7 ( )ก 7 .. 2555
ก g = 9.8 m/s2 π = 3.14159
180 = π log10(2) = 0.3010
sin 30 = cos 60 = 1
2sin 60 = cos 30 = 3
2
1. กกกกก !"m1
# กก $ g $& 'ก() #(ก '*m2
+, ก#-ก.1. 2.m2g (m2 −m1)g
3. 4.m2m1m1 −m2 g
m2m1m2 −m1 g
5. m2m1m2 +m1
g
5 : = maΣF
= m2g (m1 +m2)a
a = m2g
m1 +m2
: = mammmm1111 ΣF
T = m1
m2g
m1 +m2
T = m2m1m1 +m2
g
m2
m1
N
T
T
a
a
m g2
m g1
m1
m2
1
2. $ กก ((:; กกก ((:1
2MR2)
; ,*ก"(< <)'ก!"ก 'ก.* " กMR2)
'* #=#กก =#กกก .* $!"1. 2. 3. 4.2
3
4
3
3
2
3
4
5. 1
2
1 1 : ,* ( c $& #+"ก$ #)?I = cmR2
'ก = E1 E2
Mgh = 12Mv2 + 1
2Iω2
Mgh = 12Mv2 + 1
2cMR2
v
R2
gh = 12v2(1 + c)
v = 2gh
1+ c → vα 1
1+ c
v
v =1+ k1+ k = 1+ 1
1+ 12
= 2
3
2 :
'ก = E1 E2
Mgh = 12Mv2 + 1
2Iω2
ก ก : Mgh = 1
2Mv2 + 1
21
2MR2
v2
vกก = 4
3gh = 2
3gh (1)
ก กก : Mgh = 1
2Mv2 + 1
2(MR2)
v2
vกกก = gh (2)
(1)(2) :
v
v =
2
3gh
gh= 2
3
1
2v
hω
กกก กกกกกกก
1
2v
hω
กก กก
กกก
กกกกก
กกก
2
3. *+ก(' 3 ( *กก,*" ก(' กก*?#(ก '$* 'ก'?C:ก(ก$&ก #* 1. * 2. * 1
8
1
4
3. !#+"$& 4 4. !#+"$& 8
5. !#+"$& 8 3 .*ก('#R3 αT2
R2R1
3
= T2T1
2
= 24
32
= 82 = 64
R2R1
= 4
∴ $* 'ก'?C:ก(ก$& 4 #* 4. *+ 75 kg กก.ก#- , $ # #."*ก
ก ก.*,* 'ก$)+'?C:ก$& 100 cm 'ก$)+$& 150 cm
1. 245 N 2. 250 N3. 368 N 4. 490 N5. 735 N
1 ,*# #."*ก R $Kกก.ก# # ก R
M = M O mg100 = 2R15075(9.8)(100) = 2R150 → R = 245 N
O
mg2RN
100 cm150 cm
3
5. ก กN ก(ก$ !" $ OPM1 M2
กก " ก )+กก ,* <$# '*M2
###+ก T2T1
1. 2.M2
M1
M1
M2
3. 4.1 +M2
M11 +
M1
M2
5. 1 −M2
M1
3 1
: = maΣF
= T2 − µm1g − µm2g (m1 +m2)a
= T2 (m1 +m2)a + (m1 +m2)µg
= T2 (m1 +m2)(a + µg) (1)
m1 : = maΣF
= T1 − µm1g m1a
= T1 m1a + µm1g
= T1 m1(a + µg) (2)
(1)(2) :
T2
T1=
(m1 +m2)(a+ µg)m1(a+ µg)
=m1 +m2m1
2
"ก#$ % T2
m1 +m2=
T1m1
→T2T1
=m1 +m2m1
M2M1T1 T2
+
a a
m mT11 2
T2
f = m g1 1µ f = m g2 2µ
a a
m mT11 2
T2
4
6. .."* !"R# A !? #";กกก.!=ρ
(< ก '*##.."ก. ก.!ν
1. 2. 3. 4.ρAν ρAν2 ρAν3 νρA
5. ν2ρA
2'ก ΣF ⋅ t = m(v − u)
ΣF ⋅ t = ρVv
l vΣF ⋅ t = ρA
ΣF = ρAtv = ρAv2
7. !'-##*: N , <$" )."*ก#)?ก*: N #,$&.*:."*ก#)?'ก<$ก<&'ก
ก(ก Cก(ก
(ก 1 1
!T* 318 11.2
14.5 4.0
1. (ก < < !T* 2. !T* < < (ก3. !T* < (ก < 4. < !T* < (ก5. < (ก < !T* 5,* M(ก = M , R(ก = R'ก g = GM
R2
g(ก = GMR2
g!T* = G(318M)(11.2R)2
= 2.54GM
R2
g = G(14.5M)(4R)2
= 0.91GM
R2
" mg < mg(ก < mg!T*
l
l
A v
ก.!
v
5
8. *+ 50 kg #+"< 5.0 m , 5.0 s ),ก#+"< $ OR!ก.# ก?X: 20% ! Z <$"* ,$#! '*ก;# ก"1. 98 J/s 2. 392 J/s 3. 490 J/s 4. 1960 J/s5. 2450 J/s 2
Q = 80% # p() E
Q = 80
100mgh
= Q
t
80100mgh
t=
80100
(50)(9.8)(5)
5= 392
Js
∴ ก;# ก = 392Js
9. ?R*+, (ก x) ) !O:#= <ก` ( #=$&ก )+ก<$#*,C+x '*?R,,ก<$.* ก+ก* .* - t = 0 .* - $&"กt = T1
1. 2.1
3T1
1
2T1
3. 4.2
2T1
2− 2
2T1
5. 2+ 2
2T1
4'กก`(':ก.*,*'< #ก`(a) = v − t = −
V0
T1
: S = !",ก`'ก t = 0 − t =T02
1
2t = 0 − t =
T02
S = 12
1
2V0T
= 1
4V0t
'ก S = ut + 1
2gt2
= 1
4V0T1 V0t +
1
2
−V0
T1
t2
=
V0
T1
6
. - ก4T1
'< = T12 4T1t − 2t2
= 02t2 − 4T1t + T12
t = − (−4T1) ± (−4T1 )2 − 4(2)(T1
2)
2(2)
= 4T1 ± 8T1
2
4
= 4T1 ± 2 2 T1
4=
2± 2
2
T1
, t = 2− 2
2
T1
10. ?R`?R($#<$,- *=ก . = ก";กก *=ก '* C(#ก#?R` C(#ก#($1. 2. 3. 1 4. 21
4
1
2
5. 4 2
'ก r = mv
qB
v B rα mq
rαrp =
mαmp
qpqα
=
4
1
1
2 = 2
7
11. $*+ $ ก 300 N/m 50.0 cm ", . 1.00 kg <$<$ $ !กก<$'ก $?<$ 10.0 cm $ '*)?#+"<$< ?*!"1. 46.5 cm 2. 50.0 cm 3. 55.3 cm 4. 60.0 cm5. 65.3 cm 3
= E1 E2
= mgh1
2kx2
= 1
2(300)(0.1)2 1(9.8)h
h = 0.153 m = 15.3 cm
∴ )?#+"<$< ?*!" = 40 cm + 15.3 cm
= 55.3 cm
m
h
10 cm
40 cm
50 cm(1)
(2)
!"
8
12. * #R#."#= R#" ,."920 kg/m3
* $ *."$& ,#1030 kg/m3
$"*#R#1. 11% 2. 21% 3. 50% 4. 79%5. 89% 1
R#."#= ?#+" =
= mgFB
* V' g = )? V)? gρ ρ
ก$& = V
V
ρρ
'< V' = V)? ρρ
= )? = 0.89 V)?920
1030V
= 89% # V)?∴ $#R#."#=*." = 11% #$)?
13. *+ $f (double slit) * $f0.03 mm ) $f * 'ก;กR!$& 1.5 m $กK " * 'ก'?ก+ก;ก$& 5.0 cm '*# "1. 250 nm 2. 400 nm 3. 500 nm 4. 667 nm5. 1000 nm 3!'-) 2 ก'ก
= dx
Lnλ
= (0.03× 10−3)(5× 10−2)1.5
2λ
= λ 500 × 10−9m = 500 nm
mg FB )?
)?)?
'
*
*
x A2
L0
9
14. * (* # <``qC" $ Cก:* (*'ก=ก * (*" < กC# <` q =ก' #ก # g #(ก1. 0.5g 2. 1.0g 3. 1.5g 4. 2.0g5. 4.0g 4
$ก e )'qE = mg
#%* ก +%,,-'ก = maΣF
qE + mg = mamg + mg = ma
a = 2g
15. '" Z !<` qก: 6Ω
1. 6 W 2. 18 W3. 24 W 4. 36 W5. 54 W
1'ก*กก VOLT! Cก: 6 V6Ω
P6Ω = V2
R= 62
6= 6 W
qE
mg
E e
qE mg
EE
e
12V
2Ω
6Ω
3Ω
12 V
6 V 6 V
2Ω
6Ω
3Ω
= 2Ω62+1
10
16. * 'ก*Z 8.0 m *Z# 85 dB) * 'ก*Z 80 m # '$&ก dB1. 65 dB 2. 75 dB 3. 83 dB 4. 95 dB5. 105 dB 1
'ก β1 − β2 = 10 logI2I1
= 10 log R1R2
2
85 − β2 = 10 log 80
82
85 − β2 = 10 log 102 = 10(2)log 10 = 20
β2 = 65 dB
17. R$f .;#=ก=$ '?กuv?500 cm3
v+ R,R#.,*v+2.0 × 106 Pa
ก* ก.<``q'กRก<``q 15 V ! *'ก,*ก <*$& 10 s #กuv,R$<$$& 1.1 × 107 Pa
##,* ,1. 2. 3. 4.3
11Ω 1
3Ω 1
2Ω 3Ω
5. 10
3Ω
2'ก = ( ! = 0)∆Q ∆U + ∆W ∆W = 0 ∆V
E<``q = 32∆PV
= V2
Rt
3
2V∆P
R = 23
⋅ V2t
V∆P = 2
3⋅ (15)2(10)
(500× 10−6)(1.1× 107 − 2× 106)= 1
3Ω
R = 8 m2
R = 80
m1
= 85 dB1β
= ?2β
1
2
*Z
0
11
18. กกก !"#$กก 10.5 cm)ก* ก!+)ก!!,+ -+.*/0)*+$ !"#0144.5 cm $,$ก* ก!+)ก! *ก"! !#ก!$/ก* 3!.4ก546!ก! 340 m/s "3+01 1. 250 Hz 2. 500 Hz 3. 764 Hz 4. 810 Hz5. 1000 Hz 2$ก4#+?$@ก 4#0/ !
$ก#0 λ2
= 44.5 − 10.5 cm
λ2
= 34 cm = 0.34 m
λ = 0.68 m
$ก v = fλ → f = vλ = 340
0.68= 500 Hz
19. B"",[email protected],.* 2.0 cm ! / //กก 1.0 m 4$..+$ก@* "?Cก! $,$/ก/ -".0ก.*1. @#, 100 cm 2. @, 100 cm3. @#, 200 cm 4. @, 200 cm5. @, 400 cm 2
".!.@ -+$ก! / //กก 1 m@$," )4!.3B+#+!.@ ก* $ก@ 1 m
-+!.กก@014 !!ก$/0.! ก!ก* )+$B ?Cก! ) = ?Cก! = 1 m = 100 cm
10.5 cm
44.5 cm2λ
)
S = 1 m'
12
20. .!. R ก!.!ก0$B C .! !ก!#ก!ก* "-+/CCI!"3+*B !#0 ก/CCI+J.!.+ t "01ω
$"5!ก@/CCI+$B AI0sinωt
1. 2.ωCl0sinωt ωCI0sin ωt + π
2
3. 4.l
ωCI0sinωt 1
ωCI0sinωt + π
2
5. 1
ωCI0sinωt − π
2
5Vm = ImXC = I0
1
ωC
$ก!ก C I V I LC # IC VC
π2rad
!! VC = Vmsin(ωt − π2
)
VC = 1
ωCI0sin(ωt − π2
)
21. *"ก!!.!* A $.!.01 100 4$*"ก!!.!* B ? + A ",+*.01 T B ",+*.01 2Tก/$*"ก!!.! A ก! B $,$ก!) 1. 2. 3. 4.(2 log102)T (2 log210)T 4T
0.693(4 log102)T
5. (4 log210)T
5$ก?$@ = = NA NB
t
2Tlog102 log 102
= = N0
2
t2T
100N0
2
tT
t
2Tlog 2 2 log 10
= t = 2tT 100(2
t2T ) 2(2T) log10
log 2
= 100 t = 2t2T (4 log210)T
= log 2t2T log 100
ω
AR
ωI sin t0
~ C
~
AI sin t0 ω
13
22. "-+*+-ก""-+01 24 cm $B #B -ก 0.002 sก0+.$ก$B #B !.+#01",+,+ ! $ก$B B $!.4"-+-ก1. 10 m/s 2. 15 m/s 3. 20 m/s 4. 60 m/s5. 120 m/s 3
$ก#0 3)*$6$ก P→ Q
= tPQ2
3T
4 = T
6
!! = 0.002 sT
6
T = 0.012 s$ก v = λ
T
$/ v = 0.24
0.012= 20 m/s
∴ !.4"-+-ก = 20 m/s
23. 4 ./001"-+ก/CCI +ก )-+5,กU?"4J,ก? 5!ก4"-+1. !ก 2. !กV@3. *C 4. +."5. !./?. 2! x - rays 01"-+ก/CCI""-+!ก ? ! x "ก!J,ก+$ก* กλ
4"-+ก/CCI
P(t = 0s)
30_60_
A2A2
A Q
14
24. ./a? $.$.4?@ ก0+ !)!$ก!n = 3 /0!! n = 1 )!5!ก@/CCI (/)!! ) 4.0+/0 1. )*+4, 12.1 eV 2. )*+4, 24.2 eV3. 1.5 eV 4. 12.1 eV5. 24.2 eV 5
En =
E1
n2= −13.6
n2eV Epn = 2En
n = 3 E3 = − 1.51 eV , Ep3 = 2(−1.51) = − 3.02 eV
n = 2 E2 = − 3.4 eV , Ep2 = 2(−3.4) = − 6.8 eV
n = 1 E1 = − 13.6 eV , Ep1 = 2(−13.6) = − 27.2 eV
∴ -+ก0+ !)!$ก! n = 3 /0!! n = 1)!5!ก@/CCI4.0+/0 = 27.2 − 3.02 = 24.18 eV
≈ 24.2 eV
15
25. -+@!,+$ก!.3B01 x )ก* )$*4 401 3 $3 !.3B- $ก* )* 4 01ก+44 x
2
!.3B1. )$* 4 2. )$* 4 6 3
2
3. )- 4 4. )- 4 3 3
2
5. )- 4 6 4-+@$ก!.3B = x )ก* )$* m = + 3
@+.01@#$ก m = f
S− f
+3 = f
x− f
= f3x − 3f
f = 3x4
-+ !.3B- x2
$/ m = f
S− f =3x4
x2
− 3x4
=3x4
−x4
= − 3
∴ 3 !.3B- $ก* )-4 01 3 44 !.3Bx
2
16