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51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r<, 0 θ π, 0 <2π 5.2) r, θ,,, z r = 2 + 2 + z 2, cos θ = z r, tan =. 5.3) P e r = θ e θ e 5.1 e r e r = e θ e θ = e e =1, e r e θ = e θ e = e e r =0. 5.4)

52 5 A A = A r e r + A θ e θ + A e. 5.5) A A r = e r A, A θ = e θ A, A = e A 5.6) A r, A θ, A A, A, A z 5.9) A B A = A r e r + A θ e θ + A e, B = B r e r + B θ e θ + B e 5.7) A B = A r B r + A θ B θ + A B, 5.8) 5.9) e r = sin θ cos e + sin θ sin e + cos θ e θ = cos θ cos e + cos θ sin e sin θ e = sin e + cos e 5.10) e = sin θ cos e r + cos θ cos e θ cos θ e e = sin θ sin e r + cos θ sin e θ + cos θ e = cos θ e r sin θ e θ 5.9) 5.2 " e e e e θ θ e " e e 5.2: 5.9)

5.1. 53 1) z z z e e e e e cos sin 0 e e = sin cos 0 e 5.11) 0 0 1 2) θ = θ π/2) sin θ π ) = cos θ, cos θ π ) = sin θ 5.12) 2 2 e, e, ) e, e, ) e sin θ 0 cos θ e e = 0 1 0 e. 5.13) cos θ 0 sin θ 3) r z θ e r = e, e = e, e θ = e r 1 0 0 e e θ = 0 0 1 e 5.14) 0 1 0 e 5.9) 5.1.2, r, = r cos, = r sin 5.15) 0 r<, 0 <2π r = 2 + 2, tan = 5.16) e r = cos e + sin e e = cos e r sin e 5.17) e = sin e + cos e e = sin e r + cos e A A r, A A, A 5.17)

54 5 z =0 θ = π/2 P r P 5.3 θ = π/2 e r e e e r 5.3: 5.2 5.2.1 P, ) r = r e r 5.18) r 5.4 e r+dr e e r v e r e e e r e r d r r d dr 5.4: 5.17) e r = cos e + sin e, e = sin e + cos e

5.2. 55 t de r de = sin d e d + cos e = + d ) sin e + cos e = cos d e d sin e = d ) 5.19) cos e + sin e e e r de r = d e, de = d e r 5.20) 5.2.2 r = r e r dr = dr e r + r de r 5.21) 5.20) v v = dr = dr e r + r d e 5.22) r v r = dr, v = r d v r v SI m/s 5.23) 5.22) 5.20) dv = d2 r 2 e r + dr = de r + dr [ d 2 ) ] r d 2 2 r e r + d e + r d2 2 e + r d [ 2 dr ] d + r d2 2 de e 5.24) a = dv [ d 2 ) ] = r d 2 2 r e r + 1 r d r 2 d ) e 5.25)

56 5 5.2.3 m F r F = F r e r + F e 5.26) r [ d 2 ) ] r d 2 m 2 r = F r 5.27) m 1 d r 2 d ) r = F. 5.28) F r F r, t 5.27) 5.28) r 5.3 5.3.1 W = B A F dr = B A ) F d + F d + F z dz 5.29) 4.42) d d dz r θ dr r dθ r sin θ d dr θ r dθ r r sin θ d r θ z z W = B A F dr = B A ) F r dr + F θ r dθ + F r sin θ d 5.30) 5.30) θ = π/2 dθ =0 W = B A ) F r dr + F r d 5.31)

5.4. 57 5.3.2 F F Ur) F Ur) F = U e U e U z 5.32) r θ θ = π/2 F = U r e r 1 U r θ e θ 1 U rsin θ e 5.33) F = U r e r 1 U r e 5.34) 5.4 r r r = r 2 r = t dr r + r dr =0 r dr = 0 5.35) r v =dr/ = r cos, = r sin 5.36) ω =dt)/ t) ω = = ωt + 0 5.37) 0 t =0 5.36) 5.37) r = a = = a cos ωt + 0 ), = a sin ωt + 0 ) 5.38) 5.38) m F = m d2 2 = mω2, F = m d2 2 = mω2. 5.39)

58 5 { F = mω 2 Fr = mω 2 r r, 5.40) F = 0 5.27) 5.28) [ d 2 ) ] r d 2 m 2 r = mω 2 r, m 1 d r 2 d ) = 0 5.41) r h = r 2 d/) 5.40) r = ω =d/ = 5.5 5.5.1 m l S mg 5.5 e e S e r F = S e r + mg e = mg cos S) e r mg sin e. 5.42) mg e r r 5.5: ml ) d 2 = mg cos S 5.43) ml d2 2 = mg sin 5.44)

5.5. 59 r = l = 0 5.44) sin ml d2 = mg 5.45) 2 a δ = a sin ωt + δ), ω = g l. 5.46) 5.5.2 r 5.43) 5.44) S ds d = 3mg sin =3mg d cos ) 5.47) S =3mg cos + C. 5.48) 5.43) C =0 v = l d = v 0 =0 C = 2mg + m l v 2 0 5.49) 5.6 5.43) l d ) 2 = v0 2 2gl 1 cos ) =4gl k 2 sin 2 ) 2 5.50) 1 d g 2 = ± k l 2 sin 2 2 5.6:, k 2 = v 0 2 4gl 5.50) 5.51) + )

60 5 =0 v 0 k k <1 k = sin 0 /2) 0 0 0 k =1 0 = π k >1 k <1 k sin θ = sin 2 5.52) θ 0 0 θ 0 π/2 5.51) + θ dθ g = 1 k l 2 sin 2 θ 5.53) dθ g 1 k 2 sin 2 θ = l 5.54) T =4 l π/2 g Kk), Kk) = dθ 0 1 k 2 sin 2 θ 5.55) 5.7 Kk) 1.20 1.15 1.10 1.05 1.00 0.95 0.90 0 10 20 30 40 50 60 70 80 90 5.7: k Kk) 5.55) k sin θ 3.32) = k 2 sin 2 θ 1 1 k 2 sin 2 θ =1+1 2 k2 sin 2 θ + 1 3 2 4 k4 sin 4 θ + 1 3 5 2 4 6 k6 sin 6 θ + 5.56)

5.5. 61 sin θ π/2 sin θ ) 2n) dθ = T [ ) l 1 2 T =2π 1+ k 2 + g 2 0 2n 1)!! 2n)!! ) 1 3 2 k 4 + 2 4 π 2 ) 1 3 5 2 k 6 + ] 2 4 6 5.57) 5.58) k 2 k 5.7 5.5.3 5.31) r 0 mg F = mg sin 5.59) 5.8 =0 = 1 1 > 0 5.59) W = 1 0 F r d = mgl 1 0 sin d = mgl 1 cos 1 ) 5.60) l 1 cos 1 ) = 1 =0 5.60) 1 S r d r d F ds ds mg 5.8: =0 = 1 =0 = 1 5.60)