ÿþZ a s a d a p r a c p r z y g o t o w a n y c h w r u c h u r ó w n a n i e L a g r a n g e a I I - g o r o d z a j u
Z a d a n i e n r : 1
D a n e :
R 2
G 1 = G 2 = G , G 3 = 3 G , = 2 , i 2 x = R 1 2
R 1
S z u k a n e :
P r z y s p i e s z e n i e 1
v 1
( 0 . 1 ) R 2 = 2 R 1 , v 3 =
2
( 0 . 2 ) ´Õ2 = ´Õ3 , Ö = Ö3
2
Z w i z k i k i n e m a t y c z n e :
v 3
v 3 = Ö3 R 1 Ò!Ö3 =
R 1
( 0 . 3 )
v 1 v 1
v 1 = Ö3 R 2 Ò!Ö3 = =
R 2 2 R 1
P r z y j m u j e m y w s p ó Br z d n u o g ó l n i o n q = x 1
1
R ó w n a n i e L a g r a n g e a :
ëø öø
d "E k "E k
( 0 . 4 ) - = Q i
ìø
z
d t "q i ÷ø "q i
íø øø
( 0 . 5 ) ´ L = Q F ´ q k = m 1 g ´ x 1
´ L m 1 g ´ x 1
( 0 . 6 ) Q x = = = m 1 g
1
´ x 1 ´ x 1
1 1 1 1
( 0 . 7 ) E k = m 1 v 1 2 + J o Ö2 2 + J o Ö3 2 + m 3 v 3 2
2 3
2 2 2 2
2 2
2
ëø öø öø
1 1 v 1 1 1 v 1 1 v 1 2
ëø
( 0 . 8 ) E k = m 1 v 1 2 + m 2 R 1 2 + m 3 R 1 2 öøëø ÷ø + m 3 ëø öø
( ) ìø ÷ø ìø ÷ø ìø ÷ø
ìø
2 2 2 R 1 2 2 2
íø øøíø 2 R 1 øø 2 íø øø
íø øø
1 v 1 2 1 v 1 2 1 v 1 2
E k = m 1 v 1 2 + m 2 R 1 2 + m 3 R 1 2 + m 3
2 4 R 1 2 4 4 R 1 2 2 4
m 1 m 2 m 3 m 3
( 0 . 9 ) E k = v 1 2 + v 1 2 + v 1 2 + v 1 2
2 4 1 6 8
8 m 4 m 3 m 6 m 2 1 m
ëø öø
z
E k = + + + v 1 2 = x 1 2
ìø ÷ø
1 6 1 6 1 6 1 6 1 6
íø øø
d îø 2 1 m ùø 2 1 m
ëø öø
z
( 0 . 1 0 ) x
ìø ÷ø
ïø2 íø x 1 øøúø = 8 zz1
d t 1 6
ðø ûø
2 1 m
zz1
x = m g
8
( 0 . 1 1 )
8 8 g
zz1
x = m g = H" 3 , 7 3 7
2 1 m 2 1
2