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 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 2 )+ = (a 2 c 2 )(b 3 d 3 ) (a 3 d 3 )(b 2 c 2 ) (a 2 d 2 )(b 3 c 3 )+(a 3 c 3 )(b 2 d 2 )+ 2.4 gradient (i) f(x, y, z) =/ x 2 + y 2 + z 2 /r (ii) f(x, y) =/(x 2 + y 2 ) /ρ 2 (i) r = x, r3 x =(x, y, z) (ii) x 2 + y 2 = 2 x, ρ4 x =(x, y) 2
2.5 x =0 Taylor () cos x (3 ) (2) tan x (2 ) () (2) cos x = 2! x2 + 4! x4 + d dx tan x = cos 2 x d 2 dx tan x 2 = 2 cos 3 x + sin x d 3 dx tan x 3 = 6 cos 4 x sin 2 x + 2 cos 2 x tan x = x + 3 x3 + x 3 tan x = sin x cos x x 6 x3 2 x2 ( x 6 x3 )( + 2 x2 ) = x + 3 x3 + 2.6 ( + x) α x =0 ( + x) α = +αx + α(α ) x 2 + 2! 2.7 U(x) = 4 x4 3 x3 x 2 2 Taylor 3
U(x) 3 2-4 -2 2 4 - -2 U (x) =0 mb23- U (x) = x 3 x 2 2x =(x +)x(x 2) = 0 x =, 0, 2 U (x) = 3x 2 2x 2 U ( ) = 3 U (0) = 2 U (2) = 6 x =, 2 Taylor (i) U(x) = 5 2 + 3 2 (x +)2 + (ii) U(x) = 8 3 +3(x 2)2 + 2.8 de ax dx = aeax y = ax de y dx = dy de y dx dy = aey = ae ax 2.9 f(x) = ln( + x) x =0 Taylor 4
f(0) = ln = 0 f (x) = +x f (0) = f (x) = ( + x) 2 f (0) = f (3) (x) = 2 ( + x) 3 f (3) (0) = 2! f (n) n (n )! (x) = ( ) f (n) (0) = ( ) n (n )! ( + x) n ln( + x) = f(0) + f (0)x + 2! f (0)x 2 + = x 2 x2 + 2! 3! x3 n (n )! + +( ) x n + n! ( ) n = x n n n= 2.0 (e x ) y = e xy z =(e x ) y ln z = y ln e x = yx ln e = xy z = e xy 2. Euler (i) sin(α + β) = sin α cos β + cos α sin β (ii) cos(α + β) = cos α cos β sin α sin β e i(α+β) = e iα e iβ Euler = cos(α + β)+isin(α + β) = (cos α + i sin α)(cos β + i sin β) = cos α cos β sin α sin β + i(sin α cos β + cos α sin β) (i) (ii) 5
2.2 2 ( y F = A r, x ), A = const 2 r 2 (r 0, 0) (r 0,π) (r, θ) Γ a, Γ b Γa (r 0, π ) (r 0, 0) Γb ma24-3 () (2) (3) r 0 dθ () x F x = (A/r0 2 )( y, x) (x, y) =0 (2) F 2 = A 2 (( y) 2 + x 2 )/r0 4 = A2 /r0 2 = (3) r 0 dθ F A d x = r 0 dθ = A r0 Γ a, Γ b W a = A dθ = A Γ a W b = A dθ = A Γ b π 0 π 0 dθ = Aπ dθ dθ = Aπ W a 2.3 z B ( m e) 6
e v B B =(0, 0,B) () m dv x dt (2) m dv y dt (3) m dv z dt = ev y B = ev x B = 0 (3) z x-y (i) () (2) m d2 v x dt 2 = eb dv y dt = e2 B 2 m v x 0 = d2 v x dt + 2 ω2 v x, ω = eb m A, θ v x = A sin(ωt + θ) () v y = m eb dv x dt = Aω cos(ωt + θ) =A cos(ωt + θ) ω vx 2 + v2 y = A2 = x, y v x,v y x-y ω z z (ii) u v x + iv y (),(2) du dt = iωu u = ae iωt (a ) v x,v y 2.4 () F = kx V (x) V (0) = (2) m m F ( r) = m m G N r r ( r r) 3 r r = (x x) 2 +(y y) 2 +(z z) 2 7
m V ( r) = G N m m r r V ( ) =0 m m r r ma24-7 () V (x) = x 0 kx dx = 2 kx2 (2) / r r x (x x x ) 2 + = 2 ((x x ) 2 + ) /2 2(x x ) ( mm ) G N x r r = x x r r 3 = G N mm (x x) r r 3 2.5 () (2) z(t)z(t) () 2 mv2 z + mgz = mgh z =0 v z v z = 2gh 8
(2) 2 ( ) 2 dz + mgz = mgh dt dz dt = dz h z = 2g h z 2gdt 2 h z + const. = dz h z = 2 h z + const. 2gt t =0 z = h const =0 4(h z) = 2gt 2 z = h 2 gt2 2.6 m v 0 v 0 M m ma24-8 () (2) h (3) v 0 v 0 ( = 6400 km) () E = 2 mv2 0 G N 9 mm
(2) h E = 2 mv2 G N mm + h () 2 v2 = ( 2 v2 0 G NM ) + h v = v0 2 2G NMh ( + h) (3) V =0v =0E =0 mm 2 mv2 0 = G N 2GN M 2GM v 0 = = = 2g 2 g.2km/s 2.7 a M m G =6.67 0 8 cm 3 /gram sec 2 a m -rcos θ θ V () = GmM F = GmM() ˆ 2 l r r sin θ rcos θ - m θ l r sinθ r ma25-3 0
ρ = M 4πa 3 /3 l = r cos θ 2 +(r sin θ) 2 = 2 + r 2 2r cos θ V () = Gmρ φ cos θ = u a V () = Gmρ2π drr 2 0 2 + r 2 2ru = v u r 2 dr sin θdθdφ 2 + r 2 2r cos θ dv = 2rdu du = dv 2r du 2 + r 2 2ru ( r) 2 du 2 + r 2 2ru = dv (+r) 2 2r v (+r) 2 = 2r = 2r 2 v/2 (+r)2 ( r) 2 ( r) 2 dvv /2 = ( + r r ) r = { 2/ r 2/r r a V () = a Gmρ2π drr 2 2 0 = Gmρ 4π a 3 3 = Gmρ 4πa 3 3 = GmM
= ˆ F = V () = GmM 2 ˆ a V () = ( Gmρ2π drr 2 2 a 0 + drr 2 2 ) r = = GmM 3 2 + GmM 2a 2a 3 M() M() = ρ 4 3 π3 = M3 a 3 V () =const. + GmM() 2 F = V () = GmM 2a 3 2 = GmM a ˆ = Gm M3 3 a 3 = GmM() ˆ 2 ˆ 2 2.8 a F = mv 2 /a v = aω( sin ωt, cos ωt) d v dt = aω 2 ( cos ωt, sin ωt) = ω 2 x F = d v m dt = mω2 x x (2.) 2
2.9 a ρ ω ω a ma26-7 a =2cm, =0.5m, ρ =8g/cm 3, 2000 dm = ρardrdφ v(r) =rω 2π L = ρaω r 3 dr dφ 0 0 = 2 πρa4 ω L = 2 π 8 03 kg/m 3 2 0 2 m (0.5) 4 m 4 = 3.3 0 3 kgm 2 /sec 2π 2000 60sec 2.20 Kg m 2 /sec 6400km 5.5g/cm 3 ρ ω dm = ρr 2 dr sin θdφ, v(r, θ) =r sin θω L = ρω r 4 dr sin 3 θdθdφ = ρω2π 5 5 π 0 sin 3 θdθ 3
= 4π 5 ρω5 dx( x 2 ) (x = cos θ) = 8 5 πρω5 =6.4 0 3 Km =6.4 0 6 m ρ =5.5g/cm 3 =5.5 0 3 Kg/m 3 day=8.64 0 4 sec = ω =2π/day =7.27 0 5 /sec 0 L 7.2 0 33 Kgm 2 /sec 2.2 r = φ ma26-8 V = mgx = mgr cos φ φ ṙ =0 mr 2 g φ = mgr sin φ φ + sin φ = 0 (2.2) r r r 4