dynamics-solution2.dvi

1 1. (1) a + b = i +3i + k () a b =5i 5j +3k (3) a b =1 (4) a b = 7i j +1k. a = 14 l =/ 14, m=1/ 14, n=3/ 14 3. 4. 5. df (t) d [a(t)e(t)] =ti +9t j +4k, = d a(t) d[a(t)e(t)] e(t)+ da(t) d f (t) =i +18tj = da(t) e(t)+a(t) de(t), de(t) + a(t) d e(t) 1. dx(t) dy(t) dz(t) =t +1, =, =t, d x(t) d y(t) d z(t) =, =, = v() = 5i j +4k, a() = i +k. P r r x θ y =1/x x φ r = xi +(1/x)j, r = 1+x 4 /x, tan θ =1/x, tan φ = 1/x e r = x i + j 1+x 4, e θ = i + x j 1+x 4, 1/ρ =x 3 /(1 + x 4 ) 1+x 4 e t = x i j 1+x 4, e n = i + x j 1+x 4 x =1

3 1 ρ = 1, e r = i + j, e θ = i + j, e t = i j, e n = i + j (1) a t = dv/ =,a n = V /ρ v(x =1)=V e t (x =1), a(x =1)= V e n (x =1) () e r (x =1)=e n (x =1), e θ (x =1)= e t (x =1) v(x =1)= Ve θ (x =1), a(x =1)= V e r (x =1) 3. A x =.5 1 ρ = 16 17 17, e r = i +4j 17, B x = e θ = 4i + j 17, e t = i 4j 17, e n = 4i + j 17 v A = V (i 4j), a A = 16V (4i + j) 17 89 1 ρ = 16 17 17, e r = 4i + j i +4j, e θ =, e t = 4i j, e n = i +4j 17 17 17 17 v B = V (4i j), a B = 16V (i +4j) 17 89 v B/A = v B v A = 3V (i + j), a B/A = a B a A = 48V ( i + j) 17 89 4. x φ tan φ = dy 1 dx =3x 1, ρ = 6x [1 + (3x 1) ] 3/ e t = cos φi + sin φj, e n = sin φi + cos φj a = V ρ e n = 6xV [1 + (3x 1) ] 3/ e n a(x =1)= 6 5V e n, φ(x = 1) = tan 1 () 5

4 5. x φ tan φ = dy dx = βc cos βx, 1 ρ = βc sin βx [1 + β C cos βx] 3/ e t = cos φi + sin φj, e n = sin φi + cos φj β CV sin βx v = V e t, a = [1 + β C cos βx] e n, φ = tan 1 (βc cos βx) 3/ 6. r(t)θ(t) =c t 1 v = ± dr θ + r dθ = d r θ +dr dθ + r d θ = v θ = r dθ = dr θ = c r (dr ) ( ) vr + v θ = ± + c dr = ± dr r ( ) ( a r = d r dθ r = d r c dr r 3 a = ± a r + a θ = ± c 7. r =e kθ dr r = kekθ dθ kωr, dr 1+ c r ) a θ = dr dθ + r d θ = c d r r ( ) [ ( ) ] d r d + r c dr r d r dr = kω = k ω r dθ = ω, d θ = v r = dr = kωekθ, v θ = r dθ = ωekθ ( ) a r = d r dθ r =(k 1)ω e kθ

a θ = dr dθ + r d θ =kω e kθ v = ± vr + vθ = ±ω 1+k e kθ a = ± a r + a θ = ±ω (1 + k )e kθ 5 8. dr =1, d θ =4t, d r =, dz =1, dθ =4t, d z =, v = e r +4t e θ + k, a = 16t 3 e r +1te θ 9. V = v y = dy = c dθ cos θ, y = c tan θ a y = d y = c [ cos tan θ dθ ] θ + d θ 1. 1 ρ = c 4 + d 4 [c 4 y + d 4 x ], dy 3/ dx = x d c y 1 c (x = c, y =)= ρ d, 1 ρ (x =,y= d) = d c v P = V j, v Q = V j, a P = cv d i, a Q = dv c j v Q/P =v Q a P = V (i+j), a Q/P =a Q a P = V c d (c3 i d 3 j) 11. k i j Ω =Ωk, v = vi, r = Rj, g = gj a=g+ω (Ω r)+ω v =gj +(RΩ vω)j

6 1. r = e r = ( i + k), e θ = j, e φ = (i + k), R( i + k), v = v( e θ e φ )= v (j + i + k) 13. 14. a = (g +Ω R vω)i gk, v =1m/s v A = v C + v A/C = lωj rωi a A = a C + a A/C = lω i rω j a = dv = dv dx dx = v dv dx = 1 d [ ] v dx 3 1. f 1 + f + f 3 =(a 1)i +(b )j +(c 1)k = a =1, b =, c =1. f =(1 )i +( 4)j +( 3+)k = i j k r = 5 v = 5 a = 1 1 f =.1i.j.1k a=.1ti.tj.1tk 5 =.5i.1j.5k v =.5t i.1t j.5t k 5 = 1.5i.5j 1.5k v = 1. m/s, r =.9m

3. 1 x 1 x x = x 1 + x 7 f = k 1 x 1, f = k x, f = kx, x = x 1 + x 4. T T sin θ P =, T cos θ mg = P = mg tan θ, T = mg/ cos θ 5. f δ f cos θ = mg, (l + δ) sin θ = l, f = kδ 6. θ = tan 1 (µ s )=1.8rad 7. 8. P = mge πµ/4 9. T mg tan θ + kl sin θ = kl P = µm B g + (1 µ ) sin θ +µ cos θ (1 µ ) cos θ µ sin θ m Ag m A a = T µm A g, m B a = m B g T a = m B µm A g m A + m B 1. N cos θ mg =, N sin θ + mv ρ =

8 tan θ = v /ρg 11. dp/ f ρsv (a) f = dp/ =(ρsv)v = ρsv (b) f = dp/ =(ρsv)v (ρsv)v cos θ = ρsv (1 cos θ) 1. (3.97) (3.99) v B =,m A = m B = m, α = θ v A cos θ = v A cos φ, v B sin φ = φ =,v A = cos θ C φ = 13. T T sin θ = mlω sin θ, T cos θ = mg cos θ = g lω 14. V = m A v A + m B v B =, km B v A = V δ m A (m A + m B ), 1 m Av A + 1 m Bv B = 1 kδ v km A B = V + δ m B (m A + m B ) x G t x G + x G x G = m Av A t + m B v B t m A + m B =

9 V. 15. A B a Ax = m B sin θ cos θ m A + m B sin θ g, a Ay =, a Bx = m A sin θ cos θ m A + m B sin θ g, a By = (m A + m B ) sin θ m A + m B sin θ g 16. T =(M + m)(a + g), R = m(a + g) 17. m dv = mg ρc Dv mg ρcd g v(t) = tan ρc D m t h H dz = τ v(t) z = h τ v(τ) [ ( )] m ρcd g(h h) v = v(τ), τ = ρc D g cos 1 exp m 18. x v(x) p(x) m d dx [v (x)] = Sp(x) = x L [ p V v(l) = m ( ln 1+ SL V p S V + Sx )] 1/ 19. v = gh v = mv /(m + M) δ = M/k kδ / H

1. 1 (m + M)v + 1 kδ = 1 k(δ + H) (m + M)gH H = mg k + mg k [ ] 1/ kh 1+ (m + M)g X 11 = m 1 + m m m 1 C, X 14 = m m m 1 C, X 1 = X 13 = 4 1. W = C [ x ydx + xy dy ] = 1. T dz = v(t) m dv v(t )= mg c h dz = = cv mg ( ) e ct /m 1 T ( x 4 +x 6) dx = 17 35 v(t) h T h = m g c [ ct ( m ct 1 e /m)] h T

W = h = m3 g c T cvdz = c v (t) = m g [ 1 ( 1 e ct /m) T ( e ct/m 1) ] c ( 1 e ct /m) + ct m 11 3. W 1 = mgb sin θ, W = µmgb cos θ, v = gb(sin θ µ cos θ) 4. O θ f mr d θ + mg sin θ = f τ φ φ φ [ ] W = frdθ = mr d θ + mg sin θ R dθ [ τ 1 = d ( ) ] dθ mr mgr d (cos θ) [ = 1 ( ) τ dθ mr + mgr (cos θ)] t = dθ/ = θ = π/ t = τ dθ/ = ω θ = W = 1 mr ω + mgr W 1 = mgr Rω = gr θ(t) ( ) dθ mr + mg cos θ = f µ

1 5. f x y = f y x =xy U x = x x + y, U y = y x + y U(x, y) = 1 4 (x4 + y 4 ) 1 x y 6. f x y = f y x = xy (x + y ) U(x, y) = 1 ln(x + y ) + const., W = U(1, 1) U(, ) (, ) (ɛ, ɛ ) 7. B m A v A /(m A + m B ) B ( ) 1 m ma B va m A + m = 1 B kδ max δ max = m Av A m A + m B mb k 8. mv/(m + M) θ max (m + M)gl(1 cos θ max )= 1 ( ) m M v m + M [ θ max = cos 1 1 m Mv ] gl(m + M) 3

9. δ x 1 kδ = 1 mv + νmgx 13 x max = kδ µmg 1. (1) K = 1 ( m l dθ ), U = 1 k(cθ) mgl(1 cos θ) dk + du = ml d θ dθ + kc θ dθ dθ mgl sin θ = ml d θ + kc θ mgl sin θ = () sin θ θ kc mgl ω n = ml c mgl/k 5 1. ω = r 1 r ω 1. OA Ω A v A =(r + R)Ω (r + R)Ω rω = v A = rω a A = v A r + R = r ω r + R 3. A B v A v B a A a B

14 G ω = ωk α = αk v A = v G + ω r A/G, v B = v G + ω r B/G v B a B a A = a G + α r A/G + ω (ω r A/G ) a B = a G + α r B/G + ω (ω r B/G ) v A = vi, a A = ai v B = v B (cos θi + sin θj), r A/G = l ( cos φi sin φj), r a B = a B (cos θi + sin θj) B/G = l (cos φi + sin φj) ω, α, v B,a B α = ω = sin θ v cos(θ φ) l, v B = cos φ cos(θ φ) v sin θ a cos(θ φ) l + sin θ[sin φ cos(θ φ) sin θ] v cos φ cos 3 (θ φ) l a B = 4. 3 v = a = a A 5. cos φ cos(θ φ) a sin θ v cos 3 (θ φ) l v A =(Ωk) r A/O v B = v A +(ω AB k) r B/A v C = v B +(ω BC k) r C/B = a A =(Ωk) [(Ωk) r A/O ] a B = a A +(ω AB k) [(ω AB k) r B/A ] a C = v B +(ω BC k) [(ω BC k) r C/B ]= r A/O = l 1 (cos θi + sin θj) r B/A = l 1 (cos βi sin βj)

r C/B = l 3 (cos φi sin φj) 15 sin β = 1 l (l 1 sin θ l 3 sin φ), cos β = 1 l l (l 1 sin θ l 3 sin φ) β AB sin(θ + φ) Ωl 1 sin(θ + β) Ωl 1 ω AB =, ω BC =, sin(φ β) l sin(φ β) l 3 cos(θ φ) Ω l 1 1 α AB = sin(φ β) l tan(φ β) ω AB 1 ωbc l 3, sin(φ β) l cos(θ + β) Ω l 1 1 ωab α BC = l 1 + sin(φ + β) l 3 sin(φ β) l 3 tan(φ β) ω BC 6. P Q OA v Q/P a Q/P OA OP = r = l sin θ/ sin φ v Q/P a Q/P ω OA α OA a P =( Ωk) v P, v P =( Ωk) r P/C, v Q/P (cos φi + sin φj)=v Q/P, v Q =(ω OA k) r Q/O a Q =(α OA k) r Q/O +(ω OA k) v Q a Q/P (cos φi + sin φj) =a Q/P ω OA = sin φ cos(φ + θ)ω, sin θ α OA = sin φ sin(φ θ)ω sin θ 7. A (v A ) y (a A ) y v B = v O + v B/O, a B = a O + a B/O, v O =(ωr)i, a O = v A = v B + v A/B, a A = a B + a A/B ω AB = ω r l cos φ cos θ, α AB = ω r l r/l = sin θ/(1 + sin φ) sin φ r cos φ + ω cos θ l cos θ tan θ

16 8. (x, y, z) P v P = ωre y, a P = ω re x x z e y = e z e x =(m z n x m x n z )i +(n z l x n x l z )j +(l z m x m z l x )k 9. Z z C Z k Z i ω =Ωk + ωi, r P = li rk, α =(Ωk) (ωi), v P = ω r P =(Ωl + ωr)j a P = α ω (ω r P )= Ω(Ωl +ωr)i + ω rk 1. OA (O;y, z) r A = l(cos θi + sin θk), ω = ω z + ω x, α = ω z ω x v A = ω r A = ω z l cos θi ω x l sin θj + ω x l cos θk a A = α r A + ω (ω r A ) =ω x ω z l sin θi (ωx + ωz)l cos θj ωxl sin θk 6 1. ( a (a) (x G,y G )= 3, b ) ( 5a + b, (b) (x G,y G )=, 4b a ) 3 6 6. O x i + O y j

O x = ρbg(a cos θ b sin θ) f = sin(θ + φ) ρbg(a cos θ b sin θ) cos φ ρbg(a cos θ b sin θ) sin φ, O y =ρabg sin(θ + φ) sin(θ + φ) 3. A B C R A ( R B ( R C ( R A = R C = mg tan θ, R B =mg, ( ) h θ = cos 1 r 1, (r <h<4h) 4. f = h [ρ g(h z)+p ]ldz p hl = 1 ρ ghl(h h) 5. h z G z C ρ/ρ < 1 (a) 3a z G = 6, z C = (3 3a 4h)h 3a 6 3a 6h, h = (b) ( ) 1 1 ρρ 17 z G = 3a 3, z C = 3a 3 ρ ρ, h = 3a ρ ρ ρ ρ 6. O z(t) ρal d z(t) = ρagz(t), d z(t) g + ω n =, ω n = l. ρ A 1 t = z = h dz/ =

18 z(t) =h cos(ω n t) O h 7. (1) W x = x G W = l c w (x c)dx = w (l c), x G = 1 (l + c) 3 A x = s A O y + A y W mg =, ( ) l co y W (x G c) mg c = O y = (l c) 3c W l c mg, c A y = l + c W + l 3c c mg O y > (l c) 3 w +3(l c)mg < () W x = x G W = l w x dx = w 3 l3, x G = 1 W l w x 3 dx = 3l 4 AO O y + A y W mg =, ca y 3l 4 W l mg = O y = 4c 3l 4c W + c l mg, c A y = 3l 4c W + l c mg (4c 3l)l 3 w + 6(c l)mg >

19 8. C T, (1) O x =4. kn( ), O y =. kn( ), A x =4. kn( ), A y =, F OA =4.kN(T ), F AB =5.7 kn(c), F OD =.8 kn(t ), F OB =. kn(t ), F BD =.kn(c), F CD =. kn(t ), F BC =.8 kn(c) () O y =1.33 kn( ), C y =.67 kn( ), O x = C x =, F OA =1.33 kn(t ), F AB =.67 kn(t ), F BC =.67 kn(t ), F OE =1.89 kn(c), F AE =.67 kn(c), F AD =.94 kn(t ), F ED =1.33 kn(c), F BD = kn(), F CD =.94 kn(c) 9. I G = I C = πab(a + b )/4 1. 3x+ 3a/3 a/ I G = I C =ρ dx (x + y )dy = 1 3a/6 1 ma 11. O = O x i + O y j a G = a x i + a y j α = αk ma G = O mgj, I G α = r O/G O a O = a G + a O/G = α = 3g cos θ, l O x 3mg 4 cos θ sin θ, O y mg 4 (4 3 cos θ) l (1 + sin θ)mg = 1 3 ml ω

3g ω = (1 + sin θ) l 1. A A y j B B y i a G = a x i + a y j α = αk ma G = A y j B y i mgj, I G α = r A/G (A y j)+r B/G ( B x i) r A/G = 1 (c sin θ b cos θ)i 1 (c cos θ + b sin θ)j r B/G = 1 (c sin θ + b cos θ)i 1 (c cos θ b sin θ)j A B a A a B a A i = a G + α r A/G, a B j = a G + α r B/G α = 3g c sin θ b cos θ b +c 3bc sin θ cos θ a Gx = 3g bc (b +c ) sin θ cos θ 4 b +c 3bc sin θ cos θ, a Gy = 3g (c sin θ b cos θ) 4 b +c 3bc sin θ cos θ 13. G v G ω A v A v e m vl = m v l + 1 mlv G + I G ω, v A v = ev v A = lω, v G = 1 lω, I G = 1 ml ω = 3(1 + e)m v m +3m l, v = 3m em v m +3m 14. (1) O

O = O x i + O y j B B = B y j OA P AB Q a P = a Px i + a Py j a Q = a Qx i + a Qy j α OA = α OA k α AB = α AB k AB OA f = f x i + f y j ma P = O mgj + f ma Q = B mgj f I G α OA = r O/P O + r A/P f I G α AB = r A/Q f + r B/Q B O B α AB = α OA = mgl cos θ 4I G + ml (1 + 4 sin θ) () θ mgl(sin θ sin θ ) δ max = k δ max =l(1 cos θ) 15. I d θ =Tr, z(t) Md z =Mg T, z=θ, I = 1 (m 1R +m r ) d z = Q, Q Mr g I + Mr t = z =, dz/ = 1

z(t) = 1 Qt z = h (1) v = Qτ () a = Q (3) τ = h/q 16. R 1 R G R 1 + R mg =, (h z)f (b + µ s h)r 1 +(b µ s h)r = R 1 = b µ sh b mg + h z b f, R = b + µ sh b h (b + µ sh) mg f <z<h +(b µ sh) mg f mg h z f b R 1 >, R > ma x = f µ k R 1 µ k R, ma y = R 1 + R mg I G α = 1 (z h)f 1 (b + µ kh)r 1 + 1 (b µ kh)r h (b + µ kh) mg f <z<h +(b µ kh) mg f R 1 >, R > a x =(f µ k mg)/m 17. BC G 1 OA G C OA R R OA OA C C C C OA

3 BC = l, OC = l, lsin φ = l cos θ a C = α BC r C/B, a C = α OA r C /O Iα OC Iα BC = r G1/B ( mgj)+r C/B R, = r G/O ( mgj)+r C /O ( R), C C a C/C = a C a C = a C/C [cos θi + sin θj] a C/C I = ml /3 α BC = mlg I sin φ, α OA = mlg I cos θ sin(θ φ) 18. u ω I G = m(a + b )/1 b (mv) =1 (mu) a + b + I G ω, u = 1 ω a + b 3a ω = (a + b ) 19. AB G Q B N B Q N Q A R I G I O I G α = r B/G N B + r A/G R ma G = N B + R mgj a A = a G + α r A/G a B = a B i = a G + α r B/G I O α O = r A/O ( R)+r Q/O N Q Ma O = N Q R a A = a O + α O r A/O, a O = α O ri

4 φ l sin φ = r(1 + sin θ) α AB = lr cos θ cos φ mg J J = I G r cos θ + I O l sin φ + ml r 4(m + M) [(m cos φ + M) cos θ +4M sin θ cos φ sin(θ + φ)] 7 1. δθ δw =(kδ)aδθ (mg)lδθ = a = mgl/kaδ. [ (dr L = K U = 1 ) ] +(rω), (U =) m d r mrω = 3. K U dx/ =ẋ K = 1 mẋ, U = 1 bx + 1 3 cx3 L = K U = 1 mẋ 1 bx 1 3 cx3 d L ẋ L x = mẍ bx cx = 4. q = x p = L/ ẋ = mẋ q =ẋ H = p q L = 1 mẋ + 1 bx + 1 3 cx3

5. 3 f = GmM/r a = 1 h 1 ɛ GM, b = 1 h 1 ɛ GM K U = GmM/r L = K U = 1 m(ẋ +ẏ )+ H = 1 m(ẋ +ẏ ) GmM x + y GmM x + y 5 = m(ẋ + ẏ )/ 6. l (x, y) K = 1 ( ) dx m + 1 ( ) dy m U = k [ l ] (l x) + y k + [l ] (x +(l y) + k [ l ] (l + x) + y k + [l ] (x +(l + y) x/l 1, y/l 1 7. d x d +kx =, U = k (x + y ) y +ky = K = 1 ( ) m dθ 1 1 l 1 + 1 ( ) m dθ 1 l 1 + l dθ U = m 1 gl 1 (1 cos θ 1 )+m g[l 1 (1 cos θ 1 )+l (1 cos θ )]

6 8. d θ 1 m 1 l 1 +(m 1 + m )g sin θ 1 m g sin θ = ( d θ 1 m l 1 + l d ) θ + m g sin θ = L = K U = 1 ( I C ɛ dθ ) mgɛ(1 cos θ), I C = 1 mr + mɛ p = I C ɛ dθ, H = 1 I C ɛ θ + mgɛ(1 cos θ) I C ɛ θ + mgɛ sin θ = 9. q 1 q L = 1 ( m l dq 1 + dq ) 1 kq 1 mg[l (l + q ) cos q 1 ] ( p 1 = ml l dq 1 + dq ) (, p = m l dq 1 + dq ) H = 1 ( m l dq 1 + dq ) 1 kq 1 + mg[l (l + q ) cos q 1 ] ml d q 1 + d q 1 + kq 1 + mg(l + q ) sin q = ml d q 1 + md q 1 + mg cos q 1 =