γεωμετρία α λυκείου θέματα & λύσεις (Lisari) (β θέμα) 20 12 2014
Προτεινόμενα Θέματα Λύσεις Γ' Λυκείου
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Επιμέλεια: Κοντομάρης Στέλιος - sciencephysics4all.weebly.com/ http://sciencephysics4all.weebly.com/ 1 ΛΥΣΕΙΣ ΠΡΟΤΕΙΝΟΜΕΝΩΝ ΘΕΜΑΤΩΝ 2015 ΘΕΜΑ Α Α.1) α Α.2) γ Α.3) δ Α.4) α Α.5) α) Λ β) Σ γ) Λ δ) Λ ε) Σ ΘΕΜΑ Β Β.1) Σωστό το β m f 4 , 0 = = υ λ Τα σημεία που ταλαντώνονται με μέγιστο πλάτος ικανοποιούν τη σχέση: λ N r r = − 2 1 (1) με ,... 2 , 1 , 0 ± ± = N με d r r = + 2 1 (2) Προσθέτοντας τις σχέσεις (1) και (2) κατά μέλη λαμβάνουμε: ⇒ + = + + − d N r r r r λ 2 1 2 1 2 2 2 1 1 d N r d N r + = ⇒ + = λ λ Όμως ισχύει: ⇒ ≤ ≤ − ⇒ ≤ ≤ − ⇒ ≤ + ≤ ⇒ ≤ ≤ d N d d N d d d N d r λ λ λ 2 2 2 2 2 0 0 1 (όπου ,... 2 , 1 , 0 ± ± = N ) Συνεπώς, οι τιμές που παίρνει το Ν είναι: 2 , 1 , 0 , 1 , 2 − − = N , δηλαδή τα ζητούμενα σημεία είναι 5. 5 , 2 5 , 2 1 4 , 0 1 1 1 ≤ ≤ − ⇒ ≤ ≤ − ⇒ ≤ ≤ − N N Nλ
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Λύσεις των προτεινόμενων θεμάτων για τις πανελλήνιες Φυσική κατεύθυνσης Γ Λυκείου
Transcript of Προτεινόμενα Θέματα Λύσεις Γ' Λυκείου
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1
2015
.1) .2) .3) .4) .5) ) ) ) ) )
.1)
mf 4,0==
:
Nrr = 21 (1) ,...2,1,0 =N
drr =+ 21 (2)
(1) (2) :
+=++ dNrrrr 2121 222 11
dNrdNr +=+=
:
+ dNddNdddNdr 22222
00 1
( ,...2,1,0 =N )
, : 2,1,0,1,2 =N , 5.
5,25,214,0111 NNN
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.2)
x:
+= xx mmm 210 += xx 210 22110 += (1)
y:
= yy mm 210 = yy 210 22110 = (2)
(1) (2):
2121222
2122
120 2 ++= (3)
2121222
2122
1 20 += (4)
(3) (4):
( )++= 212121222120 2
( )2121222120 2 +++= (5)
. :
+= 2221
20 2
121
21
mmm 2221
20 += (6)
(5) (6) :
( ) =+ 02 2121 ( ) =+ 021 221
=+
1
2
0r
1r
y1r
x1r
y2r
2r
x2r
y
x
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: 21 += .
1pr
1 ,
2pr
2 ,
1pr
1 ,
2pr
2 ,
,
+=+ 2121 pppprrrr
211 ppp +=rrr
(1)
(1) . 211 ,, ppp
rrr:
++= 212
22
121 2 ppppp ( )( ) ++= 21222212212 2 mmmmm
++= 2122
222
122
12 2mmmm ( )++= 2122212212 2mm
( )++= 2122212212 2mm 21222121 2 ++= (1)
.
:
+=+ 2121 += 211 +=2
22
121 2
121
21
mmm
( )+= 222121 21
21
mm 222
121 += (2)
1pr
2
1pr
2pr
1
2
1pr
1pr
2pr
1
21 +=
1pr
2pr
0r
211 ppp +=rrr
2
,
0
2
x x
x x
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4
(1) = (2) :
+=++ 222
1212
22
1 2 = 02 21 = 0 = 2
221
=+
.3)
12 AA > .
.4)
:
HzMkf
5
21
0 ==
:
== Hzff262 12 Hzff
32 12 ==
.
0f
)(Hzf 1f 2f
1
2
xwr
xwr
wr
Nr
Fr
l
..
.
..
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:
lkwFwF xxx === 00 ,
l .
00 == T
..
:
MaTwFF xx == 0 (1)
== IRTI = RaMRRT 2
21
MT
aMaT 2
21
== (2)
(2) (1):
= MT
MTwF x
2 ( ) =+ TTwxlk x 2 =+ Twkxlk x 3
3kxT = (3)
xwr
xwr
wr
Nr
Fr
l
..
.
..
x ..
Tr
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, :
+= FwTF xx kxlkwTF xx += (4)
(3) (4) :
= kxkxFx 3= x
kF3
2 DxF =
32kD = .
.1)
( Steiner):
22
31
2MllMII cmA =
+=
: LLrr
= (1)
42lMl
mL == (2)
( ) =
+=+== 2
2
231 l
mMlIIIL =
+ 2281
31 MlMl
24
1183
22 MllMM =
+ (3)
r l
01 =U
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(2) (3) (1) :
=
2411
4
2MllM=
2444l
sradll
IS/6
111116
116
4424 .).(
=
===
.2) .
( )
=
=
%100
21
21
21
%100%2
22
m
mI
KKK
( )
= %100
2
22411
%2
222
M
MMl
( )( )..
2
222
%1001211
%IS
l
=
( )
= %100196
196361211
% ( ) %83%100196
19633% =
( ).
.3)
( .2). h .
h
01 =U
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,
+=+ 2211 UKUK = 21 UK = ghmI 2
21
=gm
Ih
2
21
=gM
Ml
h
2324
1121 22
=g
l
h3
2411 22
.).(22
7211 IS
glh = mh 55,0
2011
10723611
==
=
.4) lh = . lh =
0= . ,
+=+ 3311 UKUK = 21 UK
= hgmI 2
21
=
I
hgm22 =
2411
232
22
Ml
glM
.).(
2
1172 IS
lg= =
117202
srad /8
, (1) :
= 24
114
2MllM smlIS
/3
446
11 .).(==
.1) . ,
xDTF ==
( ) ( ) ADTF == maxmax
:
( ) maxF AD gmAm 2 gA
2 2 gA
0 0
0 0
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( ) 22 f
gA 224 f
gA
2
2
4104
104,0
Hzs
m
A mmA 310`25025,0 =
,
mA 3max 1025 =
.2) = 2mD ( ) = 22 fmD mNfmD /1604 22 ==
, ( ) += 2mmK += 2mDK
= 2DK
m
( )
= 22 f
DKm
kgm 25,0=
.3) 1t :
( ) +
==++
22222
21
21
21
xmm
KkAKxmm
( )= 2424 1025,01025,1
/200mm
kgmN
= sm /1075,0160 4
sm /10120 2=
, 1t , .
= adtd
, ( )maF =
.
,
+==+ 22222
21
21
21
m
xAAkKxm
+= mA 44 1012020011025,0 = mA 41085,0 = mA 61085
mA 31085 =
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AA
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:
( ) =+= 20 lKgmmFx ( )
mK
gmml20010
2 =+
=
:
mmll 025,0200
512 ===
. :
= xDgmT = xmgmT2 = xmgmN
2
= xmgmgm2 = xmgmgm
2
= xgg 2
gxg 2
=
Ax = :
=
gg 2
+
=
g
mm
Kg2
63
Ax = :
+
=
gg 2
++
=
g
mm
Kg2
23
: 23
=
: .
... 23
= .
, .
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.5) 2/Ax += :
( ) +
==++
22222
21
21
21
xmm
KkAKxmm
+
=
2
43
mm
K
s
mA 310312535 ==
(: , .
2 . )
2/Ax += :
= xDF = xmF2 +
=
xmm
KmF NF 5,2=
,
2/Ax += :
( )s
Js
mNFdtdK 33 1035,3121031255,2 =
==
2/Ax += ( ).
.6) i) :
smmm
K /25,0max =+===
( ) sm /25,0= .
:
Hzff SA 1360==
ii) , 2 .
:
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sK
mmTt
102
=+
==
st10
= .
