v = v i e i v 1 ] T v = = v 1 v n v n [ ] U [x y z] T (X,Y,Z)

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Stefan Skiba
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1 v U = e i,..., e n ) v = n v i e i i= e i i U = {X i } i=,n v T v = = v v n v n U x y z T X,Y,Z) v v v = 2 T A, ) b = 3 4 T B, ) c = + b b d = b c c d d 2 + 3b e b c = 5 3 T b d = 5 T c c = 34 d = 26 d 2 + 3b = 3 T e = 5 b = 5 b = 2 cos α = b/ b ) = b = 3 5 T XY b = 5 b = b x b y T b = = 3b x 5b y b = 5 = b 2 x + b 2 y b = T b 2 = b b b = + b = 2 T b = = x y T b = b x b y T b = = x b x + y b y x +b x = 2 y +b y = b = = b 2 x + b 2 y = 5 7 T b = 5 3 4T = T b = T

2 = T b = 4 2 T c = b c = T c c b rt) X, Y ) xt) yt) b xt) yt) x y c c rt ) rt 2 ) d t t 2 e t t 2 f rt) = 2t 2) e x + 4t 2 6t 4) e y, t =, t 2 = 2 rt) = 2 sin2t) e x + 2 cos2t) e y, t =, t 2 = π/4 rt) = 2 sin2t) e x + 2 cos4t) e y, t =, t 2 = π/4 xt) = 2t 2 yt) = 4t 2 6t 4 b xt) t x 2 yt) = t = 2 t 2 = 2 2 t +t 2 ) = 3 4 y 3 4 ) = 6 4 y 6 4 c t yx) = x 2 +x 6 c rt ) = 2 4 T rt 2 ) = 2 T d v v = t rt 2 ) rt )) = 2 2 T e v = 2 e x + 8t 6) e y = 2 e y vt ) = 2 6 T vt 2 ) = 2 T e v = dr dt dy dy dx dt / dx dt = dy dx dx dy t) = dt t) dt t)t A h ϱ ϱ w ϱ > ϱ w v η b ϱ ϱ < ϱ w m α k x b 2x

3 M α m µ b T c m M d m h v 2m 4m 2m 4 4m t m M F M α v s µ M k m v µ b c M α m m M T M F α µ k m m x m l v 2m 3 2

4 M 2 d m m M m M r = 2 + d 2 2G GMm r 2G GMm G M 2 G M mv2 v v = 2 GM ) d 2 m M R m E p = GMm/R v E l = mv 2 /2 r = GMm/R + mv 2 /2 = v II = 2GM E p r) = GMm/r M b m M m v v 2 m v b R

5 XY Z Y m M 2 = r 2 b f = 2 b 2 = b2 2 x ) 2 + y b ) 2 = y y = b x2 2 x f d = = x b2 2 y b x2 2 d 2 = x ) 2 ) b2 2 + b 2 x2 2 = x b2 2 d = x b2 2 x x = M f = r m r = r e x r 2 = f e x + b e y v = v e y v 2 = v 2 e x ) 2 J = r mv ) = mr v e z J 2 = r 2 mv 2 ) = m f e x + b e y ) v 2 e x ) = = m f e x ) v 2 e x ) + b e y v 2 e x )) = mbv 2 e z e x e x = J = J 2 mr v = mbv 2 v 2 = r ) v = b f b ) v = b2 2 b v = b ) b2 2 /b = t > ) ) b2 b 2 = t t 2 = t t 2 = v t + t 2 < t = b <

6 v 2 v b2 2 e b W = E k = 2 m v 2 v ) 2 W = E p = GMm ) r r 2 = GMm b2 2 = GMm b2 2 b2 2 ) h m v α x mx b v p d c b v p d c XY Y r) = h cos α v) = v sin α m = g = g F g = mg = m rt) = vt) = dt + v) = vt)dt + r) = cos α v cos α dt + v = g sin α gt + v sin α v cos α v t cos α dt + = gt + v sin α h 2 gt2 + v t sin α + h xt) = v t cos α yt) = gt 2 + v t sin α + h t = x/v cos α) yx) = g 2v 2 cos2 α ) x 2 + x tn α + h

7 yx mx ) = = tn 2 α + 4 ) g 2v 2 h = cos2 α cos 2 α sin 2 α + 2g = cos α v 2 sin 2 α + 2g ) v 2 h ) h x mx = tn α g v 2 cos2 α = sin α cos α cos α ) sin 2 α + 2g h v 2 g v 2 cos2 α = v2 cos α g sin α sin 2 α + 2g v 2 ) h b X Y r t) = rt) Rt) v t) = vt) V = A Rt) dr dt = V dv dt = A A = r d d ) = r) R) = = h h v cos α v p v cos α + v p ) = v) V = v) v p = v = sin α v sin α r t) = v t) = dt + v ) = v t)dt + r ) = = A = = g = g v cos α + v p v cos α + v p dt + = g v sin α gt + v sin α v cos α + v p d dt + gt + v sin α h y x ) = g 2v cos α + v p ) 2 v t cos α + v p t d = 2 gt2 + v t sin α + h x + d ) 2 + v sin α v cos α + v p x + d ) + h x = y = gd 2 2v cos α + v p ) 2 + v d sin α v cos α + v p + h =

8 v p = 2 v d sin α + 2 cos α) 2 4v 2d sin α + cos α) 2gd2 v d sin α + 2 cos α) c b X Y A = = A = = = g g b r t) = v t) = dt + v ) = v t)dt + r ) = v cos α + v p t + v cos α + v p dt + = g v sin α gt + v sin α t + v cos α + v p gt + v sin α dt + d h = 2 t 2 + v t cos α + v p t d 2 gt2 + v t sin α + h d r t) = 2 t 2 + v t cos α v p t d 2 gt2 + v t sin α + h v p ) = v α = r t) = 2 t 2 d 2 gt2 + h y x ) = h g x + d ) X, Y ) y Y v x X v 2 b x y c r t) r 2 t) r ) = y e y r 2 ) = x e x v = v e y v 2 = v 2 e x Rt) r t) = r t) Rt) = r t) r 2 t)

9 r x x ) = r ) r 2 ) = = y y v = v dr dt = v v 2 v 2 v 2 = = v v r t) = v t)dt + r ) = v 2 v dt + x y v 2 t x = v t + y x = v 2 t x y = v t + y y x ) = y v v 2 x + x ) b t = y v = x v 2 y = v v 2 x x = y = y v v 2 x = c r d 2 t) = r 2 = v 2 t x ) 2 + v t + y ) 2 = v 2 + v 2 2) t 2 2 v y + v 2 x ) t + x 2 + y 2 d dt r 2 = 2 v 2 + v2) 2 tmin 2 v y + v 2 x ) = t=tmin t min = v y + v 2 x v 2 + v2 2 d 2 min = d 2 t min ) = v y + v 2 x ) 2 v 2 + v2 2 2 v y + v 2 x ) 2 v 2 + v2 2 + x 2 + y 2 = v y + v 2 x ) 2 v 2 + v2 2 + x 2 + y 2 d min = ± v x v 2 y v 2 + v 2 2 m l ω T α

10 XY X Y T F g r F g + T = m r F r = m r r = ω 2 r = ω 2 l sin α r = l sin α F g = m g sin α T = T cos α r = ω 2 l sin α T = mω 2 l cos α = X T F g F c g ω 2 l F g + T + F c = F c m r F c = m r R H ω m x ω XY Y X F g F f α r = ω 2 x e x F g = m g sin α F f = F f cos α r = ω 2 x

11 m F g + F f = m r sin α + F f = m g cos α ω 2 x F f sin α = m ω 2 x F f cos α = m g tn α yx) dy dx = tn α = F f sin α F f cos α = ω2 g x yx) = ω2 g x dx = ω2 2g x2 + h yx, z) = ω2 2g x2 + z 2 ) + h Z X Y F g F f F c = m r F g + F f + F c = yx) h V = πr 2 H x = y min = h x = R y mx = ω2 2g R2 + h h π y mx y min xy) 2 2g dy = π ω 2 y mx ) 2g y mx y h) dy = π ω 2 2 y2 hy = y y min min = π ω2 = π 2g ω 2 y mx y min ) 2 y mx + y min ) h h π ω2 2g R4 π ω2 4g R4 = π ω2 4g R4 π ω2 4g R4 + πr 2 h = πr 2 H h = H ω2 4g R2 4g R4

12 yx) = x ω2 2 2 ) 2g R2 + H h h = ω crit = 2 gh R h < π y mx xy) 2 dy = π 2g ω 2 2 y2 hy ) y mx = π 2g ) ω 2 y mx 2 y mx h = πω ) ) ω 2 ω 2 2 2g R2 + h 2g R2 h ) ω πr 2 2 2g R2 + h π gω ) ) ) ω 2 ω 2 ω 2 2 2g R2 + h 2g R2 h = π 2g R2 + h 2 R2 + g ) ω 2 h πr 2 H h h < h = ωr ω > ω crit H g ω2 R 2 2g yx) = ω2 x 2 R 2) H + ωr 2g g M v 3 M 6v α M 2 3 M 3M 2 3M v = 6v M Mv = 3 Mv Mv 2 v = 3 M 6v cos α v sin α M v 2 = 2 3 cos α = 2 3 sin α 2 3 v 2 α = π 3 E k = ) 3 M v M v 2 2 M v2 2 2 = 3 4 Mv2 > = 6 v 2 = 3 2 v 3

13 V CM = v v CM = v V CM = v v = v CM = v cos α v 6 cos α V CM = 6v = v sin α 6 sin α v 2CM = v v v 2 V CM = v 2 = v 2 p CM = M v CM = = 3 Mv CM Mv CM 2 = 3 Mv 6 cos α 23 6 sin α M v v 2 = 3 Mv 6v cos α 2v 6v sin α 2v 2 6 cos α 3 = 3v sin α v 2 = E k = 3 M ) v 2 CM M ) v 2 2 CM 2 = 3 4 Mv2 M 2M M 2M v M 2M rt) R CM t) = 2Mrt) 3M = 2 3 rt) V CM = 2Mv 3M = 2 3 v v = 2V CM v = 4 3 v v 2 = 2V CM v 2 = 3 v

14 v CM = v V CM = 2 3 v e x,cm v CM 2 = v 2 V CM = 3 v e x,cm v CM i = v CM i i =, 2 m 3m 2m T T T ) 2 R CM = m + 3m m + 3m + 2m 7 + 2m = = r CM m = r m R CM = r CM 2 3m = = r CM 2m = 5 2 I zz = m r CM m 2 + 3m r CM 3m 2 + 2m r CM 2m 2 = 6 m2 m 2m X m Y b ω Z t m r = r 2 = R CM = m 4m 3 4 r 3 = m + 2m ) = 2 4 3

15 ) 2 ) 2 ) 2 I xx = m m m 2 3 = 3 4 m2 ) 2 I yy = 2m 2 = 2 m2 ) 2 I zz = 2m m R CM 2 = 5 4 m2 I xx + I yy = I zz b F = F e x,cm τ = dj dt J t J = I zz ω = I zz ω e z,cm τ = I zzω t e z,cm r CM 3 = 4 3 e y,cm ) τ = r CM 3 F = 4 3 e y,cm F e x,cm ) = 4 F 3 e z,cm F = 4I zzω t 3 = 4I zzω 3 3 t W = E k = 2 I zzω 2 HD d v D 2 m n m p D 2 X, Y ) Y X v = v e x

16 D 2 x HD r n = x e x + d e y x R CM = m n 2m n + 2m p d + m p d ) + m n + m p ) = m n 2 m n+m p x d 2d D 2 v = m n v = 2 m n + m p ) v m n 2 m n + m p ) v = v = m n 2 m n + m p ) v e x m n 2 m n + m p ) v V CM = v I = 2 m n + m p ) ) d 2 = 2 2 m n + m p ) d 2 X, Y ) J = r n m n v) = ) 2 d e y m n v e x ) = 2 m nvd e z J = Iω = 2 m n + m p ) ωd 2 e z ω = m n v ) m n + m p d E k = 2 2 m n + m p ) v Iω2 2 m nv 2 = ) 2 m nv 2 mn < m n + m p 2R 2r I M m ϵ ω m b M x

17 M m XY Z X Y Z X Y Z M m F g = Mg F g2 = mg T = T T 2 = T T 2 T = T T 2 = T 2 τ = R T ) = τ 2 = r T 2 ) = RT rt 2 M m = e y 2 = 2 e y, 2 > ϵ = ϵ e z ϵ = R = 2 r M m T 2 T Mg = MϵR T 2 mg = mϵr RT rt 2 = Iϵ ϵ = g mr MR) I + MR 2 + mr 2 ϵ = mr = MR b M x m x = r R x M m M m h h E p,p = Mgh + mgh E p,k = Mgh + x) + mgh x ) = E p,p + Mgx mgx

18 E k,k = 2 Mv2 + 2 mv Iω2 v = ωr v 2 = ωr E p,p = E p,k + E k,k 2 mgr MgR) x ω = R I + MR 2 + mr 2 ) M r α r m m m b m h M m I = 2 Mr2 I = 2 m r 2 m X, Y, Z) X, Y, Z ) X, Y, Z ) m sin α F g = Mg cos α F g2 = mg T 2 = T 2 R = Mg sin α F t = τ = r F t = rf t F t T = T T T 2 τ = r T ) = τ 2 = r 2 T 2 ) = rt rt 2 r i T i i =, 2 m F g + R + F t + T = M

19 τ = Iϵ τ + τ 2 = I ϵ F g2 + T 2 = m 2 = 2 = m 2 = e y > = e x ϵ = ϵ e z = ϵr ϵ = ϵ = ϵ ϵ = ϵ e z ϵ = ϵ e z sin α F t T Mg cos α + Mg sin α + + = I rf t r + = I rt rt 2 r + = m mg T 2 = M F t T T 2 = mg Mg sin α m + M + I + I r 2 r 2 = mg Mg sin α m M + m b H H 2 E p,p = MgH + mgh 2 E p,k = MgH + h sin α) + mgh 2 h) = E p,p + Mgh sin α mgh E k,k = 2 mv2 + 2 I ω Iω2 E p,p = E p,k +E k,k v = ωr v = 2hg m M sin α) hg m M sin α) m + I + I = 2 M + 2m + m r 2 r 2

20 l M I x v m α mx b α mx c Γ F Ω I + Mx 2 I = I +Mx 2 +ml 2 J = lmv = Iω = I + Mx 2 + ml 2) ω ω = lmv I + Mx 2 + ml 2 m m l x α mx h m h h = x cos α mx ) h = l cos α mx ) E p,p = Mg l x) E k,p = 2 Iω2 E p,k = Mg l x + h) + mgh = cos α mx ) Mgx + mgl) cos α mx ) = E k,k = Iω 2 2g Mx + ml) = l 2 m 2 v 2 ) 2g Mx + ml) I + Mx 2 + ml 2 m O m C l k

21 b c

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v = v i e i v 1 ] T v =

v = v i e i v 1 ] T v = v U = e i,..., e n ) v = n v i e i i= e i i v T v = = v v n v n U v v v +q 3q +q +q b c d XY X +q Y 3q r +q = r 3q = r +q = r +q = r 3q = r +q = E = E +q + E 3q + E +q = k q r+q 3 + k 3q r 3q 3 b V = kq