2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP

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1 : ( ) 2: 3: SF : 3 2 A m

2 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP j AP j 2 AP j (3 ) ( x = OP): F = mgψ(p) AP 2 AP dv = mg AP ψ(x) x OA x OA 3 dxdydz (1) A (1) Q n j=1 n ψ(p j ) j OPj j=1 ψ(p j ) j

3 3. 3 M OQ = 1 M ψ(p) OP dv ( ) M = ψ(p)dv ψ(p)( OP OA) dv = ψ(p) OP dv OA = ψ(p) O dv M OA ψ(p) dv Q M A F = mm G AQ AQ 2 AQ = = mg 1 M ( 1 mm G 1 M ψ(x)(x OA) dv ψ(x)(x 3 OA) dv M ψ(p) OP dv ) OA ψ(p) OP dv OA 3 (1) 3 1 O ψ(x) r = x ψ(x) = ρ(r) 2 (1) 1 2

4 3. 4 R x = (r cos φ cos θ)e 1 + (r sin φ cos θ)e 2 + (r sin θ)e 3 ( r R, φ < 2π, π/2 θ π/2) (2) e 1, e 2, e 3 e 3 OA e 1, e 2 A = O OA OA = OA e 3 D(x)/D(r, φ, θ) D(x) D(r, φ, θ) = = r 2 cos θ cos φ cos θ r sin φ cos θ r cos φ sin θ sin φ cos θ r cos φ cos θ r sin φ sin θ sin θ r cos θ cos φ cos θ sin φ cos φ sin θ sin φ cos θ cos φ sin φ sin θ sin θ cos θ = r 2 cos θ(cos 2 φ cos 2 θ + sin 2 φ sin 2 θ + cos 2 φ sin 2 θ + sin 2 φ cos 2 θ) = r 2 cos θ ( ) x OA = (r cos φ cos θ)e 1 + (r sin φ cos θ)e 2 + (r sin θ OA )e 3 x OA 2 = r 2 cos 2 φ cos 2 θ + r 2 sin 2 φ cos 2 θ + (r sin θ OA ) 2 = r 2 2r OA sin θ + OA 2 (1) (2) : F = mg ρ(r) (r cos φ cos θ)e 1 + (r sin φ cos θ)e 2 + (r sin θ OA )e 3 W (r 2 2r OA sin θ + OA 2 ) 3/2 r 2 cos θdrdφdθ (3)

5 3. 5 W = {(r, φ, θ); r R, φ < 2π, π/2 θ π/2} 2π cos φ dφ = 2π sin φ dφ = F e 1, e 2 e 3 φ F = 2πmGe 3 R π/2 dr π/2 ρ(r)r 2 (r sin θ OA ) cos θ (r 2 2r OA sin θ + OA 2 ) 3/2 dθ (4) f(t) (t > ) f(t) = π/2 π/2 (t sin θ 1) cos θ dθ (5) (t 2 2t sin θ + 1) 3/2 f ( ) r = π/2 OA 2 OA π/2 (r sin θ OA ) cos θ (r 2 2r OA sin θ + OA 2 ) 3/2 dθ (4) f(t) F = 2πmG ( ) R r OA e 2 3 ρ(r)r 2 f dr (6) OA f(t) u = t 2 2t sin θ + 1 t sin θ 1 = t2 + 1 u 2 t 1 f(t) f(t) = = 1 4t t 2 2t+1 t 2 +2t+1 [ 1 = t2 1 u, cos θdθ = 1 2 2t du t 2 1 u 2u 3/2 (t 2 1) 2 u 2 u ( 1 ) du = 1 (t+1) 2 2t 4t (t 1) 2 ] u=(t+1) 2 u=(t 1) 2 = 1 2t {(t 2 1)u 3/2 u 1/2 }du [ t u u ] u=(t+1)2 u=(t 1) 2

6 3. 6 = 1 ( t (t + 1) 2 t2 1 + (t 1) 2 ) 2t t + 1 t 1 = 1 ( 2t 2 ) + 2t 2t t + 1 2t2 2t = t 1 t 1 t 1 1 { (t > 1 ), = 2 ( < t < 1 ) (6) F = 4πmG OA 2 e 3 R OA ρ(r)r 2 dr (a b = min{a, b}) (7) s (> ) ˆM(s) ˆM(s) = <r<s ρ(r)dv = s π/2 2π s dr dθ ρ(r)r 2 cos θdφ = 4π ρ(r)r 2 dr π/2 (7) F = m ˆM(R OA )G OA 2 e 3 (8) A=O (r 2 2r OA sin θ + OA 2 ) 3/2 = r 3 (3) F = mg = = mg W R ρ(r)(e 1 cos φ cos θ + e 2 sin φ cos θ + e 3 sin θ) cos θdrdφdθ ρ(r)dr (e 1 + e 2 + e 3 ) = A A A (A ) A ψ(x) ρ(r) r, φ, θ r r

7 (8) A OA > R ˆM(R OA ) = ˆM(R) = M F = mm G OA 2 e 3 M ( 1 ) A OA < R ˆM(R OA ) = m ˆM( OA )G F = e 3 OA 2 ˆM( OA ) A A ( 2) R A 2: A r ˆM(s) = 4π 3 s3 ρ (ρ(r) = ρ )

8 4. 8 F = 4πρ mg OA e 3 3 OA ( ) 2 ( 2 ) ρ(r) < r < T r > R ( 3) OA < T ˆM( OA ) = (8) F = R T 3: ( 3 ) SF ( ) ( 4)

9 5. 9 4: SF 3 ( ) 2,3 5 ω r m mrω 2 rω 2 g = 9.8 [m/s 2 ] R 1 ω 1 4 [km] = [m] R 1 = π = 2. π 17 [m], ω 1 = 2π = π [rad/s] 3 SF

10 5. 1 R 1 ω 2 1 g = 2. π =.343 (9) F 2 F 1 3 F 2 M 2 m mm 2 G/R 2 1 mr 1 ω 2 1 M 2 mm 2 G R 2 1 = mr 1 ω 2 1 g M 1 (= [kg]) (9) mr 1 ω 2 1 =.343 mg =.343 mm 1G R 2 1 = mm 2G R 2 1 M 2 =.343 M 1 (1) ρ 3 = [kg/m 3 ] R 2 (1) ρ 1 = [kg/m 3 ] M 2 = ρ 3 2 = 4π 3 ρ 3R 3 2 =.343 4π 3 ρ 1R 3 1 R 2 = R ρ 1 ρ 3 =.238 R 1 1/4

11 5. 11 M 3 R 3 M 3 = M 1, R 3 = [m] = R 1 F 3 F 1 F 3 F 1 = mm 3G/R 2 3 mm 1 G/R 2 1 = M 3 M 1 ( R1 R 3 ) 2 = (1/2.36) = M 4 R 2 M 4 =.123 M 1, R 2 = [m] = R 1 F 4 F 4 = M 4 M 1 ( R1 R 4 ) 2 F 1 =.123 (1/6.3) F 1 = F F F 3 =.174 F, F 4 = F F 4 17% 1 1 ( ) [1] 2 (199).

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

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