Đề thi Học sinh giỏi môn vật lý lớp 12 (30)

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Câu 3: + '2#8o'8:q2#D*#&)#&'E.+'j#&
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b)W>'.X#$ $Q.R*8:q2#r#Ncm8a# $j,R u8t' $ZI
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Câu 4: )#&4/#J$> 3E#&, $_J\] 1`#*G  $2>#&'()*+ '$0 8[#&!)2.+#& $f2J$Pg#& 16#$
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YA BAM BM cm− =
 K9
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MNA BAM BM cm− =
Ih $Q.R*
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Câu 5: $2.2h#*h'$#- J#$P$6#$KOA$6#$YB 
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AB
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Cõu 7: Cho một bán cầu đặc đồng chất, khối lợng
m, bán kính R, tâm O.
1. Chứng minh rằng khối tâm G của bán cầu cách tâm
O của nó một đoạn là d = 3R/8.
2. Đặt bán cầu trên mặt phẳng nằm ngang. Đẩy bán
cầu sao cho trục đối xứng của nó nghiêng một góc nhỏ so với phơng thẳng đứng
rồi buông nhẹ cho dao động (Hình 1). Cho rằng bán cầu không trợt trên mặt phẳng
này và ma sát lăn không đáng kể. Hãy tìm chu kì dao động của bán cầu.
Cõu 8$2'g$"&/*'E*+ Ke #G#&'E,$-8P_#&*.P_'=4+'K923_
!;C,$H#&!~#Ko c4)1:#&1d'*+ .a4!;C=4+''-.X#$K92.R*\I
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Cõu 9E*h'$."##$P$6#$I
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Cõu 10 : $4>#$3>#&.g#3o''E=PL'3E#&

FMà*K92') H
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A B A B A B
 M
d d k d d k
π π
ϕ π λ
λ

∆ = − + = + → − = +
 $4+'.2h#\]
 

A B vwIIIwv
M
AB d d k AB k
λ
− < − = + < → = −

1`#.2h#\]'EY.R*'i' R4
…h.R* $4+'.2)#\]'>'$ 14#&.R**+ .2h#x'E$"4.PQ#&.'()
$)3E#&89
 
d d x− =
…R* $4+'.2h#\].j#&C`# $2Z*~#

 
 
 A B A BI
M M 
d d x k x k
λ
λ
− = = + → = +
ABKL
vwIIIwvk = −
…U2.E
)q
*#

 Y
Av BI ƒY„NA B
M 
 Y
A BI Y„NA B
M 
m
x cm
x cm

= + =




= + =


…€$Pg#& 16#$!)2.+#& p#&$_J h'>'$\]#$V#&.2h#!

K9!

89

   
I 23 A B I 23 A B A B
M M
M
u c d d c t d d mm
π π π π

ω
λ λ
   
= − + + + +
   
   
…).R*

K9

.^4 $4+'*+ f8J#$e#\]89* `4.R*#`#

   
AM BM AM BM b+ = + =
4C1)J !)2.+#&'()

K9

89





I
I 23 IY I 23
Y M M

I
I 23 IMN I 23

Y M M
M
M
M
M
b
u c c t
u
u
b
u c c t
π π π π
ω
λ
π π π π
ω
λ

   
= + + +

   
    
→ = −

   

= + + +
   


   

h $Q.R*


 
A B A B
M M
u mm u mm= → = −
•n$ a#3-
Nf Hz=
 ) $0C
  
AM AB MB
U U U= +
'$j#& [U
AB
K4H#&J$)KLU
MB
#`#.2h#\],$H#& $R'$j)
…@K9K6,$.EU
AM
K4H#&J$)U
MB
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
…@K9'4+# $4a#'Z*K6,$.EU
AM
K4H#&J$)U
MB
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

…'4+# $4a#'Z*K9 7."#K6,$.EU
AM
#&P_'J$)U
MB
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
…'4+#'Z*'E."# 1m $4a#K9."# 1m $4a#@K6,$.E&E'8"'$J$)&V)U
AB
K9
U
MB
89&E'#$d#IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII

U2.E.2h#\]'E $R'$j)'4+#'Z*'E."# 1m $4a#1.+ i'Z*K9 7."#I
•n$Z#z#&$+JW'$j) 7."#'$j)'4+#'Z*A1BI
n$
Nf Hz=
 ) $0C
   
 w A YB
C MB r L L C L C
U V U U U U U Z Z= = + = → < → <
!b $0C,$ z#& a#3-8`#c4>NHz $6Z
L
 z#&Z
C
&Z*.#8t'Z
L
= Z
C
 $6!:#&."#

$"4!7#&*L.h 'i'.h#&$•)89 z#& a#3-8`#c4>NHz $6I z#& 1>& I
U2.E,$Z#z#&#9C=X82hI
•n$Z#z#&$+JW'$j)'4+#'Z*A1BK9$+J'$j) 7I
…n$
Nf Hz=
 )'E$"
   
   
 Y
 Y
  Y

A B 
C
C
AM r L L
r
AB r L C
U V
U V
U U U U V
U V
U U U U


=
=


 

= + = → =
 
 
=
= + − =





Y
N Y
 lN Y A B
N Y N Y l A B
NA B
N
C
L
Z
C F
Z L H
r
r
π
π
−


= Ω
=



 
→ = Ω → =
 
 
= Ω
= Ω




…Ub $0C8t'
Nf Hz=
 $6qZC1)'+#&$Pm#&
*)q
Fl@#`##4 z#&{8`#c4>NHz
$6&Z* $2Z*~#& I
eC$+JW'$j)'4+#'Z*'E
NA Bw N Y l A Br L H
π
= Ω =
K9$+J'$j)
 7
Y
 lN YA BC F
−
=

…h F


DUu
AB
→=
*m':#D
2
.E#&


w UCqUuuuu
MMBAM
=→====→
…L
MB
uTt Ml <<
&Z* u


→U
#`#D
1
*m 7

J$E#&."#c4)

K9#&4/##$P#&,$H#&J$E#&."#
c4)D
1
.P_'  )'E



UCqq =+−
A„B
…h FlM
 =+→=
MBAMAB
uuu
A‚Bw, $_JABK9AB $6 h FlM ).P_'








>
+
=
<
+
−=






CC
UC

u
CC
UC
u
Mb
AM
AƒB#`#$).H .^4=X'0*
…)4 FlMm'$.+p#.X#$$).H .^4=X'0* )'E!:#&c4)$) 789./#&#$0 #`#
B3#AB3#ABAB3#A
B'23AB'23A

l

l


tUCCtICCtUCCqCqC
tUCCuCCuCCtUuu
MBAMMBAM
ωωϕωωω
ωω
−=++−⇔−=+→
=+→=+

23
3#
23

C C U
I
q q c t a
C C U
C C
i t
q q c t a
C C







=
= +



+
=

= +
+

=








I'23
AB
I'23
AM
MB
C U
q a
u t
C C C C
C U
q a
u t
C C C C



= = +

+

= = +

+

h FlMAB $[)*~#AB#`# ).P_'







=
+
=
+









C
a

CC
UC
C
a
CC
UC
$)CK92AB'$2 )

( )







+
+
+
=

+
=






'23

'23I
CC
UC
t
CC
UC
u
t
CC
UC
u
Mb
AM


A ) $0C
w
AM MB
u u t
#`#,$p#.X#$$)
.H .^4=X'0*B
1. Do đối xứng, G nằm trên trục đối xứng Ox. Chia bán cầu thành nhiều lớp mỏng
dày dx nhỏ.
Một lớp ở điểm có toạ độ x= R sin , dày dx= Rcos.d
có khối lợng dm = (Rcos )
2
dx với
Y
@
Y

* =
nên:

*
!3#'23@
*
q!*
q
l

YM
*





==
d =

@Y
*M
@
'23
*M
@
q
M
l

M
M

=

=

=

(đpcm)
2. Xét chuyển động quay quanh tiếp điểm M: gọi là góc hợp bởi OG và đờng
thẳng đứng
- mgd = I
M
. (1) biến thiên điều hoà với =


*&!
I
O
, I
G
, I
M
là các mômen quán tính đối với các trục quay song song qua O,G,M. Mô men
quán tính đối với bán cầu là:
I
O
=

*@
N

; I
O
= I
G
+ md
2
I
M
= I
G
+ m( MG)
2
. Vì nhỏ nên ta coi MG = R-d
I
M
=

*@
N

+m(R
2
2Rd) =

*@

Y

=
@v
&N

*&!

=
T =
&N
@v

e ';#=D#&,$'$P) >'!7#&8i'*&F,

o
l
$d# 17'<q $%#&.j#& u 1`#q4-#&I< 1S#&KL]*L,$'E8i' >'!7#&I
A
B
C
1
C
2
M
D
1
D
2
H.2

Hình 2


<


I
<
<

q
q
Hình 1
!q
h]*L•…€
2 2
8 q

,

∆ +
FAKLq
2
89,$2Z#&'>'$&V)]*L32KL]'’B
n$Ke 'E8.+q8:q2&~#
2 2
8 q∆ +
…q
•…€
2 2

8 q q

,

∆ + +
F*qˆˆ
⇒
qˆˆ…
,
M*
qF
eCKe UKLJ$Pg#& 16#$qF\'23A
ω + ϕ
B
12#&.E
,
M*
ω =
$PKeC'$4,6!)2.+#&'()Ke F
M*

,
π
I$Q&)# u8t' >'!7#&8i'.#,$Ke !u#&8h8a# $j#$0 
89
 M*
I
 ,
= = π
n$ FqF\'23A

ϕ
BFq
2
F
M•
,
F\
3#ω ϕ
F
⇒
\F
M•
,

ϕ = π
F\F
‚•
,
i' >'!7#&8`##$P$6#$KO
R*!)2.+#&.^4$293)4,$ >'!7#&8i'• $6J$Z.j#&C`#
⇔


≥
12#&c4> 16#$*'$4CR#.+#&
⇔
FP
B
®h max
(F

2

≥

⇔
&
2 2
8 q \

,

∆ + +
F&,
\
M

≥

⇒
•
≤
&
$d#c

K9c

89."# ?'$=Z# 1`#'() 7I
I






l
l

l

=++
=++
=−=
C
q
C
q
iL
uuu
qqi
CABCAB
0C.h2$9* $f2 $Q&)#
I

=+
′′
ii
ω
w
KL



II CCL
CC +
=
ω
K9
( )
ϕω
+= tAi I'23I
A…B
n$ F
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D
kx
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λ
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m
µ

B