work andenergy
TRANSCRIPT
1
4
1.
2.
3.
4.
5.
3
2
( work )
m F
rDv
qm m
F
F
rDv
F
q
q
3
( work ) F
cosW F r
W F r
q= D
= Duv v
g
N m J
4
F
F
rDv
cosW F r q= D
F
rDv
00
0cos0W F r F r= D = D =
5
2
F
F
rD
cosW F r q= D
F
rD
0180
0cos180W F r F r= D = - D =
6
3
F
F
rD
cosW F r q= D
F
rD
090
0cos90 0W F r= D = =
7
1F
2F
3F
4F
5F
1F
2F
3F
4F
5F
rD
1F
11FW F r= D
2F
22FW F r= D
8
3F
33FW F r= D
4F
5F 4
4FW F r= D
55FW F r= D
=
1 2 3 4 5F F F F FW W W W W W= å = + + + +
9
qm
F
m
F
q
s
s
km
10
qm
F
cosF q
sinF qN
mg
kf Nm=
4 F N mg
( )sink kf N mg Fm m q= = -
cos cosFW F r Fs Jq q= D =
( )
0cos cos180
sin
f
k
W f r fs fs
mg F s J
q
m q
= D = = -
= - -
11
0cos cos90 0NW N r Ns Jq= D = =
0cos cos90 0mgW mg r mgs Jq= D = =
( )
cos 0 0
cos sin
F f N mg
k k
W W W W W
Fs fs
F mg F s J
q
q m m q
= å = + + +
= - + +
= - +
12
4 kg 030
100 N
10 N 20 m
( )210 /g m s=
030
030
N 100F N=
mg
f
13
( )( )
( )( )
( )
( )( )
( )( ) ( )
0
0
0
0 0
0 0
0
cos 100 20 cos0 2,000
cos 10 20 cos180 200
cos cos30 cos
40cos30 20 cos90 0
cos 40 20 cos 90 30
800sin30 400
F
f
N
mg
W F r J
W f r J
W N r mg r
J
W mg r
J
q
q
q q
q
= D = =
= D = = -
= D = D
= =
= D = +
= - = -
2,000 200 0 400
1,400
F f N mgW W W W W
J
å = + + +
= - + -
=
14
F
a b
a
b
irD
iF
- i
irD
15
- i iiW F rD = ×D
-0
limi
i ia br i
W F r®D ®
= å ×D
a b
b
a b
a
W F dr® = ×ò
b
a b
a
W F dr® = ×ò
1 x
16
cos
b
a b
a
W F dxq® = ò
a b cosF q
a b
x
cosF q
xa b
- . . x
- . . x
17
3 6 2F i j k N= + -
5 2 9r i j k mD = - +
b
a
W F dr= ×ò
( )
bb
aa
b a
W F dr F r
F r r F r
é ù= × = ×ê úë û
= × - = ×D
ò
F
( ) ( )3 6 2 5 2 9
15
W i j k i j k
J
= + - × - +
= -
18
( power )
2
Watt
W1.
av
WP
t=
( )0tD ®
cosdW dr
P F F v Fvdt dt
q= = × = × =
19
06010 kg
rD
(a) 4
(b) 4
060
F
F
10 kg
0t s=
50F N=
20
(a) av
WP
t=
cosav
F rP
t
qD=
rD2
0 0
2
0
1
2
1
2
x x v t at
r v t at
- = +
D = +
a
x xF maå =
21
( ) ( )0
2
cos
50 cos60 10
2.5 /
F ma
a
a m s
q =
=
=
( )( ) ( )( )21
0 4 2.5 4 202
r mD = + =
( )( ) 050 20 cos60125
4avP W= =
22
(b) 4
cosP Fv q=
v
( )( )
0
0 2.5 4
10 /
v v at
v
v m s
= +
= +
=
4( )( ) 050 10 cos60 250P W= =
23
- ( work-energy theorem )
3m a b
m
Få
a b
0
x
y
z
24
Få
a bb
a b
a
b
a
W F dr
dvm vdt
dt
®å = å ×
= ×
ò
ò
( ) 2d dv dv dv
v v v v vdt dt dt dt
× = × + × = ×
( ) ( )21 1
2 2
dv d dv v v v
dt dt dt× = × =
25
( )
( )
2
2
2 2
2
2
2 2
b
a b
a
b
a
b a
m dW v dt
dt
md v
m mv v
®å =
=
= -
ò
ò
a b kW E®å = D
-21
2kE mv=
26
690F N=200 kg
0.25km =
200 kg
2 m
( )29.8 /g m s=
FN
mg
kf
2 m
27
( ) ( ) ( )( )
( )( ) ( )( )( )( )
2 2
2 1
22
2
2
2
2
2
2
1 1
2 2
1 12 2 0 0 200 200 0
2 2
2 690 0.25 200 9.8 2 100
400 100
2 /
k
F f N mg
k
W E
W W W W mv mv
F N v
v
v
v m s
m
å = D
+ + + = -
- + + = -
- =
=
=
2 m 2 /m s
28
( potential energy )2
1. (
gravitational potential energy )
2. ( elastic potential energy )
mm
F
1r
2r
1
2
0xy
z
29
F mgk= -
121 2 W® =
( ) ( )
( )
2
1
2
1
2
1
12
r
r
r
r
z
z
W F dr
mgk dxi dy j dzk
mg dz
= ×
= - × + +
= -
ò
ò
ò
30
( )
2
1
12
2 1
z
z
W mg dz
mg z z
= -
= - -
ò
(2)
( conservative force )
mg
pE mgz=
( gravitational potential energy ) pE
31
( )12 2 1p p pW E E E= - - = - D
1z2z
1h2h
1
2
0
z
( ) ( )
( ) ( )
2 1 2 1
12 2 1 2 1
z z h h
W mg z z mg h h
- = -
= - - = - -
32
pE mgh=
h
k m
33
m
m
m
kx-
kx-
x
x
( )a
( )b
( )c
x
F kx= -
34
1x
2x
( ) ( )
2
1
2 2
1 1
12
2 2
1 2
1 1
2 2
r
r
x x
x x
W F dr
kxi dxi kxdx
kx kx
= ×
= - × = -
æ ö÷ç= - - ÷ç ÷÷çè ø
ò
ò ò
35
21
2psE kx=
( elastic potential energy )psE
( )12 2 1ps ps psW E E E= - - = - D
x
36
Wå
mg springW W W W¢å = å + +
W¢å mg kx-
k
mg spring k
p ps k
W E
W W W E
W E E E
å = D
¢å + + = D
¢å - D - D = D
37
k p psW E E E¢å = D + D + D
0W¢å =
( )2 2 2 2
2 1 2 1 2 1
0
1 1 1 10
2 2 2 2
k p psE E E
mv mv mgh mgh kx kx
= D + D + D
æ ö æ ö÷ ÷ç ç= - + - + -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø
38
2 2 2 2
1 1 1 2 2 2
1 1 1 1
2 2 2 2mv mgh kx mv mgh kx+ + = + +
1 2
1 2E E=
“
“
mg
39
2 kg 40 m
10 /m s ( )210 /g m s=
40 m
0pE =
1
2
mg
40
( )( ) ( )( )( ) ( ) ( )( )( )
( ) ( )( )( )
1 1 1 2 2 2
2 2 2 2
1 1 1 2 2 2
2 2
2
22
2
2
1 1 1 1
2 2 2 2
1 12 10 2 10 40 0 2 2 10 0 0
2 2
10 2 10 40 900
30 /
k p ps k p psE E E E E E
mv mgh kx mv mgh kx
v
v
v m s
+ + = + +
+ + = + +
+ + = + +
= + =
=
41
A
A
100 /k N m=
10 kg 030
2 m
030
A
m
(a)
(b)
m
m ( )29.8 /g m s=
42
030
A
m
m
m
s
2 m 1
2
3
(a) ( )2s+
mg
0N J=
43
1 1 1 3 3 3
2 2 2 2
1 1 1 3 3 3
1 1 1 1
2 2 2 2
k p ps k p psE E E E E E
mv mgh kx mv mgh kx
+ + = + +
+ + = + +
3 ( )0pE =
( )( ) ( )( )( ) ( )( )
( )( ) ( )( )( ) ( )( )
( )
2 20
2 2
1 110 0 10 9.8 2 sin 30 100 0
2 2
1 110 0 10 9.8 0 100 2
2 2
2 4.08
s
s m
+ + +
= + +
+ =
44
(b) ( )2 4.08
2.08
s m
s m
+ =
=
1 1 1 2 2 2
2 2 2 2
1 1 1 2 2 2
1 1 1 1
2 2 2 2
k p ps k p psE E E E E E
mv mgh kx mv mgh kx
+ + = + +
+ + = + +
2 ( )0pE =
( )( ) ( )( )( ) ( )( )
( ) ( )( )( ) ( )( )
2 20
22
2
2
1 110 0 10 9.8 2.08 sin 30 100 0
2 2
1 110 10 9.8 0 100 0
2 2
4.5 /
v
v m s
+ +
= + +
=
45
mA
B
0m=
1 4km = C
3R m=
s
A
(a) B
(b)
3 m
4 kg
4 kg
( )210 /g m s=
4 kg
46
(a) A B
mg N
2 2 2 21 1 1 1
2 2 2 2
kA pA psA kB pB psB
A A A B B B
E E E E E E
mv mgh kx mv mgh kx
+ + = + +
+ + = + +
47
( )( ) ( )( )( ) ( )
( ) ( )( )( ) ( )
2 2
22
2
1 14 0 4 10 3 0
2 2
1 14 4 10 0 0
2 2
60
2 15 /
B
B
B
k
v k
v
v m s
+ +
= + +
=
=
(b) A C k p psW E E E¢å = D + D + D
( )2 2
2 2
1 1
2 2
1 1
2 2
A C C A C A
C A
W mv mv mgh mgh
kx kx
®
æ ö÷ç¢å = - + -÷ç ÷÷çè ø
æ ö÷ç+ - ÷ç ÷÷çè ø
48
( )( )
0cos180 0
14 10
4
10
kA C f N
k
k
W W W
f s
mgs
s
s
m
®¢å = +
= +
= -
= -
= -
( )( ) ( )( ) ( )( )( ) ( )( )( )( )2 21 1
10 4 0 4 0 4 10 0 4 10 32 2
12
s
s m
æ ö÷ç- = - + -÷ç ÷÷çè ø
=
49
0.6 m
60 kg
2.5 /m s
60 kg
0.12 m 0.04 m
km
20 /k kN m=29.8 /g m s=
50
mg
N
k p psW E E E¢å = D + D + D
( )0.6 0.04 0
376.3
kf N
k
k
W W W
mgm
m
¢å = +
= - + +
= -
( )( ) ( )( )
( )( )( ) ( )( )( )( )
( )( ) ( )( )
2 2
2 23 3
1 1376.3 60 0 60 2.5
2 2
60 9.8 0 60 9.8 0
1 120 10 0.12 0.04 20 10 0.12
2 2
kmæ ö
÷ç- = - ÷ç ÷÷çè ø
+ -
æ ö÷ç+ ´ + - ´ ÷ç ÷÷çè ø
0.20km =
51
1. 3.0 kg
2 33 4x t t t= - +
4
x t
528W J= +2.
2 kg x
3 0.5F x N= +
52
0x =
4x m= 4 m
16 , 4 /W J v m s= + =
3.
(a)
(b)
10 m 6 m 5.5 N
4 m 6 N
6 m 33 J
41 J
53
4.
(a) 5
(b)
1,200 kg
36 /km hr
63.069 10 J´
41.023 10 W´
5.
30
(a) ( )
(b) 10
10 N
12 /km hr
( )29.8 /g m s=
29 W
41.7 10 J´
54
6.
(a)
(b)
10 kg 600 /m s
41.84 10 m´
58 10 J´
41.02 10 m´
7.
m
mg
T 7mg= T 4mg=
( )29.8 /g m s=
55
8. m
a rA
(a)
(b)
a
2 sing
r
aw =
3 sinF mg a=
56
9.
( )
Rm
q q
( ) ( )13cos 2 , cos 2 3F mg q q -= - =
10. 2 kg 0.40 m
1,960 /k N m=
( )29.8 /g m s=
0.10 m
57
11. 20 N 16 m 0.6 m
f
553f N=