 5

6 ˆ x x 0 m t = m x = E (x U (x E (x U (x x = m x = ˆ x x 0 ˆ t t 0 = t t 0 E (x U (x x + t 0 t = g (x x = g (t E ( k k sin m (t U ex f (x, y x f (x x f = lim 0 x = lim f 0 x = lim 0 f (x + x, y f (x, y x f (x, y y f = lim 0 y f (x, y = x y + xy + 4x + 5y + 6 f x = xy + y + 4 f y = x + x + 5 f x = f y = 0 ( 6

7 4.. x x,y y f f = lim (f (x + x, y + y f (x, y 0 = f f x + x y y = lim 0 ( f (x + x, y + y f (x, y + y x + x f (x, y + y f (x, y y y 4..3 U (x, y U (x x = lim ( t Ux = lim 0 t 0 U = lim 0 = x U x + U y t ( Ux x x t + U y y y t = U x x + U y y 4.3 (x, y, z = r ( x, y, z = r ( x, y, z = r (F x, F y, F z = F F x = m x, F y = m y, F z = m z F = r m 7

8 m v x v y v z = mg 0 0 m v = mgẑ v (t = gẑtl + v 0 r (t = gẑt + v 0 t + r 0 3 F (x = U x F x (x, y, z = U x, F y (x, y, z = U y, F z (x, y, z = U z ex. r = x y z ( F = G Mm r ˆr = G Mm r r x y z r = x + y + z F x = G Mm x r r = G Mm r x r U (x, y, z = G Mm r 8

9 F x = U x =... ( gru U ( U x, U y, U z F = grau ( x, y, z ( U x, U y, U z ẋ (t x S (x (t = ˆ t t L (x (t, ẋ (t 5.. L T U = mẋ U (ẋ 5..3 f (x( ex F (f ˆ 0 (f (t f = t F (f = ˆ 0 t = 3 9

10 f = t c.f. F (f = ˆ 0 f (t t t 4 = 5 t = f (t = t = ex f (t = 4 t = 0 x = 0 t = T x = 0 x (t = At (t T S = = ˆ T 0 ˆ T 0 L (x (t, x (t ( m (At AT mgat (t T S x (t ( A x (t = g t (t T A = g 5..4 f (x x f x = 0 ( lim x 0 f = 0 f x f S (x 0 = = ˆ t t ˆ t t {L (x 0 + x, ẋ + ẋ L (x 0, x 0 } L (x 0, ẋ 0 0

11 f (x, y f = f f x + x y y f (x 0, y 0 = f x x 0 x + f y y 0 y S (x 0 = ˆ t t { L x x 0 x + L } ẋ ẋ 0 ẋ 0 ẋ = ( x ˆ t t L ẋ ẋ 0 ( x = fg = fg f g S (x 0 = ˆ t t [ ] t ˆ L t ẋ ẋ 0 x t t ( L x x 0 ( L x ẋ ( L x (= 0 ẋ ( L L ẋ x = 0 5. L (x, ẋ = T U = mv U (x ( L L ẋ x = 0

12 U (mv + x = 0 m v = U x 5.3 (r, θ x = r cos θ, y = r sin θ (r, θ (v r, v θ v θ = θ r v r = r r θ v θ = lim t 0 t = r θ (F r, F θ F r = U r F θ = U θ F r = U (rθ = U r θ

13 5.4 T = m ( vr + vθ = ( m ṙ + r θ L = m (ṙ + r θ U (r, θ ( L L ( L θ L r = 0 L θ = 0 (mṙ mr θ + U r = 0 ( mr θ + U θ = 0 mr θ mṙ mr θ ( mṙ = mr θ U r ( mr θ = U θ 6 6. P w L w 3

14 L = T U = m (ṙ + r θ U l = mr θ = mr θ r = mv θ r = P θ r = P r sin ϕ ˆ l = r P ˆ L = r P ˆ a b = a b sin ϕ 6. : ( = p : = F L = (r p = r p + r p = r F L = r F N ( Ö( L = m (ṙ + r θ U ( L θ L θ = 0 4

15 ( L θ = U θ = x U θ x y U θ y = r sin θ F x + r cos θ F y = yf x + xf y = r F 6.3 N : m i : i r: i M i m i R: M mi r i i m i r i = i F i + j F ij M R = i F i L all L i = r i F i + j (r i F ij L all = i (r i F i + i (r i F ij j = i (r i F i + i>j (r i r j F ij F ij = F ji P all = i m i v i = MV 5

16 L all = R MV + i ( r i m i ṽ i r i = R + r i v i = V + ṽ i L all = i = i {r i m i v i } {(R + r i m i (V + ṽ i } mi r i = 0 i m i r i = m i ṽ i = : l T = mv = ml θ U = mgl ( cos θ ( L θ L θ = 0 ( ml θ +Mgl sin θ = 0 ( ml θ = mgl sin θ 6

17 m x = kx m θ = kθ T = mv = ml θ U = mgl ( cos θ ( L θ L θ = 0 ( ml θ = mgl sin θ m ( l θ mglθ m θ = mglθ : ρ (x, y, z T = ρ (x, y.z x y z v = ρ (x.y.z x y z r θ T = = I θ (x + y ρ (x, y, z xyz θ U = ρ (x, y, z x y zgr ( cos θ 7

18 U = x + y ρ (x, y.z xyz g ( cos θ = Mgl G ( cos θ { l G = M x + y ρ (x, y.z xyz M = ρ (x, y.z xyz L = T U = I θ Mgl G ( cos θ ( L θ = L θ ( I θ = Mgl G sin θ m l ml ml θ mgl ( cos θ ml θ mg M l G x I I θ Mgl G ( cos θ I θ Mg l = ( x + y ρ (x, y.z xyz mẋ 6.5 l M ρ = M l I = ˆ l 0 x M l x = 3 Ml ml ml G = 4 ml l M l 8

19 I = ˆ l 0 x M ˆ l l l x + x M l x = M (l 3 (l l 3 3l = M ( l 3 3l l + 3ll 3l ( l = M 3 ll + l a ρ = M πa I = = r ρr rθ ˆ πr 3 ρr = 6.7 a ˆ a 0 = Ma πr M πa r I = r ρr rθ = Ma 6.8. a b. H 3. 9

20 6.9 α x y (r, φ θ x ( ẋ = a θ r θ sin φ + ẋ y r θ cos φ T = { ( ρ r θ ( sin φ + ẋ + r θ } cos φ r φ r z T = = ˆ ( ρ r θ + ẋr θ sin φ + ẋ rφrz ρr θ φz + ˆ ẋ ρrrϕ = I θ + Mẋ ( = I ẋ a + Mẋ = (M + Ia ẋ U = Mgx sin α (M + Ia ẋ = Mg sin α 6.0 I = Ma Mẍ = Mg sin α 0

21 I = Ma 3 Mẍ = Mg sin α 6. O I = I G + M I G 6.. G O (X, Y I = ˆ { (x X + (y Y } ρxyz 6. ( a θ = ẋ a F α x a θ = ẋ ( M + I a ẍ = Mg sin α M x = Mg sin α F ( ( = ( I θ = F a( Mẍ = Mg sin α µ Mg cos α ẍ = g (sin α µ g cos α

22 I θ = µ Mg cos α a a θ = µ Mg a I cos α a θ = ẋ g (sin α µ g cos α = µ Mg cos α a I ( g cos tan µ Ma + I = 0 I a θ < ẋ ( g cos tan α µ Ma + I > 0 I a θ = ẋ a θ = ẋ { Mẍ = F I θ = F a { Mẍ = 0 I θ = 0 ( ( 6.3 L M x = F ( = N (= r F ( L:

23 N : θ f N N ( (x y { Mg N = 0 N f = 0 y x ( =0 lmg sin θ N cos θ = 0 f = Mg tan θ 6.4 c.f

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9

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