1. The document discusses different types of potential energy, including gravitational potential energy and elastic potential energy. 2. Gravitational potential energy is related to an object's mass and height above a reference point in a gravitational field. Elastic potential energy is stored in elastic objects, like springs, that are being stretched or compressed. 3. An example problem involves calculating the final velocity of a 200 kg object being pulled by a 690 N force starting from rest over a 2 m distance, accounting for friction. The object's final velocity is calculated to be 2 m/s.

1. 4
3
1.
2.
3.
4.
1
5.

2. ( work )

m F v
q Dr
 
F F
m q m q
v
Dr

F
2

3. 
( work )F
W = F D r cos q
u v
v
W = F gD r
Nm J
3

4.  
F F
v
Dr
W = F D r cos q
 v
F Dr 00
0
W = F D r cos 0 = F D r =
4

5. 2
 
F F

Dr
W = F D r cos q
  0
F Dr 180
0
W = FD r cos180 = - FD r =
5

6. 3
 
F F

Dr
W = F D r cos q
  0
F Dr 90
0
W = F D r cos90 = 0 =
6

7.    
F4 F5 F4 F5
 
F1 F1
 
 F2  F2
F3 F3

Dr
  
F1 WF1 = F 1  r
D
  
F2 WF2 = F 2  r
D
7

8.   
F3 WF3 = F 3  r
D
  
F4 WF4 = F 4  r
D
  
F5 WF5 = F 5  r
D
=
= å W = WF1 + WF2 + WF3 + WF4 + WF5
8

9.  
F F
m q m q
s
s
mk
9

10. N F sin q F
f = mk N m q
F cos q
mg
4 F N mg
f = mk N = mk (mg - F sin q)
WF = FD r cos q = Fs cos q J
W f = f D r cos q = fs cos1800 = - fs
= - mk (mg - F sin q)s J
10

11. 0
WN = ND r cos q = Ns cos90 = 0 J
0
Wmg = mgD r cos q = mgs cos90 = 0 J
= å W = WF + W f + WN + Wmg
= Fs cos q - fs + 0 + 0
= (F cos q - mk mg + mk F sin q)s J
11

12. kg 4 30 0
100 N
10 N 20 m
( g = 10 m / s 2 )
N F = 100 N
f
30 0
0
30 mg
12

13. 0
WF = F D r cos q = (100)(20)cos 0 = 2, 000 J
W f = f D r cos q = (10)(20)cos1800 = - 200 J
WN = N D r cos q = (mg cos 300 )D r cos q
= (40 cos 300 )(20)cos 900 = 0 J
Wmg = mgD r cos q = (40)(20)cos (900 + 300 )
= - 800sin 300 = - 400 J
å W = WF + W f + WN + Wmg
= 2,000 - 200 + 0 - 400
= 1, 400 J
13

14. 
F
a b

a D ri

Fi
b
- i

D ri
14

15.  
- D Wi = F i × r i
D
 
- a b Wa® b = lim å F i × r i
 D
D ri ® 0 i
 
b
Wa® b = ò F ×d r
a
b
 
Wa® b = ò F ×d r
a
1x
15

16. b
Wa® b = ò F cos qdx
a
a b F cos q x
a b
F cos q
- . . x
- . . x
x
a b
16

17. 
F = 3 + 6  - 2k N
i j 

D r = 5 - 2  + 9k m
i j 
 
b
W = ò F ×d r
a
 b    b
W = F ×ò d r = F ×é ùrû
êú
ë a
a
    
 = F × r b - r a )= F × r
( D
F
W = 3 + 6  - 2k × 5 - 2  + 9k
(
i j  i )( j 
)
= - 15 J
17

18. ( power )
Watt
1.
W 2
W
Pav =
t
(D t ® 0)

dW  d r  
P= = F × = F ×v = Fv cos q
dt dt
18

19. F = 50 N
10 kg 60 0 60 0
Dr
10 kg
t= 0 s
(a) F
4
F
(b) 4 19

20. W
(a) Pav =
t
F D r cos q
Pav =
t
Dr
1 2
x - x0 = v0t + at
2
1 2
D r = v0t + at
2
a
å Fx = max
20

21. F cos q = ma
(50)cos 60 0 = (10 )a
a = 2.5 m / s 2
1 2
D r = (0)(4)+ (2.5)(4) = 20 m
2
0
(50)(20)cos 60
Pav = = 125 W
4
21

22. (b)
4
P = Fv cos q
v
v = v0 + at
v = 0 + (2.5)(4)
v = 10 m / s
4
P = (50)(10)cos600 = 250 W
22

23. - ( work-energy th
m a b 3
z m

a å F
b
y
0
x
23

24. 
å F a b
b
 
å Wa® b = ò å F ×d r
a

dv 
b
= ò m ×vdt
dt
a
  
d    dv dv   dv
dt
(v ×v)= v ×dt + dt ×v = 2v ×dt

 dv 1 d   1 d 2
v× =
dt 2 dt
(v ×v) = 2 dt (v )
24

25. b
m d 2
å Wa® b = ò 2 dt
(v )dt
a
b
m
= ò d (v 2 )
2 a
m 2 m 2
= vb - va
2 2
å Wa® b = D Ek
-
1 2
Ek = mv
2 25

26. 200 kg
F = 690 N
mk = 0.25
200 kg
2m
(g = 9.8 m/ s 2
)
N
F
fk
mg 2 m
26

27. å W = D Ek
1 2 1 2
WF + W f + WN + Wmg = mv2 - mv1
2 2
1 2 1 2
2 F - mk N (2)+ 0 + 0 = (200)v2 - (200)(0)
2 2
(2)(690)- (0.25)(200)(9.8)(2) = 100v22
2
400 = 100v2
v2 = 2 m / s
2m 2 m/s
27

28. 2
( potential energy )
1. (
gravitational potential energy )
2. ( elastic
potential energy )
m
z m
 2
1 F

r1 
r2
y
x 0
28

29.  
F = - mgk
W = 2 ®1 21

 
r2
W12 = ò F ×d r

r1

r2
= ò( - mgk × dxi + dy  + dzk

)(  j 
)

r1
z2
= ò (- mg )dz
z1
29

30. z2
W12 = - mg ò dz
z1
= - mg (z2 - z1 )
mg
(2)
( conservative force )
Ep = mgz
Ep ( gravit
30

31. W12 = - (E p 2 - E p1 )= - D E p
z
2
1
h2 z2
z1 h1
0
(z2 - z1 )= (h2 - h1 )
W12 = - mg (z2 - z1 )= - mg (h2 - h1 )
31

32. E p = mgh
h
k m
32

33. (a) m
- kx
(b) m
m x
(c) - kx
x
x
F = - kx
33

34. x1
x2


r2
 
W12 = ò F ×d r

r1
x2 x2
 
= ò( )( )
- kxi × dxi = - ò kxdx
x1 x1
æ1 2 1 2 ÷ ö
= - ç kx1 - kx2 ÷
ç ÷
ç2
è 2 ø
34

35. 1 2
E ps = kx
2
E ps ( elast
W12 = - (E ps 2 - E ps1 ) = - D E ps
x
35

36. å W
å W = å W ¢+ Wmg + Wspring
å W¢ mg - kx
å W = D Ek
å W ¢+ Wmg + Wspring = D Ek
å W ¢- D E p - D E ps = D Ek
36

37. å W ¢= D Ek + D E p + D E ps
å W ¢= 0
0 = D Ek + D E p + D E ps
æ 1 2 1 2ö æ 2 1 2ö
ç mv2 - mv1 ÷+ (mgh2 - mgh1 )+ ç1 kx2 - kx1 ÷
0= ç ÷ ç ÷
ç2
è 2 ÷
ø ç2
è 2 ÷
ø
37

38. 1 2 1 2 1 2 1 2
mv1 + mgh1 + kx1 = mv2 + mgh2 + kx2
2 2 2 2
1 2
E1 = E2
“ mg
38

39. 2 kg 40 m
10 m / s ( g = 10 m / s 2 )
1
40 m
2
Ep = 0
mg
39

40. Ek 1 + E p1 + E ps1 = Ek 2 + E p 2 + E ps 2
1 2 1 2 1 2 1 2
mv1 + mgh1 + kx1 = mv2 + mgh2 + kx2
2 2 2 2
1 2 1 2
(2)(10) + (2)(10)(40)+ 0 = (2)v2 + (2)(10)(0)+ 0
2 2
2 2
v = (10) + (2)(10)(40) = 900
2
v2 = 30 m / s
40

41. A k = 100 N / m
10 kg 30 0
A 2m
m
A
30 0
(a) m
(b) m ( g = 9.8 m / s 2 )
41

42. s
m
2m 1
m
2
A m
0
3
30
(a) (s + 2)
mg
N= 0 J
42

43. Ek1 + E p1 + E ps1 = Ek 3 + E p 3 + E ps 3
1 2 1 2 1 2 1 2
mv1 + mgh1 + kx1 = mv3 + mgh3 + kx3
2 2 2 2
3
(E p = 0)
1 2 0 1 2
(10)(0) + (10)(9.8)(s + 2)sin 30 + (100)(0)
2 2
1 2 1 2
= (10)(0) + (10)(9.8)(0)+ (100)(2)
2 2
(s + 2) = 4.08 m
43

44. (b) (s + 2) = 4.08 m
s = 2.08 m
Ek1 + E p1 + E ps1 = Ek 2 + E p 2 + E ps 2
1 2 1 2 1 2 1 2
mv1 + mgh1 + kx1 = mv2 + mgh2 + kx2
2 2 2 2
2 (E p = 0)
1 2 0 1 2
(10)(0) + (10)(9.8)(2.08)sin 30 + (100)(0)
2 2
1 2 1 2
= (10)v2 + (10)(9.8)(0)+ (100)(0)
2 2
v2 = 4.5 m / s
44

45. A m
R= 3 m
m= 0 s
B mk = 1 4 C
4 kg A
3 m ( g = 10 m / s 2 )
(a) 4 kg B
(b) 4 kg
45

46. (a) A B
mg N
EkA + E pA + E psA = EkB + E pB + E psB
1 2 1 2 1 2 1 2
mvA + mghA + kxA = mvB + mghB + kxB
2 2 2 2
46

47. 1 2 1 2
(4)(0) + (4)(10)(3)+ k (0)
2 2
1 2 1 2
= (4)vB + (4)(10)(0)+ k (0)
2 2
2
vB = 60
vB = 2 15 m / s
(b) A C
å W ¢= D Ek + D Ep + D Eps
æ 1 2 1 2ö
å WA¢ C = ç mvC - mvA ÷+ (mghC - mghA )
ç ÷
÷
® ç2
è 2 ø
æ 2 1 2ö
+ç ç1 kxC - kxA ÷
÷
ç2
è 2 ÷
ø 47

48. å WA¢ C = W f k + WN
®
0
= f k s cos180 + 0
= - mk mgs
1
= - (4)(10)s
4
= - 10 s
æ 1 1 2ö
- 10s = ç (4)(0) - (4)(0) ÷+ ((4)(10)(0)- (4)(10)(3))
2
ç
ç2 ÷
÷
è 2 ø
s = 12 m
48

49. 2.5 m / s
60 kg
0.6 m
60 kg
0.12 m 0.04 m
mk
k = 20 kN / m g = 9.8 m / s 2
49

50. mg
N
å W ¢= D Ek + D Ep + D Eps
å W ¢= W fk + WN
= - mk mg (0.6 + 0.04 )+ 0
= - 376.3mk
æ1 1 2ö
- 376.3mk = ç (60)(0) - (60)(2.5) ÷
2
ç
ç2 ÷
÷
è 2 ø
+ ((60)(9.8)(0)- (60)(9.8)(0))
æ1 ö
ç (20´ 103 )(0.12 + 0.04)2 - 1 (20´ 103 )(0.12)2 ÷
+ç ÷
ç2
è 2 ÷
ø
mk = 0.20 50

51. 1. 3.0 kg
x = 3t - 4t 2 + t 3
x t
4
W = + 528 J
2.
2 kg x
F = 3 + 0.5x N
51

52. x= 0
x= 4 m 4m
W = + 16 J , v = 4 m / s
3.
10 m 6 m 5.5 N
4 m 6 N
(a) 6 m 33 J
(b) 41 J
52

53. 4. 1, 200 kg
36 km / hr ( g = 9.8 m / s 2 )
(a) 5 6 J
3.069´ 10
(b) 1.023´ 104 W
5. 10 N
30
12 km / hr
(a) 29 W (
(b) 10 1.7´ 10 J
4
53

54. 6. 10 kg 600 m / s (g = 9.8 m / s 2 )
(a)
1.84´ 104 m
(b) 8´ 105 J
7.
1.02´ 104 m
m
mg
T = 7mg T = 4mg
54

55. 8. m
A
a r
a
2 g sin a
(a) w=
r
(b)
F = 3mg sin a
55

56. 9. m R
q q
(F = mg (3cos q - 2) , q = cos
- 1
(2 3)
10. 2 kg 0.40 m
k = 1,960 N / m
( g = 9.8 m / s 2 )
0.10 m
56

57. 11.
20 N 16 m 0.6 m
f
f = 553 N
57