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

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Pm#&IG K92$).a4.2h#*h'$*+ ."#>Jq2)C'$^4 I 23 AB u U c t = I =R4 $j''()."#>J$).a4*y 7,$$"m 1h#& $>p#.X#$I 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. 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…L ) LF< ),$t'qh<•'$?#$89=>#,?#$'().PQ#& 1:##`# $%#&&E'KL*G 'a4 h•IU2.E ) <• 14C^# $%#&c4)*G  17 u.X#$84e ,$t'qh>#$3>#&#  3#F#  3#1 4C1)3#1FN 1FY  …E'8E h•1),$[*G 'a4=D#&#`#&E'8"'$'() )8E32KL ) L<89 UF‡1FMN  ‡Y  FN  4.R*•mn 14#&.R*'4#& 1:#\] ),$t'qh L*G  17KL&E'1FY   #  3#1F#  3#ˆ 3#ˆF   ˆFFMN  I …n$.E )8E32#&32#&KL ) L#`#&E'8"'$ 1"  `4IR*mKX 1?  I)'E <  F<n )#1F@ )#Y  F@ Y Y I Y…4&E' L*G  178L#$g#&E' L&L$h# $6>#$3>#&3OJ$Z#qh 29#J$a#,$H#&'E )3>#&8E1),$[*G 17I )'E3# &$ F   34C1) &$ FMN  …n 84e#n$ )3>#& L*G J$%#&'(),$-KL&E' LMN  '$‰'E )3>#&8E,$m*G  17#4.R* Lm 1`# .2h#    I …)4,$Ku)K)'$h*Ke 'EKe# -'K $)#$'EKe# -'&E' ω I …]Z2 29#*H*f#.+#&8P_#& *K   l  F*  l K… ω    lm  ⇒  K  FK… ω l v  AB …]Z2 29##z#&8P_#&   *K   F      ω lm …   *K   ⇒ K   F    ω l …K  AB uABK9AB l v  Y =⇒ ω AYB †J!7#&.X#$8Š.+#&#z#&      ω F\ *3       *)q Y   Y A B   M  v v l ml mg l gl ⇔ =µ ϕ ⇒ ϕ = µ €$Pg#& 16#$!)2.+#& I 23A Bx A c t ω ϕ = + 12#&.E A l B K rad s m ω = =  A B 23 A B    3#  A B x cm Ac cm t v A cm ϕ ϕ π ϕ = − = − =    = → →    = = =    eC I 23A BA Bx c t cm π = + …) $0C8:q2#r#Ncm '>'8a#'$‹#8`# J'>'$#$)4*+ '$4,6!2.E8:q2#r# 8a# $j h $Q.R*     I  t t T − = + KLt 2 89 $Q.R*8:q2#r#Ncm 8a# $jI …)q>'.X#$ $Q.R*8:q2#r#Ncm8a# $j$)35!7#&JJKf' gc4)C )'E,R u $Q.R*=)#.a4.#8t'8:q2#r#Ncm8a# $j $6Kf' gc4)C*+ &E'    Œ I  l Y N l YM OM t ω π π π = = − =  N A B v t s π → =  …U2.E $Q.R*8:q2#r#Ncm8a# $j89  N  vƒ MI A B v  v t s π π π = + = + t''E*)3>  h ]'()Ke 8: q2=#!h#&*+ .2h# NA B mg l m K µ ∆ = = …) $0C'E$)]'()Ke J$7 $4+'K92'$^4'$4CR#.+#&'()Ke #4Ke . …n$'2#=d=:.P_',$2Z#&q*2*f#c4># ?#$'() $)#$K9'2#=dc4)#$'$- c4)C\89         A B Y Y Y I ml mx m l x= + = + …€$Pg#& 16#$'$4CR#.+#&'()'2#8o'89   A •B  3#  3#  Y θ = θ θ d l I mg mgx dt )C     A B •• • • 3# Y Y  Y   + θ + θ = − θ +  ÷   l x m l x mxx mg …L'>'!)2.+#&#$[#E 1m $9#$      Y A B  • •  ••  + θ θ θ + + = + + g x l xx l x l x …4'2#=d=:10 '$e* $63i $)C.pq 12#&*+ '$4,6!)2.+#&89,$H#&.>#&,R )=[c4)3-$h#& $j'() J$Pg#& 16#$K9K 8h     A Y B ••  A B + θ θ + = + g x l l x …U2.E a#3-&E''()!)2.+#&89    A Y B A B + ω = + g x l l x     N   Ž Ž Ž O C 1 C 2 x 3)#&J$Z8t'8:q2#r#N** $6]89=`# 1><AKX 1?  B8t'Ke .3)#& 1>*9 8:q2&~#N** $6]89=`#J$Z<AKX 1?  B …†J!7#&.#$84e =Z2 29##z#&8P_#& ) ?#$.P_'.+&Z* 2h.+'i'.h3)4 *y8a#c4)<89$D#&3-K9=D#& )q  NA B m mg x m K µ ∆ = =  …) -''()Ke .p'$^48a# $jMj#&KLKe .c4)]   $f2'$^43)#& 1>8a# $j>J!7#&.X#$84e =Z2 29##z#&8P_#& ).P_'  [ ]   M )q )q )q )q A B A B    A B A  B A Y B A Y B m m m m mvKA K l mg A A x A x A x A x l µ ∆ − + = = + −∆ + − ∆ + − ∆ + − ∆ − ∆ M vNA l Bv m s→ = …+8"'$J$)'()$)3E#& h*+ .R*'>'$\]#$V#&.2h#!  K9!  89    A B  d d π π ϕ λ ∆ = − + KL Y YA B  v cm f λ = = = …h89'i'.h&)2 $2)#4       A B  A B  M d d k d d k π π ϕ π λ λ ∆ = − + = → − = −  $4+'\]#`#    A B vwIIIwv M AB d d k AB k λ − < − = − < → = −  1`#.2h#\]'EY.R*'i'.h …h89'i' R4&)2 $2)       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 3#3#IIII 3#II '23I  〈⇒−==−= ′ −= ′ == ϕϕω ϕω ϕ UUUALiL Ai Ai AB 4C1)  π ϕ −= K9 ω I  L UU A − = eC       − − =  II I  π ω ω tCos L UU i KL   II CCL CC + = ω …†J!7#&J$Pg#& 16#$\#$q )#$ AK UeA hc I  += λ F}\F„v‚I ƒ •Ff …†J!7#&J$Pg#& 16#$\#$q )#$  \W    M mvA hc += λ F}  W    MAAK mvUe hchc +−= λλ …>J!7#&.X#$8Š.+#&#z#& AKMM Uemvmv +=  \W  \W     F} B  A   W λλ −= m hc v MA  $)C3- smv MA lIMN v W = …n$2Z#&K;#FY**F} D ai = λ $)C3- m µλ v = )BX 1?&a#K;# 14#& ;*#$0 *9 h.E#$V#&=j'qh'()>#$3>#& 1o#&'$2K;#3>#& 1S#&#$)489K;#.[ =e' 1S#&K;# ?*=e' … ) U  dtd xx λ == $)C3-qFY‚** =B$V#&=j'qh'()>#$3>#& 1o#&'$2K;#3>#& h qF„'* $2Z*~# BA MNI m ka D kx µλ λ =⇒= …)'E BA„vBAY‚ mm µλµ ≤≤ M„ ≤≤⇒ k w ,#&4C`#F},F‚ƒIIM eC'E„=j'qh'$2K;#3>#& hKX 1?qF„'*I …u.E ) ?#$.P_'=PL'3E#&'>'=j'qh = λ v„NwvwNMwMƒwMNwMNwY‚vA m µ B . `4I$2$D#&3-8z#&$FvvNI YM 3 -'.+>#$3>#& 12# &'$;#,$H#&'FYI *l3,$-8P_#&f8f' 12# * f F I Y ,&.+8L#."# ?'$'()f8f' 12# fFvI I I ?#$'H#& $2>. ,$-./#&'$0  12# &34- #5)$6#$ 1789 #5)$6#$ 1:# ;*<=>#,?#$@A6#$B,$-#9C89*=D#&'$0 'E'$  340 #F  .G  12# &,$H#&,$?I)3>#&#D*. $&V))#H K9') H 1`# $6 -' .+8L##$0 '()f8f' 12# c4)#&."#,$ L)#H 'E&> 1X=D#&=)2#$`4x Cõu 11: 12# & $?#&$"*'();#&K^&)2 $2)>#$3>#&,$2Z#&'>'$&V)$),$f$sJ 89)F**,$2Z#& '>'$