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"",+.@.3,.* 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