理论力学第7章答案.pdf

1、7.1 AB a b A x AB 1 = f a = ctg A h x = & & 2 A csc h x h x / sin 2 A = & & h h x x / ) / 2 sin ( sin 2 A A 2 = & & & & & b = sin / A r x = 2 A sin / cos & & r x r x / tg sin A & & = r x x x / ) sec sin sin tg sin ( A 2 A A & & & & & & & & + + = r r x x / / sin ) sec 1 ( tg sin 2 A A + = & & &7.2 AB

2、 a b AB C AB cos A v CA A C v v v + = a x 0 sin sin A CA A x C = + = = & & CA x v v vy = cos A y C v v b x 0 sin sin A CA A x C = + = + = & & CA x v v v y = cos A y C v v 7.3 OA r AB l R O = AOB AB 1 4 2 3 3 = = f = sin sin l r 1 BA A B v v v + = ) ( A r v = ) sin( ) sin( ) sin( B 2 A 2 BA + = = v v

3、 v = cos cos A BA v v = = cos cos BA AB l r l v + = cos ) sin( A B v v + = = cos ) sin( B B R r R v 1 = = cos cos cos cos l r l r & & + = cos cos B l r x + = = cos ) sin( sin sin B r l r x & & & + = = cos ) sin( / B B R r R x &7.4 ABCD B OC D CE A E BC/AD AB/CD 1 AD BC = = 2 CD AB = = A AB B A v v v

4、 + = EC C E v v v + = Q OB OC 2 = BA CE 2 = B 1 1 C 2 2 v v = = = OB OC AB 2 2 EC 2 2 v v = = = BA CE A E 2v v =7.5 BF BE = l AC = v v v = o 60 cos F v v 2 F = v v v 2 F E = = v v v = = o 60 cos E D l v AD v 2 D = = 7.6 7.7 1 F 2 F C GBJ ICJ GBI ICJ 7.8 AB O A OA r O O OA = = AB O O v A v OA C C AB

5、) , 2 ( r ) 2 , ( r r 2 r 7.9 a CD AB 2 = = c BD AC 2 = = c a BD BD AB CD P APC DPB P BD ) , ( APC DP AP = = 0 sin ) 2 ( sin = ac a 2 cos ) 2 ( cos = + 2 2 2 ) 2 ( ) cos 2 ( ) sin ( = + a c cos ) / ( 1 / ) ( 2 2 = a c a c a 7.11 AB BC CD AD a BC AB = = a CD 2 = AD CD a 0 BC = B C v v = 2 / 2 / C CD

6、= = a v + = + CB n B n C C a a a a 0 C = a 0 CD = b 0 = C v = = a v / B BC 0 CD = n CB CB n B C a a a a + + = x n CB n B c cos a a a = Q 4 / 7 sin cos = = a a 2 n B = a a 2 n CB = 7 / 8 2 C a a = 7 / 7 4 2 / 2 C CD = = a a7.12 CD/EG B DG O A B C D E G cm 20 = =EG CD cm 50 = DG cm 40 = OA CD OA/CD cm

7、/s 20 A = v B 2 Bx cm/s 10 = a 3 . 0 tan = 1 CD OA 2 B 3 OA 0 AB = B A v v =rad/s 5 . 0 / A OA = = OA v2 2 OA n A cm/s 10 = = OA a + + = + AB By Bx A n A a a a a a x = sin cos cos sin By Bx A n A a a a a 2 By n A Bx A cm/s 1 tg ) ( = + + = a a a a 2 A OA rad/s ) 40 / 1 ( / = = OA a7.13 OA r AB l R O

8、 AB n BA BA n A B a a a a + + = y + = sin cos sin 0 n BA BA n A a a a = = tg ) ( cos / ) sin sin ( 2 2 2 2 BA & & l r l a x n BA n A B ) cos( cos a a a + + = + + = cos / ) cos( 2 2 B & l r a = = tg ) ( / 2 2 BA AB & l a + + = = cos / ) cos( / 2 2 B B R l r R a & = + = = tg ) ( cos / sin cos cos / si

9、n 2 2 2 2 AB & & & l r l r + + + + = = 2 B B cos / sin ) sin( cos / ) )( cos( R r R r & & & + + = cos / ) cos( 2 2 R l r &7.14 r O O v AB l B A 1 = f ) cos 1 ( sin = r l ) 1 ( ) 2 / sin( 2 ) / ( O O B = = v BP r v v = = cos / ) 2 / cos( ) 2 / sin( 2 / O l v CB v B AB = cos / sin O l v 2 cos / ) 2 /

10、sin( ) cos / sin ( O + = = l l v CA v AB A) 2 / sin( ) 2 / sin( 2 ) cos / ( O + = v cos / ) cos( 1 O + =v n BO BO O B a a a a + + = 0 O = a BO a n AB AB B A a a a a + + = x n AB B A ) 2 cos( cos a a a + = + + = 3 2 2 2 A cos sin cos ) sin( l v r v a O O 1 = cos / sin O l v & = cos sin O A l r x x )

11、cos / sin ( sin cos O O O A A + = = l v l v v x v & cos / ) cos( 1 O + =v + + + = = 2 O A A cos / sin ) cos( cos / ) )( sin( & & & & v v a + + = 3 2 2 O 2 O cos sin cos ) sin( l v r v7.15 10cm B OA AB 40cm OA 10cm AB 40cm OA rad/s 10 = B 0 AB = = = r v v A B rad/s 10 / B B = = = r v cm r 10 = + = +

12、BA n A n B B a a a a x = sin sin cos n A n B B a a a 75 / 15 4 tg ) 5 / ( 2 2 2 2 B = = r r r r a 2 rad/s 7 . 20 75 / 15 4 / 2 B B = = = r a A x A y B x B y 0 A = x r y = A r x 15 B = 0 B = y 2 2 A B 2 A B ) 4 ( ) ( ) ( r y y x x = + 0 ) )( ( ) )( ( A B A B A B A B = + y y y y x x x x & & & & 0 ) )(

13、 ( ) ( ) )( ( ) ( A B A B 2 A B A B A B 2 A B = + + + y y y y y y x x x x x x & & & & & & & & & & & & 0 B A = = y y & & 2 A = r y & & r x y 5 / 2 B B & & & = 0 A = x & & = = r x x B A & & 75 / 15 4 2 B = r x & & rad/s 10 / B B = = r x & 2 B B rad/s 7 . 20 / = = r x & &7.16 r l A O 2 1 O v 2 O 1 O v

14、v v 2 2 1 O A = = A O 2 v v v 2 A O 2 = = 0 A O 2 = 2 O r v r v / 2 / 2 2 O O = = 1 O n AO AO O A 1 a a a a + + = 0 1 O = a 0 AO = a 2 O n A O A O A O 2 2 1 a a a a + + = x = sin cos A O 2 a a r v a / tg 2 O 2 = 2 2 2 O O / / 2 2 r l r v r a = = A O 2 l , 2 2 O A 2 O A ) ( ) ( 2 2 l y y x x = + 0 )

15、)( ( ) )( ( 2 2 2 2 O A O A O A O A = + y y y y x x x x & & & & 0 ) )( ( ) ( ) )( ( ) ( 2 2 2 2 2 2 O A O A 2 O A O A O A 2 O A = + + + y y y y y y x x x x x x & & & & & & & & & & & & 0 B A = = y y & & r v y / 2 A = & & 0 2 O = y & & 0 A = x & & v x x 2 A O 2 = = & & 2 2 2 O / 2 r l v x = & & r v r

16、x / 2 / 2 2 O O = = & 2 2 2 O O / / 2 2 r l r v r x = = & &7.17 C D AB A A m/s 3 = v 2 m/s 2 = a C B A BA A B v v v + = o o o 105 sin 15 sin 60 sin A BA B v v v = = m/s 70 . 2 B = v m/s 80 . 0 BA = v rad/s 8 . 1 5 . 1 / B C = = vrad/s 1 . 0 8 / BA AB = = v n BA BA A n B B a a a a a + + = + 5 . 4 / 2

17、 B v 8 2 AB x n BA A n B B 30 cos 15 sin 15 cos a a a a + = o o o 2 B m/s 31 . 2 = a 2 B C rad/s 54 . 1 5 . 1 / = = a7.18 AB A D B D AB m 5 . 0 = = ) 3 / 4 ( ) 2 5 . 3 /( B AB v = = ) 3 / 16 ( 4 B A v = = 3 . 5 1 / A A & v n AB AB n B n A A a a a a a + + = + 4 / 2 A v 2 2 AB AB x n AB n B n A A 45 c

18、os cos sin a a a a + = + o AB ) 45 sin( 5 . 1 sin 4 45 sin + = = o o o 377 . 15 = ) m 10 ( 92 . 4 1 = AB 2 A 45 . 12 = a 2 A 45 . 12 1 / = = a A 7.1 9 CD C B CD 0 = v 2 mm/s 600 = a A (1) O, C OC C O a a a + =0 Ox = a 0 Oy = a 0 O = a( 2) PO P a a = 2 C P m/s 8 . 0 = = OP OC a a ) / ( 8 . 0 2 P s m

19、a a = =7.20 mm/s 75 = v A O rad/s 1 C = = OC v n OC OC C O a a a a + + = 2 C rad/s 8 75 600 / = = = OC a n PO PO O P a a a a + + = 2 O PO Px m/s 725 . 0 075 . 0 8 . 0 = = = a a a 2 Px A m/s 725 . 0 = =a a7.21 O O O O = ) ( 2 1 O r r v = = ) 1 / ( / 2 1 2 O r r r v a O O O e r a + = = = ) / ( ) ( 2 1

20、 a r r r O O r = e e r / = O C CO = ) / ( 2 1 r r r7.22 I II AB III II III AB II III r r AB 2 = = r v B 2 = = 2 / B 2 r v = = r r v 4 2 2 P = = 4 / P 3 r v e r a + = = = 3 4 r 0 3 = = = & & r r7.23 cm 40 1 = r cm 20 2 = r cm 30 3 = r cm 90 4 = r min / r 1800 1 = n 2 n A a 2 1 A ) ( n r r v + = 2 1 r

21、 v A / 2 = 3 2 2 3 = BA A B v v v + = b 4 n r 3 3 r a 2 3 2 1 3 3 A B ) / 1 )( ( n r r r r r v v + + = + = rpm 3000 / ) / 1 )( ( / 4 a 2 3 2 1 4 b = + + = = r n r r r r r v n B rpm 7.25 OA = e B = 4 r OA r 3r = OP B r e e r / / = CP CO r CP = e r a + = = 3 a = = r r v 2 3 2 a B BP P B v v v + = Q =

22、= r r v 3 3 e P = = r r v 3 a BP = + = r v v v 2 3 ) ( 2 / 1 2 BP 2 P B e r a v v v + = Q = = r r v 4 r r = = r r v 10 10 e e = + = r v v v v v 2 3 ) cos 2 ( 2 / 1 e r 2 e 2 r a7.26 OA = e B = 4 r OA r 3r = OP OA P k r n e e P a a a a a + + + = e r 3 2 3 e r 2 r r Q 0 = r v x 2 n e x P 3 = = r a a y

23、 2 2 2 r e y P 13 ) 4 ( 3 = + = + = r r r a a a7.27 AB O cm 5 O A cm/s 30 AB O AB rad/s 3 / AB = = PA v A AB C =PC v rad/s 3 / AB C O = = = CO v AB r O + = rad/s 6 r = AB r O + = AB O r + = k e r O a a a a + + = O n OA OA A e a a a a + + = 0 O = a 0 A = a 0 k n OA OA r = + + + a a a a r r AB OA 2 AB

24、 OA r AB 2 v r r =r v x 0 60 cos 30 cos k n OA OA = + a a a o o 2 AB rad/s 3 9 = y 0 60 sin 30 sin n OA OA = o o a a a r 2 O rad/s 3 9 = ) 2 / ( ctg =r x A 2 / ) 2 / ( csc 2 = & r v A rad/s 3 | / ) 2 / ( sin 2 60 2 A = = = o & r v 2 60 A rad/s 3 9 | / sin = = = o & & & r v = = cos / A A C v PA PC v

25、v rad/s 3 | / cos / 60 A C O = = = = o r v r v 2 60 A O rad/s 3 9 | / sin = = = o & r v7.28 OB OA C AB r OA = r AB 4 = OA CD A C v v = 0 AB = n BA BA n A B a a a a + + = 0 n BA = a y + = cos 0 BA n A a a 15 / 4 / 2 BA AB = = r a e r a v v v + = 15 / CD r v =k e r a a a a a + + = 0 k = a C n CA CA n

26、A e a a a a + + = 0 n CA = a + + = CA n A r a a a a a x CA n A a a a = cos cos CD 15 AB / r r r a 2 2 CD 7 cos / 2 = =7.29 C 1m A m/s 5 . 0 = v DA 2rad/s = & C AD o 90 = e r a v v v + = 1 k e r n a a a a a a a + + = + 2 C CA A e v v v + = 3 n CA CA a e a a a a + + = 4 1 3 CA A r a v v v v + + = = &

27、AC CA v m/s 8 = a v m/s 4 r = v 2 4 k n CA r n a a a a a a a + + = + 1 / 2 a v 2 & AC r 2 v & y o o o 30 sin 30 cos 30 cos k n CA r n a a a a a = m/s 6 . 53 30 tg 30 cos / k n CA n r = = o o a a a a a7.30 = OA 0 OA = =l v CD 0 CD = a AB e r a v v v + = 1 k e r a a a a a + + = 0 a = a 2 P A P A e v v

28、 v + = 3 n PA PA n a e a a a a + + = 4 1 3 PA e r a v v v v + + = CD v OA l AB 2 l x PA A a 45 cos 45 cos v v v + = o o = + = = l v l l v 2 / ) ( 2 / CD OA PA AB y o o 45 cos 45 cos A r a v v v = = l v 2 r 2 4 0 k n PA PA n A r = + + + + a a a a a2 OA l AB 2 l 2 AB 2 l r AB 2 v x 0 45 cos k PA n A = + a a a o 2 2 2 PA AB 5 . 2 2 2 / 2 / = = = l l l l a7.31 cm 20 = r B D BD C AC A A cm/s 60 A = v AC C BD