(w) F (3) (4) (5)??? p8 p1w Aさんの 背 中 が 壁 を 押 す 力 垂 直 抗 力 重 力 静 止 摩 擦 力 p8 p

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1 F p38 p1w A A A p38 p1w p38 p1w () (1) ()??

2 (w) F (3) (4) (5)??? p8 p1w Aさんの 背 中 が 壁 を 押 す 力 垂 直 抗 力 重 力 静 止 摩 擦 力 p8 p11w (1) () 3n F n p8 p11w

3 (3w) p114 p11w h R z (3.13) p13 R z h ( ρ V π R z ) h p114 p11w 3 z z z ρ V π(r z ) p114 p1w (1) α θ F F β π + θ () 1 (3) p114 p1w x y (x G, y G ) p158 p1w m M

4 (4w) F m T mg p159 p13w (1) () M l m (3) () p159 p13w mg 静 止 している 時 加 速 している 時? p159 p13w mg K v = K m mg K V V v = K m 1 V v = 1 (V + v)(v v) = 1 V ( 1 V + v + 1 ) V v p174 p14w mg e z N e r y y y ϕ 3 ϕ y z = r cos θ, x = r sin θ

5 (5w) (( m r r( θ) ) ( e r + r θ + ṙ θ ) ) e θ = mg e z + N e r (F.1) r = R, ṙ =, r = p174 p15w (1) F sin θ + f = mg F cos θ = v (F.) (F.3) () x + h t (3) f f > = p193 p15w p193 p15w r v r v p193 p15w L ρ L Lg ρ L Lv L + v p193 p15w (6.19) p p9 p15w 6-3 F = ρ L gl + ρ L v L L 1 p15w 1 ρ LL 1 v + ρ L L 1 m m g L 1 }{{} h p9 p16w

6 (6w) F (1) () (3) p9 p16w 1 mv 1 (m + )(v + ) + 1 ( )(v w) ϵ (6.37) p p9 p16w p9 p16w π T > = p64 p17w ρ R r R x a R y b R z c R x = a R r sin θ cos ϕ, y = b R r sin θ sin ϕ, z = c R r cos θ abc abc R π π r sin θ R3 R p64 p19w

7 (7w) p64 p19w m = v + f(r) e r (F.4) x x e r x m = x v (F.5) p64 p19w p64 p19w (1) Iω I ω I + I () m : m ( ) m ( m I + m m + m l + I + m ) m + m l p87 pw x(t) A sin ωt x(t) A sin ωt l mg p88 p1w m d x(t) = x(t) K d x(t) x(t) = Ae iωt A p36 p1w p36 p1w (1.8) mω v e z p99 π p31 pw d ( L µr GMm ) = (F.6) r

8 (8w) F p31 pw ma = GMm m p31 pw m m p31 pw GM 1 m (r 1 ) 3 r 1 + GM m (r ) 3 r + mω r = (F.7) r r = M 1 r 1 + M r M 1 + M ( GM1 m M ) ( 1m ω GM m r (r 1 ) M ) m ω r M 1 + M (r ) 3 = M 1 + M (F.8) r 1 r θ π p3 p3w α ϕ x r cos ϕ = a cos α aϵ 1 + ϵ cos ϕ r cos ϕ = (a cos α aϵ)(1 + ϵ cos ϕ) a(1 ϵ ) cos ϕ (a cos α aϵ)ϵ cos ϕ = a cos α aϵ r cos ϕ = cos α ϵ 1 ϵ cos α (F.9) 1 ϵ cos α + ϵ cos α ϵ 1 + ϵ cos ϕ = 1 ϵ cos α = 1 ϵ 1 ϵ cos α (F.1) ϵ + cos ϕ = ϵ ϵ cos α + cos α ϵ 1 ϵ cos α = (1 ϵ ) cos α 1 ϵ cos α (F.11)

9 (9w) b sin α = r sin ϕ 1 + ϵ cos ϕ ( ) b cos α = r cos ϕ 1 + ϵ cos ϕ b cos α = r (ϵ + cos ϕ) (1 + ϵ cos ϕ) + r ϵ sin ϕ (1 + ϵ cos ϕ) (F.1) ϵ + cos ϕ = (1 ϵ ) cos α 1 ϵ cos α bcos α = (1 ϵ )cos α 1 ϵ cos α r (1 + ϵ cos ϕ) (F.13) 1 r = L µ r (1 ϵ cos α) (1 ϵ ) br (1 ϵ cos α) (1 ϵ ) = 1 (r ) (1 + ϵ cos ϕ) r (F.14) p3 p3w

10 (1w) F p38 p1w B A A B A B p38 p1w 壁 にくくりつけて 張 力 で 引 っ 張 る 両 手 で 各 々 張 力 で 引 っ 張 る T p38 p1w (1) m g = N N = mg + N N = (m + m )g m + m () N = mg + F m + F g (3) N + T = mg m T g (4) N + ρv g = mg m ρv ρv > m (5) T +ρvg = mg Mg +ρvg = N N = Mg + mg T M + m T g p8 pw A F f

11 (11w) F f p8 pw (1) 3n = mg tan θ, 4n = mg cos θ n = mg 4 cos θ mg tan θ, n = 3 () 3n + F = mg tan θ n mg tan θ F = 3 F < = mg tan θ tan θ p8 pw T 1 mg T 1 T mg T 1 T 3 y 1 : y = T : T 1 y 1 : y = 1 : 3 mg y 1 : y : y 3 : y 4 : = 1 : 3 : 5 : 7 : p114 p3w z G = 1 ( π R z M h h ρ V ) z = 1 M ρ Vπ ( ) R h h z 3 h 4 4 = ρ VπR h 4M (F.15) M ρ VπR h h :1 3: p114 p3w ρ V π(r z ) z R ρ V π(r z )z = ρ V π R [ R (R z z 3 z ) = ρ V π ] R z4 = ρ VπR (F.16) πr 3 M = ρ V z 3R 3 8

12 (1w) F p114 p3w (1) F F α > θ F β < π + θ () tan α 1 = tan θ tan β (3) p114 p3w mg = W 1 + W x G mg = LW 1 mg = W 3 + W 4 (x G cos θ y G sin θ)mg = L cos θw 3 x G y G x G = L mg W 1, y G = L mg cot θ(w 1 W 3 ) p158 p3w ma = T mg Ma = Mg T (F.17) (F.18)

13 (13w) (M + m)a = (M m)g (F.19) a = M m M + m g p159 p4w (1) T = Mg T = mg M = m () M l m l a (3) Ma = Mg T m a = T mg ( T M + m ) a = 4 ( M m ) g a = M m 4M m M + m g = 4M + m g p159 p4w (B) p159 p4w V ( 1 V v + 1 ) = K V + v m ( ) 1 V ( log V v + log V + v ) = K m t + C ( ) log V v V + v = KV m t + C ( exp ) V v V + v = C e KV t m (F.) t = v = v V v V + v = V v KV V + v e t m (F.1) t = v = V V V = v t = V = v V v

14 (14w) F V v V v v V v = (V + v) V v e KV t m ( V + v v 1 V v ) ( e KV t m = V 1 + V v e KV V + v V + v v = V 1 V v e KV m V +v 1 + V v V +v e KV t m t ) KV t m = V V + v (V v )e t m V + v + (V v )e KV m t (F.) p174 p4w m ( R( θ) ) e r + R θ e θ = mg e z + N e r (F.3) r e r e r e z = cos θ mr( θ) = mg cos θ + N θ e θ e z = sin θ mr θ = mg sin θ (F.4) (F.5) θ θ = R θ = v 1 ( ) mr θ = mg cos θ + C (F.6) 1 R m (v ) = mg + C (F.7) C = 1 R m (v ) + mg 1 ( ) mr 1 θ = mg(1 cos θ) + R m(v ) ( θ) g = R (1 cos θ) + 1 (F.8) R (v ) r ( g mr R (1 cos θ) + 1 ) R (v ) = mg cos θ + N ( g mr R ( 3 cos θ) + 1 ) R (v ) = N (F.9) N = cos θ = gR (v )

15 (15w) p174 p5w (1) F = v cos θ θ π cos θ F d () x x + h = = v cos θ x + h (3) F = v cos θ v tan θ + f = mg f = mg v tan θ (F.3) tan θ mg < v tan θ f p193 p5w p193 p5w D v = v (r ) 3 D 4π(r ) 3 v = D 4πr3 3 3 v (F.31) r p193 p5w = F F = ρ L gl + ρ L v (F ρ L gl) = ρ L (L + v )v ρ L Lv (F.3) p193 p5w p9 p5w L1 ] (ρ L gl + ρ L v ) = [ρ L g L L1 ( + ρ g(l1 ) ) LLv = ρ L + v L 1

16 (16w) F h L 1 m = ρ LL 1 mgh + mv 1 mv v p9 p5w (1) () (3) rot F p9 p6w 1 mv ϵ = 1 (m + )(v + ) + 1 ( )(v w) (F.33) 1 mv ϵ = 1 m(v + v ) + 1 ϵ = mv + vw 1 v + 1 ( )(v vw + w ) w (F.34) (6.37) m + w = p191 ϵ = 1 w (F.35) ϵ = 1 w 6.5 w p19 w p9 p6w 1 mv = 1 m(v 1) + 1 m(v ) ((mv) = (mv 1 ) + (mv ) ) p9 p6w l π v mgl = mgr + 1 mv v = g (l 4r) (F.36)

17 (17w) m v r = T + mg (F.37) T > = T = m v r mg = mg(l 4r) r mg = mg(l 5r) r (F.38) 5 r p64 p6w I xx I yy, I zz (x, y, z)(a, b, c) I xx = a a = ρ V a = ρ V a a b b b b [ y 3 c 3 + y c3 1 c c [y z + z3 ] b b 3 ] c b ρ V (y + z ) = ρ V a b c = ρ V a b = ρ V a b ( b 3 ) c 1 + bc3 1 c c ) (y c + c3 1 = ρ Vabc 1 (b + c ) (y + z ) (F.39) ρ V abc = M I xx = M(b + c ) 1 I xy = I yx, I yz = I zy, I zx = I xz

18 (18w) F π h R I xx = ρ V ρ((ρ sin ϕ) + z ) r h y R π [ ] h = ρ V zρ 3 sin ϕ + z3 ρ r 3 h R π ) = ρ V (hρ 3 sin ϕ + h3 r 1 ρ R ) = ρ V (πhρ 3 + πh3 r 6 ρ [ hρ 4 ] R = πρ V 4 + h3 1 ρ r = πρ V ( h 4 (R4 r 4 ) + h3 1 (R r ) I yy sin θ cos θ ) ( ) 1 = πρ V (R r )h 4 (R + r ) + h 1 M (F.4) I zz = ρ V R r = ρ V R r π R 4 r 4 4 π ρ 3 h π h h h h ρ(x + y ) ρ = πρ V (R4 r 4 )h = 1 πρ V(R r )h(r + r ) M (F.41) I zz = abc R R 3 = abc R R 5 = abc R 5 = 4 R 3 πabc 5 π π R 5 5 π r 4 π π sin 3 θ 4 3 r sin θ(( a r sin θ cos ϕ) + ( b r sin θ sin ϕ) ) R R x y r 4 sin 3 θ(a cos ϕ + b sin ϕ) π (a + b ) = M 5 (a + b ) (a cos ϕ + b sin ϕ) π(a +b ) (F.4) I xx, I yy a, b, c I xx = M 5 (b +c ), I yy = M 5 (a +c )

19 (19w) p64 p6w 腕 が 反 時 計 周 り の 角 運 動 量 を 持 っている 間 その 他 の 部 分 が 時 計 回 りの 角 運 動 量 を 持 つ (1) () 9 (3) 9 (4) (3) z p64 p7w x m = x m v m L (F.43) L L L L (F.43) L = L e m t (F.44) p64 p7w p64 p7w (1) ω 1 Iω +I ω = (I +I )ω 1 ω 1 = Iω + I ω I + I () ω ( ( m Iω + I ω ) ( ) ) m = I + m m + m l + I + m m + m l ω ω = Iω + I ω I + I + mm m+m l

20 (w) F p87 p7w m d x(t) = (x(t) A sin ωt l) + mg (F.45) x(t) = C sin ωt + l + mg ( m d C sin ωt + l + mg ) = (C sin ωt + l + mg A sin ωt l) + mg mcω sin ωt = (C A)sin ωt ( mω )C = A (F.46) A C = mω m d y(t) = y(t) (F.47) y(t) = D sin m t + α A x(t) = sin ωt + D sin mω m t + α + l + mg t = A sin ωt x = l + mg v(t) = Aω cos ωt + D mω m cos m t + α (F.48) l + mg mg = D sin α + l + = Aω mω + D m cos α (F.49) (F.5) D sin α = cos α = ±1 D cos α = Aω m mω A x(t) = sin ωt Aω m mω mω sin m t + l + mg

21 (1w) p88 p7w x(t) = Ae iωt m d Aeiωt = Ae iωt K d Ae iωt + F e iωt mω Ae iωt = Ae iωt iωkae iωt + F e iωt mω A F = A iωka F mω + iωk = A (F.51) 9.4 p77 ( C 1 e K i 4m K m x(t) = (C 3 t + C 4 )e K t m ( C 5 e K K 4m m F + mω + iωk eiωt ( )t + C e K+i ( )t + C6 e K+ 4m K m K 4m m ) t ) t K 4m < K 4m = K 4m > (F.5) = mω p36 p7w mgh W = N V t = NV t cos ( ) π θ = NV t sin θ (F.53) N = mg cos θ mgv t sin θ sin θ g sin θ t = v W = mvv cos θ = m v V (F.54) m v V p36 p7w.5m/s m.5 π = m (F.55) m/s m

22 (w) F p31 p7w r ( L ) ( ) d 1 µ GMm = (F.56) r r = p31 p8w ma = GMm m a = GM r p31 p8w v m v + ( m) v = 1 m v + 1 ( m) v = r (F.57) p31 p8w ( GM1 m M ) ( 1m ω GM m r (r 1 ) M ) m ω r M 1 + M (r ) 3 = ( ) ( M 1 + M ) G M 1 (r 1 ) 1 G ω r M M 1 + M (r ) 1 ω r 3 = M 1 + M (F.58) G (r 1 ) = G 3 (r ) = ω (F.59) 3 M 1 + M ( ) 1 G(M1 + M ) 3 r 1 = r = ω G(M 1 + M ) ω = R 3 r 1 = r = R (F.6) θ = ± π 3

23 (3w) p3 p8w (11.16) 1 (r ) p313 r = (F.14) (1 + ϵ cos ϕ) p9w 1 r = br (1 ϵ cos α) (1 ϵ ) = L µ (F.61) a = r 1 ϵ ab(1 ϵ cos α) = L µ (F.6) ab(α ϵ sin α) = L µ t + C (F.63) α ϵ sin α = L µab t + C p3 p9w GMm A r = Mm A M + m A GMm B r = Mm B M + m B (F.64) (F.65) G(M + m A ) = r (F.66) G(M + m B ) = r (F.67) M + m A M + m B

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =

1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v = 1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v

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