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 s τ = r F

m A F = m A a m B F = G Mm B R 2 = m B g g = GM/R 2 a = m B m A g a = g m A = m B Kg Kg 1m/s 2 N = Kg m/s 2

v = d r dt r v = 0 m F = m a = m d2 r dt 2 p = m v m

F = d dt (m v) = d ( p) = p dt F = m d dt ( v) = md2 r dt 2 = m a d 2 r dt 2 = F m x d 2 x dt 2 = F m d x dt = F m t + c v(0) = v 0 c = v 0 x(t) = F 2m t2 + v 0 t + x 0 F = 0 0 = m a m S

S S S v 0 S m S v 0 S S S m S F 21 F 12 F 21 F 21 F 21 F 12 = F 21 a I = ω 2 R ω = 2π/T T = 24 3600sec R 6.45 10 6 m a I g/290

F 1 = p 1 = F 12 + F ext 1 F 2 = p 2 = F 21 + F ext 2 F i = p 1 p = p 1 + p 2 p = p 1 + p 2 = F 12 + F 1 ext + F 21 + F 2 ext F 12 = F 21 p = F 1 ext + F 2 ext p = 0 F ext = 0 p = constant

q 1 x v 1 q 2 y v 2 q 1 B 1 q 2 B 1 q 2 z q 2 F 21 = q 2 v 2 B 1 x B 2 q 2 z F 12 = q 1 v 1 B 2 q 1 y F21 + F 12 0 z B 1 x 1 2 F21 12 B F 2 y B

(x, y) x = x(t), y = y(t) x, y x = r ϕ y = r ϕ r = (x 2 + y 2 ) 1 2 ϕ = 1 y r = 1 x r r = r(t), ϕ = ϕ(t) ϕ ϕ [ π, π] ê r, ê ϕ r(t) r(t) = rê r ê r = î ϕ + ĵ ϕ ê ϕ ê ϕ = î ϕ + ĵ ϕ ê r, ê ϕ î, ĵ ϕ dê r dϕ = î ϕ + ĵ ϕ ê ϕ dê ϕ dϕ = î ϕ + ĵ ϕ ê r

t v(t) = d v dt d r(t) dt = d{r(t)ê r(t)} dt = dr(t) ê r (t) + r(t) dê r(t) dt dt dê r(t) dt = dϕ dt êϕ v(t) = dr dt êr + r dϕ dt êϕ [ d 2 ( ) ] r dϕ 2 [ ] α(t) = dt 2 r ê r + r d2 ϕ dt dt 2 + 2dr dϕ ê ϕ dt dt α(t) = α r ê r + α ϕ ê ϕ F = F r ê r + F ϕ ê ϕ ( ) F r = m d2 r dϕ 2 dt 2 mr dt F ϕ = mr d2 ϕ dt 2 + 2mdr dϕ dt dt F r F ϕ = 0

x, y, z r, θ, ϕ x = r θ ϕ, 0 r < y = r θ ϕ, 0 ϕ 2π z = r θ, 0 θ π r = r = x 2 + y 2 + z 2 r = r(t), θ = θ(t), ϕ = ϕ(t) r = rê r ê r = ϕ θ î + ϕ θ ĵ + θ ˆk r ê r dê r dt = 0 dê r ê r dt dê r dt = θ ê r θ + ϕ ê r ϕ = θ ê θ + ϕ θê ϕ

ê θ = ( ϕ î + ϕ ĵ) θ θ ˆk ê ϕ = ϕ î + ϕ ĵ ê r ê r, ê θ, ê ϕ ê θ r (xy) z ê ϕ (xy) x 1, x 2, x 3 θ ê r θ ê θ = ˆk

ê ϕ dê θ dt = θ ê θ θ + ϕ ê θ ϕ = θ ê r + ϕ θê ϕ dê ϕ dt = ϕ( ϕî + ϕĵ) θ dê ϕ dt θ dê ϕ dt = ϕ θ ( ϕî + ϕĵ) = ϕ(ê r θˆk) = ϕ(ê r θ( θê r θê θ )) θ dê ϕ dt = ϕ( 2 θê r + θ θê θ ) ˆv = d r dt = ṙê r + r dê r dt = ṙê r + r θê θ + r ϕ θê ϕ v = v r ê r + v θ ê θ + v ϕ ê ϕ v(t) = ṙê r + r θê θ + r ϕ θê ϕ

a = a r ê r + a θ ê θ + a ϕ ê ϕ a r = r r θ 2 r ϕ 2 2 θ a θ = r θ + 2ṙ θ r ϕ 2 θ θ a ϕ = r ϕ θ + 2ṙ ϕ θ + 2r ϕ θ θ F = m a = ma r ê r + ma θ ê θ + ma ϕ ê ϕ

m v 0 δv 0 R t δt = 2π R δv 0 ; t l 0 = v 0 t v 0 +δv 0 l = (v 0 + δv 0 )t = l 0 + δl t δt δl = δv 0 t δl 0 = (δv 0 2)π R δv 0 = 2πR! m A F = m A a m B F = G Mm B R 2 a = G Mm B R 2 = m B g m A m A a = g m A = m B

F = ke x/λ m x 0 = 0 v 0 > 0 x v = 0 x m d2 x = k e x/λ dt2 F (x) d 2 x dt 2 = d dx dv (v) = dt dt dx = v dv dx v dv = k e x/λ dx v 2 v 2 0 = 2kλ m ( ) e x/λ 1 v 2 = 2kλ m v 0 > 0 v = v 20 ( v2 1 e x/λ) x v = v 2 0 v2 v 0 = v dx dt = v = v0 2e x/λ = v 0 e x/2λ x(t) = 2λ (1 + v 0t 2λ ) v(t) = v 0 1 + v 0t 2λ

µ m m l θ ϕ = ω r = λ e ωt F N m M m 1 m 2 > m 1 F

m M F m 1 M T T m 2 B F T ` T T B 1 B 2 F = k k > 0 x 3 x 0 v 0 = 0

x = 0 2α l v 0 = gl α. dê ϕ dt = ϕ( θê r + θê θ ) θ( ϕî + ϕĵ) = ê r θˆk m R T e er B v 0 v0 2 /g = 1m (x, y) y (x, y) ) y = v2 0 (1 g2 2g v0 4 x 2 y = 1 2 (1 x2 )

m v 0 C 0 < ϕ 0 < π 2 ϕ ϕ 0 C D = 1 f(v) = bv cv 2 v b, c b = β D, β = 1.6 10 4 Ns/m 2 c = γ D 2, γ = 0.25Ns/m 4 t = 0 v 0 = 0.625 m/s M = bτ 0 τ 0 = 1 s t v 0 /2 AB m ϕ OA F = kx k x m x = a (ω 1 t), y = b (ω 2 t) y T A 0 P O Q x O B

x, y m v M ṙ = r = 0 ϕ = 0 f r = r r ϕ 2 = r ϕ 2 T = R ϕ 2 ϕ = ω = v/r T = mrv 2 /R 2 = mv 2 /R a r = r r ϕ 2 = ( ω) 2 r 0 e ωt rω 2 = 0 a ϕ = r ϕ + 2ṙ ϕ = 2rω 2 F = (Nm)a a = F Nm a T 1 T 1 = ma = m F Nm = F N m M

T 2 T 1 T 2 T 1 = ma T 2 = T 1 + ma = 2F N n T n = nf N m 2 T = M 2 = m 2 g m 1 a T = m 1 a a = m 2 m 1 g M F = (M + m 1 + m 2 )a = (M + m 1 + m 2 ) m 2 m 1 g θ = a r = l F r F r = m ( r r θ 2 r ϕ 2 2 θ ) 2 α θ=a,r=l r mg α g ϕ α = l α v = ωr ω = ϕ, R = l α v = gl α

F r = (T + B ϕ) F ϕ = B ϕ F r = m( r r ϕ 2 ) F ϕ = m(r ϕ + 2ṙ ϕ) r = R ṙ = r = 0 mr ϕ 2 = T + mg ϕ R ϕ = g ϕ T 0 v gr R ϕ + g ϕ = 0 ϕ = ϕ d dϕ ϕ ( d ϕ 2 + 2g ) dϕ R ϕ = 0 ϕ 2 + 2g R ϕ = C dϕ C 2 ϕ = g R dt

C ϕ = π/2 v 1 = gr C = 3g/R dϕ g = 3 2 ϕ R dt ϕ E( π 2ϕ g, 4) = 4 R t x = v 0 t ϕ y = v 0 t ϕ 1 2 gt2 y = x ϕ 1 2 g v 2 0 x 2 2 ϕ ϕ = 0 y = 0 x = v 0 t ϕ 0 x dy dϕ = x ( 2 1 gx ) ϕ v0 2 ϕ = 0 ϕ = v2 0 gx ) y = v2 0 (1 g2 2g v0 4 x 2 y = 1 2 (1 x2 ) y = v 0 t ϕ 1 2 gt2 x = v 0 t ϕ

0 C 0 ϕ 0 ϕ 0 C y x = ϕ 0 t τϵλ. = = 2v 0 (ϕ ϕ 0 ) g ϕ 0 l l = x = v 0t τϵλ. ϕ = 2v2 0 ϕ (ϕ ϕ 0) ϕ 0 ϕ 0 2 ϕ 0 dl dϕ = 0 ϕ = ϕ 0 2 + π 4 l = v 2 0 g(1 + ϕ 0 )

f( v) f( v) b v c v v + = bvê v cv 2 ê v + ê v v b β d, c = γ d 2 d α, β γ/β 1.6 10 3 s/m 2 f 2 1.6 10 3 (sec/m 2 ) d v f 1 v = 10m/s d = 5cm f 2 /f 1 800 v 5 10 5 m/s d 10 6 m f 2 /f 1 10 7

f 2 /f 1 R = ρ η d v R R f = b v, b > 0 b Kg/s m v = bv dv dt = m b v m b τ = m b, b 0 dv v = dt τ, τ 0 t 0 = 0 v 0 b 0 v v 0 = t τ v = v 0 e t/τ t 0 = 0 x 0 = 0 x(t) = v 0 τ (1 e t/τ )

x 0.50 x v 0 t 0.45 0.40 0.35 0.5 1.0 1.5 2.0 2.5 3.0 t x(t) t = 0 x(0) = 0 t x(t ) = v 0 τ(1 0) = v 0 τ x = v 0 τ τ v 0 m g y 0 v 0 f(v) = bv m dv dt = mg bv v > 0 v 0 > 0 t 0 = 0 τ = m/b b 0 τ dv dt = gτ v

-b mg bv v 0 < gτ v oρ = gτ v oρ dv v oρ v = d(v oρ v) = dt v oρ v τ v oρ v v oρ v 0 = t τ v = v oρ + (v 0 v oρ ) e t/τ v 0 > v oρ v 0 = v oρ y(t) = y 0 + v oρ t + (v 0 v oρ ) τ (1 e t/τ ) y

v v 0 v ΟΡ. 1.4 1.2 1.0 0.8 0.6 1 2 3 4 t mg f(u) m dv dt = mg bv τ 0, τ v = v oρ + (v 0 + v oρ ) e t/τ y(t) = y 0 v oρ t + (v 0 + v oρ ) τ(1 e t/τ ) v 0 = v 0x î + v 0y ĵ t 0 = 0 x 0 = y 0 = 0 x, y x(t) = v 0x τ(1 e t/τ ) y(t) = v oρ t + (v 0y + v oρ ) τ(1 e t/τ )

y = y(x) x(t) 1 e t/τ = x v 0x τ t = τ (1 x v 0x τ ) y(t) y(x) = v 0y + v oρ v 0x x + v oρ τ (1 x v 0x τ ) x y = 0 y = κx + λ (1 x µ ) κ = v 0y + v oρ v 0x λ = v oρ τ µ = v 0x τ x max y = 0 x κx + λ (1 x µ ) = 0 x 2 x 3 (1 x µ ) x µ 1 2 µ 2 1 3 µ 3... (1 x µ ) x µ 1 2 x 2 µ 2

x = 2µ( κµ λ 1) κ, µ, λ x = 2 v 0xv 0y g ( x κx = λ µ + 1 x 2 2 µ 2 + 1 x 3 ) 3 µ 3 x 2µ ( κµ ) x = 2µ λ 1 2 x 2 3 µ 2 g x 2 + x 2 v 0xv 0y = 0 3 v 0x v oρ g x ( ) x max = 3 v 0x v oρ 1 + 1 + 16 v 0y 4 g 3 v oρ 1 + x 1 + 1 2 x + 1 8 x2 x max = 2 v ( 0xv 0y 1 4 ) v 0y g 3 v oρ v 2 t 0 = 0 x 0 = 0 v 0 > 0

x 1.5 1.0 Linear Quadratic 0.5 1 2 3 4 t x(t) f(v) = cv 2 m dv dt = cv2 τ = m cv 0 v(t) = v 0 1 + t τ t x(t) = v 0 τ (1 + t/τ) x = v 0 τ

v 2 cv 2 m dv = mg cv2 dt mg = cv 2 mg v oρ = c dv 1 v2 v 2 oρ dv 1 v + dv v oρ 1 + v = gdt v oρ v 0 = 0 v oρ v v oρ + v = 2 gt v oρ = 2gdt v = v oρ gt v oρ z z z 3 /3 ( v(t) gt 1 1 ( ) ) gt 2 3 y(t) = v oρ 2 g v oρ gt ( ) v oρ ( z) z 2 /2 z 4 /12 ( y(t) 1 2 gt2 1 1 ( ) ) gt 2 6 v oρ

ρ Q ρ a v τ E v τ Q = (ρ ρ a)v g (1 + v τ ) E v τ D = 1 f(v) = bv cv 2 v b, c b = β D, β = 1.6 10 4 Ns/m 2 c = γ D 2, γ = 0.25Ns/m 4 t = 0 v 0 = 0.625 m/s M = bτ 0 τ 0 = 1 s t v 0 /2 i m bv+cv 2 c = 2b 2 /(mg) m/b = τ. m v = mg bv cv 2 v = 0 mg = bv t + cv 2 t v t = b ± b 2 + 4mgc c v t = 1 2 gτ m n F res = kv n Newton m dv dt + kvn = 0

n t v(0) = v 0, x(0) = 0 R ρ ρ m f = π 5 ρ mr 2 v 2 h λv 3 /2 x v 0 t(1 1 t 4 τ ) v 0 f = cv 2 dv dt = g(1 + v2 /v 2 ρ) h = v2 ρ 2g ) (1 + v2 v 2 ρ dv dt = dy dv dt dy = v dv dy = 1 dv 2 2 dy