m i N 1 F i = j i F ij + F x

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2 N m i i = 1,..., N m i Fi x N 1 F ij, j = 1, 2,... i 1, i + 1,..., N m i F i = j i F ij + F x i mi Fi j Fj i mj O

3 P i = F i = j i F ij + F x i, i = 1,..., N P = i F i = N F ij + i j i N i F x i, i = 1,..., N j F ij j i i F ij F ji Fij + F ji j > i F ij = F ij + F ji i j>i i j i m i, m j F ij, F ji F ij + F ji = 0 P = i F x i F x oλ. F x oλ. = 0 P = 0 P = mi v i = constant N r c.m. = N i=1 m i r i N i=1 m i = 1 M N m i r i i=1

4 M = N N i=1 m i r cm = x cm î + y cm ĵ + z cmˆk x cm = 1 M N m i x i, y cm = 1 M i=1 N m i y i, z cm = 1 M i=1 N m i z i i=1 r cm = m 1 r 1 + m 2 r 2 = m 1 r 1 + m 2 r 2 m 1 + m 2 M M = m 1 + m 2 r cm = r 1 + m 2 M ( r 2 r 1 ) = r 2 + m 1 M ( r 1 r 2 ) x c.m. = xρ(x)d x ρ(x)d x ρ(x, y) S rρ(x, y)dxdy r c.m. = S ρ(x, y)dxdy x c.m. = y c.m. = Sxρ(x, y)dxdy, S ρ(x, y)dxdy Syρ(x, y)dxdy ρ(x, y)dxdy S

5 z ρ h R x cm = y cm = 0 z z cm = 1 zρdv M M = dm r < R z < h dz dm = ρdv dv M = πρ R2 dv = πr 2 dr h 2 r = R h z R zdm = πρ R2 h 2 0 R 0 z 2 dz = πρ 3 R2 h z 3 dz = πρ 4 R2 h 2 z cm r R z h x y

6 z cm = 3 4 h dm/dt t v M δm t M + δm P (t) = (M + δm) v t t+δt δm v κ v v + δ v t + δt v κ + v + δ v t + δt P (t + δt) = M( v + δ v) + δm ( v κ + v + δ v) δ P = P (t + δt) P (t) = Mδ v + δm v κ + δmδ v dp dt = δp δt 0 δt ( = M δ v δt 0 δt + v κ = δt 0 ( M δ v δt + v δ m κ = M d v dt + v dm κ dt δ m δt + δ mδ v ) δt δt + δ m δt δ v δt δt )

7 M dm dt = dm dt F x d P dt F x = M d v dt v dm κ dt F x = 0 M d v dt = v dm κ dt v κ = c t t 0 d v dt = v κ t t 0 1 M dm dt dt v(t) = v 0 v κ M 0 M(t) M R M A M 0 = m R + m A x v = vî, v 0 = v 0 î, v κ = v κ ( î) = v κ î v τ = v 0 + (1 + m A m R )

8 F x = m g m g = m d v dt v dm κ dt d v = gdt + v κ dm m v τ = v 0 + v κ m(t 0) m(t) g(t t 0) y g = gĵ, v 0 = v 0 ĵ, v κ = v κ ĵ v τ = v 0 + v κ m(t 0) m(t) g(t t 0)

9 m r O L = r p d dt L = d dt ( r p) = r p + r p = r p r p = m r r = 0 τ = d L dt = r F p = m v F = p L = r p τ = r F F = G Mm r 2 êr τ = r F = 0 τ = r F = 0 d L dt = 0 L = c L r, v

10 P, P r, r P P = r = v t OP P A = 1 2 r r = 1 r v t 2 t dt p = m v A t 0 t = d A dt = 1 r v 2 da dt = 1 1 r p = L 2m 2m r = rê r, v = r = ṙ ê r + r ϕê ϕ L = m r v = mr 2 ϕ êr ê ϕ = mr 2 ϕ êz da dt = 1 2 r2 ϕ = 1 2 r2 ω N (m i, r i, v i ) L i = m r i v i = r i p i L = i L i = i r i p i

11 L = i r i F i F i = j i F ij + F x i F ij i j L = i j i r i F ij + i r i F x i i j i r i F ij = i ( r i F ij + r j F ji ) j>i i F ij = F ji j i r i F ij = i ( r i r j ) F ij r i r j r i r j F ij + F ji = 0 r ij = ( r i r j ) Fij F ij r ij j>i ( r i r j ) F ij = 0

12 L = i r i F x i N τ ex = L τ ex = 0 τ ex = 0 L = c O O ϕ ϕ ˆk ω = ω ˆk, ω(t) = ϕ ω(t) dt dϕ ds = ρ dϕ v = ds dt = ωρ ρ = r θ v = ω r

13 w r sinq v r q O ρ = r θ z v = ω r m v L i L = m r v = m r ( ω r) L i = mϵ ijk x j v k = mϵ ijk x j ( ω r) k = mϵ ijk x j ϵ kln ω l x n = m(δ il δ jn δ in δ jl )ω l x n x j = m(ω i x j x j ω j x i x j ) = mω j (δ ij r 2 x i x j ) L 1 L i = I ij ω j I ij = m(δ ij r 2 x i x j ) L 1 = m(ω 1 r 2 ω 1 x 2 1 ω 2 x 2 x 1 ω 3 x 3 x 1 ) = I 11 ω 1 + I 12 ω 2 + I 13 ω 3 I 11 = m(r 2 x 2 1) = m(x x 2 3), I 12 = mx 1 x 2, I 13 = mx 1 x 3

14 L 2 L 3 I ij, (i, j = 1, 2, 3) L ω L 1 I 11 I 12 I 13 ω 1 L 2 = I 21 I 22 I 23 ω 2 L 3 I 31 I 32 I 33 ω 3 N I ij (n) = m(n)[r 2 (n)δ ij x i (n)x j (n)] x i (n) i n I ij = δ ij ρ( r)r 2 dv ρ( r)x i x j dv V ρ( r) I ij 3 3 I 11, I 22, I 33 I ij (i j) I ij z L z = Iω τ = dl z dt V

15 α = dω dt τ = I dω dt = Iα τ = Iα F = m a K = i = i 1 2 m iv 2 i 1 2 m iρ 2 i ω 2 = 1 2 I ω2 n m n I n I = m n ρ 2 n n ρ dρ σ di(ρ) = ρ 2 dm = σ2πρ 2 dρ [0, R] I = R 0 σ2πρ 2 dρ = 1 2 σπr4 m = dm = σπr 2 I = 1 2 mr2

16 M R T R T ` T 1 2 B T 1 1 T B 2 2 B 1 > B 2 B 1 a T 1, T 2 B 1 T 1 = m 1 a T 2 B 2 = m 2 a (m 1 + m 2 )a + T 1 T 2 = B 1 B 2 R τ = (T 1 T 2 ) R τ = d dt L = I dω dt = 1 dω MR2 2 dt

17 T 1 T 2 = 1 2 MRdω dt v = ωr a = dv dt = Rdω dt dω dt a = m 1 m 2 m 1 + m 2 + M/2 g M M/2 m 1, m 2 M z d I M z I = I cm + M d 2 I cm z R = x cm î + y cm ĵ + z cmˆk R = x cm î + y cm ĵ z m n r n z m n r n = x n î + y n ĵ

18 z r x m n O ` r R CM y y cm x cm z m n ρ n = x nî + y nĵ r n = ρ n + R z I = n = n m n r 2 m n ( ρ n + R ) ( ρ n + R ) = n m n ρ 2 n + n m n R ρ n + n m n R 2 m n ρ n = n n m n ( r n R ) = n m n r n M R n m n r n = M R

19 R 2 = d 2 I = m n ρ 2 n + m nr 2 n n = I cm + Md 2 M, R a f R N. B I c = 1 2 MR2 I = I c + MR 2 = 3 2 MR2 a Ma = B θ f f N N B f Rf = Iα

20 a = Rα f f = Iα R = I a R R = I R 2 a = 1 2 Ma Ma = Mg θ 1 2 Ma a = 2 3 g θ A = b v dm/dt = λ v = λ b v κ (1 (M/M 0 ) b/λ) r A( r) = n m i ( r i r) i=1 A( rc.m. ) = 0 m 1, m 2, m 3 A( r 2 ) = 0 (e, m) M m (e, m) m r = ge r r r 3

21 J = L + eg r r A ( B C) = ( A C) B ( A B) C m i I = i m ir 2 i L (O) O L (C.M.) L (C.M.) = i r i p i = = Iω ˆk O L (O) = i m i ( r i v i ) = i m i ( r cm + r i ) v i = r cm i m i v i +L (C.M.) v i C.M. r i m i r i ` r cm O

22 r cm ( m i v i = r cm d m i r i = d r cm dt dt i i i m i r i ) = 0 L (O) = L (C.M.) r, θ, ϕ ρ a m i L (C.M.) = Iω I = i m i r 2 i L (O) O L (C.M.) a I = 1 2 Ma2 I = 1 4 Ma2 a I = 2 3 Ma2 I = 2 5 Ma2 a > b I = 2 5 M a5 b 5 a 3 b 3

23 σ i i = 1,..., N m i, p i F ij σ i F i x p = i F x i L = i r i p i (ρ, ϕ, z) F ρ = m( ρ ρ ϕ 2 ), F ϕ = m(ρ ϕ + 2 ρ ϕ), F z = m z v(t) v(t) v 0 ϕ 0 s max l = s max /2 m n F res = kv n m dv dt + kvn = 0 n t v(0) = v 0, x(0) = 0

24 A, B l v b v a M m v v v 1, v 2 m 1, m 2 v 1 m 1, m 2 l 4l/9 a y > 0 (x, y) (a, b) m v = v x î + v y ĵ m v 0 t = t 0 x 0 = 0 a A l B b

25 F = ±kx k > 0 dq 1 Q 2 = (Q) 1 1 Q 2 = 1 ( Q + 1 ) 1 + Q T aylor (x), ( (x)) h v ϕ h = 10/3 m v = 36 Km/h g 10m/s 2 m n F res = kv n Newton m dv dt + kvn = 0

26 n t v(0) = v 0, x(0) = 0 Euler α = π 2, β = π 2, γ = π 4 ; F (t) T (t) = F (t) F (t) T (t) ; r(t) = (t t) î + (1 t) ĵ + t 2 ˆk t = [0, 4π] t = [0, 2π] r(t) = (t t) î + (1 t) ĵ + t 2 ˆk t = [0, 4π] t = [0, 2π] r 2 = 1 ( (t) 2)2 2 s = 2π 0 r dt = 2 2π

27 R µ v T, N, B T N B T, N v = ṡ T, v = at T + an N a n, a t y = x 2 ϕ = 2v 2 /(g(1 + 4x 2 ) 3/2 ) R v ; v ; A B ϕ r s A = r + ϕ s B 1 + r ϕ m v r = 2abϕ, v ϕ = a r a, b

28 m v 0 C 0 < ϕ 0 < π 2 C l = v 2 0 g(1 + ϕ 0 ) ϕ ϕ 0 r(t) v(t) = d r dt t s(t) = t 0 v(t ) dt = t 0 d r dt dt O

29 ṡ = ds(t) dt = d r dt v(t) ê t = d r dt d r dt = d r dt ds(t) dt = d r ds v = d r dt = ds d r dt ds = ṡê t ê t = d r ds v = ṡê t ê t ê t dêt ds ê n dê t ds = k ê n k ρ k = 1 ρ ê b = ê t ê n ê b

30 ê b dê b ds dê b ds ê b dê b ds = d ds (ê t ê n ) = ê t dê n ds dê b ds ê t ê b ê t ê b ê n dê b ds = τ ê n τ dê n ds = k ê t + τ ê b a(t) = d v dt = d dt (ṡ ê t) = s ê t + ṡ d dtêt = s ê t + ṡ ds d dt dsêt = s ê t + ṡ 2 kê n = s ê t + ṡ2 ρ ên

31 a(t) = a t ê t + a n ê n a t = s a n = ṡ2 ρ a t a n (ê t, ê n, ê b ) (ê t, ê n ) ρ a t = s = 0 a n = ṡ2 ρ = v2 ρ k = 0. dê t dt = 0 ê t = c d r ds = c r = c s + d c, d k = 0 ρ τ = 0 dê b ds = 0 ê b = c c (ê t, ê n ) ê b

32 k = r r r 3 τ = r r r r r 2 r(t) = x(t)î + y(t)ĵ y = y(x) k = τ = 0 ẋÿ ẏẍ ẋ 2 + ẏ 2 3/2 = y (1 + (y ) 2 ) 3/2 ẏ = dy dt, y = dy dx k = r2 + 2ṙ 2 2ṙ r (r 2 + ṙ 2 ) 3/2 r = x(t)î + y(t)ĵ y = dy dx r = ẋî + ẏĵ = ẋ(î + dy dxĵ) r = ẋ(î + y ĵ) r = ẍî + (y + ẋ 2 y )ĵ

33 r r = ẋ 3 y r 3 = ẋ 3 (1 + (y ) 2 ) 3/2 y = x 2 v 0 N N = N ϕ = mg N = N ϕ a = a t ê t + a n ê n ê t ê n F n = ma n = m v2 0 ρ = mkv2 0 tanφ x y = x 2 x

34 k k = y (1 + (y ) 2 ) 3/2 = 2 (1 + 4x 2 ) 3/2 ϕ = F n /B 2v0 2 ϕ = g(1 + 4x 2 ) 3/2 x

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A 1 A 2 A 3 B 1 B 2 B 3

16 0 17 0 17 0 18 0 18 0 19 0 20 A A = A 1 î + A 2 ĵ + A 3ˆk A (x, y, z) r = xî + yĵ + zˆk A B A B B A = A 1 B 1 + A 2 B 2 + A 3 B 3 = A B θ θ A B = ˆn A B θ A B î ĵ ˆk = A 1 A 2 A 3 B 1 B 2 B 3 W = F