ÿþ S N y : B i o t e c h n o l o g i a S t u d e n c k i e N o t a t k i C y f r o w e w w w . s n y . o n e . p l
F i z y k a I w i c z e n i a
n o t a t k i z e s t u d i ó w
n a k i e r u n k u B i o t e c h n o l o g i a
n a W y d z i a l e C h e m i c z n y m
P o l i t e c h n i k i W r o c Ba w s k i e j
A u t o r :
M a t e u s z J d r z e j e w s k i
m a t e u s z . j e d r z e j e w s k i @ o n e . p l
w w w . j e d r z e j e w s k i . o n e . p l
W r o c Ba w , 2 0 0 6
S N y : B i o t e c h n o l o g i a
S p i s t r e [c i
I n f o r m a c j e o k u r s i e _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3
Z E S T A W 1 W e k t o r y _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4
Z E S T A W 2 K i n e m a t y k a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 2
Z E S T A W 3 R u c h o b r o t o w y _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 8
Z E S T A W 4 D y n a m i k a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 4
K o l o k w i u m I Z a d a n i a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 9
Z E S T A W 5 P r a c a , e n e r g i a , m o c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3 7
Z E S T A W 6 S p r \y s t o [, p d _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 5
Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 0
K o l o k w i u m I I Z a d a n i a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 6
Z E S T A W 8 R u c h d r g a j c y _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 1
Z E S T A W 9 E l e k t r o s t a t y k a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 6 7
Z E S T A W 1 0 M a g n e t y z m _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 7 4
K o l o k w i u m I I I Z a d a n i a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 8 3
S N y : B i o t e c h n o l o g i a
W s t p
o t a t k a j e s t c z [c i p r o j e k t u S N y ( S t u d e n c k i e N o t a t k i
N
C y f r o w e ) . N o t a t k i s s a m o d z i e l n i e s p o r z d z a n e
i o p r a c o w y w a n e p r z e z s t u d e n t ó w P o l i t e c h n i k i . U d o s t p n i a n e
s w I n t e r n e c i e . K a \d y m o \n e z n i c h k o r z y s t a d o w o l i
w c e l a c h e d u k a c y j n y c h .
a k o n k r e t n a n o t a t k a d o t y c z y k u r s y F Z C 1 0 0 3 c , c z y l i
T
w i c z e D d o w y k Ba d ó w z F i z y k i I . N o t a t k a z a w i e r a
p r z y k Ba d o w e o p r a c o w a n i e s z k i c ó w r o z w i z a D d o z e s t a w ó w
z a d a D. N i e s t o w i c p e Bn e r o z w i z a n i a d o z a d a D.
Z a d a n i a p o c h o d z z l i s t o m a w i a n y c h n a w i c z e n i a c h o r a z
z k o l o k w i ó w . S z k i c e b d , w m i a r m o \l i w o [c i , o p a t r y w a n e
d o d a t k o w y m i r y s u n k a m i i k o m e n t a r z a m i .
w a g a n a b Bd y ! M i m o s t a r a n n o [c i j a k w Bo \y l i a u t o r z y
U
w o p r a c o w a n i e t e j n o t a t k i m o g z d a r z y s i b Bd y .
K a \d y w i c k o r z y s t a z t y c h m a t e r i a Bó w n a w Ba s n
o d p o w i e d z i a l n o [. W s z e l k i e z a u w a \o n e b Bd y p r o s z
z g Ba s z a a u t o r o w i n o t a t k i ( n a j l e p i e j d r o g e l e k t r o n i c z n ) .
[y c z w s z y s t k i m s k u t e c z n e g o k o r z y s t a n i a z n o t a t e k .
M a t e u s z J d r z e j e w s k i
( a u t o r s t r o n y w w w . s n y . o n e . p l )
S z c z e g ó Bo w e i n f o r m a c j e o n o t a t c e
N a z w a p l i k u : e - n o t a t k a - F i z y k a I - c w i c z e n i a . p d f
N a z w a k u r s u : F i z y k a I ( F Z C 1 0 0 3 C )
P r o w a d z c y k u r s : d r E l \b i e t a B r o n i e k
S e m e s t r / r o k : 0 6 z ( r o k 1 , I s e m e s t r )
K i e r u n e k : B i o t e c h n o l o g i a
W y d z i a B: W y d z i a B C h e m i c z n y
U c z e l n i a : P o l i t e c h n i k a W r o c Ba w s k a
A u t o r n o t a t k i : M a t e u s z J d r z e j e w s k i
S t a t u s : n i e p e Bn a
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 0
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : I n f o r m a c j e o k u r s i e . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 0
I n f o r m a c j e o k u r s i e
N a z w a k u r s u : F i z y k a I w i c z e n i a
K o d k u r s u : F Z C 1 0 0 3 C
P r o w a d z c y : d r E l \b i e t a B r o n i e k
K o n s u l t a c j e : c z w a r t k i , g o d z i n y 1 1 : 1 5 - 1 5 : 0 0 , b u d . F - 3 p o k . 2 3 3 ( p o k . 2 2 8 ) ,
P r z e b i e g z a j : l i s t y z a d a D d o s t p n e s n a s t r o n i e h t t p : / / a r . c h . p w r . w r o c . p l
W a r u n k i z a l i c z e n i a : z a l i c z e n i e ( m i n i m u m 5 0 % p k t . ) w s z y s t k i c h t r z e c h k o l o k w i ó w
k o l o k w i u m I : 3 l i s t o p a d a 2 0 0 6 r .
k o l o k w i u m I I : 8 g r u d n i a 2 0 0 6 r . ( e w e n t u a l n i e 1 g r u d n i a 2 0 0 6 r . )
k o l o k w i u m I I I : 1 2 s t y c z n i a 2 0 0 7 r .
k o l o k w i u m p o p r a w k o w e ( z c a Bo [c i ) : 1 9 s t y c z n i a 2 0 0 7 r .
i n f o r m a c j e d o d a t k o w e : n a k o l o k w i a c h m o \n a u \y w a k a l k u l a t o r ó w
w a \n e : z a l i c z e n i e k o l o k w i ó w z w i c z e D j e s t w a r u n k i e m K O N I E C Z N Y M
d o p r z y s t p i e n i a d o e g z a m i n u z F i z y k i I . T e n e g z a m i n b d z i e w f o r m i e
t e s t o w e j z u j e m n y m i p u n k t a m i z a b Bd n o d p o w i e d z ( 0 , 2 5 p k t . ) .
P r z e l i c z a n i e o c e n n a p u n k t y ( m a k s i m u m t o 6 0 p k t . ) :
o d 3 0 p k t . 3
o d 3 6 p k t . 3 , 5
o d 4 2 p k t . 4
o d 4 8 p k t . 4 , 5
o d 5 4 p k t . 5
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
Z E S T A W 1 W e k t o r y
6 . 1 0 . 2 0 0 6 r .
S Bo w n i c z e k
k o l i n e a r n y r ó w n o l e g By ,
o r t o g o n a l n y p r o s t o p a d By ,
w e k t o r j e d n o s t k o w y w e k t o r o d Bu g o [c i 1 ,
w e r s o r w e k t o r j e d n o s t k o w y r ó w n o l e g By d o j e d e n z o s i u k Ba d w s p ó Br z d n y c h i m a j c y
t e n s a m z w r o t c o o n a .
D a n e s d w a w e k t o r y
r r
u = [ u x , u y ] v = [ v x , v y ]
D Bu g o [ w e k t o r a
r
u = u x 2 + u y 2
K t p o m i d z y w e k t o r e m a d a n o s i w s p ó Br z d n y c h
u y
u x r r
= c o s "( u , O X ) = c o s "( u , O Y )
r r
u u
I l o c z y n s k a l a r n y
r r r r r r r r
u Å"v = u Å" v Å"c o s "( u , v ) u Å"v = u x Å" v x + u y Å"u y
w i c :
r r r r
u Å"v = 0 Ô! u ¥" v
I l o c z y n w e k t o r o w y
r r r r r r
u × v = u Å" v Å"s i n "( u , v )
w i c :
r r r r
u Å"v = 0 Ô! u v
z a d . 1 .
D a n e : A ( - 1 , 0 , 3 ) , B ( 0 , - 2 , 5 ) ,
r r r r r
S z u k a n e : a = A B , a = ? , "( a , O X ) = ? , "( a , O Y ) = ? , "( a , O Z ) = ? ,
r r
n = ? ( w e k t o r j e d n o s t k o w y k o l i n e a r n y d o a i o z w r o c i e p r z e c i w n y m ) ,
v
a = [ 0 - ( - 1 ) , - 2 - 0 , 5 - 3 ] = [ 1 , - 2 , 2 ]
v
a = 1 2 + ( - 2 ) 2 + 2 2 = 9 = 3 = a
a x r r a x
= c o s "( a , O X ) Ò! "( a , O X ) = a r c o s
a a
r a x 1
"( a , O X ) = a r c o s = a r c o s H" 7 0 , 5 °
a 3
a y
r - 2
"( a , O Y ) = a r c o s = a r c o s H" 1 3 1 , 8 °
a 3
r a z 2
"( a , O Z ) = a r c o s = a r c o s H" 4 8 , 2 °
a 3
r r
w e k t o r m ( j e d n o s t k o w y i k o l i n e a r n y ) d o w e k t o r a a
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
r
îø
r a a x a y a z ùø
m = = , ,
r
ïø úø
a a a a
ðø ûø
r 1 - 2 2
îø ùø
1 2 2
m = , , = [ , - , ]
3 3 3
ïø3 3 3 úø
ðø ûø
r r
w e k t o r n j e s t p r z e c i w n y d o w e k t o r m
r r
n = - m
r
1 2 2 1 2 2
n = - [ , - , ] = [ - , , - ]
3 3 3 3 3 3
z a d . 2 .
D a n e : &
r
S z u k a n e : a = ? ,
x 1 = 0 k m
ñø
òø
= - 3 , 5 k m
óøy 2
1
ñø
2
ôøx = 8 , 2 k m Å" H" 5 , 8 k m
2
òø
ôøy 2 = 8 , 2 k m Å" 1 2 H" 5 , 8 k m
óø
x 1 =
ñø - 1 5 k m
òø
= 0 k m
óøy 2
r
2 2
a = ( 0 + 5 , 8 - 1 5 ) + ( - 3 , 5 + 5 , 8 + 0 ) H" 9 , 4 8 k m
z a d . 3 .
D a n e : r 1 = 1 5 0 c m , ±1 = 1 2 0 ° , r w = 1 4 0 c m , ±w = 3 5 ° ,
r
S z u k a n e : r 2 = ? ,
r r r
r w = r 1 + r 2
r r r
r 2 = r w - r 1 = [ r w c o s ±w , r w s i n ±w ] - [ r 1 c o s ±1 , r 1 s i n ±1 ]
r
r 2 = [ 1 4 0 c o s 3 5 ° - 1 5 0 c o s 1 2 0 ° , 1 4 0 s i n 3 5 ° - 1 5 0 s i n 1 2 0 ° ] H" [ 1 8 9 , 7 ; - 4 9 , 6 ]
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
z a d . 4 .
r
r
D a n e : a = ( 3 , - 4 , 4 ) , b = ( 2 , 3 , - 7 ) ,
r r
r r
S z u k a n e : c = ? , d = ? , "( c , d )
r
r r
c = a + b = [ 3 + 2 , - 4 + 3 , 4 - 7 ] = [ 5 , - 1 , - 3 ]
r r
r
d = 2 a - b = [ 2 Å" 3 - 2 , 2 Å" ( - 4 ) - 3 , 2 Å" 4 + 7 ] = [ 4 , - 1 1 , 1 5 ]
r r r
r r r
c o d = c Å" d Å"c o s "( c , d )
r
r
r
c x Å" d x + c y Å" d y + c z Å" d z
r c o d
c o s "( c , d ) = r = r
r r
c Å" d c Å" d
r
c = c x 2 + c y 2 + c z 2 = 5 2 + ( - 1 ) 2 + ( - 3 ) 2 H" 5 , 9 2
r
d = d x 2 + d y 2 + d z 2 = 4 2 + ( - 1 1 ) 2 + 1 5 2 H" 1 9 , 0 3
r
c x Å" d x + c y Å" d y + c z Å" d z
r
"( c , d ) = a r c c o s r
r
c Å" d
r
r 5 Å" 4 + ( - 1 ) Å"( - 1 1 ) + ( - 3 ) Å"1 5
"( c , d ) H" a r c c o s
5 , 9 2 Å"1 9 , 0 3
r
r - 1 4
"( c , d ) H" a r c c o s H" 9 7 , 1 °
1 1 2 , 6 6
z a d . 5 .
r r r r
D a n e : F 1 = 5 , F 2 = 8 , "( F 1 , F 2 ) = 1 2 0 °
r
S z u k a n e : F w = ? ,
n i e c h p u n k t P b d z i e w ( 0 , 0 ) ,
r
n i e c h w e k t o r F 1 b d z i e k o l i n e a r n y o o s i O X , a j e g o z w r o t z g o d n y z t o s i .
r r r
F w = F 1 + F 2 = [ F 1 + F 2 c o s 1 2 0 ° , F 2 s i n 1 2 0 ° ] H" [ 1 ; 6 , 9 3 ]
r
F w = F w = F w x 2 + F w y 2 H" 1 2 + 6 , 9 3 2 H" 7
z a d . 6 .
r
r r
D a n e : a ( 1 , - 1 ) , b ( 4 , 3 ) , c ( - 1 0 , - 1 1 ) ,
S z u k a n e : x , y ,
r
r r
c = x Å" a + y Å"b
c x = x Å" a x + y Å"b x
ñø
òøc = y Å" a y + y Å"b y
y
óø
ñø- 1 0 = x + 4 y
òø- 1 1 = - x + 3 y
óø
- 2 1 = 7 y Ò! y = - 3
x = 3 y + 1 1 Ò! x = - 3 Å"3 + 1 1 = 2
r
r r
c = 2 a + 3 b
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
z a d . 7 .
r
r
D a n e : a , b ,
r
r
S z u k a n e : "( a , b ) = ? ,
r
W e k t o r a l e \y n a d w u s i e c z n e j k t a m i d z y o s i a m i O X i O Y , w i c j e g o p o s t a
r
m o \e b y n a s t p u j c a : a = [ 1 , 1 , 0 ] .
r
W e k t o r b l e \y n a d w u s i e c z n e j k t a m i d z y o s i a m i O Y i O Z , w i c j e g o p o s t a
r
m o \e b y n a s t p u j c a : b = [ 0 , 1 , 1 ] .
r r r
r r r
a o b = a Å" b Å"c o s "( a , b )
r
a = 1 2 + 1 2 + 0 2 = 2
r
b = 0 2 + 1 2 + 1 2 = 2
r
r
r
r a o b 1 Å"0 + 1 Å"1 + 0 Å"1 1
"( a , b ) = a r c c o s = a r c c o s = a r c c o s = 6 0 °
r
r
2
2 Å" 2
a Å" b
z a d . 8 .
r
r
D a n e : a ( 2 , 1 , 1 ) , b ( 1 , - 1 , 2 ) ,
r r
S z u k a n e : a b = ? , a ¥"b = ? ,
r
a = 2 2 + 1 2 + 1 2 = 6
r
b = 1 2 + ( - 1 ) 2 + 2 2 = 6
r r r
a b = a b Å" n b
r
r b îø 1 1 2 ùø
n b = r = , - ,
ïø úø
6 6 6
b ðø ûø
r
r r
a b
r r r r
= c o s "( a , b ) Ò! a b = a Å"c o s "( a , b )
r
a
r
r
r r r r
r r r r a o b
a o b = a Å" b Å" c o s "( a , b ) Ò! c o s "( a , b ) = r
r
a Å" b
r
r
r
a x b x + a y b y + a z b z
r r r r r a o b r r
a b = a Å"c o s "( a , b ) Å" n b = a Å" r Å" n b = r Å" n b
r
a Å" b b
r 2 Å"1 + 1 Å"( - 1 ) + 1 Å" 2 r 3 îø 1 1 2 ùø
1 1
a b = Å" n b = Å" , - , = [ , - , 1 ]
2 2
ïø úø
6 6 6 6 6
ðø ûø
r
1 1
a b = [ , - , 1 ]
2 2
r
2 2
6
1 1
a b = ( ) + ( - ) + 1 2 =
2 2 2
2 2
r r r r r r
2 2
6 3 2
a ¥"b = a - a b Ò! a ¥"b = a - a b = 6 - ( ) =
2 2
r r r r
3 2
a ¥"b = a ¥"b Å" n ¥"b = Å" n ¥"b
2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
r
w i e m , \e w e k t o r n ¥"b = [ x , y , z ] s p e Bn i a n a s t p u j c e w a r u n k i :
r r
r r r r
ñøn ¥"b o b = n ¥"b Å" b Å"c o s "( n ¥"b , a b )
ôør r r r
r r
ôøn o a = n ¥"b Å" a Å"c o s "( n ¥"b , a )
òø
¥"b
r
ôø
n ¥"b = 1
ôø
óø
r r
"( n ¥"b , a b ) = 9 0 ° Ò! c o s 9 0 ° = 0
r
a b 2 6
r r 1 r r
c o s "( a , a b ) = = = Ò! "( a , a b ) = 6 0 °
r
a 2
6
r r r r r r
"( n ¥"b , a ) = "( n ¥"b , a b ) - "( a , a b )
r r
3
"( n ¥"b , a ) = 9 0 ° - 6 0 ° = 3 0 ° Ò! c o s 3 0 ° =
2
r r
r
r r
r
ñøn ¥"b o b = n ¥"b Å" b Å"c o s 9 0 ° ñøn ¥"b o b = 0
ôør r r r
r
ôøn o a = n ¥"b Å" a Å"c o s 3 0 ° Ò! ôør o a = 1 Å" 6 Å" 3
ôøn
òø òø
¥"b ¥"b 2
r
ôø ôøx + y 2 + z 2 = 1 2
2
n ¥"b = 1
ôø
ôø
óø
óø
x Å"b x + y Å"b y + z Å"b z = 0
ñø
x
ñø - y + 2 z = 0
ôø
ôø
òøx Å" a x + y Å"a y + z Å" a z = 3 2 Ò! òø2 x + y + z = 3 2
2 2
ôø ôøx + y 2 + z 2 = 1
2
2
óø
óøx + y 2 + z 2 = 1
2 z = y
ñø - x 2 z = y 2 z = y
ñø - x
ñø - x
ôø ôø ôø
òø4 x + 2 y + 2 z = 3 2 Ò! òø4 x + 2 y + y - x = 3 2 Ò! òø3 x + 3 y = 3 2
ôøx + y 2 + z 2 = 1 ôøx + y 2 + z 2 = 1 ôøx + y 2 + z 2 = 1
2 2 2
óø óø óø
2
ñøy = 2 - y + 2 z ñø -
y = x + 2 z z = y
ñø
2
ôø
ôø
ôøx = 2 - y Ò! ôø = 2 - y
ôøx
òøx = 2 - y Ò! òø òø
ôøx + y 2 + z 2 = 1 ôøx + y 2 + z 2 = 1 ôøx + y 2 + z 2 = 1
2 2 2
ôø ôø
óø
óø óø
2 2
ñø - z = y
ñø -
z = y
2 2
ôø ôø
ôøx = 2 - y ôøx
Ò! = 2 - y
òø òø
ôø ôø2 - 2 2 y + y 2 + y 2 + y 2 - y 2 + 1 = 1
2
2
2
ôø( 2 - y ) + y 2 + ( y - ) = 1 ôø
2
óø 2 óø
1
2 - 2 2 y + y 2 + y 2 + y 2 - y 2 + = 1
2
3 2
3 y 2 - 3 2 y + = 0 / Å"
2 3
2 y 2 - 2 2 y + 1 = 0
" = 8 - 8 = 0
- 2 2 2
y = - =
4 2
2
ñø -
z = y
z = 0
ñø
2
ôø
ôø r
ôøx = 2 - y Ò! = 2 Ò! n ¥"b = 2 , 2 , 0 ]
[
òø òøx
2 2 2
ôøy = 2 ôøy = 2
ôø
2 óø 2
óø
r r r r
3 2 2 2 3 3
a ¥"b = a ¥"b Å" n ¥"b Ò! a ¥"b = Å"[ , , 0 ] = [ , , 0 ]
2 2 2 2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 9
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
z a d . 9 .
D a n e : A ( 2 , 0 ) , B ( 1 , 3 ) , C ( - 5 , 1 ) ,
r
S z u k a n e : n = ? ,
r
a = B A = [ 2 - 1 , 0 - 3 ] = [ 1 , - 3 ]
a = 1 2 + ( - 3 ) 2 = 1 0
r
b = B C = [ - 5 - 1 , 1 - 3 ] = [ - 6 , - 2 ]
b = ( - 6 ) 2 + ( - 2 ) 2 = 2 1 0
r
n = [ n x , n y ]
r
r r r
c o s ± = c o s "( n , a ) = c o s "( n , b )
r
r r r
ñøn o a n o b
=
ôø
n Å" a n Å"b
òø
r
ôø
n = n = 1
óø
n x Å"a x + n y Å" a y n x Å"b x + n y Å"b y
ñø
=
ôø
1 Å" a 1 Å"b
òø
ôø
n x 2 + n y 2 = 1
óø
n x Å"1 + n y Å"( - 3 ) n x Å"( - 6 ) + n y Å"( - 2 )
ñø
=
ôø
1 Å" 1 0 1 Å" 2 1 0
òø
ôøn 2 + n y 2 = 1
x
óø
( n x
ñø - 3 n y ) = - 6 n x - 2 n y
ôø2
òø
2 2
ôø
óøn x + n y = 1
ñø
y
ôøn = 2 n x
òø
2 2
ôø ( 2 n x ) = 1
+
óøn x
ñø
y
ôøn = 2 n x Ò! ñø = 2 n x Ò! ñøn y = ± 2 1
ôøn y ôø 5
òø òø òø
2
1
1
ôø = 1 ôø
ôø
óø5 n x 5
óøn x = ± 5
óøn x = ±
r
1 1
n = [ - , - 2 ] H" [ - 0 , 4 4 7 ; - 0 , 8 9 4 ]
5 5
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 0
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
z a d . 1 0 .
r
D a n e : a = ( 3 , 6 , 8 ) ,
r
S z u k a n e : n = ? ,
r r r r
n ¥" a Ô! n o a = 0
r r r
n ¥" O X Ô! n ¥" î Ô! n o i Æ = 0
î = [ 1 , 0 , 0 ]
r
ñø n = 1
ôør r
òøn o a = 0
r
ôøn o î = 0
óø
ñø
n x 2 + n y 2 + n z 2 = 1
ôø
ôø
òøn Å" a x + n y Å" a y + n z Å" a z = 0
x
ôøn Å"1 + n y Å"0 + n z Å"0 = 0
x
ôø
óø
ñø
n x 2 + n y 2 + n z 2 = 1
2 2 2 2
ñø ñø
+ n z = 1 + n z = 1
ôø
ôøn y ôøn y
Ò!
òø3 n + 6 n y + 8 n z = 0 Ò! òø òø
x
4
ôø
ôøn = 0
óø6 n y + 8 n z = 0 ôø = - 3 n z
óøn y
x
óø
4 9
ñø ñø ñø
( - n z ) 2 + n z 2 = 1 n z 2 = n z = ± 0 , 6
3 2 5
ôø ôø ôø
Ò! = - n z Ò! = m 0 , 8
òøn = - 4 n z òøn 4 òøn
y 3 y 3 y
ôøn = 0 ôøn = 0 ôøn = 0
óø x óø x óø x
r r
n = [ 0 ; - 0 , 8 ; 0 , 6 ] l u b n = [ 0 ; 0 , 8 ; - 0 , 6 ]
z a d . 1 1 .
r
r r
D a n e : a = ( 1 , 2 , 1 ) , b = ( 2 , - 1 , 2 ) , c = 2 ,
r
S z u k a n e : c = ? ,
r r r
c = c Å" n
r r r r r r
c ¥" a Ô! n ¥" a Ô! n o a = 0
r r
r r
n ¥" b Ô! n o b = 0
r
ñø n = 1
ôør r
òøn o a = 0
r
r
ôøn o b = 0
óø
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 1
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 1
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 W e k t o r y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 9 : 2 3
ñø
n x 2 + n y 2 + n z 2 = 1
ôø
ôøn Å" a x + n y Å"a y + n z Å" a z = 0
òø
x
ôøn Å"b x + n y Å"b y + n z Å"b z = 0
x
ôø
óø
ñø ñø
n x 2 + n y 2 + n z 2 = 1 n x 2 + n y 2 + n z 2 = 1
ñø
n x 2 + n z 2 = 1
ôø ôø
ôø
Ò! = - n z
òø1 Å"n + 2 Å" n y + 1 Å" n z = 0 Ò! òøn = - 2 Å"n y - n z òøn
x x x
ôø2 Å" n x - 1 Å" n y + 2 Å" n z = 0 ôø- 4 Å"n y - 2 Å" n z - n y + 2 Å" n z = 0 ôøn = 0
y
óø
óø óø
1
2
ñø
n z = ±
ñø
( - n z ) + n z 2 = 1
2
ôø
ôø
Ò! = m
òøn = - n z òøn 1 2
x x
ôøn = 0 ôøn = 0
y
óø y
óø
r
1 1
n = [ ± , 0 , m ]
2 2
r r
v v v
c = c Å" n Ò! c = [ 1 , 0 , - 1 ] l u b c = [ - 1 , 0 , 1 ]
z a d . 1 2 .
a )
r
r
D a n e : a = ( 1 , 2 , 3 ) , b = ( 0 , - 2 , 5 ) ,
S z u k a n e : P = ? ( p o l e r ó w n o l e g Bo b o k u ) ,
2 2
2
r
a y a z a x a y
a z a x
r
P = a × b = + +
b y b z b x b y
b z b x
2 2 2
P = ( a y b z - a z b y ) + ( a z b x - a x b z ) + ( a x b y - a y b x )
2 2 2 2 2
P = ( 2 Å"5 - 3 Å" ( - 2 ) ) + ( 3 Å"0 - 1 Å"5 ) + ( 1 Å"( - 2 ) - 3 Å"0 ) = 1 6 2 + ( - 5 ) + ( - 2 ) = 2 8 5
b )
D a n e : A ( 1 , - 1 , 3 ) , B ( 0 , 2 , - 3 ) , C ( 2 , 2 , 1 ) ,
S z u k a n e : P = ? ( p o l e t r ó j k t a ) ,
r
a = A B = [ 0 - 1 , 2 - ( - 1 ) , - 3 - 3 ] = [ - 1 , 3 , - 6 ]
r
b = A C = [ 2 - 1 , 2 - ( - 1 ) , 1 - 3 ] = [ 1 , 3 , - 2 ]
r
r
2 2 2
1 1
P = a × b = ( a y b z - a z b y ) + ( a z b x - a x b z ) + ( a x b y - a y b x )
2 2
2 2 2
1
P = ( 3 Å"( - 2 ) - ( - 6 ) Å"3 ) + ( - 6 Å"1 - ( - 1 ) Å"( - 2 ) ) + ( - 1 Å"3 - 3 Å"1 )
2
2 2
1 1 1 2 4 4
P = 1 2 2 + ( - 8 ) + ( - 6 ) = 2 4 4 = 2 4 4 = = 6 1
2 2 4 4
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 2
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
Z E S T A W 2 K i n e m a t y k a
1 3 . 1 0 . 2 0 0 6 r .
P r z y s p i e s z e n i e
r
a w e k t o r p r z y s p i e s z e n i e c a Bk o w i t e g o ,
a w a r t o [ p r z y s p i e s z e n i a c a Bk o w i t e g o ,
r
a n w a r t o [ p r z y s p i e s z e n i a n o r m a l n e g o ( p r o s t o p a d Be g o d o v ) ,
r
a s w a r t o [ p r z y s p i e s z e n i a s t y c z n e g o ( r ó w n o l e g Be g o d o v ) ,
r r r
a = a n + a s
a 2 = a n 2 + a s 2
v 2
a n = R p r o m i e D k r z y w i z n y t o r u ,
R
t o p r z y s p i e s z e n i e w p By w a n a z m i a n k i e r u n k u p r d k o [c i ,
d v
a s = d v p r d k o [ c h w i l o w a ,
d t
t o p r z y s p i e s z e n i e w p By w a n a z m i a n w a r t o [c i p r d k o [c i ,
z a d . 1 .
D a n e : x ( t ) = 2 + 3 t - 4 t 2 ,
S z u k a n e : t z = ? ( c z a s z a w r a c a n i a ) , a = ? , v ( x ( t = 0 ) ) = ? ,
x ( t ) = 2 + 3 t - 4 t 2
d x
v ( t ) = = 3 - 8 t b o ( a x n ) 2 = a n x n - 1
d t
d v
a ( t ) = = - 8
d t
w i c j e s t t o r u c h j e d n o s t a j n i e o p ó zn i o n y
3
v ( t z ) = 0 Ò! 3 - 8 t z = 0 Ò! t z = [ s ]
8
2
3 3
x ( t z ) = 2 + 3 Å" - 4 Å"( ) H" 2 , 5 6 [ m ]
8 8
x ( t = 0 ) = 2
3
2 = 2 + 3 t - 4 t 2 Ò! 0 = t ( 3 - 4 t ) Ò! t = 0 [ s ] (" t = [ s ]
4
3 3
v ( t = ) = 3 - 8 Å" = - 3 [ m ]
4 4 s
z a d . 2 .
D a n e : v ( t ) = 4 0 - 5 t 2 ,
S z u k a n e : a [r = ? ( w p r z e d z i a l e 0 - 2 s e k . ) , a ( t = 2 ) = ? ,
v ( t ) = 4 0 - 5 t 2
d v
a ( t ) = = - 1 0 t
d t
a ( t = 0 ) = 0
a ( t = 2 ) = - 2 0
a ( t = 0 ) + a ( t = 2 ) 0 - 2 0
a [r = = = - 1 0
2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 3
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
z a d . 3 .
m m
D a n e : H = 5 0 m , v 0 = 2 0 , t D = 5 s , g = 9 , 8 1
s
s 2
S z u k a n e : t B = ? , y ( t B ) = ? , t C = ? , v ( t C ) = ? ,
v ( t D ) = ? , x ( t D ) = ? , t E = ? ( c z a s c a Bk o w i t y ) , v ( t E ) = ? ,
1
y ( t ) = y 0 + v 0 t - g t 2
2
v y ( t ) = v 0 - g t
a )
v 0 2 0
v y ( t B ) = 0 Ò! 0 = v 0 - g t B Ò! t B = H" H" 2 , 0 4 s
g 9 , 8 1
b )
1
y ( t B ) = v 0 t B - g t 2
2
2
ëø öø
v 0 1 v 0 v 0 2 v 0 2 v 0 2
y ( t B ) = v 0 - g ìø ÷ø = - =
2
ìø ÷ø
g g g 2 g 2 g
íø øø
v 0 2 2 0 2
y ( t B ) = H" = 2 0 , 4 m
2 g 2 Å" 9 , 8 1
c )
y ( t C = 0 ) = 0
1
v 0 t C - g t C 2 = 0
2
1
t C ( v 0 - g t C ) = 0
2
2 v 0 2 Å" 2 0
t C = 0 (" t C = H" H" 4 , 0 8 s
g 9 , 8 1
m
v ( t C ) = v 0 - g t C H" 2 0 - 9 , 8 1 Å" 4 , 0 8 = - 2 0
s
d )
1
y ( t D ) = v 0 t D - g t D 2
2
1
y ( t D ) H" 2 0 Å"5 - Å"9 , 8 1 Å"5 2 H" - 2 2 , 6 m
2
v ( t D ) = v 0 - g t D
m
v ( t D ) H" 2 0 - 9 , 8 1 Å"5 = - 2 9
s
e )
y ( t E ) = - H = - 5 0 m
1
- H = v 0 t E - g t E 2
2
g t E 2 - 2 v 0 t E - 2 H = 0
" = 4 v 0 2 + 8 g H
" = 2 v 0 2 + 2 g H
2 v 0 + 2 v 0 2 + 2 g H v 0 + v 0 2 + 2 g H
t E > 0 Ò! t E = =
2 g g
2 0 + 2 0 2 + 2 Å"9 , 8 1 Å"5 0 2 0 + 3 7 , 1 6
t E H" H" H" 5 , 8 3 s
9 , 8 1 9 , 8 1
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 4
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
f )
v y ( t E ) = v 0 - g t E
v 0 + v 0 2 + 2 g H
v y ( t E ) = v 0 - g Å"
g
v y ( t E ) = v 0 - v 0 - v 0 2 + 2 g H
v y ( t E ) = - v 0 2 + 2 g H
m
v y ( t E ) H" - 2 0 2 + 2 Å"9 , 8 1 Å"5 0 H" - 3 7 , 2
s
z a d . 4 .
r
D a n e : r ( 1 8 t , 4 t - 4 , 9 t 2 ) ,
S z u k a n e : v 0 = ? , ± = ? , v ( t = 2 ) = ? ,
v H = ? ( p r d k o [ w n a j w y \s z y m p u n k c i e t o r u ) ,
r x = x ( t ) = 1 8 t
r y = y ( t ) = 4 t - 4 , 9 t 2
x ( t ) = v x t
0
v x = v x = c o n s t .
0
d x
v x ( t = 0 ) = v x = = 1 8 [ m ]
s
0
d t
v y ( t ) = v y - g t
0
d y
v y ( t = 0 ) = v y = = 4 - 9 , 8 t = 4 [ m ]
s
0
d t
r
v = [ v x , v y ]
0 0
v 0 = v x 2 + v y 2 = 1 8 2 + 4 2 = 1 8 , 4 [ m ]
s
0 0
v x = v 0 c o s ±
0
v y = v 0 s i n ±
0
v y v y
4
0 0
t g ± = Ò! ± = a r c t g = a r c t g H" 1 2 , 5 °
v x v x 1 8
0 0
w n a j w y \s z y m p u n k c i e t o r u v y = 0
r
v H = [ v x , v y ] = [ v x , 0 ] = [ 1 8 , 0 ]
0
v H = 1 8 2 + 0 2 = 1 8 [ m ]
s
t = 2 s
r
v = [ v x , v y ] = [ 1 8 ; 4 - 9 , 8 t ] = [ 1 8 ; 4 - 9 , 8 Å" 2 ] = [ 1 8 ; - 1 5 , 6 ]
v = 1 8 2 + ( - 1 5 , 6 ) 2 H" 2 3 , 8 [ m ]
s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 5
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
z a d . 5 .
m
D a n e : ± = 3 0 ° , v 0 = 2 0 [ m ] , H = 4 5 m , g = 9 , 8 1 [ s ]
2
s
S z u k a n e : t c = ? , v k = ? , a s = ? ( w c h w i l i t = 0 , 5 s ) ,
z = ? ( z a s i g o d l e g Bo [ z p o z i o m i e p r z e b y t a p r z e z k a m i e D) ,
a )
v y = v 0 s i n ± = 2 0 s i n 3 0 ° = 1 0 [ m ]
s
0
v x = v x = v 0 c o s ± = 2 0 c o s 3 0 ° H" 1 7 , 3 2 [ m ]
s
0
1
y = H + v y t c - g t c 2
2
0
0 = 4 5 + 1 0 t c - 4 , 9 0 5 t c 2
" = 1 0 0 + 8 8 2 , 9 = 9 8 8 , 2
" H" 3 1 , 3 5
1 0 + 3 1 , 3 5
t c > 0 Ò! t c = H" 4 , 2 1 s
2 Å" 4 , 9 0 5
b )
v x = c o n s t . H" 1 7 , 3 2
v y = v y - g t c
0
v y H" 1 0 - 9 , 8 1 Å" 4 , 2 1 H" - 3 1 , 3
v k = v x 2 + v y 2 H" 1 7 , 3 2 2 + ( - 3 1 , 3 ) 2 H" 3 5 , 8 [ m ]
s
c )
m
v x = c o n s t . H" 1 7 , 3 2
s
z = v x t c
z H" 1 7 , 3 2 Å" 4 , 2 1 H" 7 3 m
d )
d v
a s =
d t
v = v x 2 + v y 2 = v x 2 + ( v y - g t ) 2
0 0
d v x 2 + ( v y - g t ) 2
2 ( v y - g t ) Å" g g Å" ( v y - g t )
0 0
0 0
a s = = =
d t
2 v x 2 + ( v y - g t ) 2 v x 2 + ( v y - g t ) 2
0 0 0 0
1
b o ( x ) 2 = { f [ g ( x ) ] } 2 = f ' [ g ( x ) ] Å" g ' ( x )
2 x
t = 0 , 5
9 , 8 1 Å"( 1 0 - 9 , 8 1 t ) 9 , 8 1 Å"( 1 0 - 9 , 8 1 Å"0 , 5 ) 4 9 , 9 8
m
a s ( t ) = = = H" 2 , 7 7 [ s ]
2
1 7 , 3 2 2 + ( 1 0 - 9 , 8 1 t ) 2 1 7 , 3 2 2 + ( 1 0 - 9 , 8 1 Å"0 , 5 ) 2 1 8 , 0 5
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 6
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
z a d . 6 .
m
D a n e : v 0 = 7 , 5 5 [ m ] , t = 0 , 5 s , g = 9 , 8 1 [ s ]
2
s
S z u k a n e : v = ? , a s = ? , a n = ? ,
v x = v 0 = c o n s t .
x ( t ) = v 0 Å"t
v y ( t ) = - g t
1
y ( t ) = - g t 2
2
d v
a s =
d t
v = v x 2 + v y 2 = v 0 2 + ( - g t ) 2 = v 0 2 + g 2 t 2
d v 0 2 + g 2 t 2 v y
2 g 2 t g t
a s = = = g = g
d t
2 v 0 2 + g 2 t 2 v 0 2 + g 2 t 2 v
1
b o ( x ) 2 = { f [ g ( x ) ] } 2 = f ' [ g ( x ) ] Å" g ' ( x )
2 x
t = 0 , 5
g 2 t 9 , 8 1 2 Å"0 , 5 4 8 , 1 1 8
m
a s ( t ) = H" = H" 5 , 3 4 [ s ]
2
7 , 5 5 2 + 9 , 8 1 2 Å" 0 , 5 2 9 , 0 0 3
v 0 2 + g 2 t 2
a n 2 = g 2 - a s 2
2
v y 2 g 2 v 2 - g 2 v y 2 g
g
a n = g 2 - a s 2 = g 2 - g 2 = = v 2 - v y 2 = v x 2 + v y 2 - v y 2
v 2 v 2 v v
g g v 0
a n = v x 2 + v y 2 - v y 2 = v x 2 = g
v v v
v 0 9 , 8 1 Å"7 , 5 5 7 4 , 0 6 5 5
m
a n ( t ) = g H" H" H" 8 , 2 3 [ s ]
2
7 , 5 5 2 + 9 , 8 1 2 0 , 5 2 9 , 0 0 3 4
v 0 2 + g 2 t 2
z a d . 7 .
D a n e : R = 3 , 6 4 m v = 1 7 , 4 [ m ] , ± = 2 2 ° ,
s
S z u k a n e : a c = ? , a s = ?
a c 2 = a n 2 + a s 2
v 2 1 7 , 4 2
a n = = H" 8 3 , 1 7 6
R 3 , 6 4
r r r
± = "( a c , R ) a" "( a c , a n )
a s
m
t g ± = Ò! a s = a n Å" t g ± Ò! a s H" 8 3 , 1 7 6 Å"0 , 4 0 4 H" 3 3 , 6 [ s ]
2
a n
m
a c = a n 2 + a s 2 H" 8 3 , 1 7 6 2 + 3 3 , 6 2 H" 8 9 , 7 [ s ]
2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 7
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 2 K i n e m a t y k a . Z m o d y f i k o w a n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 3
z a d . 8 .
m
D a n e : R = 0 , 2 m a s = 0 , 0 5 , ± = 2 2 ° ,
s 2
S z u k a n e : a s = a n Ò! t 1 = ? , 2 a s = a n Ò! t 2 = ? ,
a )
a n = a s
v 2
a n = Ò! v = a n Å" R
R
a n Å" R
v
v = a s Å" t Ò! t = =
a s a s
a n Å" R a s Å" R
R 0 , 2
t 1 = = = = = 2 s
a s a s a s 0 , 0 5
b )
1
a n = 2 a s Ò! a s = a n
2
a n Å" R 2 a s Å" R
2 R 2 Å"0 , 2
t 2 = = = = H" 2 , 8 3 s
a s a s a s 0 , 0 5
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 8
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
Z E S T A W 3 R u c h o b r o t o w y
2 0 . 1 0 . 2 0 0 6 r .
P o r ó w n a n i e w z o r ó w w r u c h u o b r o t o w y m i p o s t p o w y m
r u c h o b r o t o w y r u c h p o s t p o w y
R p r o m i e D
S d Bu g o [ Bu k
S
x p o Bo \e n i e ,
Õ k t
Õ = [ r a d ]
( s d r o g a )
R
d Õ r a d d x m
îø ùø îø ùø
É p r d k o [ k t o w a É = v =
v p r d k o [
ïø úø ïø úø
d t s d t s
ðø ûø ðø ûø
v = É Å" R
d É r a d d v m
µ p r z y s p i e s z e n i e îø ùø a s p r z y s p i e s z e n i e îø ùø
µ = a s =
2
ïø ïøs úø
k t o w e
d t s 2 úø s t y c z n e d t
ðø ûø ðø ûø
a s = µ Å" R
a n p r z y s p i e s z e n i e
v 2 m
îø ùø
a d =
2
ïøs úø
n o r m a l n e ( d o [r o d k o w e )
R
ðø ûø
a d = É2 Å" R = É Å"v
d l a r u c h u j e d n o s t a j n i e z m i e n n e g o
1 1
Õ = Õ0 + É0 t + µt 2 x = x 0 + x 0 t + a t 2
2 2
É = É0 + µt v = v 0 + a t
z a d . 1 .
r r
D a n e : t = 4 s , "( a s , v ) = 5 8 ° = ± ,
S z u k a n e : µ = ? ,
D l a r u c h u j e d n o s t a j n i e z m i e n n e g o p r a w d z i w e j e s t : É = µt
v 2
2
2
a d = = É2 R = ( µt ) R = µ t 2 R
R
a s = µR
2
a d µ t 2 R t g ± îø ùø
r a d
t g ± = = = µt 2 Ò! µ =
a s µR t 2 ïø s 2 úø
ðø ûø
t g 5 8 ° 1 , 6 r a d
îø ùø
µ = H" = 0 , 1
ïø úø
4 2 1 6 s 2
ðø ûø
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 1 9
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
z a d . 2 .
D a n e : R = 2 0 c m = 0 , 2 m , ¸( t ) = 3 - t + 0 , 1 t 3 [ r a d ] ( t w s e k . ) , t = 1 0 s ,
S z u k a n e : a s = ? , a n = ? ,
d ¸
É = = 0 , 3 t 2 - 1 b o ( a Å" x n ) 2 = a Å" n Å" x n - 1 c z y l i ( 3 - t 1 + 0 , 1 t 3 ) 2 = 1 Å"3 0 - 1 Å"t 0 + 0 , 1 Å"3 Å"t 2
d t
d É
µ = = 0 , 6 t b o ( a Å" x n ) 2 = a Å" n Å" x n - 1 c z y l i ( 0 , 3 t 2 - 1 ) 2 = 0 , 3 Å" 2 Å"t 1 - 1 Å"1 0
d t
µ( t ) = 0 , 6 t µ( t = 1 0 ) = 0 , 6 Å"1 0 = 6
m
îø ùø
a s = µR = 6 Å"0 , 2 = 1 , 2
2
ïøs úø
ðø ûø
2 2 m
îø ùø
a d = É2 R = ( 0 , 3 t 2 - 1 ) R = ( 0 , 3 Å"1 0 2 - 1 ) Å"0 , 2 = 1 6 8 , 2
2
ïøs úø
ðø ûø
P r z y s p i e s z e n i a d l a p u n k t ó w l e \c y c h , n p . w p o Bo w i e p r o m i e n i a l i c z y s i t a k s a m o .
1
2
R = R = 0 , 1 m
2
2
2 2 2 2 2
a s ( R ) = 6 Å" R a d ( R ) = ( 0 , 3 Å"1 0 2 - 1 ) Å" R = 8 4 1 Å" R
m m
îø ùø îø ùø
2 2
a s ( R ) = 6 Å"0 , 1 = 0 , 6 a d ( R ) = 8 4 1 Å"0 , 1 = 8 4 , 1
2 2
ïøs úø ïøs úø
ðø ûø ðø ûø
z a d . 3 .
m m
D a n e : v 1 = 3 , v 2 = 2 , "R = 1 0 c m = 0 , 1 m
s s
S z u k a n e : n = ? ( n p r d k o [ o b r o t o w a ) ,
2 1 É
É = = = 2 n Ò! n =
T f 2
v
É = = c o n s t . Ò! É1 = É2
R
v 1 v 1 v 1
É1 = Ò! R = Ò! R =
R É1 É
v 2 v 2 v 2 v 1
É2 = Ò! É = Ò! É = Ò! Éëø - "R öø = v 2
ìø ÷ø
v 1
R - "R R - "R É
íø øø
- "R
É
v 1 v 1 - v 2 3 - 2 r a d
îø ùø
É - É"R = v 2 Ò! É = = = 1 0
ïø úø
É "R 0 , 1 s
ðø ûø
É 1 0 o b r .
îø ùø
n = = H" 1 , 5 9
ïø úø
2 2 s
ðø ûø
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 0
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
z a d . 4 .
D a n e : µ = 3 , 5 [ r a d ] , t = 0 s Ò! É0 = 2 [ r a d ] ,
s
s 2
S z u k a n e : Õ( t = 2 s ) = ? , É( t = 2 s ) = ? , "Õ = ? ,
1
Õ = Õ0 + É0 t + µt 2
É = É0 + µt
2
3 , 5
Õ( t ) = 2 t + t 2 É( t ) = 2 + 3 , 5 t
2
r a d
3 , 5
É( 2 ) = 2 + 3 , 5 Å" 2 = 9 [ ]
Õ( 2 ) = 2 Å" 2 + Å" 2 2 = 1 1 [ r a d ]
s
2
"Õ = Õ( 3 ) - Õ( 2 )
3 , 5
Õ( 3 ) = 2 Å"3 + Å"3 2 = 2 1 , 7 5 [ r a d ]
2
Õ( 2 ) = 1 1 [ r a d ]
"Õ = Õ( 3 ) - Õ( 2 ) = 2 1 , 7 5 - 1 1 = 1 0 , 7 5 [ r a d ]
z a d . 5 .
D a n e : t = 3 s , h = 1 , 5 m , R = 4 c m = 0 , 0 4 m ,
S z u k a n e : µ = ? ,
1
D l a r u c h u j e d n o s t a j n i e z m i e n n e g o p r a w d z i w e j e s t : x = x 0 + v 0 t + a t 2
2
W i a d o m o , \e v 0 = 0 o r a z z a Bó \m y , \e x 0 = 0 w t e d y :
1
h = a t 2 b o x = h
2
2 h
a =
t 2
P r z y s p i e s z e n i e s t y c z n e o b r a c a j c e g o s i k r \k a j e s t r ó w n e p r z y s p i e s z e n i u s p a d a j c e g o
m a Be g o o d w a \n i k a b o t e n d w a c i a Ba t p o Bc z o n e l e k k n i e r o z c i g l i w n i c i .
a s = a a = µ Å" R
2 h
a s =
t 2
a s 2 h 2 Å"1 , 5 r a d
îø ùø
µ = = = = 8 , 3 3
ïø úø
R R t 2 0 , 0 4 Å"3 2 s 2
ðø ûø
z a d . 6 .
m
D a n e : R = 0 , 1 m , N = 5 Ò! v 1 = 0 , 1 , t 2 = 2 0 s ,
s
S z u k a n e : a n = ? ,
Õ = 2 N
1 1
Õ = É0 + µt 2 = µt 2
2 2
4 N
1
2 N = µt 1 2 Ò! µ =
2
t 1 2
v = a s Å"t
v 1
v 1 = a s Å"t 1 Ò! t 1 =
a s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 1
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
4 N a s 2 v 1 2
a s = µ Å" R = R = 4 R N Ò! a s =
4 R N
t 1 2 v 1 2
v 2 = a s Å"t 2
v 2 2 a s 2 Å"t 2 2 t 2 2 v 1 4 t 2 2 v 1 4 Å"t 2 2
a n = = = a s 2 Å" = Å" =
2 2
R R R 1 6 R 2 N R 1 6 R 3 N
v 1 4 Å"t 2 2 0 , 1 4 Å" 2 0 2 0 , 0 4
m
a n = = H" H" 0 , 0 1 [ s ]
2
2
1 6 Å" R 3 Å" N 1 6 Å"0 , 1 3 Å"5 2 3 , 9 5
z a d . 7 .
r a d
D a n e : É0 = 1 , 4 4 , N = 4 2 , 3 Ò! É = 0 , µ = c o n s t . µ < 0 ,
s
S z u k a n e : t z = ? ( c z a s z a t r z y m a n i a ) , µ = ? ( o p ó zn i e n i e k t o w e ) ,
t 1 = ? ( c z a s w y k o n a n i a p i e r w s z e j p o Bo w y o b r o t ó w )
r u c h o b r o t o w y j e d n o s t a j n i e o p ó zn i o n y , w i c :
É = É0 - µt
É0
0 = É0 - µt z Ò! µ =
t z
1
Õ = É0 t - µt
2
É0 4 N
1
2 N = É0 t z - t z 2 Ò! 4 N = t z ( 2 É0 - É0 ) Ò! t z =
2
t z É0
4 N 4 Å" 4 2 , 3
t z = = H" 3 6 9 [ s ]
É0 1 , 4 4
É0 É0 É0 2 1 , 4 4 2
r a d
µ = = É0 = = H" 0 , 0 0 3 9 [ ]
s 2
t z 4 N 4 N 4 Å" 4 2 , 3
1 1
¸ = Õ = 2 N = N
2 2
É0 - É1
É1 = É0 - µt 1 Ò! t 1 =
µ
1
¸ = É0 t 1 - µt 1 2
2
2
É0 - É1 1 É0 - É1
ëø öø
N = É0 - µ
ìø ÷ø
2
µ µ
íø øø
É0 2 - É0 É1 É0 2 - 2 É0 É1 + É1 2
N = 2 -
µ µ
µ 2 N = 2 É0 2 - 2 É0 É1 - É0 2 + 2 É0 É1 - É1 2
µ 2 N = É0 2 - É1 2 Ò! É1 = É0 2 - 2 N µ = 1 , 4 4 2 - 2 Å" 4 2 , 3 Å"0 , 0 0 3 9 H" 1 , 0 1 8 [ r a d ]
s
É0 - É1 É0 - É1 1 , 4 4 - 1 , 0 1 8
t 1 = = = H" 1 0 8 [ s ]
µ µ 0 , 0 0 3 9
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 2
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
z a d . 8 .
o b r o b r
D a n e : N = 4 2 , 3 , n 1 = 2 4 0 , n 2 = 3 6 0 , µ = c o n s t . µ > 0 ,
m i n . m i n .
S z u k a n e : µ = ? ( p r z y s p i e s z e n i e k t o w e ) ,
Õ = 2 N
o b r 1 o b r o b r o b r 1 o b r o b r
n 1 = 2 4 0 = 2 4 0 = 4 n 1 = 3 6 0 = 3 6 0 = 6
m i n . 6 0 s s m i n . 6 0 s s
É1 = 2 n 1 É2 = 2 n 2
É = É0 + µt
É2 - É1
É2 = É1 + µt Ò! t =
µ
1
Õ = É1 t + µt 2
2
2
É2 - É1 1 ( É2 - É1 )
Õ = É1 + µ
2
2
µ µ
É1 É2 - É1 2 É2 2 - 2 É1 É2 + É1 2
2 Õ = 2 +
µ µ
2 Õµ = 2 É1 É2 - 2 É1 2 + É2 2 - 2 É1 É2 + É1 2
2 2
É2 2 - É1 2 ( 2 n 2 ) - ( 2 n 1 ) 4 2 ( n 2 2 - n 1 2 )
µ = = =
2 Õ 4 N 4 N
µ = ( n 2 2 - n 1 2 ) = ( 6 2 - 4 2 ) H" 1 , 2 6 [ r a d ]
s
N 5 0
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 3
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 3 R u c h o b r o t o w y . Z m o d y f i k o w a n a : 1 . 1 1 . 2 0 0 6 1 8 : 4 0
D o d a t k o w e z a g a d n i e n i a :
a ) C i a Bo p r z e b y Bo p o Bo w d r o g i z e s t a B p r d k o [c i v 1 o r a z d r u g p o Bo w z e s t a B
p r d k o [c i v 2 . W y z n a c z [r e d n i p r d k o [ t e g o c i a Ba .
s s c
v = v [r =
t t c
s c = s 1 + s 2 = 2 s
t c = t 1 + t 2
s s
t 1 = t 2 =
v 1 v 2
s c s 1 + s 2 2 s 2 s 2 v 1 v 2
v [r = = = = =
t c t 1 + t 2 s + s s ( v 1 + v 2 ) v 1 + v 2
v 1 v 2 v 1 v 2
2 v 1 v 2
v [r =
v 1 + v 2
b ) C i a Bo p r z e z p o Bo w c z a s u p o r u s z a Bo s i z e s t a B p r d k o [c i v 1 o r a z d r u g p o Bo w z e s t a B
p r d k o [c i v 2 . W y z n a c z [r e d n i p r d k o [ t e g o c i a Ba .
s s c
v = v [r =
t t c
s c = s 1 + s 2
t c = t 1 + t 2 = 2 t Ò! t = t 1 = t 2
s 1 = v 1 t 1 s 2 = v 2 t 2
s c s 1 + s 2 v 1 t 1 + v 2 t 2 t ( v 1 + v 2 ) v 1 + v 2
v [r = = = = =
t c 2 t 2 t 2 t 2
v 1 + v 2
v [r =
2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 4
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 4 D y n a m i k a . Z m o d y f i k o w a n a : 2 . 1 1 . 2 0 0 6 2 1 : 3 5
Z E S T A W 4 D y n a m i k a
2 7 . 1 0 . 2 0 0 6 r .
N a j w a \n i e j s z e w z o r y t e g o d z i a Bu
r
r F w
a =
m
Q = m Å" g
T = F n Å" f
d l a r ó w n i p o c h y Be j o p o c h y Bo [c i ± :
Q ¥" = Q Å"c o s ±
Q = Q Å"s i n ±
z a d . 1 .
D a n e : F 1 = 5 N , F 2 = 8 N , m = 0 , 3 k g ,
r r
"( F 1 , O X ) = 6 0 ° , "( O X , F 2 ) = 2 0 ° ,
r
r
S z u k a n e : a = ? , "( a , O X ) = ? , F 3 = ? ,
r
r F w
a =
m r r
r
F w = F 1 + F 2
r
F 1 = [ F 1 Å" c o s ( 3 6 0 ° - 2 0 ° ) , F 1 Å" s i n ( 3 6 0 ° - 2 0 ° ) ] H" [ 4 , 6 9 8 ; - 1 , 7 1 ]
r
F 2 = [ F 2 Å" c o s 6 0 ° , F 2 Å"s i n 6 0 ° ] H" [ 4 ; 6 , 9 2 8 ]
r
F w H" [ 4 , 6 9 8 + 4 ; - 1 , 7 1 + 6 , 9 2 8 ] H" [ 8 , 6 9 8 ; 5 , 2 1 8 ] H" [ 8 , 7 0 ; 5 , 2 2 ]
F w = 8 , 7 0 2 + 5 , 2 2 2 H" 1 0 , 1 5
F w 1 0 , 1 5
m
a = H" H" 3 3 , 8 [ s ]
2
m 0 , 3
F w 8 , 7 0
x
a x = H" H" 2 9
m 0 , 3
r a x 2 9 r 2 9
c o s "( a , O X ) = H" Ò! "( a , O X ) = a r c c o s = 3 0 , 9 °
a 3 3 , 8 3 3 , 8
r r
F 3 = - F w
r
F 3 = [ - 8 , 7 0 ; - 5 , 2 2 ]
z a d . 2 .
D a n e : m 1 = 1 k g , m 2 = 2 k g , m 3 = 3 k g ,
F 1 = 1 2 N , F 2 = 6 N ,
S z u k a n e : a = ? , F 1 2 = ? , F 2 3 = ? ,
r
r
r F w r
a = Ò! F w = m Å" a
m
F 1 - F 2 1 2 - 6
m
a = = = 1 [ s ]
2
m 1 + m 2 + m 3 1 + 2 + 3
F 1 - F 1 2 = m 1 Å" a Ò! F 1 2 = F 1 - m 1 Å" a = 1 2 - 1 Å"1 = 1 1 N
F 2 3 - F 2 = m 3 Å" a Ò! F 2 3 = F 2 + m 3 Å" a = 6 + 3 Å"1 = 9 N
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 5
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 4 D y n a m i k a . Z m o d y f i k o w a n a : 2 . 1 1 . 2 0 0 6 2 1 : 3 5
z a d . 3 .
m
D a n e : m 1 = 1 k g , m 2 = 2 k g , H = 3 m , g = 9 , 8 1 ,
s 2
S z u k a n e : H m a x = ? , t = ? ( c z a s p o k t ó r y m m a s a 1 b d z i e n a w y s o k o [c i H m a x ) ,
S y t u a c j p r z e d s t a w i o n w z a d a n i u m o \n a p o d z i e l i n a d w i e f a z y .
P i e r w s z a f a z a z a c z y n a s i g d y m a s a 2 j e s t n a w y s o k o [c i 3 m . M a s a 2 z a c z y n a s p a d a
( p o r u s z a s i w d ó B) z e s t a By m p r z y s p i e s z e n i e m a . W t y m s a m y m c z a s i e m a s a 1 p o r u s z a
s i z t a k i m s a m y m p r z y s p i e s z e n i e m , a l e s k i e r o w a n y m p r z e c i w n i e ( t j . d o g ó r y ) . M a s a 2 p o
p r z e b y c i u d r o g i H z a t r z y m u j e s i w p o d Bo \u . M a s a 1 w t e j c h w i l i j e s t n a w y s o k o [c i H .
D r u g a f a z a t o r u c h j e d n o s t a j n i e o p ó zn i o n y z o p ó zn i e n i e m r ó w n y m g m a s y 1 . D r o g
p r z e b y t p r z e z m a s 1 d o p o Bo \e n i a m a k s y m a l n e g o o z n a c z j a k o h ( g d z i e v 2 = 0 ) .
F w Q 2 - Q 1 m 2 g - m 1 g m 2 - m 1 2 - 1
m
a = = = = g = 9 , 8 1 = 3 , 2 7 [ s ]
2
m m 1 + m 2 m 1 + m 2 m 1 + m 2 1 + 2
a t 1 2 2 H 2 Å" 3
H = Ò! t 1 = = H" 1 , 3 5 4 6 s
2 a 3 , 2 7
2 H
v 1 = a Å" t 1 Ò! v 1 = a Å" = 2 H a
a
v 2 = v 1 - g t 2
v 1 2 H a 2 Å" 3 Å" 3 , 2 7
0 = v 1 - g t 2 Ò! t 2 = = = H" 0 , 4 5 1 5 s
g g 9 , 8 1
t = t 1 + t 2 H" 1 , 3 5 4 6 + 0 , 4 5 1 5 H" 1 , 8 0 6 s
g t 2 2
h = v 1 t 2 -
2
2
2 H a
g ( )
2 H a 2 H a H a 3 Å" 3 , 2 7
g
2 H a
h = 2 H a Å" - = - = = = 1 m
g
2 g 2 g g 9 , 8 1
H m a x = H + h = 3 + 1 = 4 m
z a d . 4 .
m
D a n e : m = 5 , 1 k g , f = 0 , F = 1 2 N , ± = 2 5 ° , g = 9 , 8 1 ,
s 2
S z u k a n e : a = ? , F m a x = ? ( m a k s y m a l n a w a r t o [ s i By o k i e r u n k u z g o d n y m z s i B F ,
d l a k t ó r e j k l o c e k j e s z c z e p o r u s z a s i p o p o w i e r z c h n i ) ,
F x = F c o s ± = 1 2 c o s 2 5 ° H" 1 0 , 8 8 N
F y = F s i n ± = 1 2 s i n 2 5 ° H" 5 , 0 7 N
r r r
r
F + F R + Q = m Å" a
F x 1 0 , 8 8
m
x : F x = m a Ò! a = H" H" 2 , 1 3 [ s ]
2
m 5 , 1
y : F R + F y - Q = 0
O d e r w a n i e k l o c k a o d p o d Bo \a n a s t p i g d y F N = 0 ( s i Ba n a c i s k u ) , w t e d y F R = 0 ( s i Ba r e a k c j i ) .
F m a x = F m a x s i n 2 5 °
y
m g 5 , 1 Å"9 , 8 1
0 + F m a x s i n 2 5 ° - Q = 0 Ò! F m a x = H" H" 1 1 8 , 4 N
s i n 2 5 ° 0 , 4 2 2 6
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 6
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 4 D y n a m i k a . Z m o d y f i k o w a n a : 2 . 1 1 . 2 0 0 6 2 1 : 3 5
z a d . 5 .
m
D a n e : m 1 = 9 , 5 k g , m 2 = 2 , 6 k g , ± = 3 4 ° , f = 0 , 1 5 , g = 9 , 8 1 ,
s 2
S z u k a n e : a = ? , N = ? ( s i Ba n a p r \e n i a n i c i ) ,
G d y t a r c i a m a n i e b y t o f = 0 p o d p u n k t a ) z a d a n i a .
Q 1 ¥" = Q 1 c o s ± = m 1 g c o s ±
Q 1 = Q 1 s i n ± = m 1 g s i n ±
2 ) N - m 2 g = m 2 a
1 ) Q 1 - T - N = m 1 a
F R 1 - Q 1 ¥" = 0 Ò! F R 1 = Q 1 ¥"
T = f Å" m 1 g c o s ±
" 1 ) + 2 ) "
N - m 2 g + Q 1 - T - N = m 2 a + m 1 a
- m 2 g + m 1 g s i n ± - f Å" m 1 g c o s ± = a ( m 2 + m 1 )
m 1 ( s i n ± - f c o s ± ) - m 2 9 , 5 ( s i n 3 4 ° - 0 , 1 5 Å"c o s 3 4 ° ) - 2 , 6
m
a = g = 9 , 8 1 H" 1 , 2 4 [ s ]
2
m 1 + m 2 9 , 5 + 2 , 6
N = m 2 g + m 2 a H" 2 , 6 ( 9 , 8 1 + 1 , 2 4 ) H" 2 8 , 7 [ N ]
d l a f = 0
9 , 5 ( s i n 3 4 ° - 0 Å"c o s 3 4 ° ) - 2 , 6
m
a = 9 , 8 1 H" 2 , 2 0 [ s ]
2
9 , 5 + 2 , 6
N = m 2 g + m 2 a H" 2 , 6 ( 9 , 8 1 + 2 , 2 0 ) H" 3 1 , 2 [ N ]
z a d . 6 .
D a n e : Q s = 6 0 N ( c i \a r s a n e k ) , v = c o n s t . , F = 2 5 N ,
m
±1 = 1 5 ° , ±2 = 3 5 ° g = 9 , 8 1 ,
s 2
S z u k a n e : f = ? , a = ? ,
m s , m c h ( m a s a s a n e k i m a s a c h Bo p c a t o t y l k o z m i e n n e p o m o c n i c z e )
±3 = ±2 - ±1 = 3 5 ° - 1 5 ° = 2 0 °
Q ¥" = Q c o s ±1
Q = Q s i n ±1
F x = F c o s ±3
F y = F s i n ±3
y ) F R + F y - Q ¥" = 0
F R = Q c o s ±1 - F s i n ±3
T = f Å" F N = f Å" F R
T = f Å"( Q c o s ±1 - F s i n ±3 )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 7
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 4 D y n a m i k a . Z m o d y f i k o w a n a : 2 . 1 1 . 2 0 0 6 2 1 : 3 5
r u c h j e d n o s t a j n y , w i c d z i a Ba j c e s i By s i r ó w n o w a \:
x ) F x - Q - T = 0
F c o s ±3 - Q s i n ±1 - f Å"( Q c o s ±1 - F s i n ±3 ) = 0
F c o s ±3 - Q s i n ±1 2 5 c o s 2 0 ° - 6 0 s i n 1 5 ° 7 , 9 6
f = = H" H" 0 , 1 6 1
Q c o s ±1 - F s i n ±3 6 0 c o s 1 5 ° - 2 5 s i n 2 0 ° 4 9 , 4
g d y c h Bo p i e c z j e \d \a n a s a n k a c h :
F w Q + Q - T m s g s i n ±1 + m c h g s i n ±1 - F N f
s a n e k c h
a = = =
m m s + m c h m s + m c h
g s i n ±1 ( m s + m c h ) - ( Q ¥" s a n e k + Q ¥" c h ) f
g s i n ±1 ( m s + m c h ) - g c o s ±1 ( m s + m c h ) f
a = =
m s + m c h m s + m c h
m
a = g Å"( s i n ±1 - f c o s ±1 ) H" 9 , 8 1 Å"( s i n 1 5 ° - 0 , 1 6 1 Å" c o s 1 5 ° ) H" 1 , 0 1 3 [ s ]
2
z a d . 7 .
m
D a n e : m = 1 k g , a = c o n s t . , F = 1 0 N , f = 0 , 2 3 , g = 9 , 8 1 ,
s 2
±1 = 6 0 ° , ±2 = 0 ° , ±3 = - 6 0 ° ,
S z u k a n e : a = ? , F R = ? ,
Z a d a n i e b d z i e r o z w a \a n e d l a d o w o l n e g o k t a ± , g d z i e :
r
± = "( F , O X )
n a k o Dc u z o s t a n p o d s t a w i o n e m o \l i w e w a r t o [c i : ±1 , ±2 , ±3 .
F = F c o s ±
F ¥" = F s i n ±
T = F N Å" f Ò! T = ( Q - F ¥") Å" f = f m g - f F s i n ±
F w F - T F c o s ± - f m g + f F s i n ± 1 0 c o s ± - 0 , 2 3 Å"1 Å"9 , 8 1 + 0 , 2 3 Å"1 0 Å"s i n ±
a = = = =
m m m 1
a ( ±) = 1 0 c o s ± + 2 , 3 s i n ± - 2 , 2 5 6 3
r r
F R = - F N
F R = Q - F ¥" = m g - F s i n ± = 1 Å"9 , 8 1 - 1 0 s i n ±
F R ( ±) = 9 , 8 1 - 1 0 s i n ±
m
a ( ±1 ) = 1 0 c o s 6 0 ° + 2 , 3 s i n 6 0 ° - 2 , 2 5 6 3 H" 4 , 7 3 [ s ]
2
F R ( ±1 ) = 9 , 8 1 - 1 0 s i n 6 0 ° H" 1 , 1 5 [ N ]
m
a ( ±2 ) = 1 0 c o s 0 ° + 2 , 3 s i n 0 ° - 2 , 2 5 6 3 H" 7 , 7 4 [ s ]
2
F R ( ±1 ) = 9 , 8 1 - 1 0 s i n 0 ° H" 9 , 8 1 [ N ]
m
a ( ±3 ) = 1 0 c o s ( - 6 0 ° ) + 2 , 3 s i n ( - 6 0 ° ) - 2 , 2 5 6 3 H" 0 , 7 5 [ s ]
2
F R ( ±1 ) = 9 , 8 1 - 1 0 s i n ( - 6 0 ° ) H" 1 8 , 4 7 [ N ]
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 8
U t w o r z o n a : 2 9 . 1 0 . 2 0 0 6 1 6 : 4 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 4 D y n a m i k a . Z m o d y f i k o w a n a : 2 . 1 1 . 2 0 0 6 2 1 : 3 5
z a d . 8 .
m m
D a n e : v 0 = 2 0 , ¸ = 5 ° , f = 0 , 1 8 , g = 9 , 8 1 ,
s
s 2
S z u k a n e : s ( d r o g a , k t ó r p r z e b d z i e n a r c i a r z d o c h w i l i z a t r z y m a n i a s i ) ,
Q = Q s i n ¸ = m g s i n ¸
Q ¥" = Q c o s ¸ = m g c o s ¸
T = F N Å" f = Q ¥" Å" f = f Å" m g c o s ¸
F w Q - T m g s i n ¸ - f Å" m g c o s ¸
a = = = = g ( s i n ¸ - f c o s ¸ )
m m m
m
a = 9 , 8 1 ( s i n 5 ° - 0 , 1 8 Å"c o s 5 ° ) H" - 0 , 9 0 4 [ s ]
2
C z y l i t o c i a Bo p o r u s z a s i r u c h e m j e d n o s t a j n i e o p ó zn i o n y m , z p r z y s p i e s z e n i e m a .
v = v 0 + a t
v 0
0 = v 0 + a t Ò! t =
- a
2
v 0 1 v 0 v 0 2 v 0 2 v 0 2
1
s = v 0 t + a t 2 = v 0 + a ëø öø = - + = -
ìø ÷ø
2 2
- a - a a 2 a 2 a
íø øø
2 0 2
s H" - H" 2 2 1 m
2 Å"( - 0 , 9 0 4 )
z a d . 9 .
r
m
D a n e : m = 3 k g , F , ± = 5 0 ° , f = 0 , 2 5 , g = 9 , 8 1 ,
s 2
S z u k a n e : F m i n = ? o r a z F m a x = ? ( d l a k t ó r e g o k l o c e k p o z o s t a n i e w b e z r u c h u ) ,
F = F s i n ±
F ¥" = F c o s ±
T = F ¥" Å" f Ò! T = F Å" f c o s ±
F m i n : F + T = Q Ò! F m i n s i n ± + F m i n Å" f c o s ± = m g
m g 3 Å"9 , 8 1
F m i n = H" H" 3 1 , 8 N
s i n ± + f c o s ± 0 , 7 6 6 + 0 , 2 5 Å"0 , 6 4 3
F m a x : F - T = Q Ò! F m a x s i n ± - F m a x Å" f c o s ± = m g
m g 3 Å"9 , 8 1
F m a x = H" H" 4 8 , 6 N
s i n ± - f c o s ± 0 , 7 6 6 - 0 , 2 5 Å"0 , 6 4 3
F " ( 3 1 , 8 ; 4 8 , 6 )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 2 9
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
K o l o k w i u m I Z a d a n i a
3 . 1 1 . 2 0 0 6 r .
P r z y k Ba d o w e z e s t a w y z a d a D d o r o z w i z a n i a n a k o l o k w i u m .
T r e [c i z a d a D Z e s t a w A
z a d . 1 .
r
W y z n a c z w e k t o r c o d Bu g o [c i 2 l e \c y n a d w u s i e c z n e j k t a p r z y w i e r z c h o Bk u A w t r ó j k c i e A B C .
D a n e s w s p ó Br z d n e w i e r z c h o Bk ó w : A ( 2 , 0 ) , B ( 1 , 3 ) , C ( - 4 , 2 ) .
r
O d p . c = [ - 2 , 2 ] .
z a d . 2 .
D a n y j e s t u k Ba d k l o c k ó w p r z e d s t a w i o n y n a r y s u n k u 1 . o r a z m a s y p o s z c z e g ó l n y c h k l o c k ó w :
m 1 = 1 , 5 k g , m 2 = 2 k g , m 3 = 0 , 5 k g . W s p ó Bc z y n n i k t a r c i a k l o c k a 2 . o p o d Bo \e w y n o s i 0 , 1 .
W y z n a c z p r z y s p i e s z e n i e t e u k Ba d y , n a p r \e n i e n i e w a \k i e g o i n i e r o z c i g l i w e g o s z n u r k a , w a r t o [
s i By d z i a Ba j c e j p r z e z k l o c e k 1 . n a k l o c e k 3 .
2 .
3 .
1 .
r y s . 1 .
m
O d p . a = 4 , 4 1 , N = 1 0 , 7 8 N , F 1 3 = 2 , 7 N .
s 2
z a d . 3 .
W y r z u c o n o p o z i o m o c i a Bo ( w j e d n o r o d n y m p o l u g r a w i t a c y j n y m ) w t a k i s p o s ó b , \e z a s i g t e g o
r z u t u w y n o s i 5 m , a c z a s 0 , 5 s . O p ó r p o w i e t r z a n a l e \y z a n i e d b a . O b l i c z w y s o k o [ z j a k i e j
w y r z u c o n o c i a Bo , p r d k o [ p o c z t k o w , p r d k o [ t u \ p r z e d u d e r z e n i e m w z i e m i . D o d a t k o w o
w y z n a c z w s p ó Br z d n e p o Bo \e n i a c i a Ba i w s p ó Br z d n e w e k t o r a p r d k o [c i d l a c z a s u 0 , 3 s
( o d m o m e n t y w y r z u c e n i a c i a Ba ) .
r
m m m
O d p . v 0 = 1 0 , h = 1 , 2 3 m , v k = 4 , 9 0 5 , A ( 3 ; 0 , 4 4 ) , v 1 = 1 0 , 4 2 .
s s s
z a d . 4 .
C i a Bo w y k o n u j e r u c h p o o k r g u o p r o m i e n i u R = 0 , 5 m . D r o g a p o k o n y w a n a p r z e z t o c i a Bo
z a d a n a j e s t f u n k c j s ( t ) = 2 t + 0 , 1 6 t 2 + 0 , 2 t 3 ( g d z i e d r o g a j e s t w m e t r a c h , a c z a s w s e k u n d a c h ) .
m
W y z n a c z c z a s d l a k t ó r e g o p r z y s p i e s z e n i e s t y c z n e b d z i e w y n o s i Bo 1 , 2 8 . D l a t e g o c z a s u
s 2
w y z n a c z p r d k o [ l i n i o w c i a Ba o r a z p r z y s p i e s z e n i a : n o r m a l n i e , c a Bk o w i t e i k t o w e .
m m m r a d
O d p . t = 0 , 8 s , v = 2 , 6 4 , a n = 1 3 , 9 4 , a = 1 4 , 1 9 , µ = 2 , 5 6 .
s
s 2 s 2
s 2
S z k i c e r o z w i z a D d o z a d a D z n a j d u j s i n a k o l e j n y c h s t r o n a c h .
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 0
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
S z k i c e r o z w i z a D Z e s t a w A
z a d . 1 .
r
D a n e : c = 2 , A ( 2 , 0 ) , B ( 1 , 3 ) , C ( - 4 , 2 ) ,
r
S z u k a n e : c = ? ,
r r r
c = n Å" c
r
n = [ x , y ]
r
n = 1
r
a = A B = [ 1 - 2 , 3 - 0 ] = [ - 1 , 3 ]
r
a = ( - 1 ) 2 + 3 2 = 1 0
r
b = A C = [ - 4 - 2 , 2 - 0 ] = [ - 6 , 2 ]
r
b = ( - 6 ) 2 + 2 2 = 4 0 = 2 1 0
r r
r r r r r r
"( a , c ) = "( a , n ) = "( b , c ) = "( b , n )
r r r r r r r
r r r
ñøa o n = a Å" n Å"c o s "( a , n )
ñø a x Å" x + a y Å" y b x Å" x + b y Å" y
ñø
a o n b o n
= r = r
ôø r r
r r r
ôø
r ôø r
r r r
ôør r ôø
a Å" n a Å" n
b Å" n b Å" n
Ò!
òøb o n = b Å" n Å"c o s "( b , n ) Ò! òø òø
ôø ôø r ôø
r
2
n = 1
ôø
n = 1
ôø
óøx + y 2 = 1
óø
óø
ñø- 1 Å" x + 3 Å" y - 6 Å" x + 2 Å" y
=
ñø- 2 x + 6 y = - 6 x + 2 y x =
ñø - y
ôø
1 0 Å"1 2 1 0 Å"1 Ò! Ò!
òø òø òø
2
2 óøx + y 2 = 1 óø( - y ) 2 + y 2 = 1
ôøx + y 2 = 1
óø
x =
ñø - y ñø
ôøx = m 2
2
Ò!
òø òø
2
1
2
ôøy = ±
óøy = 2
óø 2
r r r r
2 2 2 2
n 1 = [ - , ] Ò! c 1 = c Å" n 1 = 2 Å"[ - , ] = [ - 2 , 2 ]
2 2 2 2
r r r r
2 2 2 2
n 2 = [ , - ] Ò! c 2 = c Å" n 2 = 2 Å"[ , - ] = [ 2 , - 2 ]
2 2 2 2
w y z n a c z o n e w e k t o r y m u s z s p e Bn i a j e s z c z e d o d a t k o w y w a r u n e k :
m u s z l e \e n a d w u s i e c z n e j k t a , w i c n a p e w n o :
r
r r
"( a , O X ) < "( c , O X ) < "( b , O X )
a x r b x
a r c c o s < "( c , O X ) < a r c t a n
a b
- 1 r - 6
a r c c o s < "( c , O X ) < a r c c o s
1 0 2 1 0
r
7 1 , 5 6 ° < "( c , O X ) < 1 6 1 , 5 6 °
r c 1 x
- 2
"( c 1 , O X ) = a r c c o s = a r c c o s = 1 3 5 °
c 1 2
r c 2 x 2
"( c 2 , O X ) = a r c c o s = a r c c o s = 4 5 ° 3 6 0 - 4 5 ° = 3 1 5 °
c 2 2
r r
o s t a t e c z n i e : c = c 1 = [ - 2 , 2 ]
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 1
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
z a d . 2 .
D a n e : m 1 = 1 , 5 k g , m 2 = 2 k g , m 3 = 0 , 5 k g , f = 0 , 1
S z u k a n e : a = ? , N = ? , F 1 3 = ? ,
F w Q 1 + Q 3 - T m 1 g + m 3 g - f Å" F N m 1 + m 3 - f Å" m 2
a = = = = g
m m 1 + m 2 + m 3 m 1 + m 2 + m 3 m 1 + m 2 + m 3
1 , 5 + 0 , 5 - 0 , 1 Å" 2
m
a = 9 , 8 1 Å" = 9 , 8 1 Å"0 , 4 5 H" 4 , 4 1
s 2
1 , 5 + 2 + 0 , 5
r o z p a t r u j s i By d z i a Ba j c e n a k l o c e k 2 .
F w N - T N - f g m 2
a = = = Ò! N = m 2 ( a + f g )
m m 2 m 2
N H" 2 Å"( 4 , 4 1 + 0 , 1 Å"9 , 8 1 ) H" 1 0 , 7 8 N
r o z p a t r u j s i By d z i a Ba j c e n a k l o c e k 3 .
F w Q 3 - F 1 3
a = = Ò! F 1 3 = m 3 ( g - a )
m m 3
F 1 3 H" 0 , 5 Å"( 9 , 8 1 - 4 , 4 1 ) H" 2 , 7 N
z a d . 3 .
D a n e : z = 5 m , t = 0 , 5 s , t 1 = 0 , 3 s ,
r r
S z u k a n e : h = ? , v 0 = ? , v k = ? , r 1 = ? , v 1 = ? ,
[ + r y s . z u k Ba d e m w s p ó Br z d n y c h ]
R u c h p o z i o m o r z u c o n e g o c i a Ba r o z k Ba d a m n a d w i e s k Ba d o w e , s k Ba d o w r ó w n o l e g B d o
p o w i e r z c h n i z i e m i x , g d z i e c i a Bo p o r u s z a s i r u c h e m j e d n o s t a j n y m o r a z s k Ba d o w y
p r o s t o p a d B d o p o w i e r z c h n i z i e m i , g d z i e c i a Bo p o r u s z a s i r u c h e m j e d n o s t a j n i e
p r z y s p i e s z o n y m b e z p r d k o [c i p o c z t k o w e j z p r z y s p i e s z e n i e m g .
z 5
m
v x = v 0 = = = 1 0
s
t 0 , 5
v y = v y + g Å"t
0
g t 2 9 , 8 1 Å"0 , 5 2
h = = H" 1 , 2 3 m
2 2
m
v k = g Å"t = 9 , 8 1 Å"0 , 5 = 4 , 9 0 5
s
o b l i c z e n i a d l a t 1 :
r
1 1
r 1 = [ v 0 Å"t 1 , g Å"t 1 2 ] = [ 1 0 Å"0 , 3 ; Å"9 , 8 1 Å"0 , 3 2 ] H" [ 3 ; 0 , 4 4 ]
2 2
r
v 1 = [ v 0 , g Å"t 1 ] = [ 1 0 ; 9 , 8 1 Å"0 , 3 ] H" [ 1 0 ; 2 , 9 4 ]
r
m
v 1 = v x 2 + v y 2 = 1 0 2 + 2 , 9 4 2 H" 1 0 , 4 2
s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 2
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
z a d . 4 .
m
D a n e : R = 0 , 5 m , s ( t ) = 2 t + 0 , 1 6 t 2 + 0 , 2 t 3 , a s = 1 , 2 8 ,
s 2
S z u k a n e : t = ? , v = ? , a n = ? , a = ? , µ = ? ,
2
s ( t ) = 2 t + 0 , 1 6 t 2 + 0 , 2 t 3 ( a Å" x n ) = a Å" n Å" x n - 1
d s
v ( t ) = = 2 + 0 , 3 2 t + 0 , 6 t 2
d t
d v
a ( t ) = = 0 , 3 2 + 1 , 2 t
d t
a s = a ( t )
1 , 2 8 = 0 , 3 2 + 1 , 2 t Ò! t = 0 , 8 s
m
v ( 0 , 8 ) = 2 + 0 , 3 2 Å"0 , 8 + 0 , 6 Å" ( 0 , 8 ) 2 = 2 , 6 4
s
v 2 2 , 6 4 2
m
a n = = H" 1 3 , 9 4
s 2
R 0 , 5
m
a = a s 2 + a n 2 H" 2 , 6 4 2 + 1 3 , 9 4 2 H" 1 4 , 1 9
s 2
a s 1 , 2 8
r a d
µ = = = 2 , 5 6
s 2
R 0 , 5
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 3
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
T r e [c i z a d a D Z e s t a w B
z a d . 1 .
C z s t k a w y k o n u j e r u c h p o o k r g u o p r o m i e n i u R = 3 m . D r o g a p o k o n y w a n a p r z e z t o c i a Bo
z a d a n a j e s t f u n k c j s ( t ) = t 3 ( g d z i e d r o g a j e s t w m e t r a c h , a c z a s w s e k u n d a c h ) . W y z n a c z c z a s d l a
k t ó r e g o p r z y s p i e s z e n i e s t y c z n e b d z i e r ó w n e p r z y s p i e s z e n i u n o r m a l n e m u . J a k i e b d z i e w ó w c z a s
p r z y s p i e s z e n i e c a Bk o w i t e .
m
O d p . t = 1 , 2 6 s , a = 5 , 3 4 5 .
s 2
z a d . 2 .
P i Bk a k o p n i t a u k o [n i e p o d k t e m ± = 6 0 ° d o p o d Bo \a . W i a d o m o , \e p i Bk a b y Ba w p o w i e t r z u
3 s . Z j a k p r d k o [c i z o s t a Ba w y k o p a n a ? N a j a k m a k s y m a l n i e w y s o k o [ w z n i e s i e s i ? J a k i e
b d w s p ó Br z d n e ( w z g l d e m p u n k t w y k o p u p i Bk i ) p o Bo \e n i a i p r d k o [c i p i Bk i p o c z a s i e 2 s o d
w y k o p u .
r r
m
O d p . v = 1 6 , 9 9 , h = 1 1 , 0 3 m , r 1 = [ 1 6 , 9 9 ; 9 , 8 1 ] , v 1 = [ 8 , 4 9 5 ; - 4 , 9 0 6 ]
s
z a d . 3 .
?
O d p . ? .
z a d . 4 .
o b r
K o Bo o [r e d n i c y 1 m o b r a c a s i z e s t a B p r d k o [c i o b r o t o w n = 1 8 0 . W p e w n y m
m i n .
m o m e n c i e k o Bo z a c z y n a h a m o w a n i e r u c h e m j e d n o s t a j n i e o p ó zn i o n y m z o p ó zn i e n i e m k t o w y m
r a d
3 . P o j a k i m c z a s i e t o k o Bo z a t r z y m a s i ? I l e w y k o n a o b r o t ó w d o m o m e n t u z a t r z y m a n i a s i ?
s 2
O b l i c z p r d k o [ l i n i o w p u n k t ó w n a o b w o d z i e k o Ba w c h w i l i r o z p o c z c i a h a m o w a n i a .
m
O d p . t = 6 , 2 8 s , N = 9 , 4 2 , v = 9 , 4 2 .
s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 4
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
S z k i c e r o z w i z a D Z e s t a w B
z a d . 1 .
D a n e : R = 3 m , s ( t ) = t 3 , a s = a n ,
S z u k a n e : t = ? , a = ? ,
2
s ( t ) = t 3 ( a Å" x n ) = a Å" n Å" x n - 1
d s
v ( t ) = = 3 t 2
d t
d v
a ( t ) = = 6 t
d t
a s = a ( t ) = 6 t
2
v 2 ( 3 t 2 ) 9 t 4
a n = = =
R R R
v 2 9 t 4
a s = a n Ò! 6 t = Ò! 6 t =
R R
9 t 4 - 6 R t = 0
3 t ( 3 t 3 - 2 R ) = 0
2 2
3 3
( ( t = 0 (" t = R ) '" t > 0 ) Ò! t = R
3 3
3
2 2
3 3
t = R = 3 = 2 H" 1 , 2 6 s
3 3
v 2
a s = a n =
R
2 2
2
ëø öø ëø öø
v 2 v 2 v 2 ( 3 t 2 ) 9 2 t 4
a = a s 2 + a n 2 = ìø ÷ø ìø ÷ø 2 =
+ = 2 =
ìø ÷ø ìø ÷ø
R R R R R
íø øø íø øø
9 2 Å"1 , 2 6
m
a H" H" 3 2 Å"1 , 2 6 H" 5 , 3 4 5
s 2
3
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 5
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
z a d . 2 .
D a n e : ± = 6 0 ° , t = 3 s , t 1 = 2 s
r r
S z u k a n e : v 0 = ? , h = ? , r 1 = ? , v 1 = ? ,
[ + r y s . z u k Ba d e m w s p ó Br z d n y c h ]
R u c h u k o [n i e r z u c o n e g o c i a Ba r o z k Ba d a m n a d w i e s k Ba d o w e , s k Ba d o w r ó w n o l e g B d o
p o w i e r z c h n i z i e m i x , g d z i e c i a Bo p o r u s z a s i r u c h e m j e d n o s t a j n y m o r a z s k Ba d o w y
p r o s t o p a d B d o p o w i e r z c h n i z i e m i , g d z i e c i a Bo p o r u s z a s i r u c h e m p r z y s p i e s z o n y m .
v x = v x = v Å" c o s ±
0
v y = v Å"s i n ±
0
c z a s w z n o s z e n i a t o p o Bo w a c z a s u l o t u p i Bk i , a c z a s o p a d a n i a t o t e \ p o Bo w a c z a s u l o t u :
1 1
t 0 , 5 = t = Å"3 = 1 , 5 s
2 2
g d y c i a Bo n a m a k s y m a l n y m w z n i e s i e n i u m a v y = 0 w i c :
v y = v y - g Å"t 0 , 5
0
0 = v s i n ± - g Å" t 0 , 5
g Å" t 0 , 5
9 , 8 1 Å"1 , 5
m
v = = H" 1 6 , 9 9
s
s i n ± s i n 6 0 °
c i a Bo w z n o s i s i n a w y s o k o [c i h ( c z y l i t a m a k s y m a l n a ) w p i o n i e r u c h e m j e d n o s t a j n i e
o p ó zn i o n y m z o p ó zn i e n i e m g :
1
h = v y t 0 , 5 - g t 0 , 5 2
2
0
1
h = t 0 , 5 v s i n ± - g t 0 , 5 2
2
1
h H" 1 , 5 Å"1 6 , 9 9 Å"s i n 6 0 ° - Å"9 , 8 1 Å"1 , 5 2 H" 1 1 , 0 3 m
2
o b l i c z e n i a d l a t 1 = 2 s :
r
r 1 = [ r x , r y ]
r x = v x Å"t 1 = v Å"t 1 Å"c o s ± H" 1 6 , 9 9 Å" 2 Å"c o s 6 0 ° H" 1 6 , 9 9 m
1 1
r y = v y Å"t 1 - g t 1 2 = v Å"t 1 Å"s i n ± - g t 1 2 H" 1 6 , 9 9 Å" 2 Å"s i n 6 0 ° - 9 , 8 1 Å" 2 2 H" 9 , 8 1 m
2 2
0
r
r 1 = [ 1 6 , 9 9 ; 9 , 8 1 ]
r
v 1 = [ v x , y y ]
1 m
v x = v Å"c o s ± H" 1 6 , 9 9 Å" H" 8 , 4 9 5
2 s
m
v y = v y - g t 1 = v s i n ± - g t 1 H" 1 6 , 9 9 Å"s i n 6 0 ° - 9 , 8 1 Å" 2 H" - 4 , 9 0 6
s
0
r
v 1 = [ 8 , 4 9 5 ; - 4 , 9 0 6 ]
z a d . 3 .
?
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 6
U t w o r z o n a : 4 . 1 1 . 2 0 0 6 0 : 0 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I Z a d a n i a . Z m o d y f i k o w a n a : 8 . 1 1 . 2 0 0 6 2 3 : 2 3
z a d . 4 .
o b r r a d
D a n e : d = 1 m , n = 1 8 0 , µ = 3
m i n .
s 2
S z u k a n e : t = ? , N = ? , v = ? ,
1
R = d
2
Õ = 2 N
o b r 1 o b r o b r
n = 1 8 0 = 1 8 0 = 3
m i n . 6 0 s s
É0 = 2 n
É = É0 - µt
É0 2 n
0 = É0 - µt Ò! É0 = µt Ò! t = =
µ µ
2 Å"3
t = H" 6 , 2 8 s
3
1
Õ = É0 t - µt 2
2
2
2 n 2 n
ëø öø
1
2 N = 2 n Å" - µ
ìø ÷ø
2
µ µ
íø øø
4 2 n 2 4 2 n 2
2 N = -
µ 2 µ
2 2 n 2 2 2 n 2 n 2
2 N = Ò! N = =
µ 2 µ µ
3 2
N = = 3 H" 9 , 4 2
3
v = É0 Å" R
1 m
v = 2 n Å" d = n d H" 9 , 4 2
2 s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 7
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
Z E S T A W 5 P r a c a , e n e r g i a , m o c
1 0 . 1 1 . 2 0 0 6 r .
W a \n e w z o r y :
R U C H O B O T O W Y
v 2
a d =
R
P R A C A
r r
r r
W = F o s = F Å" s Å"c o s "( F , s )
E N E R G I A
E n e r g i a p o t e n c j a l n a ( z a w s z e o k r e [l a n a w z g l d e m j a k o [ p o z i o m u o d n i e s i e n i a E p = 0 ) :
E p = m g h
E n e r g i a k i n e t y c z n a ( r u c h u p o s t p o w e g o ) :
m v 2
E k = K =
2
W = "E
M O C
d E
P =
d t
d l a s t a Be j s i By w c z a s i e : P = F Å"v
z a d . 1 .
m
D a n e : ¸ = 2 0 ° , R = 8 0 m , g = 9 , 8 1
s 2
S z u k a n e : v m a x = ? ,
N a l e \y r o z p a t r z e s y t u a c j n a s t y k , w t e d y g d y c i a Bo p o r u s z a s i p o o k r g u r u c h e m
j e d n o s t a j n y m . O z n a c z a t o , \e w y p a d k o w a s i Ba d z i a Ba j c y c h n a s a m o c h ó d t o s i Ba
d o [r o d k o w a , s k i e r o w a n a p o z i o m o w z d Bu \ p r o m i e n i a R .
r r
r
Q + F R = m a
v 2
F R = m a d = m
R
F R ¥" - Q = 0 Ò! F R ¥" = m g
F R
t g ¸ =
F R ¥"
v 2
m
R
t g ¸ = Ò! v = R Å" g Å" t g ¸
m g
m
v = 8 0 Å"9 , 8 1 Å" t g 2 0 ° H" 2 8 5 , 6 4 H" 1 6 , 9
s
1
1 0 0 0
m k m k m k m k m
v = 1 6 , 9 = 1 6 , 9 Å" = 1 6 , 9 Å"3 , 6 = 1 6 , 9 Å"3 , 6 H" 6 0 , 8
s h h h h
1
3 6 0 0
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 8
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 2 .
m
D a n e : L = 3 0 c m , R = 1 5 c m , v = c o n s t . , g = 9 , 8 1
s 2
o b r .
S z u k a n e : n = ? [ m i n ] ,
² k t w y c h y l e n i a n i t k i o d p i o n u ( t y l k o z m i e n n a p o m o c n i c z a )
r r
r
N + Q = m a
v 2
N = m a d = m
R
N ¥" - m g = 0 Ò! N ¥" = m g
N
t g ² =
N ¥"
v 2
m
R
t g ² = Ò! v = R Å" g Å" t g ²
m g
R R 1 5
s i n ² = Ò! ² = a r c s i n = a r c s i n = 3 0 °
L L 3 0
É
É = 2 n Ò! n =
2
v
v = ÉR Ò! É =
R
v
R Å" g Å" t g ²
v R Å" g Å" t g ² 1 g
R
n = = = = = t g ²
2 2 R 2 R 4 2 R 2 2 R
R = 1 5 c m = 0 , 1 5 m
1 9 , 8 1 1
o b r .
n = t g 3 0 ° H" 3 7 , 7 6 H" 0 , 9 7 8
s
2 0 , 1 5 2
1
o b r . o b r . o b r .
n = 0 , 9 7 8 = 0 , 9 7 8 Å" H" 5 8 , 7
s m i n m i n
1
6 0
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 3 9
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 3 .
D a n e : m , N = 2 Q ,
S z u k a n e : ± = ? ,
L d Bu g o [ s z n u r k a ( t o t y l k o z m i e n n a p o m o c n i c z a ) ,
h w y s o k o [ n a j a k z n o s i s i k u l k a w c z a s i e m a k s y m a l n e g o w y c h y l e n i a w z g l d e m
p o Bo \e n i a r ó w n o w a g i
N = 2 Q = 2 m g
r r
r
N + Q = m a
v 2
N - m g = m a d = m
L
m v 2
N = + m g
L
m v 2
2 m g = + m g
L
v 2 = g L
z z a s a d y z a c h o w a n i a e n e r g i i :
m v 2
m g h = Ò! v 2 = 2 g h
2
w i c p o r ó w n u j c w y r a \e n i a n a v 2 o t r z y m u j e m y :
g L = 2 g h Ò! L = 2 h
s k d i n d w i a d o m o , \e :
L - h 2 h - h
1
c o s ± = = =
2
L 2 h
1
± = a r c c o s = 6 0 °
2
z a d . 4 .
D a n e : R , f = 0 ,
S z u k a n e : H = ? ,
C i a Bo p o r u s z a s i p o p t l i z e s t a B p r d k o [c i l i n i o w v . N a l e \y z n a l e z t a k i e v m i n a b y
c i a Bo z d o Ba Bo p o k o n a n a j b a r d z i e j n e w r a l g i c z n y p u n k t p t l i , t j . s a m s z c z y t p t l i .
r r
r
Q + F R = m a
v 2
m g + F R = m
R
d l a F R = 0 w y z n a c z y m y v m i n :
v m i n 2
m g + 0 = m
R
v m i n 2 = g R
z z a s a d y z a c h o w a n i a e n e r g i i :
m v m i n 2
m g H = + m g Å" 2 R
2
2 g H - 4 R g = v m i n 2 Ò! v m i n 2 = 2 g ( H - 2 R )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 0
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
w i c p r z y r ó w n u j c w y r a \e n i a v m i n 2 o t r z y m u j e m y :
g R = 2 g ( H - 2 R )
R = 2 H - 4 R
5
H = R = 2 , 5 R
2
z a d . 5 .
D a n e : R ,
S z u k a n e : H = ? ,
r r
r
Q + F R = m a
Q = m g c o s ±
v 2
m g c o s ± - F R = m
R
w m o m e n c i e o d e r w a n i a F R = 0 :
v m a x 2
m g c o s ± + 0 = m
R
v m a x 2 = g R c o s ±
z z a s a d y z a c h o w a n i a e n e r g i i :
m v m a x 2
m g R = m g H +
2
2 g R - 2 g H = v m a x 2
s k d i n d w i a d o m o , \e :
H
c o s ± =
R
p r z e z p o r ó w n a n i e w y r a \e D n a v m a x 2 o t r z y m u j e m y :
2 g R - 2 g H = g R c o s ±
H
2 R - 2 H = R
R
2
H = R
3
z a d . 6 .
m
D a n e : m = 2 k g , H = 1 m , W = 3 0 J , g = 9 , 8 1
s 2
S z u k a n e : a = ? ,
r
r
W = F Å" s Å" c o s "( F , s ) = F Å" s Å"c o s 0 ° = F Å" s
W
W = F Å" H Ò! F =
H
F w F - m g F W
a = = = - g = - g
m m m H m
3 0
m
a = - 9 , 8 1 = 5 , 1 9
s 2
1 Å" 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 1
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 7 .
m
D a n e : m = 2 k g , ± = 3 0 ° , W = 2 1 6 J , g = 9 , 8 1
s 2
S z u k a n e : E p = ? , E k = ? ,
Z a d a n i e zl e s f o r m u Bo w a n e , p y t a n i e o w a r t o [ e n e r g i i p o t e n c j a l n e j w n a j w y \s z y m p u n k c i e
n i e m a s e n s u f i z y c z n e g o . D l a n a d a n i a s e n s u t e m u z a d a n i u z a k Ba d a m , \e w a r t o [ e n e r g i i
p o t e n c j a l n e j b d l i c z y B w z g l d e m p o z i o m u w y r z u c e n i a k u l i ( E p = 0 ) .
v x = v x = v 0 c o s ±
0
v y = v 0 s i n ±
m v 0 2 2 W
W = Ò! v 0 =
2 m
r ó w n a n i a d l a r u c h u j e d n o s t a j n i e o p ó zn i o n e g o w p i o n i e :
ñø - g t
ôø0 = v y 0
òø
ôøH = v y t - 1 g t 2
2
óø 0
v y
ñø
0
ôøt = g
ôø
òø
2
ôøH = v y v y - 1 g v y
0 0
2
0
ôø
g g 2
óø
v y 2
0
H =
2 g
z z a s a d y z a c h o w a n i a e n e r g i i :
2
m ( v y 2 + v x 2 ) m v
x 0
0 0
= m g H +
2 2
v y 2 1
2 W
0
1
E p = m g H = m g = m v 0 2 s i n 2 ± = m s i n 2 ± = W s i n 2 3 0 ° = 2 1 6 Å"0 , 2 5 = 5 4 J
2 2
2 g m
m v x 2 m v 0 2 c o s 2 ± m Å" 2 W Å"c o s 2 ±
0
E k = = = = W Å"c o s 2 3 0 ° H" 2 1 6 Å"0 , 7 5 = 1 6 2 J
2 2 2 Å"m
b o :
W = E p + E k = 5 4 + 1 6 2 = 2 1 6 J
M a s a k l o c k a w z a d a n i u z o s t a Ba p o d a n a n i e p o t r z e b n i e !
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 2
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 8 .
D a n e : m = 1 5 k g , v 0 = 0 s = 5 m , F = 7 0 N , ¸ = 2 0 ° ,
m
f = 0 , 3 , g = 9 , 8 1
s 2
2
S z u k a n e : W F = ? , E k = ? , W T = ? , F ,
W F = F Å" s Å" c o s ¸
W F = 7 0 Å" 5 Å" c o s 2 0 ° H" 3 2 8 , 9 J
W T = T Å" s Å" c o s 1 8 0 ° = - T s = - F N Å" f Å" s = - ( Q - F ¥") Å" f Å" s
W T = - ( m g - F s i n ¸ ) Å" f Å" s = f s ( F s i n ¸ - m g )
W T = 0 , 3 Å"5 Å" ( 7 0 Å" s i n 2 0 ° - 9 , 8 1 Å"1 5 ) H" - 1 8 4 , 8 J
"E k = W F + W T
"E k = 3 2 8 , 9 - 1 8 4 , 8 H" 1 4 4 , 1 J
P r z e s u n i c i e b d z i e o d b y w a Bo s i r u c h e m j e d n o s t a j n y m g d y "E k = 0 b o w t y m r u c h u
"v = 0 , w i c ( u w a g a : z m i e n i s i w a r t o [ s i By t a r c i a j a k z m i e n i s i w a r t o [ s i By F ) :
2 2
W F = F Å" s Å" c o s ¸
2 2
"E k = W F + W T = 0
2 2
W F = - W T
2 2
F Å" s Å" c o s ¸ = - f s ( F s i n ¸ - m g )
2 2
F c o s ¸ + f F s i n ¸ = f m g
f m g 0 , 3 Å"1 5 Å"9 , 8 1 4 4 , 1 4 5
2
F = - = H" H" 4 2 , 3 5 N
c o s ¸ + f s i n ¸ c o s 2 0 ° + f s i n 2 0 ° 1 , 0 4 2 2 9 9
z a d . 9 .
m
D a n e : m = 1 0 k g , v 0 = 1 , 5 ¸ = 2 0 ° , F = 1 0 0 N ,
s
m
f = 0 , 4 , s = 5 m , g = 9 , 8 1 ,
s 2
S z u k a n e : W F = ? , W T = ? , "E k = ? , v k = ? ,
W F = F Å" s Å"c o s 0 °
W F = 1 0 0 Å"5 Å"c o s 0 ° = 5 0 0 J
W T = T Å" s Å"c o s 1 8 0 ° = - T s = - F N Å" f Å" s = - Q ¥" Å" f Å" s
W T = - m g c o s ¸ Å" f Å" s = - 1 0 Å"9 , 8 1 Å"c o s 2 0 ° Å"0 , 4 Å"5 H" - 1 8 4 , 4 J
"E = W F + W T = 5 0 0 - 1 8 4 , 4 = 3 1 5 , 6
"E = "E p + "E k
"E k = "E - "E p = "E - m g Å""h = "E - m g Å" s s i n ¸
"E k = 3 1 5 , 6 - 1 0 Å"9 , 8 1 Å"5 Å"s i n 2 0 ° H" 3 1 5 , 6 - 1 6 7 , 7 6 H" 1 4 7 , 8 4 J
m v k 2 m v 0 2 2 "E k
"E k = - Ò! v k = + v 0 2
2 2 m
2 Å"1 4 7 , 8 4
m
v k = + 1 , 5 2 H" 3 1 , 8 1 8 H" 5 , 6 4
s
1 0
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 3
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 1 0 .
m
D a n e : m = 6 5 0 k g , a 1 = c o n s t . v 1 = 1 , 7 5 t 1 = 3 s ,
s
k
m m
a 2 = 0 , v 2 = 1 , 7 5 , g = 9 , 8 1 ,
s
s 2
S z u k a n e : P [r . 1 = ? , P 2 = ? ,
d E
P =
d t
"E 1
P [r . 1 =
t 1
v 1
k
a 1 =
t 1
v 1 1
k
1 1
h = a 1 t 1 2 = t 1 2 = v 1 t 1
2 2
k
t 1 2
m v 1 2 2 m g h m v 1 2
k k
m g h + +
1
2 m g h + m v 1 2 2 m g Å" v 1 t 1 + m v 1 2
2
2 2 2 k k k
P [r . 1 = = = =
t 1 t 1 2 t 1 2 t 1
m v 1 ( g Å"t 1 + v 1 )
6 5 0 Å"1 , 7 5 Å"( 9 , 8 1 Å"3 + 1 , 7 5 )
k k
P [r . 1 = = H" 5 9 1 1 , 2 W H" 5 , 9 1 k W
2 t 1 2 Å"3
"v 2 = 0
"h = v 2 "t
m "v 2 2
"E 2 m g "h + 2 m g "h m g Å" v 2 "t
P 2 = = = = = m g Å"v 2
"t "t "t "t
P 2 = 6 5 0 Å"9 , 8 1 Å"1 , 7 5 H" 1 1 1 5 8 , 8 7 5 W H" 1 1 , 1 6 k W
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 4
U t w o r z o n a : 1 2 . 1 1 . 2 0 0 6 1 5 : 5 3
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 5 P r a c a , e n e r g i a , m o c . Z m o d y f i k o w a n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 5 0
z a d . 1 1 .
k m m
D a n e : m = 1 0 0 0 k g , v = 7 2 , f = 0 , 1 , g = 9 , 8 1 ,
h
s 2
±a = 0 ° , ±b = 3 ° , ±c = - 3 ° ,
S z u k a n e : P a = ? , P b = ? , P c = ? ,
P o n i \s z y s z k i c r o z w i z a n i a b d z i e d l a d o w o l n e g o k t a ± , a n a k o Dc u b d t y l k o
p o d s t a w i o n e m o \l i w e w a r t o [c i t e g o k t a i o b l i c z o n e o d p o w i e d n i e m o c e .
k m 1 0 0 0 m m m
v = 7 2 = 7 2 Å" = 7 2 : 3 , 6 = 2 0
h 6 0 Å"6 0 s s s
"s = v Å" "t
Q ¥" = m g Å"c o s ±
Q = m g Å"s i n ±
W T + W Q Q ¥" f Å" "s + m g Å"s i n ± Å" "s m g v Å" "t
P = = = ( f c o s ± + Å"s i n ±) =
"t "t "t
P ( ± ) = m g v ( f c o s ± + Å"s i n ± ) = 1 0 0 0 Å"9 , 8 1 Å" 2 0 Å"( 0 , 1 Å"c o s ± + s i n ± )
P ( ± ) = 1 9 6 , 2 Å"( 0 , 1 c o s ± + s i n ±) Å"1 0 3 [ W ] = 1 9 6 , 2 Å" ( 0 , 1 c o s ± + s i n ± ) [ k W ]
P ( ± ) = 1 9 , 6 2 c o s ± + 1 9 6 , 2 s i n ±) [ k W ]
P a = P ( ±a ) = P ( 0 ° ) = 1 9 , 6 2 c o s 0 ° + 1 9 6 , 2 s i n 0 ° [ k W ] = 1 9 , 6 2 [ k W ]
P b = P ( ±b ) = P ( 3 ° ) = 1 9 , 6 2 c o s 3 ° + 1 9 6 , 2 s i n 3 ° H" 2 9 , 8 6 [ k W ]
P c = P ( ±c ) = P ( - 3 ° ) = 1 9 , 6 2 c o s ( - 3 ° ) + 1 9 6 , 2 s i n ( - 3 ° ) H" 9 , 3 2 [ k W ]
D o d a t k o w e z a g a d n i e n i e :
Z j a k m a k s y m a l n p r d k o [c i l i n i o w m o \e s a m o c h ó d r u c h e m j e d n o s t a j n y m p r z e j e c h a
m o s t o p r o m i e n i u k r z y w i z n y r ó w n y m R .
T r z e b a r o z p a t r z y s y t u a c j w n a j w y \s z y m p u n k c i e n a m o [c i e .
m v 2
F d =
R
Q = m g
F d = Q
m v m a x 2
= g m
R
v m a x = g R
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 5
U t w o r z o n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 4 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 6 S p r \y s t o [, p d . Z m o d y f i k o w a n a : 2 3 . 1 1 . 2 0 0 6 2 0 : 0 9
Z E S T A W 6 S p r \y s t o [, p d
1 7 . 1 1 . 2 0 0 6 r .
W a \n e w z o r y :
S I AA S P R [Y S T O ZC I
F s = - k Å" x
g d z i e :
N
k w s p ó Bc z y n n i k s p r \y s t o [c i [ m ] ,
x w y c h y l e n i e z p o z i o m u r ó w n o w a g i .
k o m e n t a r z :
m i n u s w p o w y \s z y m w z o r z e o z n a c z a , \e s i Ba s p r \y s t o [c i j e s t s k i e r o w a n a
p r z e c i w n i e d o k i e r u n k u r u c h u , j e s t s k i e r o w a n a z a w s z e w s t r o n p u n k t u r ó w n o w a g i .
E N E R G I A P O T E N C J A L N A S P R [Y S T O ZC I
k Å" x 2
s
E p = E s =
2
k o m e n t a r z :
w p o Bo \e n i u r ó w n o w a g i w a r t o [ t e j e n e r g i i r ó w n a s i 0 , w a r t o [ t e j e n e r g i i j e s t
m a k s y m a l n a w m a k s y m a l n y m w y c h y l e n i u .
P D
r r
p = m Å"v
z a d . 1 .
o b r . N m
D a n e : m = 0 , 5 k g , n = 2 , ± = 3 0 ° , k = 6 0 0 , g = 9 , 8 1 ,
s m
s 2
S z u k a n e : l 0 = ? ,
r a d
É = 2 n = 2 Å" 2 = 1 2 , 5 7
s
v = ÉR
k l u c z o w y m e l e m e n t e m j e s t z a u w a \e n i e , \e :
r r
N = F s
g u m k a r o z c i g a s i , w i c :
"l = l - l 0
F s = k "l
N = k "l
v 2
N = m = m É2 R = m É2 l s i n ±
R
R
= s i n ±
l
N ¥" - m g = 0 Ò! N ¥" = m g
N
m É2 l s i n ± g t g ± g
= t g ± = Ò! l = =
N ¥" m g É2 s i n ± É2 c o s ±
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 6
U t w o r z o n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 4 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 6 S p r \y s t o [, p d . Z m o d y f i k o w a n a : 2 3 . 1 1 . 2 0 0 6 2 0 : 0 9
N ¥" m g m g
c o s ± = = Ò! "l =
N k "l k c o s ±
l 0 = l - "l
g m g g 1 m
ëø öø
l 0 = - =
ìø - ÷ø
É2 c o s ± k c o s ± c o s ± É2 k
íø øø
9 , 8 1 ëø 1 0 , 5 öø
l 0 = ìø - ÷ø = 0 , 0 6 2 2 5 m = 0 , 0 6 2 3 c m
2
c o s 3 0 °
íø1 2 , 5 7 6 0 0 øø
z a d . 2 .
m
D a n e : "x 0 = 0 , 1 2 m , m = 0 , 0 3 5 k g , H = 2 0 m , H 1 = 1 0 m g = 9 , 8 1 ,
s 2
S z u k a n e : k = ? , v 0 = ? , v 1 = ? ,
z z a s a d y z a c h o w a n i a e n e r g i i :
k ( "x 0 ) 2 m v 0 2 m v 1 2
= + m g "x 0 = + m g H 1 = m g H
2 2 2
w i c :
k ( "x 0 ) 2 2 m g H 2 Å"0 , 0 3 5 Å"9 , 8 1 Å" 2 0
N
= m g H Ò! k = = = 9 5 3 , 7 5
m
2 ( "x 0 ) 2 0 , 1 2 2
m v 0 2
m
+ m g "x 0 = m g H Ò! v 0 = 2 g ( H - "x 0 ) = 2 Å"9 , 8 1 Å"( 2 0 - 0 , 1 2 ) = 1 9 , 7 5
s
2
m v 1 2
m
+ m g H 1 = m g H Ò! v 1 = 2 g ( H - H 1 ) = 2 Å"9 , 8 1 Å"( 2 0 - 1 0 ) = 1 4 , 0 1
s
2
z a d . 3 .
N m
D a n e : k = 4 2 7 , ± = 2 8 ° , m = 3 , 2 2 k g , "x = 2 1 , 4 c m , g = 9 , 8 1 ,
m
s 2
2
S z u k a n e : s = ? , "x = ? ( d l a v m a x ) ,
H
= s i n ± Ò! H = s s i n ±
s
z z a s a d y z a c h o w a n i a e n e r g i i ( p o z i o m e m o d n i e s i e n i a s i j e s t p u n k t z a t r z y m a n i a s i k l o c k a ) :
k ( "x ) 2
m g H =
2
w i c :
k ( "x ) 2 k ( "x ) 2 4 2 7 Å"0 , 2 1 4 2
m g s s i n ± = Ò! s = = = 0 , 6 5 9 m = 6 5 , 9 c m
2 2 m g s i n ± 2 Å"3 , 2 2 Å"9 , 8 1 Å"s i n 2 8 °
m a k s y m a l n a p r d k o [ j e s t w c h w i l i g d y :
F s = Q
w i c :
2
k "x = m g s i n ±
m g s i n ± 3 , 2 2 Å"9 , 8 1 Å"s i n 2 8 °
2
"x = = = 0 , 0 3 4 7 m = 3 , 4 7 c m
k 4 2 7
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 7
U t w o r z o n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 4 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 6 S p r \y s t o [, p d . Z m o d y f i k o w a n a : 2 3 . 1 1 . 2 0 0 6 2 0 : 0 9
z a d . 4 .
m m
D a n e : m = 3 k g , v = 1 0 , ± = 3 0 ° , "t = 0 , 2 s , g = 9 , 8 1 ,
s
s 2
S z u k a n e : F [r = ? ,
r
r
d p
F =
d t
"p
F [r =
"t
v 1 = v 2 = v
r
v 1 = [ - v c o s ±, v s i n ± ]
r
v 2 = [ v c o s ±, v s i n ± ]
r r
p = m v
r r r r r r r
"p = p 2 - p 1 = m v 2 - m v 1 = m ( v 2 - v 1 ) = m Å"[ v c o s ± + v c o s ±, v s i n ± - v s i n ± ]
r
"p = [ 2 m v c o s ±, 0 ]
"p = 2 m v c o s ±
"p 2 m v c o s ± 2 Å"3 Å"1 0 Å"c o s 3 0 °
F [r = = = = 2 5 9 , 8 N
"t "t 0 , 2
z a d . 5 .
m
D a n e : m p = 0 , 1 % Å" m k , L = 1 m , ¸ = 1 0 ° , g = 9 , 8 1 ,
s 2
S z u k a n e : v = ? ,
m p = 0 , 0 0 1 Å" m k
p p = p k
m p v p = ( m p + m k ) Å"v k
( m p + m k ) Å"v k ( 0 , 0 0 1 Å" m k + m k ) Å"v k
v p = = = 1 0 0 1 Å" v k
m p 0 , 0 0 1 Å" m k
z z a s a d y z a c h o w a n i a e n e r g i i :
m v k 2
= m g h Ò! v k = 2 g h
2
L - h
= c o s ¸ Ò! h = L ( 1 - c o s ¸ )
L
v k = 2 g h = 2 g L ( 1 - c o s ¸ )
m
v p = 1 0 0 1 v k = 1 0 0 1 2 g L ( 1 - c o s ¸ ) = 1 0 0 1 2 Å"9 , 8 1 Å"1 Å" ( 1 - c o s 1 0 ° ) = 5 4 5 , 9
s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 8
U t w o r z o n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 4 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 6 S p r \y s t o [, p d . Z m o d y f i k o w a n a : 2 3 . 1 1 . 2 0 0 6 2 0 : 0 9
z a d . 6 .
r r
D a n e : m 1 = 3 k g , v 1 ( 5 , 0 ) [ m ] , m 1 = 2 k g , v 1 ( 0 , - 3 ) [ m ] ,
s s
r
S z u k a n e : u = ? , x = ? [ % ] ( u t r a c o n y p r o c e n t e n e r g i i k i n e t y c z n e j )
r r r
p 1 + p 2 = p u
r r r
m 1 v 1 + m 2 v 2 = ( m 1 + m 2 ) Å" u
r r
r m 1 v 1 + m 2 v 2 3 Å"[ 5 , 0 ] + 2 Å"[ 0 , - 3 ]
1
u = = = [ 1 5 , - 6 ] = [ 3 ; - 1 , 2 ]
5
m 1 + m 2 3 + 2
r
m
u = u = 3 2 + ( - 1 , 2 ) 2 = 3 , 2 3
s
ëø öø
E 1 + E 2 - E u E 1 + E 2 E u E u
x = Å"1 0 0 % = ìø - ÷ø = 1 0 0 % - Å"1 0 0 %
ìø
E 1 + E 2 E 1 + E 2 E 1 + E 2 ÷øÅ"1 0 0 % E 1 + E 2
íø øø
( m 1 + m 2 ) Å"u 2
( m 1 + m 2 ) Å"u 2
2
x = 1 0 0 % - Å"1 0 0 % = 1 0 0 % - Å"1 0 0 %
m 1 Å"v 1 2 m 2 Å"v 2 2 m 1 Å"v 1 2 + m 2 Å"v 2 2
+
2 2
ëø öø ëø öø
( 3 + 2 ) Å"3 , 2 3 2 ( 3 + 2 ) Å"3 , 2 3 2
x = ìø
ìø1 - 3 Å"5 2 + 2 Å"( - 3 ) 2 ÷ø Å"1 0 0 % = ìø 3 Å"5 2 + 2 Å"( - 3 ) 2 ÷ø = 1 0 0 % - 5 6 , 0 9 % = 4 3 , 9 %
÷ø ìø1 - ÷øÅ"1 0 0 %
íø øø íø øø
z a d . 7 .
m m
D a n e : m 1 = m 2 , v = 5 , v 1 = 4 , 3 3 , ±1 = 3 0 ° ,
s s
S z u k a n e : v 2 = ? , ±2 = ? ,
m = m 1 = m 2
r r r
p = p 1 + p 2
r r r
m v = m 1 v 1 + m 2 v 2
r r r
v = v 1 + v 2
[ v , 0 ] = [ v 1 Å" c o s ±1 , v 1 Å" s i n ±1 ] + [ v 2 Å" c o s ±2 , v 2 Å" s i n ±2 ]
5 = 4 , 3 3 Å" c o s 3 0 ° + v 2 Å" c o s ±2
ñø
òø
óø0 = 4 , 3 3 Å" s i n 3 0 + v 2 Å" s i n ±2
1 , 2 5
ñøv Å" c o s ±2 = 1 , 2 5 Ò! v 2 =
ôø 2
c o s ±2
òø
ôøv Å" s i n ±2 = - 2 , 1 6 5
óø 2
1 , 2 5
Å" s i n ±2 = - 2 , 1 6 5
c o s ±2
- 2 , 1 6 5 2 , 1 6 5
t g ±2 = Ò! ±2 = a r c t g - = - 6 0 °
1 , 2 5 1 , 2 5
1 , 2 5 1 , 2 5
m
v 2 = = = 2 , 5
s
c o s ±2 c o s ( - 6 0 ° )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 4 9
U t w o r z o n a : 2 0 . 1 1 . 2 0 0 6 1 3 : 4 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 6 S p r \y s t o [, p d . Z m o d y f i k o w a n a : 2 3 . 1 1 . 2 0 0 6 2 0 : 0 9
z a d . 8 .
m m m
D a n e : v 0 = 3 0 0 , v 1 = 4 5 0 , v 2 = 2 4 0 ,
s s s
r
S z u k a n e : v 3 = ? ,
r r r r
p 0 = p 1 + p 2 + p 3
r r r r
m m m
m v 0 = v 1 + v 2 + v 3
3 3 3
1 1 1
[ 0 , 3 0 0 ] = [ 0 , 4 5 0 ] + [ 2 4 0 , 0 ] + [ v x , v y ]
3 3 3
3 Å"[ 0 , 3 0 0 ] = [ 2 4 0 , 4 5 0 ] + [ v x , v y ]
[ v x , v y ] = [ 0 , 9 0 0 ] - [ 2 4 0 , 4 5 0 ] = [ 0 - 2 4 0 , 9 0 0 - 4 5 0 ] = [ - 2 4 0 , 4 5 0 ]
r
v 3 = [ - 2 4 0 , 4 5 0 ]
D o d a t k o w e z a g a d n i e n i a :
a ) K u l k a o m a s i e m i p r d k o [c i v p o r u s z a j c a s i p o l i n i i p r o s t e j o d b i Ba s i i d e a l n i e
s p r \y [c i e o d [c i a n y u s t a w i o n e j p r o s t o p a d l e d o k i e r u n k u r u c h u . J a k z m i e n i B s i p d k u l k i ?
P d t o w i e l k o [ w e k t o r o w a , w i c t r z e b a j e d n a k u w z g l d n i n i e t y l k o w a r t o [, a l e t a k \e
k i e r u n e k i z w r o t .
r r
p = m Å"v
r r
"p = p p - p k = m Å"v - m Å"( - v ) = 2 m v
r
b ) K u l k a o m a s i e m m a p d p . J a k a j e s t w a r t o [ e n e r g i i k i n e t y c z n e j t e g o c i a Ba ?
p = m v
m v 2 m v 2 m m 2 v 2 ( m v ) 2 p 2
E k = = Å" = = =
2 2 m 2 m 2 m 2 m
c ) A b y r o z c i g n s p r \y n z p o Bo \e n i a r ó w n o w a g i o x t r z e b a w y k o n a j a k [ p r a c . I l e r a z y
w i k s z d o d a t k o w p r a c t r z e b a w y k o n a a b y r o z c i g n s p r \y n j e s z c z e o x ( c z y l i d o
p o Bo \e n i a 2 x o d p o Bo \e n i a r ó w n o w a g i ) ?
W = "E
k Å" x 2
W 1 =
2
k Å"( 2 x ) 2 k Å"( 2 x ) 2 k Å" x 2 k Å" 4 x 2 k Å" x 2 3 k x 2
W 2 = - W 1 = - = - =
2 2 2 2 2 2
3 k x 2
W 2 2
= = 3
W 1 k x 2
2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 0
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o
2 4 . 1 1 . 2 0 0 6 r .
W a \n e w z o r y :
ZR O D E K M A S Y
n n n
m i x i m i y i m i z i
x [r = y [r = z [r =
" " "
£m £m £m
i i i
M O M E N T B E Z W AA D N O ZC I
a n a l o g i a d o r u c h u p o s t p o w e g o : m m a s a c i a Ba
I m o m e n t b e z w Ba d n o [c i c i a Ba
k o m e n t a r z : r ó \n e c i a Ba m o g m i e r ó \n e m o m e n t y b e z w Ba d n o [c i ; a l e t a k \e
w a r t o [ m o m e n t u b e z w Ba d n o [c i z a l e \y o d w y b r a n e j o s i o b r o t u ( k t ó r a m o \e b y
d o w o l n i e w y b r a n a ) .
I 0 = I [r m o m e n t b e z w Ba d n o [c i w z g l d e m o s i p r z e c h o d z c e j p r z e z [r o d e k m a s y
n n n
I x = ( y i 2 + z i 2 ) I = ( x i 2 + z i 2 ) I z = ( x i 2 + y i 2 )
"m i y "m i "m i
i i i
T w i e r d z e n i e S t e i n e r a
2
I = I 0 + m d
g d z i e :
d o d l e g Bo [ m i d z y d w i e m a r ó w n o l e g By m i o s i a m i o b r o t u
k o m e n t a r z : s t o s u j e s i d o o b l i c z e n i a m o m e n t u b e z w Ba d n o [c i w z g l d e m w y b r a n e j o s i
r ó w n o l e g Be j d o o s i p r z e c h o d z c e j p r z e z [r o d e k m a s y c i a Ba ( z a k Ba d a m y , \e z n a m y m o m e n t
b e z w Ba d n o [c i c i a Ba , w z g l d e m t e j o s i p r z e c h o d z c e j p r z e z [r o d e k m a s y ) .
L i s t a p o d s t a w o w y c h m o m e n t ó w
b e z w Ba d n o [c i d l a o s i o b r o t u
M O M E N T S I AY
p r z e c h o d z c y c h p r z e z [r o d e k m a s y :
r
r
a n a l o g i a d o r u c h u p o s t p o w e g o : F = m Å" a s i Ba
" p u n k t u m a t e r i a l n e g o
I = m R 2
M = I Å"µ
" w a l c a ( k r \k a )
g d z i e : µ p r z y s p i e s z e n i e k t o w e
1
I = m R 2
2
r r " c i e n k o [c i e n n e g o w a l c a
r
M = r × F
( o b r c z y , r u r y )
g d z i e : r r a m i s i By F , o d l e g Bo [ m i d z y o s i o b r o t u a o s i
I = m R 2
w y z n a c z o n p r z e z k i e r u n e k s i By F
" k u l i
2
I = m R 2
5
M O M E N T P D U
r r
" p o w i e r z c h n i k u l i ( s f e r y )
a n a l o g i a d o r u c h u p o s t p o w e g o : p = m Å"v p d
2
I = m R 2
3
" c i e n k i e g o p r t a
L = I Å"É
1
I = m L 2
g d z i e : É p r d k o [ k t o w a
1 2
r
r r
L = r × p
g d z i e : r o d l e g Bo [ m i d z y o s i o b r o t u a o s i
r
w y z n a c z o n p r z e z k i e r u n e k w e k t o r a p d u p
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 1
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
z a d . 1 .
D a n e : m = 0 , 1 k g , M = 0 , 3 k g , a = 2 0 c m , b = 4 0 c m ,
S z u k a n e : I x = ? , I y = ? , I z = ? ,
" m = m + m + M + M = 0 , 1 + 0 , 1 + 0 , 3 + 0 , 3 = 0 , 8 k g
m Å"0 + m Å" a + M Å" 0 + M Å" a 0 , 1 Å"0 , 2 + 0 , 3 Å" 0 , 2
x [r = = = 0 , 1 m
" m 0 , 8
m Å"b + m Å"b + M Å"0 + M Å"0 2 Å" 0 , 1 Å"0 , 4
y [r = = = 0 , 1 m
" m 0 , 8
z [r = 0
m 1 ( - 0 , 1 ; 0 , 3 ) m 2 ( 0 , 1 ; 0 , 3 ) M 1 ( - 0 , 1 ; - 0 , 1 ) M ( 0 , 1 ; - 0 , 1 )
2
I x = 2 Å"0 , 1 Å"( 0 , 3 2 + 0 2 ) + 2 Å"0 , 3 Å"( ( - 0 , 1 ) 2 + 0 2 ) = 0 , 0 2 4 [ k g Å" m 2 ]
I = ( 0 , 1 + 0 , 3 ) Å"( ( - 0 , 1 ) 2 + 0 2 ) + ( 0 , 1 + 0 , 3 ) Å"( ( 0 , 1 ) 2 + 0 2 ) = 0 , 0 0 8 [ k g Å" m 2 ]
y
I z = 0 , 1 Å"( ( - 0 , 1 ) 2 + 0 , 3 2 ) + 0 , 1 Å"( 0 , 1 2 + 0 , 3 2 ) + 0 , 3 Å"( ( - 0 , 1 ) 2 + ( - 0 , 1 ) 2 ) +
+ 0 , 3 Å"( 0 , 1 2 + ( - 0 , 1 ) 2 ) = 0 , 0 3 2 [ k g Å" m 2 ]
z a d . 2 .
D a n e : m a = 4 k g , m b = 2 k g , m c = 3 k g ,
r a d
y a = 3 m , y b = - 2 m , y a = - 4 m , É = 2 ,
s
2
S z u k a n e : E k = ? ( d l a o b r o t u w o k ó B o s i O X ) , E k = ? ( d l a o b r o t u w o k ó B [r . m a s y ) ,
I x = m a y a 2 + m b y b 2 + m c y c 2 = 4 Å"3 2 + 2 Å"( - 2 ) 2 + 3 Å"( - 4 ) 2 = 9 2 [ k g m 2 ]
I Å"É2 9 2 Å"2 2
E k = = = 1 8 4 J
2 2
" m = m a + m b + m c = 4 + 2 + 3 = 9
m a y a + m b y b + m c y c 4 Å"3 + 2 Å"( - 2 ) + 3 Å"( - 4 )
4
y [r = = = - H" - 0 , 4 4 4 4
9
" m 9
k o r z y s t a m z t w i e r d z e n i a S t e i n e r a
2 2
I = I 0 + m d Ò! I 0 = I - m d
I 0 = I [r
4 8 1 2
I [r = I x - £m Å" y [r 2 = 9 2 - 9 Å"( - ) 2 = H" 9 0 , 2 2 [ k g m 2 ]
9 9
I [r Å"É2 9 0 , 2 2 Å" 2 2
2
E k = = = 1 8 0 , 4 4 J
2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 2
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
z a d . 3 .
m
D a n e : l = 1 m , m 1 = 3 k g , m 2 = 4 k g , v = 5 ,
s
r
S z u k a n e : L = ? ,
l
r =
2
2
l l 2
ëø öø
I = m 1 r 2 + m 2 r 2 = r 2 ( m 1 + m 2 ) = ( m 1 + m 2 ) = ( m 1 + m 2 )
ìø ÷ø
2 4
íø øø
v 2 v
v = É Å" r Ò! É = =
r l
l 2 2 v
k g Å"m 2
1 1
L = I Å"É = ( m 1 + m 2 ) Å" = v l ( m 1 + m 2 ) = 5 Å"1 Å"( 3 + 4 ) = 1 7 , 5
2 2 s
4 l
m o m e n t p d u j e s t w e k t o r e m , p r o s t o p a d By m d o p Ba s z c z y z n y o b r o t u , w i c :
r
2
Æ Æ
L = L Å" k = 1 7 , 5 k [ k g Å"m ]
s
Æ
g d z i e k j e s t w e r s o r e m o s i O Z .
I I s p o s ó b r o z w i z a n i a z a d a n i a
r
r r
L = r × p
p = p 1 + p 2 = m 1 v + m 2 v = v ( m 1 + m 2 )
r r l
1
L = r Å" p Å" s i n "( r , p ) = Å" v ( m 1 + m 2 ) Å" s i n 9 0 ° = v l ( m 1 + m 2 )
2
2
w i c o s t a t e c z n i e d o s t a j e m y t o s a m o , j a k m o \n a b y Bo p r z y p u s z c z a .
z a d . 4 .
m
D a n e : m 1 = 8 k g , m 2 = 2 k g , t = 2 s , g = 9 , 8 1 ,
s 2
S z u k a n e : a = ? , N = ? , E k = ? ( p o c z a s i e t ) ,
m o m e n t b e z w Ba d n o [c i k r \k a w z g l d e m o s i p r z e c h o d z c e j p r z e z [r o d e k m a s y w y n o s i :
1
I 0 = m 1 Å" R 2
2
r o z p a t r u j c s i By d z i a Ba j c e n a k l o c e k o t r z y m u j e m y :
m 2 g - N = m 2 a
m 2 g - N
a =
m 2
r o z p a t r u j e m y s i By d z i a Ba j c e n a k r \e k , k t ó r e s i s t o t n e , t j . i c h m o m e n t y s n i e z e r o w e
M = N Å" R Å"s i n 9 0 ° = N Å" R
s k d i n d w i a d o m o , \e :
M = I 0 Å"µ
o r a z
a
a = µ Å" R Ò! µ =
R
w i c
a I 0 Å" a
M = I 0 Å" = N Å" R Ò! N =
R R 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 3
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
p o d s t a w i a j c o t r z y m u j e m y :
1
I 0 Å" a m 1 Å" R 2 Å" a
2 1
N = = = m 1 a
R 2 R 2 2
m 2 a = m 2 g - N
2 m 2 2 Å" 2
1 m
m 2 a = m 2 g - m 1 a Ò! a = g = Å"9 , 8 1 = 3 , 2 7
2
s 2
m 1 + 2 m 2 8 + 2 Å" 2
1 1
N = m 1 a = Å"8 Å"3 , 2 7 = 1 3 , 0 8 N
2 2
k l o c e k p o r u s z a s i r u c h e m j e d n o s t a j n i e p r z y s p i e s z o n y m ( c o z o s t a Bo p o k a z a n e p o w y \e j ) ,
w i c i p u n k t y n a o b r z e \u k r \k a p o r u s z a s i z t a k i m p r z y s p i e s z e n i e m , w i c :
v = a t
v = ÉR
a t
ÉR = a t Ò! É =
R
2
a t
1
m 1 R 2 Å"ëø öø
ìø ÷ø
I 0 Å"É2 2 íø øø m 1 a 2 t 2 8 Å"3 , 2 7 2 Å" 2 2
R
E k = = = = = 8 5 , 5 J
2 2 4 4
d o d a t k o w o :
z a s a d a z a c h o w a n i a e n e r g i i d l a s y t u a c j i w t y m z a d a n i u w y g l d a n a s t p u j c o :
I 0 Å"É2 m Å"v 2
m g H = +
2 2
g d z i e H d r o g j a k p o k o n a B k l o c e k
z a d . 5 .
1 m
D a n e : L = 1 m , L 1 = L , ± = 9 0 ° , g = 9 , 8 1 ,
3
s 2
S z u k a n e : É = ? , v = ? ( p r d k o [ k o Dc a p r t a ) ,
Z a d a n i e j e s t n i e p r e c y z y j n e : t r z e b a o b l i c z y p r d k o [ k o Dc a p r t a v , a p r z e c i e \ p r t
p o d o b n i e j a k k i j , m a d w a k o Dc e . W i c o b l i c z z a r ó w n o v 1 j a k i i v 2 , c z y l i p r d k o [
j e d n e g o i p r d k o [ d r u g i e g o k o Dc a p r t a .
M o m e n t b e z w Ba d n o [c i p r t a w z g l d e m o s i p r z e c h o d z c e j p r z e z [r o d e k c i \k o [c i w y n o s i :
1
I 0 = m L 2
1 2
D l a o b l i c z e n i a o s i o b r o t u w z g l d e m p u n k t u O s t o s u j t w i e r d z e n i e S t e i n e r a
1 1
I = m L 2 + m ( 1 L ) 2 = m L 2
1 2 6 9
N i e m a \a d n y c h s i B t a r c i a , w i c s k o r z y s t a m z z a s a d y z a c h o w a n i a e n e r g i i .
W A [N E : z p u n k t u w i d z e n i a f i z y k i c a By p r t m o g z a s t p i p u n k t e m m a t e r i a l n y m o t a k i e j
s a m e j m a s i e u m i e s z c z o n y m w [r o d k u c i \k o [c i p r t a .
I É2
1
( L - L - L 1 ) g m =
2
2
L L m g L m g 3 g
1 r a d
g m = I É2 Ò! É = = = = 5 , 4 2
2 s
1
6 3 I 3 m L 2 L
9
v = É Å" r
1 2 2 m 1 1 1 m
v 1 = É Å"( L - L ) = L É = Å"5 , 4 2 = 3 , 6 2 v 2 = É Å" L = L É = Å"5 , 4 2 = 1 , 8 1
3 3 3 s 3 3 3 s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 4
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
z a d . 6 .
m
D a n e : H = 1 m , v 0 = 0 , g = 9 , 8 1 ,
s 2
S z u k a n e : v a = ? , v b = ? v = ? ( d l a f = 0 ) ,
J e \e l i j e s t t o s t a c z a n i e b e z p o [l i z g u t o m o \n a s k o r z y s t a z z a l e \n o [c i :
v = ÉR
Z z a s a d y z a c h o w a n i a e n e r g i i :
m v 2 I Å"É2
m g H = +
2 2
m v 2 I Å"v 2
m g H = +
2 2 R 2
m g H 2 R 2 m g H
v = =
m R 2 + I m R 2 + I
2 R 2
d l a k r \k a :
1
I a = m R 2
2
2 R 2 m g H 2 R 2 m g H
4 4 m
v a = = = g H = Å"9 , 8 1 Å"0 , 5 = 2 , 5 6
3 s
1
m R 2 + I a m R 2 + m R 2 3
2
d l a k u l i :
2
I b = m R 2
5
2 R 2 m g H 2 R 2 m g H
1 0 1 0 m
v b = = = g H = Å" 9 , 8 1 Å" 0 , 5 = 2 , 6 5
7 s
2
m R 2 + I b m R 2 + m R 2 7
5
g d y f = 0 w t e d y É = 0
m v 2
m g H =
2
m
v = 2 g H = 2 Å"9 , 8 1 Å"0 , 5 = 3 , 1 3
s
z a d . 7 .
D a n e : ¸ , g ,
S z u k a n e : a = ? , f = ? ,
r u c h p o s t p o w y :
Q - T = m a
r u c h o b r o t o w y :
M = T Å" R
1 a 1
M = I Å"µ = m R 2 Å" = m R a
2 R 2
1 1
m R a = T R Ò! T = m a
2 2
1 2
Q - T = m a Ò! m g s i n ¸ - m a = m a Ò! a = g s i n ¸
2 3
1 1 2 1 s i n ¸ 1
T = m a Ò! f Å" m g c o s ¸ = m g s i n ¸ Ò! f = = t g ¸
2 2 3 3 c o s ¸ 3
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 5
U t w o r z o n a : 2 5 . 1 1 . 2 0 0 6 1 6 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 7 D y n a m i k a r u c h u o b r o t o w e g o . Z m o d y f i k o w a n a : 7 . 1 2 . 2 0 0 6 1 9 : 3 7
z a d . 8 .
o b r
D a n e : n 0 = 9 0 0 , N = 7 5 , Ék = 0 , W T = 4 4 , 4 J ,
m i n .
S z u k a n e : I = ? , M T = ? ,
o b r 1 o b r o b r
n 0 = 9 0 0 = 9 0 0 Å" = 1 5
m i n . 6 0 s s
É0 = 2 n 0
r u c h o b r o t o w y j e d n o s t a j n i e o p ó zn i o n y w i c :
É0
Ék = É0 - µt Ò! 0 = É0 - µt Ò! µ =
t
É0 2 2 Õ 2 Å" 2 N 2 N
2
1 1 1
Õ = É0 t - µt Ò! Õ = É0 t - t Ò! Õ = É0 t Ò! t = = =
2 2 2
t É0 2 n 0 n 0
W T = "E
I Å"É0 2 2 W T 2 W T W T 4 4 , 4
W T = Ò! I = = = = = 0 , 0 1 k g m 2
2 ( 2 n 0 ) 2 2 Å" 2 Å"1 5 2
É0 2 2 2 n 0 2
M = I Å"µ
É0 n 0 É0 n 0 Å" 2 n 0 W T W T 4 4 , 4
M T = I Å" = Å" I = Å" = = = 0 , 0 9 4 2 N m
t 2 N 2 N 2 N 2 Å"7 5
2 2 n 0 2
I I s p o s ó b m o \n a p o l i c z y p r a c z p r o s t e g o w z o r u , b o d z i a Ba s t a By m o m e n t s i By T :
W T W T 4 4 , 4
W = M Å"Õ Ò! M T = = = = 0 , 0 9 4 2 N m
Õ 2 N 2 Å"7 5
z a d . 9 .
o b r
D a n e : m s t o l i k a = m 1 = 1 0 0 k g , m s t u d e n t a = m 2 = 6 0 k g , n 1 = 1 0 ,
m i n .
o b r
S z u k a n e : n 2 = ? ,
m i n .
T r z e b a s k o r z y s t a z z a s a d y z a c h o w a n i a m o m e n t u p d u :
L 1 = L 2
I 1
I 1 Å"É1 = I 2 Å"É2 Ò! I 1 Å" 2 n 1 = I 2 Å" 2 n 2 Ò! I 1 Å" n 1 = I 2 Å" n 2 Ò! n 2 = Å" n 1
I 2
1
I 1 = m 1 R 2 + m 2 R 2 = R 2 ( 1 m 1 + m 2 )
2 2
1 1
I 2 = m 1 R 2 + m 2 Å"0 2 = m 1 R 2
2 2
R 2 ( 1 m 1 + m 2 ) m 1 + 2 m 2
I 1 m 2
2 6 0 o b r
n 2 = Å"n 1 = Å" n 1 = Å" n 1 = ( 1 + 2 ) Å" n 1 = ( 1 + 2 ) Å" n 1 = 2 2
I 2 m 1 1 0 0 m i n .
1
m 1 R 2 m 1
2
D o d a t k o w e z a g a d n i e n i e :
J a k i j e s t s t o s u n e k e n e r g i i k i n e t y c z n e j r u c h u o b r o t o w e g o d o e n e r g i i c a Bk o w i t e j e n e r g i i
k i n e t y c z n e j d l a w a l c a w y k o n u j c e g o r u c h b e z p o [l i z g u p o l i n i i p r o s t e j .
1
I 0 = m R 2 v = ÉR
2
o b r I 0 É2
1 1
E k I 0 É2 m R 2 É2 1
2 2 2
= = = = =
c a Ba .
E k I 0 É2 + m v 2 I 0 É2 + m v 2 1 m R 2 É2 + m É2 R 2 3 3
2 2
2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 6
U t w o r z o n a : 8 . 1 2 . 2 0 0 6 2 0 : 5 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I Z a d a n i a . Z m o d y f i k o w a n a : 1 4 . 1 2 . 2 0 0 6 1 2 : 1 1
K o l o k w i u m I I Z a d a n i a
8 . 1 2 . 2 0 0 6 r .
P r z y k Ba d o w e z e s t a w y z a d a D d o r o z w i z a n i a n a k o l o k w i u m .
T r e [c i z a d a D Z e s t a w A
z a d . 1 .
W p i o n o w o p o d w i e s z o n y z a j e d e n z k o Dc ó w p r t o m a s i e m i d Bu g o [c i l u d e r z a p o c i s k o m a s i e
m 1 i p r d k o [c i v 1 . P o z d e r z e n i u c i a Ba s k l e i By s i . J a k a j e s t p r d k o [ k t o w a t u \ p o z d e r z e n i u ?
z a d . 2 .
K o Bo z a m a c h o w e p o s i a d a j c e e n e r g i k i n e t y c z n E w i r u j e z p r d k o [c i o b r o t o w n . P o j a k i m
c z a s i e p r z y Bo \o n y d o t e g o k o Ba s t a By m o m e n t s i By M z w i k s z y j e g o p r d k o [ o b r o t o w
d o d w u k r o t n i e w i k s z e j ?
z a d . 3 .
J a k a j e s t p r d k o [ k o Dc o w a ( u p o d n ó \a r ó w n i ) [r o d k a m a s y w a l c a s t a c z a j c e g o s i b e z p o [l i z g u
z r ó w n i p o c h y Be j z w y s o k o [c i H ? Z a Bó \, \e p r d k o [ p o c z t k o w a w a l c a w y n o s i Ba 0 .
z a d . 4 .
C i a Bo o m a s i e m i p r d k o [c i v o d b i j a s i i d e a l n i e s p r \y [c i e o d [c i a n y u s t a w i o n e j p r o s t o p a d l e
d o k i e r u n k u r u c h u . J a k a j e s t [r e d n i a w a r t o [ s i By p o d c z a s z d e r z e n i a t e g o c i a Ba z e [c i a n , j e \e l i
c z a s z e t k n i c i a t y c h c i a B w y n o s i B "t ?
z a d . 5 .
C h Bo p i e c r o z c i g n B s p r \y n o "x z p o Bo \e n i a r ó w n o w a g i . J e g o o j c i e c r o z c i g n B s p r \y n j e s z c z e
( d o d a t k o w o ) o 2 "x . J a k p r a c w y k o n a B o j c i e c , j e \e l i w s p ó Bc z y n n i k s p r \y s t o [c i s p r \y n y w y n o s i k ?
z a d . 6 .
S a m o c h ó d o m a s i e m i d y s p o n u j c y m o c s i l n i k a P j e d z i e w g ó r p o r ó w n i o k c i e n a c h y l a n i a ±
i w s p ó Bc z y n n i k u t a r c i a f . Z j a k m a k s y m a l n p r d k o [c i m o \e p o r u s z a s i t e n s a m o c h ó d ?
z a d . 7 .
R z u c o n o p o z i o m o k a m i e D w j e d n o r o d n y m p o l u g r a w i t a c y j n y m . P o c z a s i e t j e g o e n e r g i a
k i n e t y c z n a s t a n o w i 1 , 5 e n e r g i i k i n e t y c z n e j w c h w i l i t = 0 . Z j a k p r d k o [c i p o c z t k o w
w y r z u c o n o k a m i e D?
z a d . 8 .
C i a Bo p o r u s z a s i w p Ba s z c z y zn i e p o z i o m e j p o o k r g u z e s t a B p r d k o [c i . J e s t p o d c z e p i o n e
d o n i e r o z c i g l i w e j i n i e w a \k i e j n i c i o d Bu g o [c i L o d c h y l o n e j o d p i o n u o k t ± . J a k a j e s t
c z s t o t l i w o [ t y c h o b r o t ó w ?
S z k i c e r o z w i z a D d o z a d a D z n a j d u j s i n a k o l e j n y c h s t r o n a c h .
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 7
U t w o r z o n a : 8 . 1 2 . 2 0 0 6 2 0 : 5 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I Z a d a n i a . Z m o d y f i k o w a n a : 1 4 . 1 2 . 2 0 0 6 1 2 : 1 1
S z k i c e r o z w i z a D Z e s t a w A
z a d . 1 .
D a n e : m , l , m 1 , v 1 ,
S z u k a n e : É = ? ,
v v 1
É1 = =
R l
l i c z e n i e m o m e n t ó w b e z w Ba d n o [c i :
I 1 = I = m R 2 = m 1 l 2
p o c i s k u
1
I 0 = m l 2
1 2
2
I = I 0 + m d
1 1
I = m l 2 + m ( 1 l ) 2 = m l 2
p r e t a
1 2 2 3
1
I 2 = I + I = m l 2 + m 1 l 2 = ( 1 m + m 1 ) l 2
p r e t a p o c i s k u
3 3
z z a s a d y z a c h o w a n i a m o m e n t u p d u :
L 1 = L 2
I 1 m 1 l 2 v 1 m 1 v 1
I 1 É1 = I 2 É Ò! É = É1 = =
I 2 ( 1 m + m 1 ) l 2 l ( 1 m + m 1 ) l
3 3
z a d . 2 .
D a n e : E , n , M = c o s n t . , n k = 2 n ,
S z u k a n e : t = ? ,
É0 = 2 n
Ék = 2 n k = 2 Å" 2 n = 4 n
I É0 2 2 E 2 E E
E = Ò! I = = =
2 ( 2 n ) 2 2 2 n 2
É0 2
Ék = É0 + µ Å"t
Ék - É0 4 n - 2 n 2 n
µ = = =
t t t
E 2 n E E
M = I Å"µ = Å" = Ò! t =
2 2 n 2 t n Å"t M n
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 8
U t w o r z o n a : 8 . 1 2 . 2 0 0 6 2 0 : 5 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I Z a d a n i a . Z m o d y f i k o w a n a : 1 4 . 1 2 . 2 0 0 6 1 2 : 1 1
z a d . 3 .
D a n e : H , g , v 0 = 0 ,
S z u k a n e : v = ? ,
m o m e n t b e z w Ba d n o [c i w a l c a :
1
I = m R 2
2
w a l e c s t a c z a s i b e z p o [l i z g u :
v
v = ÉR Ò! É =
R
z z a s a d y z a c h o w a n i a e n e r g i i :
m v 2 I É2
m g H = +
2 2
m v 2 1 m R 2 Å" v 2
2
m g H = +
2 2 R 2
4 g H = 2 v 2 + v 2
4 g H g H
v 2 = Ò! v = 2
3 3
z a d . 4 .
D a n e : m , v , "t ,
S z u k a n e : F [r = ? ,
p = m v
"p = m v - m ( - v ) = 2 m v
v
a =
t
v m v p
F = m a = m = =
t t t
"p 2 m v
"p = F [r Å" "t Ò! F [r = =
"t "t
z a d . 5 .
D a n e : k , "x , "x ' = 2 "x ,
S z u k a n e : W o j c a = ? ,
k x 2
E =
2
W = "E
k ( "x ) 2
W c h =
2
k ( "x + "x ' ) 2 k ( 3 "x ) 2 k ( "x ) 2 8 k ( "x ) 2
W o j c a = - W c h = - = = 4 k ( "x ) 2
2 2 2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 5 9
U t w o r z o n a : 8 . 1 2 . 2 0 0 6 2 0 : 5 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I Z a d a n i a . Z m o d y f i k o w a n a : 1 4 . 1 2 . 2 0 0 6 1 2 : 1 1
z a d . 6 .
D a n e : m , P , ± , f , g ,
S z u k a n e : v m a x = ? ,
T = f m g c o s ±
Q = m g s i n ±
F = T + Q = f m g c o s ± + m g s i n ± = m g ( f c o s ± + s i n ±)
D l a r u c h u j e d n o s t a j n e g o :
s
v =
t
W = F Å" s
W F Å" s P
P = = = F Å"v Ò! v =
t t F
P
v m a x =
m g ( f c o s ± + s i n ±)
z a d . 7 .
D a n e : t , E k 2 = 1 , 5 E k 1 , g ,
S z u k a n e : v 0 = ? ,
m v 0 2
E k 1 =
2
3 m v 0 2 3 m v 0 2
E k 2 = 1 , 5 E k 1 = =
2 2 4
C i a Bo w p i o n i e p o r u s z a s i r u c h e m j e d n o s t a j n i e p r z y s p i e s z o n y m z v y 0 = 0 :
g t 2
h =
2
Z z a s a d y z a c h o w a n i a e n e r g i i :
m v 0 2 3 m v 0 2
m g h + =
2 4
g t 2 m v 0 2
m g = Ò! v 0 2 = 2 g 2 t 2 Ò! v 0 = g t 2
2 4
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 0
U t w o r z o n a : 8 . 1 2 . 2 0 0 6 2 0 : 5 5
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I Z a d a n i a . Z m o d y f i k o w a n a : 1 4 . 1 2 . 2 0 0 6 1 2 : 1 1
z a d . 8 .
D a n e : L , ± , g ,
S z u k a n e : f = ? ,
1
f =
T
2
T =
É
r r
N a c i a Bo d z i a Ba j d w i e s i By : n a c i g n i t k i N o r a z c i \a r Q :
N ¥" = Q = m g
v 2
N = m a d = m
R
Z g e o m e t r i i u k Ba d u w i e m , \e :
R
s i n ± = Ò! R = L s i n ±
L
v 2
m
N
v 2 v 2 g L
R
t g ± = = = = Ò! v = g L s i n ± Å" t g ± = s i n ±
N ¥" m g g R g L s i n ± c o s ±
v
v = ÉR Ò! É =
R
g L
v
s i n ±
1 É 1 g L 1 g
c o s ±
R
f = = = = = =
T 2 2 2 L s i n ± 2 L 2 c o s ± 2 L c o s ±
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 1
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
Z E S T A W 8 R u c h d r g a j c y
1 . 1 2 . 2 0 0 6 r .
W a \n e w z o r y :
O S C Y L A T O R H A R M O N I C Z N Y P R O S T Y
x ( t ) = A c o s ( Ét + Õ) x m a x = A
d x
v ( t ) = = - A É s i n ( Ét + Õ) v m a x = A É
d t
2
d v d x
a ( t ) = = = - É2 A c o s ( Ét + Õ) = - É2 Å"x ( t ) a m a x = A É2
d t d t 2
F ( t ) = m Å" a ( t ) = - m É2 Å"x ( t ) = - k Å" x ( t )
k
k = m É2 Ò! É =
m
g d z i e :
A a m p l i t u d a r u c h u ( m a k s y m a l n e w y c h y l e n i e ) ,
Õ s t a Ba f a z o w a , p o Bo \e n i e p o c z t k o w e r u c h u ,
( Ét + Õ) f a z a r u c h u .
E N E R G I A D R G A C H A R M O N I C Z N Y C H
E c = E p + E k
1 1 1
E p = k x 2 = k A 2 c o s 2 ( Ét + Õ) E p = k A 2
2 2 2
m a x
1 1 1 1
E k = m v 2 = m A 2 É2 s i n 2 ( Ét + Õ) = k A 2 s i n 2 ( Ét + Õ) E k = k A 2
2 2 2 2
m a x
1 1 1
E c = k A 2 c o s 2 ( Ét + Õ) + k A 2 s i n 2 ( Ét + Õ) = k A 2 ( c o s 2 ( Ét + Õ) + s i n 2 ( Ét + Õ) )
2 2 2
c o s 2 ( Ét + Õ) + s i n 2 ( Ét + Õ) = 1
1
E c = k A 2
2
P o p o d s t a w i e n i u z a k = m É2 :
1 1
E p = k x 2 = m É2 x 2
2 2
1 1
E k = m v 2 = m É2 ( A 2 - x 2 )
2 2
p o n i e w a \:
E c = E p + E k
1 1 1
k A 2 = k x 2 + m v 2
2 2 2
k A 2 - k x 2
v 2 =
m
k
v 2 = ( A 2 - x 2 )
m
v 2 = É2 ( A 2 - x 2 )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 2
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
z a d . 1 .
D a n e : m 0 = 1 , 6 5 k g , x 0 = 7 , 3 3 c m ,
m
A = 1 1 , 6 c m , m 1 = 2 , 4 3 k g , x 1 = 8 c m , t 2 = 0 , 2 1 5 s , g = 9 , 8 1 ,
s 2
S z u k a n e : E = ? , v m a x = ? , a m a x = ? ,
v 1 = ? , E k 1 = ? , E p 1 = ? ,
x 2 = ? , v 2 = ? , a 2 = ? ,
N a c i a Bo o m a s i e m 0 p o d w i e s z o n e d o s p r \y n y d z i a Ba j d w i e s i By , k t ó r e s i r ó w n o w a \:
Q = F s
m 0 g 1 , 6 5 Å"9 , 8 1
N
m g = k x Ò! k = = = 2 2 0 , 8
m
x 0 0 , 0 7 3 3
1 1
E = k A 2 = Å" 2 2 0 , 8 Å" 0 , 1 1 6 2 = 1 , 4 8 5 5 J
2 2
k
É =
m
k 2 2 0 , 8
m
v m a x = A É = A = 0 , 1 1 6 Å" = 1 , 1 0 6
s
m 1 2 , 4 3
k
m
a m a x = A É2 = v m a x É = v m a x = 1 0 , 5 4
s 2
m 1
k
v 2 = ( A 2 - x 2 )
m
k 2 2 0 , 8
m
v 1 = ( A 2 - x 1 2 ) = Å"( 0 , 1 1 6 2 - 0 , 0 8 2 ) = 0 , 8 0
s
m 1 2 , 4 3
1 1
E k = m 1 v 2 = Å" 2 , 4 3 Å" 0 , 8 2 = 0 , 7 7 8 J
2 2
1
1 1
E p = k x 1 2 = Å" 2 2 0 , 8 Å" 0 , 0 8 2 = 0 , 7 0 7 J
2 2
1
k 2 2 0 , 8
É = = = 9 , 5 3 2
m 1 2 , 4 3
t 2 É = 0 , 2 1 5 Å"9 , 5 3 2 = 2 , 0 4 9
x ( t ) = A c o s ( Ét )
x ( t 2 ) = 0 , 1 1 6 Å"c o s ( 2 , 0 4 9 ) = - 0 , 0 5 3 4 m
v ( t ) = - A É s i n ( Ét )
m
v ( t 2 ) = - 0 , 1 1 6 Å"9 , 5 3 2 Å"s i n ( 2 , 0 4 9 ) = - 0 , 9 8 2
s
a ( t ) = - É2 A c o s ( Ét ) = - É2 Å"x ( t )
m
a ( t 2 ) = - É2 Å"x ( t 2 ) = - 9 , 5 3 2 2 Å" - 0 , 0 5 3 4 = 4 , 8 5
s 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 3
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
z a d . 2 .
N
D a n e : k = 8 , m = 0 , 5 k g , A = 1 0 c m ,
m
x = 6 c m , x 1 = 0 c m , x 2 = 8 c m ,
S z u k a n e : v m a x = ? , a m a x = ? , v x = ? , a x = ? , "t = ? ,
k 8
É = = = 1 6 = 4
m 0 , 5
m
v m a x = A É = 0 , 1 Å" 4 = 0 , 4
s
m
a m a x = A É2 = 0 , 1 Å" 4 2 = 1 , 6
s
1 x 1 6
x = A c o s ( Ét ) Ò! t = a r c c o s ( ) = a r c c o s ( 1 0 ) = 0 , 2 3 2 s
É A 4
m
v ( t ) = - A É s i n ( Ét ) Ò! v x = - 0 , 1 Å" 4 Å"s i n ( 4 Å"0 , 2 3 2 ) = - 0 , 3 2
s
m
a ( t ) = - É2 A c o s ( Ét ) Ò! a x = - 4 2 Å"0 , 1 Å"c o s ( 4 Å"0 , 2 3 2 ) = - 0 , 9 6
s 2
x 1 1
1 0
x 1 = A c o s ( Ét 1 ) Ò! t 1 = a r c c o s ( ) = a r c c o s ( 1 0 ) = 0 , 3 9 3 s
É A 4
x 2 1
1 8
x 2 = A c o s ( Ét 2 ) Ò! t 2 = a r c c o s ( ) = a r c c o s ( 1 0 ) = 0 , 1 6 1 s
É A 4
"t = t 1 - t 2 = 0 , 3 9 3 - 0 , 1 6 1 = 0 , 2 3 2 s
z a d . 3 .
1
D a n e : A = 3 c m , v = v m a x ,
2
S z u k a n e : x = ? c m ,
v m a x = A É
1 1
v = v m a x = A É
2 2
v ( t ) = - A É s i n ( Ét )
1
A É = - A É s i n ( Ét )
2
1 1
s i n ( Ét ) = - Ò! Ét = a r c s i n ( - ) = -
2 2 6
x ( t ) = A c o s ( Ét )
3
x = A c o s ( - ) = 3 Å" H" 2 , 6 0 c m
6 2
z a d . 4 .
1
D a n e : E k = E p , x b = A ,
2
E k
1
S z u k a n e : x a ( d l a E k = E p ) , = ? ( d l a x b = A ) ,
E c 2
1 1
E k = E p Ò! k A 2 s i n 2 ( Ét ) = k A 2 c o s 2 ( Ét ) Ò! s i n ( Ét ) = c o s ( Ét ) Ò! Ét =
2 2 4
A 2
x ( t ) = A c o s ( Ét ) Ò! x a = A c o s ( ) =
4
2
1 1
E c - E p
E k k A 2 - k x b 2 A 2 - ( 1 A ) 2
2 2 2
Å"1 0 0 % = Å"1 0 0 % = Å"1 0 0 % = Å"1 0 0 % = 7 5 %
1
E c E c k A 2 A 2
2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 4
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
1 5 . 1 2 . 2 0 0 6 r .
z a d . 5 .
m
D a n e : m = 0 , 0 5 k g , R = 3 , 2 m m , T = 2 , 7 s , g = 9 , 8 1 ,
s 2
g
S z u k a n e : Á = ? ( g s t o [ c i e c z y ) ,
c m 3
m m
T = 2 Ò! k = 4 2 T
2
k
C i a Bo t o p By w a w c i e c z y w i c d z i a Ba n a n i e s i Ba w y p o r u .
G d y c i a Bo z a n u r z y s i g Bb i e j w c i e c z y t o d z i a Ba n a n i e s i Ba w y p o r u w i k s z a o :
"F w = "m Å" g
g d z i e "m t o m a s a d o d a t k o w o w y p a r t e j w o d y ,
m
Á = Ò! m = ÁV Ò! "m = Á"V
V
V = S Å" h Ò! V = R 2 x Ò! "V = R 2 "x
P r z y r o s t s i By w y p o r u p e Bn i r o l s i By s p r \y s t o [c i ( p r z y w r a c a r ó w n o w a g u k Ba d o w i ) :
F s = - k Å" x Ò! "F s = k "x
m
4 2 T "x
2 4 m
"F w = "F s Ò! "m g = k "x Ò! Á"V g = k "x Ò! Á = =
2
R 2 "x g T R 2 g
4 m 4 Å" 0 , 0 5
k g g g
1 0 0 0
Á = = = 8 5 8 = 8 5 8 Å" = 0 , 8 5 8
2
m 3 1 0 0 3 c m 3 c m 3
T R 2 g 2 , 7 2 Å" 0 , 0 0 3 2 2 Å"9 , 8 1
z a d . 6 .
D a n e : S , l , g ,
S z u k a n e : T = ? ,
Á g s t o [ c i e c z y ( j a k o d a n a p o m o c n i c z a ) ,
m
T = 2
k
m = ÁV = Á Å" S l
"m = Á"V = Á Å" S Å" ( x + x ) = 2 ÁS Å" x
F = a "m = - g "m = - g Å" 2 ÁS Å" x = - 2 ÁS g Å" x
F = - k x Ò! - 2 ÁS g Å" x = - k x Ò! k = 2 ÁS g
Á Å"S l
l
T = 2 = 2
2 ÁS g 2 g
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 5
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
z a d . 7 .
m
D a n e : x = 0 , 0 9 8 m , t = 1 0 s , A n = A 1 Å"1 % , g = 9 , 8 1 ,
s 2
S z u k a n e : ´ = ? ( w s p ó Bc z y n n i k t Bu m i e n i a ) ,
a )
b t
2 m
x = A e -
A n = 0 , 0 1 Å" A 1
b Å"0
2 m
b Å"t
A 1 A e - A 1 1
2 m b Å"t b l n 1 0 0
= Ò! = Ò! 1 0 0 = e Ò! l n 1 0 0 = Ò! =
b Å"t b Å"t
2 m 2 m t
2 m 2 m
A n A e - 0 , 0 1 Å" A 1 e -
w i e m , \e :
b
´ =
2 m
l n 1 0 0 4 , 6
1
´ = H" = 0 , 4
s
t 1 0
b )
w i e m , \e [ ? ] :
k
É =
m
b
É = = ´
2 m
F s = Q
g
k
k x = m g Ò! =
m x
g 9 , 8 1
k 1
= ´ Ò! ´ = = H" 1 0
m x 0 , 0 9 8 s
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 6
U t w o r z o n a : 1 . 1 2 . 2 0 0 6 2 2 : 5 6
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 8 R u c h d r g a j c y . Z m o d y f i k o w a n a : 1 5 . 1 2 . 2 0 0 6 1 8 : 4 1
z a d . 8 .
D a n e : l = 9 9 , 4 c m , A 1 = 5 c m , A 2 = 4 c m ,
S z u k a n e : A 3 = ? , Ä = ? ( c z a s r e l a k s a c j i ) ,
a )
Z a u w a \y Be m , \e [ ? ] :
A 1 - A 2 A 2 - A 3
=
A 1 A 2
A 1 A 2 - A 2 A 1 + A 2 A 2 A 2 2
A 2 A 1 - A 2 A 2 = A 1 A 2 - A 1 A 3 Ò! A 3 = =
A 1 A 1
4 2 1 6
A 3 = = = 3 , 2 c m
5 5
b )
W i a d o m o , \e :
l
T = 2
g
o r a z :
1
Ä =
´
T ´ = ›
A n
› = l n
A n + 1
g d z i e : Ä c z a s r e l a k s a c j i , ´ w s p ó Bc z y n n i k t Bu m i e n i a , › d e k r e m e n t l o g a r y t m i c z n y ,
w i c :
›
T ´ = › Ò! ´ =
T
0 , 9 9 4
l
2 2
1 T 2
g 9 , 8 1
Ä = = = = H" H" 8 , 9 6 9 s
A 1 0 , 0 5
´ › 0 , 2 2 3
l n l n
A 2 0 , 0 4
z a d . 9 .
1
D a n e : › = 0 , 0 0 9 9 , E 2 = E 1 ,
2
S z u k a n e : N = ? ( l i c z b a d r g a D u k Ba d u ) ,
b
1 m
E = k A 2 e - t
2
b b
m m
b
E 1 1 k A 2 e - t E 1 e - t
N T
2 m b
= Ò! = Ò! 2 = e Ò! l n 2 = N T
b b
m
1
m m
E 2 1 k A 2 e - ( t + N T ) E 1 e - ( t + N T )
2
2
b
´ =
2 m
2 m ›
› = T ´ Ò! T =
b
b m l n 2 1 m l n 2 b l n 2 l n 2
l n 2 = N T Ò! N = Å" = Å" = = H" 3 5
m b T b 2 m › 2 › 2 Å"0 , 0 0 9 9
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 7
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
Z E S T A W 9 E l e k t r o s t a t y k a
2 2 . 1 2 . 2 0 0 6 r .
W a \n e w z o r y :
P R A W O C O U L O M B A
q 1 Å" q 2
F = k
r 2
g d z i e :
1
k =
4 ˵0
µ0 p r z e n i k a l n o [ e l e k t r y c z n a p r ó \n i
q 1 , q 2 Ba d u n k i e l e k t r y c z n e
r o d l e g Bo [ m i d z y Ba d u n k a m i
N A T [E N I E P O L A P O T E N C J A A P O L A
E p
F
V =
E =
q 0
q 0
Q
w z ó r r o b o c z y : F e = E Å" q
V = k
r
Q
E = k
g d z i e :
r 2
q Q
g d z i e :
E p e n e r g i a p o t e n c j a l n a E p = k
r
q 0 n i e w i e l k i p r ó b n y Ba d u n e k d o d a t n i
P o l e e l e k t r o s t a t y c z n e z n a c z y t o s a m o c o p o l e e l e k t r y c z n e .
P R A C A W P O L U E L E K T R O S T A T Y C Z N Y M
W = q U
g d z i e :
U r ó \n i c a p o t e n c j a Bó w , c z y l i "V = V b - V a ,
K O N D E N S A T O R P AA S K I
U Q µ0 µS
E = C = =
d U d
g d z i e :
d o d l e g Bo [ m i d z y o k Ba d a m i k o n d e n s a t o r a ,
S p o w i e r z c h n i a o k Ba d e k k o n d e n s a t o r a ,
C p o j e m n o [ e l e k t r y c z n a k o n d e n s a t o r a ( j e d n o s t k j e s t f a r a d 1 F ) ,
W A [N E S T A AE
C 2
µ0 H" 8 , 8 5 Å"1 0 - 1 2 N Å"m 2
k H" 8 , 9 9 Å"1 0 9 N Å"m 2
C 2
e H" 1 , 6 0 2 1 7 7 3 Å"1 0 - 1 9 C ( Ba d u n e k e l e k t r o n u , Ba d u n e k e l e m e n t a r n y )
m e H" 9 , 1 0 9 3 8 9 7 Å"1 0 - 3 1 k g ( m a s a e l e k t r o n u )
m p H" 1 , 6 7 2 6 2 1 3 Å"1 0 - 2 7 k g ( m a s a p r o t o n u )
m n H" 1 , 6 7 4 9 2 8 6 Å"1 0 - 2 7 k g ( m a s a n e u t r o n u )
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 8
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
z a d . 1 .
D a n e : q 1 = + 5 µ C , q 2 = - 2 µ C , q 3 = + 5 µ C , a = 0 , 1 m ,
S z u k a n e : F w 3 = ? ,
[ + r y s . ]
R o z p a t r u j j e d y n i e s i By d z i a Ba j c e n a Ba d u n e k q 3 .
S i By e l e k t r o s t a t y c z n e d z i a Ba j w z d Bu \ l i n i i Bc z c e j Ba d u n k i p u n k t o w e .
q Q
F = k
r 2
F w 3 = F 1 3 + F 3 2
5 Å"1 0 - 6 Å"5 Å"1 0 - 6
q 1 Å" q 3
F 1 3 = k = 8 , 9 9 Å"1 0 9 Å" H" 1 1 , 2 3 7 5 N
2 2
( a 2 ) ( 0 , 1 Å" 2 )
F 1 3 = [ F 1 3 c o s 4 5 ° , F 1 3 s i n 4 5 ° ]
5 Å"1 0 - 6 Å"( - 2 Å"1 0 - 6 )
q 3 Å" q 2
F 3 2 = k = 8 , 9 9 Å"1 0 9 Å" H" 8 , 9 9 N
a 2 0 , 1 2
F 3 2 = [ - F 3 2 , 0 ]
F w 3 = [ F 1 3 c o s 4 5 ° - F 3 2 , F 1 3 s i n 4 5 ° + 0 ]
2 2
F w 3 = [ 1 1 , 2 3 7 5 Å" - 8 , 9 9 Å"1 0 - 3 ; 1 1 , 2 3 7 5 Å" ] = [ - 1 , 0 4 ; 7 , 9 5 ] N
2 2
z a d . 2 .
C
D a n e : Ã = 4 Å"1 0 - 5 m 2 , q = 1 0 - 9 C , m = 1 0 - 3 k g ,
S z u k a n e : ± = ? ,
[ + r y s . ]
à g s t o [ p o w i e r z c h n i o w a Ba d u n k u ,
Q
à =
S
Q c a Bk o w i t y z g r o m a d z o n y Ba d u n e k ,
S p o l e p o w i e r z c h n i ,
D l a n a Ba d o w a n e j p Ba s z c z y z n y p r a w d j e s t , \e :
Ã
E =
2 µ0 µ
µ w z g l d n a p r z e n i k a l n o [ o [r o d k a ( d l a p r ó \n i c z y p o w i e t r z a m o \n a p r z y j , \e µ = 1 ) ,
w a r u n e k r ó w n o w a \e n i a s i s i By d l a k u l k i b d c e j w b e z r u c h u :
F e + N + Q = 0
à à Å" q 4 Å"1 0 - 5 Å"1 0 - 9
F e = E Å" q = Å" q = = = 2 , 2 6 Å"1 0 - 3 N
2 µ0 µ 2 µ0 Å"1 2 Å"8 , 8 5 Å"1 0 - 1 2
Q = m g = 1 0 - 3 Å"9 , 8 1 = 9 , 8 1 Å"1 0 - 3
z g e o m e t r i i u k Ba d u w y n i k a , \e :
F e 2 , 2 6 Å"1 0 - 3
t g ± = = = 0 , 2 3 Ò! ± H" 1 3 °
Q 9 , 8 1 Å"1 0 - 3
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 6 9
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
z a d . 3 .
D a n e : Á1 ( g s t o [ c i e c z y ) , µ ( p r z e n i k a l n o [ d i e l e k t r y c z n a c i e c z y ) ,
S z u k a n e : Á = ? ( g s t o [ k u l k i ) ,
[ + r y s . ]
S y t u a c j a p r z e d z a n u r z e n i e m ( c z y l i k u l k i w p o w i e t r z u ) :
Q = m g = ÁV g
F F
t g ± = =
Q ÁV g
q 1 q 2 1 q 1 q 2
F = k =
r 2 4 µ0 r 2
S y t u a c j a p o z a n u r z e n i u ( c z y l i k u l k i w c i e c z y ) d z i a Ba d o d a t k o w o s i Ba w y p o r u :
F w = Q w c = m c g = Á1 V g
F 1 F 1 F 1
t g ± = = =
Q - F w ÁV g - Á1 V g V g ( Á - Á1 )
q 1 q 2 1 q 1 q 2 1 q 1 q 2 1 F
F 1 = k ' Å" = = Å" Ò! F 1 =
r 2 4 µ0 µ r 2 4 µ0 r 2 µ µ
k t y m a j b y t a k i e s a m e w i c w a r t o [c i f u n k c j i t a n g e n s t e \ m a j b y t a k i e s a m e
( p o n i e w a \ f u n k c j a t a n g e n s j e s t r ó \n o w a r t o [c i o w a n a ( - , ) ) , z a t e m :
2 2
F
F F 1 F µÁ1
µ
= Ò! = Ò! µ ( Á - Á1 ) = Á Ò! Á =
ÁV g V g ( Á - Á1 ) Á Á - Á1 µ - 1
z a d . 4 .
&
z a d . 5 .
V
D a n e : v 0 = 1 , 2 Å"1 0 5 m , E = 3 0 , l = 1 0 c m ,
s c m
v k
S z u k a n e : x = ? ( g d z i e x = ) ,
v 0
D o d a t k o w e z a Bo \e n i e : p o r u s z a j c y s i p r o t o n n i e z d e r z y s i z o k Ba d a m i k o n d e n s a t o r a .
M a m y d o c z y n i e n i a z e z w y k By m r z u t e m p o z i o m y m , t y l e , \e n i e w p o l u g r a w i t a c y j n y m ,
a l e w p o l e e l e k t r o s t a t y c z n y m .
v x = v 0 = c o n s t .
0
s s l
v = Ò! t = Ò! t =
t v v 0
l
v y = a Å" t Ò! v y = a
v 0
v k 2 = v 0 2 + v y 2
P r z y s p i e s z e n i e d z i a Ba j c e n a c i a Bo m o \n a o b l i c z y z I I z a s a d y d y n a m i k i o r a z z e w z o r u n a
s i B d z i a Ba j c n a Ba d u n e k w p o l u e l e k t r o s t a t y c z n y m .
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 0
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
F = m p Å" a
F = q Å" E
q E
m p Å" a = q Å" E Ò! a =
m p
o s t a t e c z n i e :
2
ëø öø
l v 0 4 a 2 l 2
ìø ÷ø
v 0 2 + + 2
ìøa v 0 ÷ø
v 0 2 + v y 2 ëø öø
v 0 2 v 0 2
1 q E
íø øø
ìø ÷ø
x = = = = v 0 4 + l 2
2
ìø ÷ø
v 0 v 0 v 0 m p
v
0
íø øø
2
- 1 9
1 4 ëø öø 2
x = ( 1 , 2 Å"1 0 5 ) + ìø1 , 6 0 2 Å"1 0 Å"3 0 Å"1 0 2 ÷ø ( 1 0 Å"1 0 - 2 ) =
2
ìø
1 , 6 7 3 Å"1 0 - 2 7 ÷ø
( 1 , 2 Å"1 0 5 )
íø øø
1 3 , 2 1 Å"1 0 1 0
= 2 , 0 7 4 Å"1 0 2 0 + 8 , 2 5 3 Å"1 0 2 2 Å"0 , 0 1 = H" 2 , 2 3
1 , 4 4 Å"1 0 1 0 1 , 4 4 Å"1 0 1 0
z a d . 6 .
V
D a n e : p = 6 , 2 Å"1 0 - 3 0 C m , E = 1 5 0 , Õ = 1 8 0 ° ,
c m
S z u k a n e : W = ? ,
D i p o l o b r a c a s i j e \e l i d z i a Ba n a n i e g o n i e z e r o w y m o m e n t s i By M .
Z n a n a j e s t z a l e \n o [ ( w y s t p u j e t u i l o c z y n w e k t o r o w y ) :
M = p × E
r
g d z i e : p w e k t o r m o m e n t u d i p o l o w e g o p = q Å"l .
E n e r g i a p o t e n c j a l n a d i p o l a w y n o s i ( w y s t p u j e t u i l o c z y n s k a l a r n y ) :
E o = - p o E
S w o b o d n e p o Bo \e n i e d i p o l a t o t a k i e , \e w e k t o r y m o m e n t u d i p o l o w e g o o r a z n a t \e n i a
p o l a s r ó w n o l e g Be i m a j t a k i e s a m e z w r o t y ( w i c k t ± = 0 ° ) .
W = "E p = E ' p - E p = - p E c o s Õ - ( - p E c o s ±) = - p E c o s 1 8 0 ° + p E c o s 0 °
W = - p E Å"( - 1 ) + p E Å"1 = 2 p E
W = 2 Å"6 , 2 Å"1 0 - 3 0 Å"1 5 0 Å"1 0 2 = 1 , 8 6 Å"1 0 - 2 5 J
z a d . 7 .
&
z a d . 8 .
&
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 1
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
z a d . 9 .
D a n e : a = 1 0 c m , q = 0 , 0 1 µ C ,
S z u k a n e : E p 1 = ? , E p 2 = ? ,
P r z y p a d e k 1 º P r z y p a d e k 2 º
+ q + q
+ q - q
a
a
- q - q
- q + q
q Q 1
E p = k = k q 2
r r
ëø 1 1 1 1 1 1 öø 2 k q 2
E p 1 = k q 2 - + - - - ÷ø -
=
ìø
a a a a
a 2 a 2 a 2
íø øø
2 Å"8 , 9 9 Å"1 0 9 Å" ( 0 , 0 1 Å"1 0 - 6 ) 2
E p 1 = - = - 1 2 , 7 µ J
0 , 1 Å" 2
ëø öø
ëø 1 1 1 1 1 1 öø ëø 4 2 öø 2 - 4 2
÷ø
E p 2 = k q 2 - - - + + = k q 2 + = k q 2 ìø
ìø-
÷ø ìø-
÷ø
ìø ÷ø
a a a a a
a 2 a 2 a 2 a 2
íø øø íø øø
íø øø
ëø öø
2 - 4 2
÷ø
E p 2 = 8 , 9 9 Å"1 0 9 Å"( 0 , 0 1 Å"1 0 - 6 ) 2 ìø = - 2 3 , 2 5 µ J
ìø ÷ø
0 , 1 Å" 2
íø øø
T r w a l s z y u k Ba d j e s t t e n , k t ó r e g o e n e r g i a p o t e n c j a l n a j e s t n i \s z a , p o n i e w a \ t r z e b a b y Bo b y
w y k o n a w i k s z p r a c a b y t a k i u k Ba d z n i s z c z y .
E p 2 < E p 1
T r w a l s z y j e s t w i c w a r i a n t n u m e r d w a .
z a d . 1 0 .
D a n e : C 1 = 1 2 µ F , C 2 = 5 , 3 µ F , C 3 = 4 , 5 µ F , U = 1 2 , 5 V ,
S z u k a n e : C 1 2 3 = ? , q 1 = ? ,
Q
C =
U
P o Bc z e n i e r ó w n o l e g Be k o n d e n s a t o r ó w ( n a p i c i e t a k i e s a m o n a k a \d y m z k o n d e n s o r ó w ) :
q 1 = C 1 U q 2 = C 2 U
q 1 2
q 1 2 = q 1 + q 2 = C 1 U + C 2 U = U ( C 1 + C 2 ) Ò! = C 1 + C 2 Ò! C 1 2 = C 1 + C 2
U
n
O g ó l n i e : C = .
"C i
i = 1
w i c : C 1 2 = C 1 + C 2 = 1 2 + 5 , 3 = 1 7 , 3 µ F .
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 2
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
P o Bc z e n i e s z e r e g o w e k o n d e n s a t o r ó w ( Ba d u n e k n a k a \d y m z k o n d e n s o r ó w j e s t t a k i s a m ) :
q q
U 1 2 = U 3 =
C 1 2 C 3
q q U 1 2 3 1 1 1 1 1
U 1 2 3 = U 1 2 + U 3 = + Ò! = + Ò! = +
C 1 2 C 3 q C 1 2 C 3 C 1 2 3 C 1 2 C 3
n
1 1
O g ó l n i e : = .
"C
C
i = 1
i
1 1 1 C 1 2 + C 3 C 1 2 C 3 1 7 , 3 Å" 4 , 5
w i c : = + = Ò! C 1 2 3 = = = 3 , 5 7 µ F .
C 1 2 3 C 1 2 C 3 C 1 2 C 3 C 1 2 + C 3 1 7 , 3 + 4 , 5
D l a o b l i c z e n i a q 1 n a l e \y z a u w a \y , \e :
q 1 2 = q 3 = q
q = C 1 2 3 U Ò! q 3 = C 1 2 3 U
q 3 C 1 2 3 U
U 3 = =
C 3 C 3
C 1 2 3 C 1 2 3
C 1 2 3
U = U 3 + U 1 2 = U + U 1 2 Ò! U 1 2 = U - U = U ( 1 - )
C 3
C 3 C 3
U 1 2 = U 1 = U 2
q 1 q 1
C 1 2 3
C 1 = = Ò! q 1 = C 1 U 1 2 = C 1 Å"U ( 1 - )
C 3
U 1 U 1 2
3 , 5 7
q 1 = 1 2 Å"1 2 , 5 Å"( 1 - ) = 3 1 µ C
4 , 5
z a d . 1 1 .
D a n e : C 1 = 3 , 5 5 µ F , U 1 = 6 , 3 V , C 2 = 8 , 9 5 µ F ,
S z u k a n e : U = ? , E = ? , E ' = ? ,
Q
o g ó l n i e : C =
U
W c z a s i e Ba d o w a n i a n a k o n d e n s a t o r p i e r w s z y p r z e p By w a Ba d u n e k :
Q = C 1 U 1
P o o d Bc z e n i u k o n d e n s a t o r a o d b a t e r i i i p r z y Bc z e n i u d o d r u g i e g o k o n d e n s a t o r a
s u m a r y c z n y Ba d u n e k p o z o s t a j e b e z z m i a n ( t j . Ba d u n e k n i e u c i e k a , a n i n i e p r z y b y w a ) , w i c :
Q = Q 1 + Q 2 Ò! C 1 U 1 = C 1 U ' 1 + C 2 U ' 2
P o p o d Bc z e n i u r ó w n o l e g l e k o n d e n s a t o r ó w i c h n a p i c i a s s o b i e r ó w n e : U = U ' 1 = U ' 2
C 1 3 , 5 5
C 1 U 1 = C 1 U + C 2 U Ò! U = U 1 = Å" 6 , 3 = 1 , 7 9 V
C 1 + C 2 3 , 5 5 + 8 , 9 5
C 1 2 = C 1 + C 2
2
C U
E n e r g i a k o n d e n s a t o r a w y r a \a s i w z o r e m : E =
2
2
C 1 U 1 2 3 , 5 5 Å"6 , 3 2 C 1 2 U ( 3 , 5 5 + 8 , 9 5 ) Å"1 , 7 9 2
E = = = 7 0 , 4 µ J E ' = = = 2 0 µ J
2 2 2 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 3
U t w o r z o n a : 2 1 . 1 2 . 2 0 0 6 1 2 : 5 2
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 9 E l e k t r o s t a t y k a . Z m o d y f i k o w a n a : 1 2 . 0 1 . 2 0 0 7 2 3 : 4 9
D o d a t k o w e z a g a d n i e n i a :
a ) J a k i j e s t p o t e n c j a B V p o l a o r a z w a r t o [ n a t \e n i a E p o l a w y t w o r z o n y p r z e z p o n i \s z e
u k Ba d y Ba d u n k ó w w z a z n a c z o n y c h m i e j s c a c h ?
( 1 ) D w a Ba d u n k i j e d n o i m i e n n e ( n p . d o d a t n i e ) :
+ q + q
r
q 1 q 2 2 q 2 q 4 q
V 1 = k V 2 = k q 1 = q 2 V = V 1 + V 2 = k + k = k
r r
r r r
2 2
q 1 q 2 4 q 4 q
E 1 = k E 2 = k q 1 = q 2 E = E 1 - E 2 = k - k = 0
2 2
r r
r 2 r 2
( ) ( )
2 2
( 2 ) D w a Ba d u n k i r ó \n o i m i e n n e :
- q + q
r
q 1 q 2 2 q 2 q
V 1 = k V 2 = k q 1 = - q 2 V = V 1 + V 2 = k - k = 0
r r
r r
2 2
q 1 q 2 4 q - 4 q 8 q
E 1 = k E 2 = k q 1 = - q 2 E = E 1 - E 2 = k - k = k
2 2
r r
r 2 r 2 r 2
( ) ( )
2 2
b ) U m i e s z c z o n o e l e k t r o n p r z y u j e m n i e n a Ba d o w a n e j o k Ba d c e k o n d e n s a t o r a p Ba s k i e g o .
R ó \n i c a p o t e n c j a Bó w p o m i d z y o k Ba d a m i w y n o s i U , a i c h o d l e g Bo [ w y n o s i d .
Z j a k i m p r z y s p i e s z e n i e m b d z i e p o r u s z a B s i t e n e l e k t r o n ?
U
D l a k o n d e n s a t o r a p Ba s k i e p r a w d z i w e j e s t : E =
d
U
N a e l e k t r o n d z i a Ba s i Ba e l e k t r o s t a t y c z n a : F = q E = e
d
F e U
Z d r u g i e j z a s a d y d y n a m i k i : F = m a Ò! a = =
m e m e d
c ) C z s t k a o m a s i e m i Ba d u n k u q z o s t a Ba u m i e s z c z o n a w j e d n o r o d n y m p o l u
e l e k t r o s t a t y c z n y m . J a k i b d z i e p d t e j c z s t k i p o p o k o n a n i u r ó \n i c y p o t e n c j a Bu U ?
P r a c a w p o l u e l e k t r o s t a t y c z n y m : W = q U
W y k o w y w a n a n a d c z s t k p r a c a w p By w a n a z m i a n j e j e n e r g i i k i n e t y c z n e j :
m v 2 m v 2 m ( m v ) 2 p 2
q U = Ò! q U = Å" Ò! q U = Ò! q U = Ò! p = 2 m q U
2 2 m 2 m 2 m
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 4
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
Z E S T A W 1 0 M a g n e t y z m
5 . 0 1 . 2 0 0 7 r .
W a \n e w z o r y :
B d s i p o j a w i a w p o n i \s z y c h z a d a n i a c h w o d p o w i e d n i m k o n t e k [c i e , o p r ó c z :
B w e k t o r i n d u k c j i p o l a m a g n e t y c z n e g o , j e j w a r t o [ w y r a \a s i w t e s l a c h ( 1 T ) .
W z o r y r o b o c z e n a w a r t o [ i n d u k c j i m a g n e t y c z n e j :
d l a n i e s k o Dc z e n i e d Bu g i e g o p r z e w o d n i k a p r o s t o l i n i o w e g o :
µ 0 i
B =
2 À d
g d z i e :
µ 0 p r z e n i k a l n o [ m a g n e t y c z n a p r ó \n i ,
N
µ 0 = 4 À Å"1 0 - 7 A 2
d l a p r z e w o d n i k a w k s z t a Bc i e o k r g u :
µ 0 i
B =
2 r
g d z i e :
r p r o m i e D o k r g u ,
z a d . 1 .
D a n e : i 1 = 3 A , i 2 = 3 A , r = 1 3 c m , r 1 = 5 c m , r 2 = 1 2 c m ,
S z u k a n e : B w = ? ( w a r t o [ w y p a d k o w a i n d u k c j i m a g n e t y c z n e j w p u n k c i e P ) ,
M a m y d o c z y n i e n i e z e s z c z e g ó l n y m p r z y p a d k i e m ( z t w i e r d z e n i a P i t a g o r a s a ) :
r 1 2 + r 2 2 = r 2 Ô! 5 2 + 1 2 2 = 1 3 2 Ô! 2 5 + 1 4 4 = 1 6 9 Ô! 1 6 9 = 1 6 9
w i c o d c i n k i r 1 , r 2 , r t w o r z t r ó j k t p r o s t o k t n y .
Z r e g u By p r a w e j d Bo n i n a l e \y w y z n a c z y k i e r u n k i w e k t o r a i n d u k c j i m a g n e t y c z n e j
o d p o s z c z e g ó l n y c h p r z e w o d n i k ó w w p u n k c i e P .
B w = B 1 + B 2
J e s t t o t r ó j k t p r o s t o k t n y c h w i c B 1 ¥" B 2 , s t d :
B w = B 1 2 + B 2 2
Z g o t o w y c h w z o r ó w :
µ 0 i 1 4 À Å"1 0 - 7 3
B 1 = = = 1 , 2 Å"1 0 - 5 T
2 À r 1 2 À 0 , 0 5
µ 0 i 2 4 À Å"1 0 - 7 3
B 2 = = = 0 , 5 Å"1 0 - 5 T
2 À r 2 2 À 0 , 1 2
B w = B 1 2 + B 2 2 = ( 1 , 2 Å"1 0 - 5 ) 2 + ( 0 , 5 Å"1 0 - 5 ) 2 = 1 , 3 Å"1 0 - 5 T
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 5
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
z a d . 2 .
D a n e : i , R , B o = 0 ,
S z u k a n e : ± = ? ,
z p r a w a B i o t a - S a v a r t a :
µ 0 i s i n ± d ±
d B =
4 À R
N a p u n k t w [r o d k u o k r g u n a w a r t o [ i n d u k c j i m a g n e t y c z n e j w p By w a j : i n d u k c j a
m a g n e t y c z n o d j e d n e g o ( B p 1 ) i d r u g i e g o ( B p 2 ) p r o s t o l i n i o w e g o f r a g m e n t u d r u t u o r a z
i n d u k c j a m a g n e t y c z n a o d f r a g m e n t y Bu k u ( B Bu k u ) . Z r e g u By p r a w e j d Bo n i w y z n a c z a m y
k i e r u n k i t y c h i n d u k c j i , w i c w y p a d k o w y w e k t o r i n d u k c j i ( B w ) m a p o s t a :
B w = B p 1 + B p 2 - B Bu k u
Z t r e [c i z a d a n i a :
B w = B o Ò! B o = B p 1 + B p 2 - B Bu k u Ò! 0 = B p 1 + B p 2 - B Bu k u Ò! B p 1 + B p 2 = B Bu k u
D l a i n d u k c j i m a g n e t y c z n e j j e d n e g o p r z e w o d n i k a w p o s t a c i p ó Bp r o s t e j s t y c z n e j :
À
À
2
2
µ 0 i µ 0 i µ 0 i µ 0 i
À
B p 1 =
2
+"s i n ± d ± = 4 À R ( - c o s ±) = 4 À R ( - c o s - ( - c o s 0 ) ) = 4 À R
4 À R
0
0
r
r
K t 0 r a d ( w d o l n e j g r a n i c y c a Bk i ) b o w + " w e k t o r d l r .
r
r
À
K t r a d ( w g ó r n e j g r a n i c y c a Bk i ) b o w p u n k c i e s t y c z n o [c i z o k r g i e m w e k t o r d l ¥" r .
2
D l a i n d u k c j i m a g n e t y c z n e j d r u g i e g o p r z e w o d n i k a w p o s t a c i p ó Bp r o s t e j s t y c z n e j , p o d o b n i e :
À
À
µ 0 i µ 0 i µ 0 i µ 0 i
À
B p 2 =
2
+"s i n ± d ± = 4 À R ( - c o s ±) = 4 À R ( - c o s À - ( - c o s ) ) = 4 À R
4 À R À
À
2
2
r
r
À
K t r a d ( w d o l n e j g r a n i c y c a Bk i ) b o w p u n k c i e s t y c z n o [c i z o k r g i e m w e k t o r d l ¥" r .
2
r
r
K t À r a d ( w g ó r n e j g r a n i c y c a Bk i ) b o w + " w e k t o r d l r o r a z t e w e k t o r y
s p r z e c i w n i e s k i e r o w a n e .
D l a i n d u k c j i m a g n e t y c z n e j p r z e w o d n i k a w p o s t a c i f r a g m e n t u Bu k u :
g d z i e ±R t o d Bu g o [ t e g o Bu k u
±R
±R
µ 0 i µ 0 i µ 0 i µ 0 i ±
B Bu k u = ( l ) = ±R =
+"d l =
4 À R 2 0 4 À R 2 0 4 À R 2 4 À R
w i c :
B p 1 + B p 2 = B Bu k u
µ 0 i µ 0 i µ 0 i ± µ 0 i µ 0 i
+ = Ò! 2 = ± Ò! ± = 2 r a d
4 À R 4 À R 4 À R 4 À R 4 À R
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 6
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
z a d . 3 .
D a n e : i = 1 0 A , R = 2 0 c m ,
S z u k a n e : B = ? ( w a r t o [ i n d u k c j i p o l a m a g n e t y c z n e g o w p u n k c i e O ) ,
Z r e g u By p r a w e j r k i :
B w = B Bu k u + B p 1 + B p 2 + B p 3 + B p 4
O b l i c z a m w i c k o l e j n e w a r t o [c i t y c h i n d u k c j i p o l a m a g n e t y c z n e g o :
µ 0 i ±
B Bu k u = ( w y p r o w a d z e n i e p a t r z z a d . 2 . )
4 À R
3
µ 0 i Å" À 3 µ 0 i 3 Å" 4 À Å"1 0 - 7 1 0
3 2
± = À Ò! B Bu k u = = = = 2 3 , 5 6 Å"1 0 - 6 T
2
4 À R 2 R 8 0 , 2
B p 1 = 0 p o n i e w a \ o d l e g Bo [ t e g o p r z e w o d n i k a o d p u n k u O j e s t r ó w n a z e r o .
z p r a w a B i o t a - S a v a r t a :
µ 0 i s i n ± d ±
d B =
4 À R
3
3
À
À
4
4
µ 0 i µ 0 i µ 0 i µ 0 i 2
3 À
B p 2 =
4 2
+"s i n ± d ± = 4 À 2 R ( - c o s ± ) = 4 À 2 R ( - c o s À - ( - c o s ) ) = 4 À 4 R
4 À 2 R À
À
2
2
À
À
2
2
µ 0 i µ 0 i µ 0 i µ 0 i 2
À À
B p 3 =
2 4
+"s i n ± d ± = 4 À 2 R ( - c o s ±) = 4 À 2 R ( - c o s - ( - c o s ) ) = 4 À 4 R
4 À 2 R À
À
4
4
µ 0 i 2 4 À Å"1 0 - 7 1 0 Å" 2
B p 2 = B p 3 = = = 1 , 7 7 Å"1 0 - 6 T
4 À 4 R 4 À 4 Å"0 , 2
B p 4 = 0 p o n i e w a \ o d l e g Bo [ t e g o p r z e w o d n i k a o d p u n k u O j e s t r ó w n a z e r o .
O s t a t e c z n i e :
B w = B Bu k u + B p 1 + B p 2 + B p 3 + B p 4
B w = 2 3 , 5 6 Å"1 0 - 6 + 0 + 1 , 7 7 Å"1 0 - 6 + 1 , 7 7 Å"1 0 - 6 + 0 = 2 7 , 1 Å"1 0 - 6 T = 2 7 , 1 µ T
z a d . 4 .
D a n e : B = 0 , 0 0 9 T , R = 1 c m ( p r o m i e D s p i r a l i ) , h = 7 , 8 c m ( s k o k s p i r a l i ) ,
S z u k a n e : f = ? , v = ? ,
N a c i a Bo p o r u s z a j c e s i w j e d n o r o d n y m p o l u m a g n e t y c z n y m d z i a Ba s i Ba L o r e n z a :
r r
r
F = q v × B
r
r
F = q v B s i n "( v , B )
g d z i e :
q w a r t o [ Ba d u n k u c i a Ba ,
v p r d k o [ p o r u s z a j c e g o s i c i a Ba ,
B w a r t o [ i n d u k c j i p o l a m a g n e t y c z n e g o ,
M o \n a z a u w a \y , \e g d y :
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 7
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
r
r
"( v , B ) = 0 Ò! s i n 0 = 0 Ò! F = 0
w i c c i a Bo t a k i e p o r u s z a j s i r u c h e m j e d n o s t a j n y m ( l u b w o g ó l e s i n i e p o r u s z a ) .
Z a Bó \m y , \e c i a Bo ( e l e k t r o n ) w p a d a p o d j e d n o r o d n e g o p o l a m a g n e t y c z n e g o p o d p e w n y m
r
r
k t e m ± ( g d z i e ± = "( v , B ) ) . J e g o r u c h m o \e m y r o z Bo \y n a d w a r u c h s k Ba d o w e : z g o d n y
( r ó w n o l e g By ) z k i e r u n k i e m l i n i i p o l a ( v = c o n s t . ) o r a z p r o s t o p a d By d o l i n i i p o l a ( v ¥" ) .
M o \n a w i c t e r a z z a p i s a , \e :
2
v = v + v ¥"2
g d z i e :
v = v c o s ±
v ¥" = v s i n ±
R o z k Ba d a j c p r d k o [c i ( r u c h y ) n a s k Ba d o w e , m o \n a m i e t e \ n a m y [l i s k Ba d o w e s i By ,
a w i c : s k Ba d o w a p r o s t o p a d Ba s i By L o r e n z a p e Bn i r o l s i By d o [r o d k o w e j ( c o j e s t o c z y w i s t e ,
b o j e s t t o s t a Ba s i Ba s k i e r o w a n a z a w s z e p r o s t o p a d l e d o k i e r u n k u r u c h u ) .
F ¥" = m a d
v ¥"2 q B R
q v ¥"B = m Ò! v ¥" =
R m
d l a e l e k t r o n u :
e B R 1 , 6 Å"1 0 - 1 9 Å" 0 , 0 0 9 Å" 0 , 0 1
v ¥" = = = 1 , 5 8 Å"1 0 7 m
s
m e 9 , 1 Å"1 0 - 3 1
N a t o m i a s t w z d Bu \ l i n i i p o l a e l e k t r o n p o r u s z a s i r u c h e m j e d n o s t a j n y m :
s = v Å"t
h
h = v T Ò! v =
T
g d z i e :
T o k r e s o b i e g u e l e k t r o n u ,
C z y o k r e s z a l e \y o b i e g u e l e k t r o n u z a l e \y o d j e g o p r d k o [c i ?
S p r a w d zm y , o g ó l n i e o k r e s d e f i n i u j e m y :
2 ÀR
T =
v
c i a Bo p o r u s z a s i p o o k r g u ( c o b y Bo w y j a [n i a n i e p o w y \e j ) , w i c
v 2 m v
q v B = m Ò! R =
R q B
p o p o d s t a w i e n i u :
2 À m v 2 Àm
T = =
v q B q B
w i c o k r e s o b i e g u n i e z a l e \y o d p r d k o [c i .
D l a e l e k t r o n u :
2 Àm p 2 Å"9 , 1 Å"1 0 - 3 1
T = = = 0 , 3 9 7 Å"1 0 - 8 s
e B 1 , 6 Å"1 0 - 1 9 Å"0 , 0 0 9
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 8
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
1 1
f = = = 2 , 5 2 Å"1 0 8 H z = 2 5 2 M H z
T 0 , 3 9 7 Å"1 0 - 8
M o \n a t e r a z w y l i c z y :
h 0 , 0 7 8
v = = = 1 , 9 6 Å"1 0 7 m
s
T 0 , 3 9 7 Å"1 0 - 8
O s t a t e c z n i e :
2
v = v + v ¥"2 = ( 1 , 9 6 Å"1 0 7 ) 2 + ( 1 , 5 8 Å"1 0 7 ) 2 = 2 , 5 2 Å"1 0 7 m
s
z a d . 5 .
W
D a n e : i 1 = i 2 , = 4 , 3 Å"1 0 - 5 ,
l
S z u k a n e : i = ? ,
W z ó r n a s i B e l e k t r o d y n a m i c z n d z i a Ba j c n a p r z e w o d n i k z p r d e m :
r
F = i l × B
w i c :
µ 0 i 1 µ 0 i 1 i 2 l
F 2 1 = i 2 l B 1 = i 2 l =
2 À d 2 À d
b o p r z e w o d n i k p r o s t o l i n i o w y :
µ 0 i 1
B 1 =
2 À d
p o b o d n i e :
µ 0 i 2 µ 0 i 1 i 2 l
F 1 2 = i 1 l B 2 = i 1 l =
2 À d 2 À d
b o p r z e w o d n i k p r o s t o l i n i o w y :
µ 0 i 2
B 2 =
2 À d
w i c :
µ 0 i 1 i 2 l
F 1 2 = F 2 1 =
2 À d
µ 0 i 1 i 2 l µ 0 i 2 l 1
F z ( x ) = = Å"
2 À x 2 À x
p r a c a p o d w p By w e m z e w n t r z n e j s i By F z ( x ) t o :
3 d
3 d 3 d
µ 0 i 2 l 1 µ 0 i 2 l µ 0 i 2 l µ 0 i 2 l 3 d
W = ( x ) d x = d x = ( l n x ) = ( l n 3 d - l n d ) = l n
z
+"F +"
2 À x 2 À 2 À 2 À d
d d
d
µ 0 i 2 l W 4 À Å"1 0 - 7 Å"i 2
W = l n 3 Ò! = l n 3 = i 2 Å"1 0 - 7 Å" 2 l n 3
2 À l 2 À
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 7 9
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
W
= 4 , 3 Å"1 0 - 5
l
4 , 3 Å"1 0 - 5 = i 2 Å"1 0 - 7 Å" 2 l n 3
4 3 0
i 2 = = 1 9 5 , 7 Ò! i = ± 1 3 , 9 9 A
2 l n 3
W y n i k n a l e \y i n t e r p r e t o w a t a k , \e t o ± n i e m a z n a c z e n i a , b o a l b o p r d y w o b y d w u
p r z e w o d n i k a c h p By n u m o w n i e w p r a w o ( c z y w g ó r ) a l b o p r d y w o b y d w u
p r z e w o d n i k a c h p By n u m o w n i e w l e w o ( c z y w d ó B ) . G r u n t , \e p By n w t y m s a m y m
k i e r u n k u .
D o k o Dc z e n i e t e j l i s t y z a d a D
S e m e s t r : 0 7 l
P r o w a d z c y : d o c . K r z y s z t o f R o l e d e r
z a d . 6 .
D a n e : i 1 = 3 0 A , i 2 = 2 0 A , a = 1 0 c m , b = 1 5 c m , l = 4 5 c m ,
S z u k a n e : F w ,
[ + r y s . ]
N a l e \y z a c z o d w y z n a c z e n i a k i e r u n k ó w s i By d z i a Ba j c y c h n a k a \d y z b o k ó w r a m k i .
P o a n a l i z i e ( r e g u Ba p r a w e j d Bo n i e ) o k a z u j e s i , \e t e s i By d z i a Ba j z a z e w n t r z r a m k i .
R o z p a t r u j p i o n o w e f r a g m e n t y r a m k i :
µ 0 i 1
B 1 =
2 À Å" a
µ 0 i 1 i 2 l
F 1 = i 2 l B 1 =
2 À Å" a
µ 0 i 1
B 2 =
2 À Å" ( a + b )
µ 0 i 1 i 2 l
F 2 = i 1 l B 1 =
2 À Å" ( a + b )
R o z p a t r u j p o z i o m e f r a g m e n t y r a m k i :
n a l e \y z a u w a \y , \e n a g ó r n y f r a g m e n t r a m k i d z i a Ba d o k Ba d n i e t a k i e s a m a s i Ba j a k
n a d o l n y c o d o w a r t o [c i i k i e r u n k u , a l e o p r z e c i w n y m z w r o c i e . W i c :
F 3 = - F 4
F 3 + F 4 = 0
S i By t e r ó w n o w a \ s i .
O s t a t e c z n i e w y p a d k o w a s i Ba d z i a Ba j c a n a r a m k b d z i e p o z i o m a i s k i e r o w a n a w p r a w o
p o n i e w a \ F 1 > F . W a r t o [ s i By w y p a d k o w e j t o :
2
F w = F 1 - F 2
µ 0 i 1 i 2 l µ 0 i 1 i 2 l µ 0 i 1 i 2 l 1 1
ëø öø
F w = - =
ìø - ÷ø
2 À Å" a 2 À Å"( a + b ) 2 À a a + b
íø øø
4 À Å"1 0 - 7 Å"3 0 Å"2 0 Å"0 , 4 5 ëø 1 1 öø
F w = ìø - ÷ø = 3 , 2 4 Å"1 0 - 4 N
2 À 0 , 1 0 , 1 + 0 , 1 5
íø øø
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 0
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
z a d . 7 .
D a n e : d = 1 0 c m , i = 2 0 A , B = 0 , 0 1 6 T ,
S z u k a n e : W ,
M o m e n t m a g n e t y c z n y t o : p B = i Å" S , g d z i e i t o n a t \e n i e p r d u , a S t o p o l e p o w i e r z c h n i .
M o m e n t s i By : Ä = p B × B
P r a c a w y r a \a s i j a k o : W = - p B o B , p o n i e w a \:
W = d ¸ = p B Å" B Å"s i n ¸ d ¸ = p B B d ¸ = - p B B c o s ¸ = - p B o B
+"Ä +" +"s i n ¸
g d z i e : ¸ t o k t p o m i d z y w e k t o r e m i n d u k c j i B a w e k t o r e m p o w i e r z c h n i S .
R o z p a t r y w a n y u k Ba d j e s t z a c h o w a w c z y w i c m o \n a r o z p a t r z e w a r t o [c i e n e r g i i
p o t e n c j a l n e j w d w ó c h p o Bo \e n i a c h . O d e j m u j c j e o d s i e b i e u z y s k a m y p r a c :
W = U ( 9 0 ° ) - U ( 0 ° ) = - p B B c o s 9 0 ° - ( - p B B c o s 0 ° ) = 0 + p B B = i Å" S Å" B
2
d
ëø öø
2
1
S = Àr 2 = À = Àd
ìø ÷ø
4
2
íø øø
2
1 1
W = i Å" Àd Å" B = 2 0 Å" Å"0 , 1 2 Å"À Å" 0 , 0 1 6 = 2 , 5 1 Å"1 0 - 3 J
4 4
z a d . 8 .
D a n e : l = 1 m , B = 5 Å"1 0 - 5 T , U = 1 0 - 3 V ,
S z u k a n e : n ( l i c z b a o b r o t ó w n a s e k u n d ) ,
P r a w o F a r a d a y a
d ¦B
• = -
d t
g d z i e : • t o w a r t o [ s i By e l e k t r o m o t o r y c z n e j , ¦B t o s t r u m i e D m a g n e t y c z n y
¦B = o d s
+"B
J e \e l i p o w i e r z c h n i a j e s t p Ba s k a , a p o l e m a g n e t y c z n e j e s t j e d n o r o d n e t o w t e d y :
¦B = B o S = B S c o s "( B , S )
W t y m z a d a n i u d r u t z a k r e [l a k o Bo o p r o m i e n i u l . Z p r o s t e j p r o p o r c j i m o \n a w y z n a c z y
z a l e \n o [ p o l a w y c i n k a k o Ba o d k t a ± :
± S
1
= Ò! S = l 2 ±
2
2 À Àl 2
1
¦B = B S c o s 0 ° = B l 2 ±
2
d ¦B d ± 2 U
1 1
U = - Ò! U = - B l 2 = - B l 2 É Ò! É =
2 2
d t d t B l 2
P o n i e w a \:
d ±
É =
d t
É
É = 2 Àn Ò! n =
2 À
2 U U 1 0 - 3
o b r
n = = = = 6 , 3 7
s
2 ÀB l 2 ÀB l 2 À Å"5 Å"1 0 - 5 Å"1 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 1
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
z a d . 9 .
o b r
D a n e : S = 0 , 0 9 m 2 , n = 6 0 , B = 0 , 5 T ,
s
S z u k a n e : •( t ) ( s i Ba e l e k t r o m o t o r y c z n a o d c z a s u ) ,
R a m k a j e s t p Ba s k a , a p o l e m a g n e t y c z n e j e s t j e d n o r o d n e , w i c :
¦B = B o S = B S c o s ±
g d z i e ± t o k t o j a k i o b r ó c i Ba s i r a m k a ( k t p o m i d z y w e k t o r a m i B i S ) .
Z p r a w a F a r a d a y a
2
d ¦B B S ( c o s ± ) d ±
• = - = - = B S s i n ± = B S É s i n ±
d t d t d t
O g ó l n i e w r u c h u o b r o t o w y m :
d ±
É =
d t
W r u c h u o b r o t o w y m j e d n o s t a j n y m :
±
É = Ò! ± = Ét
t
É = 2 Àn
W i c :
•( t ) = 2 Àn B S s i n ( 2 Àn t ) = 2 À Å"6 0 Å"0 , 5 Å"0 , 0 9 Å"s i n ( 2 À Å"6 0 Å"t ) = 1 7 s i n ( 3 7 7 t )
z a d . 1 0 .
D a n e : B 0 = 0 , 1 T , S = 2 5 c m 2 ( p o w i e r z c h n i a r a m k i ) ,
s = 1 m m 2 ( p o l e p r z e k r o j u d r u t u ) , Á = 1 , 7 Å"1 0 - 8 &!m ( o p ó r w Ba [c i w y ) ,
S z u k a n e : Q ( Ba d u n e k e l e k t r y c z n y ) ,
Z p r a w a F a r a d a y a
d ¦B d ( B S ) d B
• = - = - = - S ( * )
d t d t d t
I n d u k c j a n i e z m i e n i a s i w z a l e \n o [c i o d w y b o r u n a p o w i e r z c h n i . S w o j d r o g z m i e n i a s i
w z a l e \n o [c i o d c z a s u . U w z g l d n i a t o r ó w n a n i e ( * ) .
U
Z p r a w a O h m a ( i = ) o r a z s i Ba e l e k t r o m o t o r y c z n a ( • ) t o r ó \n i c a p o t e n c j a Bó w ( U ) :
R
• = U = R Å"i
w i a d o m o , \e :
l
R = Á
s
g d z i e : l t o d Bu g o [ p r z e w o d n i k a ; s ( m a Be ) t o p o l e p o w i e r z c h n i d r u t u ,
Á t o o p ó r w Ba [c i w y ,
u n a s : l = 4 Å" a = 4 Å" S = 4 Å" 2 5 = 2 0 c m ( j e s t t o o b w ó d k w a d r a t u , b o t a k i k s z t a Bt m a r a m k a )
d q
i = ( d e f i n i c j a n a t \e n i a p r d u )
d t
l d q
w i c : • = U = R Å"i = Á
s d t
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 2
U t w o r z o n a : 6 . 0 1 . 2 0 0 7 2 3 : 2 8
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : Z E S T A W 1 0 M a g n e t y z m . Z m o d y f i k o w a n a : 7 . 0 3 . 2 0 0 7 0 : 3 8
p o r ó w n u j d w a w y r a \e n i a n a s i B e l e k t r o m o t o r y c z n :
d B
üø
• = - S
ôø
ôø d B l d q
d t
Ò! - S = Á Ò! - S s Å" d B = Ál Å" d q Ò! - S s Å" = Ál Å"
ýø
+"d B +"d q
l d q d t s d t
ôø
• = Á
ôø
s d t þø
Q
0
- S s Å" = Ál Å"
+"d B +"d q
B 0 0
S Å" s 2 5 Å"1 0 - 4 Å"1 0 - 6
- S s Å" ( 0 - B 0 ) = Ál Å" ( Q - 0 ) Ò! Q = B 0 = Å" 0 , 1 = 7 3 , 5 m C
Á Å" l 1 , 7 Å"1 0 - 8 Å" 2 0 Å"1 0 - 2
K o n i e c u z u p e Bn i e n i a .
D o d a t k o w e z a g a d n i e n i e :
J a k i j e s t s t o s u n e k s k o k u d o p r o m i e n i a s p i r a l n e g o t o r u e l e k t r o n u p o r u s z a j c e g o s i
w j e d n o r o d n y m p o l u m a g n e t y c z n y m ? W i a d o m o , \e e l e k t r o n w p a d B w p o l e p o d k t e m ± .
O z n a c z e n i a :
R p r o m i e D s p i r a l i ,
h s k o k s p i r a l i ,
T o k r e s o b i e g u e l e k t r o n u ,
r
r
± = "( v , B )
2
v = v + v ¥"2
v = v c o s ±
v ¥" = v s i n ±
h
h = v T Ò! v = ( w r a z i e w t p l i w o [c i p a t r z z a d . 4 . )
T
v ¥"T
T =
2 À
v T v
h v c o s ±
= = 2 À = 2 À = 2 À t g ±
v ¥"T
R v ¥" v s i n ±
2 À
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 3
U t w o r z o n a : 1 2 . 0 1 . 2 0 0 7 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I I Z a d a n i a . Z m o d y f i k o w a n a : 1 3 . 0 1 . 2 0 0 7 1 7 : 0 9
K o l o k w i u m I I I Z a d a n i a
1 2 . 0 1 . 2 0 0 7 r .
P r z y k Ba d o w y z e s t a w z a d a D d o r o z w i z a n i a n a k o l o k w i u m .
T r e [c i z a d a D Z e s t a w A
z a d . 1 .
N a p Ba s k i e j p o w i e r z c h n i z n a j d u j e s i k u l k a o m a s i e M , k t ó r a j e s t p o d c z e p i o n a d o s p r \y n y
o w s p ó Bc z y n n i k u s p r \y s t o [c i k . W k u l k u d e r z a p o c i s k o m a s i e m z p r d k o [c i v 0 . Z d e r z e n i e
j e s t i d e a l n i e n i e s p r \y s t e ( t j . o b y d w a c i a Ba s k l e j a j s i ) . U k Ba d z a c z y n a d r g a . J a k i j e s t o k r e s
i a m p l i t u d a t y c h d r g a D? Z a n i e d b a m a s s p r \y n y o r a z s i By t a r c i a .
B
z a d . 2 .
+ q
O b l i c z w a r t o [ i w y z n a c z k i e r u n e k w e k t o r a w y p a d k o w e g o n a t \e n i a
p o l a w p u n k c i e P . P a t r z r y s u n e k o b o k .
P
a
S k o r z y s t a j z s y m e t r i i u k Ba d u : p u n k t P l e \y w p o Bo w i e o d c i n k a B C .
+ 2 q - q
C
A
z a d . 3 .
K o n d e n s a t o r p Ba s k i p o w i e t r z n y m a o k Ba d k i w o d l e g Bo [c i 5 m m i o p o l u p o w i e r z c h n i r ó w n y m
1 0 0 c m 2 . D o t e g o k o n d e n s a t o r a p r z y Bo \o n e n a p i c i e r ó w n e 3 0 0 V . N a s t p n i e p o m i d z y
o k Ba d k i w s u n i t o d i e l e k t r y k , e b o n i t o w z g l d n i e j p r z e n i k a l n o [c i r ó w n e j 2 , 6 . J a k i e b d z i e
w ó w c z a s n a p i c i e n a o k Ba d a c h t e g o k o n d e n s a t o r a ? J a k z m i e n i s i g s t o [ p o w i e r z c h n i o w a
Ba d u n k u t e g o k o n d e n s a t o r a p r z e d i p o w s u n i c i u d i e l e k t r y k a ?
z a d . 4 .
À À
P o Bo \e n i e ( w m e t r a c h ) c i a Ba w c h w i l i t ( w s e k u n d a c h ) o p i s u j e r ó w n a n i e : x ( t ) = 2 s i n ( t + ) .
2 4
J a k i j e s t o k r e s d r g a D o r a z p r d k o [ m a k s y m a l n a i p r z y s p i e s z e n i e m a k s y m a l n e ?
z a d . 5 .
C z s t k a o m a s i e m i Ba d u n k u q z o s t a Ba p r z y s p i e s z o n a p r z e z r ó \n i c p o t e n c j a Bó w U .
J a k i p d u z y s k a Ba t a c z s t k a ?
z a d . 6 .
P o j e d y n c z a c z s t k a w y t w a r z a p o l e e l e k t r o s t a t y c z n e . W o d l e g Bo [c i r 1 = 4 c m o d t e j c z s t k i
V
n a t \e n i e p o l a w y n o s i E 1 = 5 . J a k i j e s t p o t e n c j a B w o d l e g Bo [c i r 2 = 1 0 c m o d t e j c z s t k i ?
m
z a d . 7 .
K o n d e n s a t o r p Ba s k i m a p o j e m n o [ 2 m F ( d w a m i k r o f a r a d y ) , a e n e r g i a p o t e n c j a l n a m a w a r t o [
0 , 4 J . J a k i Ba d u n e k j e s t z g r o m a d z o n y n a t y m k o n d e n s a t o r z e ?
S z k i c e r o z w i z a D d o z a d a D z n a j d u j s i n a k o l e j n y c h s t r o n a c h .
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 4
U t w o r z o n a : 1 2 . 0 1 . 2 0 0 7 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I I Z a d a n i a . Z m o d y f i k o w a n a : 1 3 . 0 1 . 2 0 0 7 1 7 : 0 9
S z k i c e r o z w i z a D Z e s t a w A
z a d . 1 .
D a n e : M , m , k , v 0 ,
S z u k a n e : T = ? , A = ? ,
k
k = m É2 Ò! É =
m
2 À 2 À m + M
T = = = 2 À
É k
k
m + M
z z a s a d y z a c h o w a n i a e n e r g i i m o \n a o b l i c z y p r d k o [ t u \ p o z d e r z e n i u :
m
M Å"0 + m v 0 = ( M + m ) v Ò! v = v 0
M + m
p r d k o [ t a j e s t p r d k o [c i m a k s y m a l n w r u c h u d r g a j c y m :
m
v m a x M + m v 0 m v 0 m + M m v 0
v m a x = A É Ò! A = = = =
É
k ( m + M ) k
( M + m ) 2 k
m + M
z a d . 2 .
D a n e : a , q A = + 2 q , q B = + q , q C = - q ,
S z u k a n e : E P = ? ( n a t \e n i e p o l a w p u n k c i e P ) ,
q
E = k
B
r 2
E A
+ q
Aa t w o z a u w a \y , \e E B E C
P
o r a z E A ¥" E B i E A ¥" E C , w i c : a
E B
E P 2 = ( E B + E C ) 2 + E A 2 .
+ 2 q - q
E C
q A
2 q 4 q C
A
E A = k = k = k
2 2
2 2
a 2
( a ) ( a )
2 2
q B
q 2 q
E B = k = k = k
E P
2 2
2 2
a 2
( a ) ( a )
2 2
E B
E A
q C
q 2 q
E C
E C = k = k = k
2 2
2 2
a 2
( a ) ( a )
2 2
2 2
2 q 2 q 4 q k q k q
ëøk + k öø ëøk öø
E P = ( E B + E C ) 2 + E A 2 = + = 4 2 + 4 2 = 4 2
ìø ÷ø ìø ÷ø
a 2 a 2 øø íø a 2 øø a 2 a 2
íø
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 5
U t w o r z o n a : 1 2 . 0 1 . 2 0 0 7 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I I Z a d a n i a . Z m o d y f i k o w a n a : 1 3 . 0 1 . 2 0 0 7 1 7 : 0 9
z a d . 3 .
D a n e : d = 5 m m , S = 1 0 0 c m 2 , U = 3 0 0 V , µr = 2 , 6 ,
S z u k a n e : U ' = ? , Ã = ? ( g s t o [ p o w i e r z c h n i o w a Ba d u n k u ) ,
Q
C =
U
S Q S S
C = µµr Ò! = µµr Ò! Q = U µµr
d U d d
P o p r z y Bo \e n i u d o k o n d e n s a t o r a n a p i c i a z g r o m a d z i B s i n a n i m Ba d u n e k Q :
S S S
Q = U µµr = U µ Å"1 Å" = U µ
d d d
P o w s u n i c i u e b o n i t u Ba d u n e k z g r o m a d z o n y n a o k Ba d k a c h k o n d e n s a t o r a n i e z m i e n i s i :
Q = c o n s t .
S
Q = U ' µµr
d
P o r ó w n u j o b y d w a w y r a \e n i a n a w a r t o [ Ba d u n k u :
S S U 3 0 0
U µ = U ' µµr Ò! U = U ' µr Ò! U ' = = = 1 1 5 , 3 8 V
d d µr 2 , 6
Q
P r z y p o m n d e f i n i c j g s t o [c i p o w i e r z c h n i o w e j Ba d u n k u : Ã = . Aa t w o z a u w a \y ,
S
\e z a r ó w n o Ba d u n e k z g r o m a d z o n y n a k o n d e n s a t o r z e j a k i p o w i e r z c h n i a o k Ba d e k
n i e z m i e n i a s i , w i c r ó w n i e \ g s t o [ p o w i e r z c h n i a Ba d u n k u j e s t s t a Ba i n i e z a l e \y o d µr .
S
U µ
Q U µ 3 0 0 Å"8 , 8 5 Å"1 0 - 1 2
d
C
à = = = = = 5 , 3 1 Å"1 0 - 4 m
S S d 5 Å"1 0 - 6
z a d . 4 .
À À
D a n e : x ( t ) = 2 s i n ( t + ) ,
2 4
S z u k a n e : T = ? , v m a x = ? , a m a x = ? ,
Z r ó w n a n i a o d c z y t u j , \e :
A = 2
À
É =
2
À
Õ =
4
w i c :
2 À 2 À
T = = = 4 s
À
É
2
À m
v m a x = A É = 2 Å" = À
2 s
2
2
À À m
a m a x = A É2 = 2 Å"( ) =
2 2
s 2
S t u d e n c k i e N o t a t k i C y f r o w e
S N y : B i o t e c h n o l o g i a
w w w . s n y . o n e . p l s n y @ s n y . o n e . p l S t r o n a 8 6
U t w o r z o n a : 1 2 . 0 1 . 2 0 0 7 1 6 : 4 4
N o t a t k a : F i z y k a I ( F Z C 1 0 0 3 C ) w i c z e n i a .
T e m a t : K o l o k w i u m I I I Z a d a n i a . Z m o d y f i k o w a n a : 1 3 . 0 1 . 2 0 0 7 1 7 : 0 9
z a d . 5 .
D a n e : m , q , U ,
S z u k a n e : p = ? ,
p = m v
W = q U
S k o r z y s t a m z z a s a d y z a c h o w a n i a e n e r g i i :
m v 2 m v 2 m
q U = Ò! q U = Ò! 2 m q U = p 2 Ò! p = 2 m q U
2 2 m
z a d . 6 .
V
D a n e : r 1 = 4 c m , E 1 = 5 , r 2 = 1 0 c m ,
m
S z u k a n e : V 2 = ? ,
q
E = k
r 2
q
E 1 = k Ò! k Å" q = E 1 Å" r 1 2 = 5 Å" 0 , 0 4 2 = 8 Å"1 0 - 3
r 1 2
q
V = k
r
q k Å" q 8 Å"1 0 - 3
V 2 = k = = = 0 , 0 8 V
r 2 r 2 0 , 1
z a d . 7 .
D a n e : C = 2 m F , E p = 0 , 4 J ,
S z u k a n e : Q = ? ,
Q Q
C = Ò! U =
U C
Q
Q
Q U Q 2
C
E p = = = Ò! Q = 2 C E p
2 2 2 C
Q = 2 Å" 2 Å"1 0 - 3 Å"0 , 4 = 0 , 0 4 C
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