大学物理（习题与答案）：作业答案3-1~~3-19

vDt ? ¨ i? o?? p  ?? ? ? 3-1é  m=10kg￥t8 ? F x =30+4t N￥T¨/
xà?
 kp
1
 7 S 2s=N ? ￥  I

2
?   I=300N·s
N ?￥T ¨ HW ^ 
$
3
?t 8￥ ? v 1 =10m/s
 t=6.86s H
Nt8￥ ? v 2 1  
$ 3
 (1) sN68d)430(d 2 0 2 0 ?=+== ∫∫ tttFI xx (2)
300230d)430(d 2 00 =+=+== ∫∫ tttttFI tt xt s86.6=t (3)

 1212 mvmvppI ?=?= s86.6=t sN300 ?=I
 m/s20)1010300( 10 1 )( 1 12 =×?=?= mvI m v 3-2é  m=1kg￥t8 
xà?
 ? s￥ ??m 3-2 ? Ub t=0 H
 é?á?USe?
 k¨d? p ?? ? ?sYp3 t=7s HNé?￥ ?b m 3-2 10 O 75 F/N t/s 3
 (1) ? ? ? ≤≤+? ≤≤ = 75355 502 tt tt F 50 ≤≤t
 t t v m 2 d d =

 ∫∫ = 5 00 d2d 1 ttvm v (m/s)25 25 1 == m v
75 ≤≤t 355 d d +?= t t v m

 ∫∫ +?= 7 5 d)355(d 2 1 ttvm v v (m/s)35 2 =v (2) s)(N35)107( 2 1 d 7 0 ?=×== ∫ tFI
 212 mvmvmvI =?=
 (m/s)35 2 =v ?  o?? p 3-3 oé sY1 m 1 =3.0g
 m 2 =5.0g
;á ￥ ￡ü ? ?
¨°?US xOy í ??
 ? ? sY1
cm/s8 1 iv v v = cm/s)168( 2 jiv vv v +=
 ?·a o?1B8
 5·a o ? v v ￥vl 1  
$D xà￥C?1 
$ 3
 "d?  o? jivmvmvmm vv vvv 8064)( 221121 +=+=+
 jiv vv v 108 += cm/s8.12108 22 =+== vv v
 D xàC? °== 3.51 8 10 arctanα 3-4 ?m 3-4 ? U
é 1 M ￥ 1/4 ??á￠T;á￥ ￡ü ? 
B?é 1 m ￥lt81????? á?/ábp?lt8á?? H
??á￠ ￡ü ? M?￥  ?b m M m 3.4 R 3
 "d ￡üZ_?  o? 0)( =?+ VMmv
 MVmv = H??/ ?V?s ∫∫ = tt tVMtvm 00 dd 7 s? sY1 ?S m M  ￡üZ_￥M?  ?
5
∫ = t tvs 0 d

 b? ∫ = t tVS 0 d MSms = SRs ?=
 ?[ R Mm m S + = 63
 mM? M  ￡üZ_￥ ? v M Mm Vvv + =+=′ b??/ ?V?s ∫∫ + =′ tt tv M Mm tv 00 dd
 s M Mm R + =
 M  ￡üZ_￥M?  ? R Mm m sRS + =?= 8 vDt ? ¨ i? o?? p é? é??? p 3-5p??1 R￥???é e￥é?
?m 3-3 ? U
 b O y R m 3-5 x 3
 ! e é 1
 ? á1m 2 2 R m π σ = b?é s????
 é? xà b   ?1o x￥1Z|B z1 %éH
?￥é  xd xxRm d2d 22 ?= σ
?é??l π σ 3 4 d 2 d 0 220 R xxRx mm mx x R R c =?== ∫ ∫ 3-6 B?é 1 L
é  (￥è 7
+B???l￥;á?0 
?m 3-6 ? Ub 7 S H
 BC=b
 k ¨ é?￥ZE￡ ü? BC=2L/3 H
 7￥F ?1 a=g/3
 ? q1 ) 9 2 ( 2 22 bbLL L g v ?+?= b C B b m 3-6 3
 ?è 7?Z_￥ s ??d? p LayLyg λλ =?? )]([
 g L Ly a ? = 2
 ga Ly 3 1 3 2 = = y v v t y y v t v g L Ly a d d d d d d d d2 === ? =
 ∫∫ ?= L b v yLy L g vv 3 2 0 d)2(d ? ? ? ? ? ? ?+?= 22 9 22 bbLL L g v 63 (¨é? ) ? H
 y"￥é?1 bBC = L bLbL m b b bL bL y c 2 22 22 )( 22 +? = + ? ? = λλ ? LBC 3 2 = H
 y"￥é?1 Ly c 18 5 =′ ?× ?￥???t8??￥9  2 2 1 )( mvyymg cc =?′

)(2 2 cc yygv ?′= ? ? ? ? ? ? ?+?= 22 9 22 bbLL L g v ?? 
?  
# o?? p 3-7 X?é  1 m ￥|/????1 r ￥?E? ?
?? vl1 L
p ?￥??a ]??9? b
? ? ]? r mm GE p 21 ?=
 G1￡μ? ?è ?
 3

rmvL = mr L v =
 2 2 2 22 1 mr L mvE k == !1oé 
 e M r mM GE e p ?=
?d? p r v m r mM G e 2 2 =
 2 mv r mM G e =
 2 2 mr L E e p ?= 9 vDt ? ¨ i? o?? p 2 2 2 2 2 2 22 mr L mr L mr L EEE pk ?=?=+=∴ 3-8é 1 m￥é? xOyü ?=?
 ê? O 1 jtbitar vv v ωω sincos +=
 ? ωaaba 1è 
 p
1
é?? ￥vl

2
é?M?e?￥?? b 3
 (1) jtbita t r v vv v v ωωωω cossin d d +?== )cossin( jtbitamvmp vv vv ωωω +?==
 tbtampp ωωω cossin 222 +== v (2) kabmjtbitamjtbitaprL v vvvv vv v ωωωωωω =+?×+=×= )cossin()sincos( 3-9é  (1 m￥ ? lo a? b%?é1 l￥D?^é%?￥ 
? V ￡ü ? ? O?à1? ?
 ?e ?á?b CμB?é 91 m￥lo c
< °??[ ￡ü ? o v v D bo·
?m 3-9 ? U

 iTBbp
1
·- coM? O￥?? ￥vl?Z_

2
·a??? ?b 3
 (1) 0 vmrL vv v ×= Z_<°? ?_/b 00 4 3 lmvrmvL == m m 0 v v O a c b m l/4 l m 3.9 (2) "d o?￥??  o?b !·a?￥? ?1 ω
5 ) 4 1 ( 4 1 ) 4 3 ()2( 4 3 4 3 0 ωω lmllmllmv ××+××=
 l v 19 12 0 =ω ????? ? 3-10B| V 10m '￥Y? 4 ￡
 X? ￡aD ￡× 10kg
p
1
 ? 4 H
 | ?T￥?

2
[ a=0.1m/s 2 F ? 4 H
 | ?T￥?

3
 ? ￡a ? 4V??
 ￡[ 0.2kg/m￥ ? q P ￡
 5| ?T￥?1  
$ 3
 (1)
0=? mgF mgF =
 J980dd 10 0 10 0 === ∫∫ ymgyFA (2)

 mamgF =? )( agmF += J990d)(d 10 0 10 0 =+== ∫∫ yagmyFA (3) 0)2.0( =?? gymF

gymF )2.0( ?= 882d)2.0(d 10 0 10 0 =?== ∫∫ yymgyFA 3-11é  m=6kg￥t8
  ? F x =3+4x N￥T¨/
1á? 7 S
xà?  3m
?? 9 ?
p
F x ?T￥?

2
N Ht8￥ ?

3
N Ht8￥F ?b 3
 (1) J27d)43(d 3 0 3 0 =+== ∫∫ xxxFA x (2) ???? ? 2 2 2 1 3 2 2 1 2 1 2 1 mvmvmvA =+=
 m/s3 2 2 == m A v (3) ?d? p 2 m/s5.2 6 343 = ×+ == m F a x x
3-12é 1 m￥t81á??
xà?
 ! ? s? ?1 F x =bt
 b1è 
p T s=N ? ?T￥?b 3
 ?d? p t v mbtF d d ==

 ∫∫ = vt vmtbt 00 dd m bt v 2 2 =
 Tt = H
 m bT v 2 2 =
???? ?
m Tb mvmvmvA 82 1 2 1 2 1 42 22 0 2 ==?=
10 J 1
 ? vDt ? ¨ i? o?? p 63
 t m bt tvx d 2 dd 2 ==
 m Tb t m bt btxFA T x 8 d 2 d 42 0 2 === ∫∫
 o ?￥?? ]? 3-13é  1 m￥lo"é1 l￥^ 7B
 7￥ 6B %?
 ülo ?à ￡üê?
 Vá? db
 ?m 3-13 ? U
 ?lo/? θ ? H

1
 7?f ? T v loS? ?
$?? ? gmTF v vv += lo ?S￥? 1 
$
2
 NV??
 lo ]?￥9 1 
$iD
1
 ￥2T 1?

3
 ?¨??? ?pl o/? θ ? H￥ ? qb 3
 (1) rT v v d⊥
 0d =?= ∫ rTA T v v
f ? T v lo?S?b O T v gm v m 3-13 ∫ ∫∫∫ =?= +??=?=?+= 2 1 sind )d(ddd)( y y F mglymg jyixjmgrgmrgmTA θ vvv vvvv v
(2) θsin)( 12 mglyymgE p ?=?=?
 Vn× ?￥???lo ]?9 ￥μ′b
(3) ???? ? 2 2 1 sin mvmgl =θ
 θsin2glv =
3-14é  1 m ￥é?
x à?Z_ ?
 ? s? ? ?￥T¨
 B? ? ^·_e?a vl1 B ￥è ?
 6B? ?
x à?Z_ avl1 A/x 2
 AaB1è ?b
1
 k ??é?￥ü?ê?

2
 p?é?V ü?ê? ???iê ? x ) H  ?òS￥?
i?  ? ^?1 o ?

3
 [ü?ê? 1 ] ? ,?
p?iê?)é?￥ ]?b 3
 (1) B x A F ?= 2
 0=F H
 B A x = 0
(2) ) 11 (dd 0 2 11 00 xx Ax x A xFA x x x x ?=== ∫∫

)(dd 022 00 xxBxBxFA x x x x ?=?== ∫∫


a oD Sê?μ1
'  ? (1 o ?b
1 A 2 A (3) ABBx x A xxB xx AxB x A xFE x x x x p 2)() 11 (d)(d 0 0 2 00 ?+=?+?=?== ∫∫
??e ???? o? 3-15 ?m 3-15 ? U
B é 1 m’ ￥t vb?| ?￥K? A )
| ?￥`?1 α
ú 1 h
t vD| ?￥? ?y ?1 μ
?μBé 1 m ￥ 0?[ ? 0 v v 
￡üZ_ ?t vi =?
 O Pt v
| ?_ á?
pt vá? H￥ ?vlb 0 v v α h 3
 [t v? 0?1ù?`
 ·-a"d
ü?| ?Z_?  o?
0? ?t va￥ ?vl1
5
1 v 10 )(cos vmmmv ′+=α
 mm mv v ′+ = αcos 0 1
A m 3.15 || ???1 ]? ,?
t vá? H￥ ?vl1
???? ?
2 v ghmmvmmvmm h gmm )()( 2 1 )( 2 1 sin cos)( 2 2 2 1 ′+?′+?′+=′+ α αμ
11 vDt ? ¨ i? o?? p )1cot(2 cos 2 0 2 +?? ? ? ? ? ? ′+ =∴ αμ α gh mm mv v
3-16 ?" ?1 k ￥^ é?T
 B %?: 
 6B "Bé 1 m A ￥t8 A
 b;á ￡ü ? 
 ?ü?Ta ê xba
  L" A bBé 1 m B ￥t8 B
? m 3-16 ? Ub 7 S H
??? ?￥ T¨"d)?á? ?
?ü?? ?
 kp A D B ? 7 H B?￥ ?? A??r￥Kv  ?b 3
 (1)
?T?reé H 7 Sh ?
 aA A Bs ?b
x O x 0 B A !N H ?vl1 v
??? o?
22 0 )( 2 1 2 1 vmmkx BA +=
 BA mm K xv + = 0
m 3-16 (2) aA Bs ?a
 ??_·M??Kv  ? )
5
A m x 22 2 1 2 1 mA kxvm =
 BA AA m mm m x k m vx + == 0
3-17 ?m 3-17 ? U
? ó4? ?μB??1 R ￥?o?ì ?
μBb vV;áì ?￥Kú??á?
ì ?á/
? ? ? {??9bpNb v ? 7ì ?￥ê?[#?ê?￥ ?b θ R 3
 ??? o?
2 2 1 )sin1( mvmgR =? θ

)sin1(2 2 θ?= gRv b v ? 7ì ? H
?d? p
R v mmg 2 sin =θ
 3 2 sin =∴ θ
 °== 8.41 3 2 arcsinθ
gRgRv 3 2 )sin1(2 =?= θ
m 3.17 · 3-18Bé 1 m 0 [ ? q v 0 ?￥ ì0
 ·?Bé 1 2 m 0 ￥á? ì0b 2T
 é 1 m 0 ￥ ì0ê  45 0
 i μ ? v 0 /2bpé 1 2 m 0 ￥ ì0êa￥ ? q?Z_b 3

? 0 2m 0 v 2 α ·-a?  o?
? ? ? ? ? ? ? =° +°= α α sin245sin 2 cos245cos 2 0 0 0 0 0 000 vm v m vm v mvm
x y ? 0 m 0 v v 0 0 368.0225 4 v v v =?=
 °= ° = 7.28 4 45sin arcsin 0 v v α
3-19m 3-19 ? U
 Bé 1 m￥lo ADBé 1 M￥| ?8 B? 3 ????·b
1
? | ?8b? ;á￥ ￡ ü ? 
l o·a ? °?
5 ·a| ? 8?lo￥ ? ?v lò1 
$
2
 ?| ?8%? ￡ü ? 
 ·alo?￥ ?vl1 
$?Z_D ￡üZ_￥C?1 
$ 3
 (1) [lo?| ?1ù?`
 ￡üZ_?  o?b
A B θ v v !·alo?| ? ?vl1 v′aV
5
m 3-19 12 vDt ? ¨ i? o?? p MVmv =
 v M m V = b
?? ?  o?? ?
222 2 1 2 1 2 1 MVvmmv +′=
 M mM vv ? =′
(2) ? ? ? o? ? vv =′ blo D| ? · H
| ? lo￥ T¨ ?< °? | ?Z _
 ·- a ü??| ?Z_?  o? θθ ′′= coscos vmmv
 θθ =′
?[·alo?Z_D ￡üZ_￥C?1 θ2 b 13