1. Chapter 7:Potential Energy and Energy Conservation
Conservative/Nonconservative Forces
 Work along a path
(Path integral)
 Work around any closed path
Potential Energy
Mechanical Energy Conservation
Energy Conservation

2. Work Done by the Gravitational Force (I)
Near the Earth’s surface l
(Path integral)
Energy Conservation

3. Work Done by the Gravitational Force (II)
Near the Earth’s surface y
(Path integral) dl
Energy Conservation

4. Work Done by the Gravitational Force (III)
Wg < 0 if y2 > y1
Wg > 0 if y2 < y1
The work done by the gravitational
force depends only on the initial and
final positions..
Energy Conservation

5. Work Done by the Gravitational Force (IV)
Wg(ABCA)
= Wg(AB) +
Wg(BC) +
Wg(CA)
= mg(y1 – y2) +
0 +
mg(y2- y1)
= 0 C B dl A
Energy Conservation

6. Energy Conservation

7. Work Done by the Gravitational Force (V)
Wg = 0 for a closed path
The gravitational force is a conservative force.
Energy Conservation

8. Work Done by Ff (I) (Path integral) - μmg L L depends on the path. LB
Path B
Path A LA
Energy Conservation

9. Work Done by Ff (II) The work done by the friction force
depends on the path length.
The friction force:
(a) is a non-conservative force;
(b) decreases mechanical energy of the system.
Wf = 0 (any closed path)
Energy Conservation

10. Example 1 A 1000-kg roller-coaster car moves from
point A, to point B and then to point C.
What is its gravitational potential energy
at B and C
relative to
point A?
Energy Conservation

11. Wg(AC) = Ug(yA) – Ug(yC)
Wg(ABC) = Wg(AB) + Wg(BC)
= mg(yA- yB) + mg(yB - yC)
= mg(yA - yC) y B A dl B C A
Energy Conservation

12. Work-Energy Theorem  Conservation of Mechanical Energy (K+U)
Wconservative (AC) = UA – UC
If Wnet = Wconservative , then
KC – KA = Wnet (AC) = UA – UC
KC + UC = KA + UA
Energy Conservation

13. Energy Conservation

14. Example 1 X
h/2
U K X
Energy Conservation

15. Work Done by FS using Uel(x)
WS = Uel(xi ) – Uel(xf )
where
Uel(x) = (1/2) k x2
Energy Conservation

16. Glossary K: Energy associated with the motion of an object.
U: Energy stored in a system of objects
Can either do work or be converted to K.
Q: Thermal Energy (Internal Energy)
 The energy of atoms and molecules that make up a body.
Energy Conservation

17. Work Done by Fg using Ug(h)
Wg = U(hi ) – U(hf )
where:
Ug(h) = m g h
(near the Earth’s surface) h1 h3
h4 = 0 h2
Energy Conservation

18. Example 2 A roller coaster sliding without friction along
a circular vertical loop (radius R) is to remain
on the track at all times. Find
the minimum release height h. A C B
Energy Conservation

19. Energy Conservation

20. Example 2 (cont’d) A roller coaster sliding without friction along
a circular vertical loop (radius R) is to remain
on the track at all times. Find
the minimum release height h. A v C
(2) mv2/R = mg mg
FN = 0
(1) UA = UC + KC B
Energy Conservation

21. Energy Conservation

22. Example 4 Circular Motion & WT = 0 A C vC = ? B vB = ?
Energy Conservation

23. Energy Conservation

24. Energy Conservation

25. Energy Conservation

26. Work Done by FG using UG(r)
WG = UG(ri ) - UG(rf )
where:
UG(r)= - GmME / r
(dl)r =
(dl)f
Energy Conservation

27. Work Done by FS using Uel(x)
WS = Uel(xi ) – Uel(xf )
where
Uel(x) = (1/2) k x2
Energy Conservation

28. Example 2 (4) W-E Theorem to (1) F.B.D. (2) W by each force
(3) Wnet
(4) W-E Theorem to
find v2 (= 1.93 m/s).
v2 = ? FN FP
v1 = 0
motion
d=5m Ff Fg
mk= 0.100
Wg = Ug2 – Ug1 or
Wg = m g d
(FP)x = FP /cosq
Energy Conservation

29. Example 3
mk = ?
vf = 0 d
Energy Conservation

30. Example 5 How much work must the satellite’s engines perform to move
its satellite (mass
m = 300 kg) from
a circular orbit of
radius rA = 8000 km
about the Earth to
another circular orbit
of radius rC = 3 rA?
vA = ?
vC = ?
Energy Conservation

31. Gravitational Potential Energy
Near the Earth’s surface y dl
Energy Conservation