1. System: ball
Surroundings: Earth
Let distance fallen = h.
Then Wsurr = Fg x h
= mgh
Energy principle:
ΔK = Wsurr
Kf – Ki = Wsurr
(1/2)mv2 = mgh

2. Same problem, different approach:

3. System: ball + Earth
Surroundings: none
This time, there is no external force acting on the system, so Wsurr = 0.
Energy principle:
ΔK = Wsurr = 0
This can’t be right!
What did we leave out?

4. Systems with more than one particle have another type of energy, that comes from internal interactions:
Potential energy 势能

5. System: ball
Surroundings: Earth
One particle system:
Work is done by the surroundings (the Earth) on the ball.
The energy principle says:
ΔK = Wby Earth h
(1/2)mv2 = mgh

6. System: ball + Earth
Surroundings: none
Two particle system:
There is no work done by the surroundings, but work is done inside the system.
We can account for it by moving the W to the other side of the equation:
ΔK – Wby Earth = 0 h
(1/2)mv2 – mgh = 0

7. ΔK + (– Wby Earth) = 0 System: ball + Earth Surroundings: none h
We define this to be the change in potential energy,
ΔU.

8. ΔK + ΔU = 0 System: ball + Earth Surroundings: none h We find that
ΔU = mgΔy
for (object + Earth) near the surface of the Earth.

9. Definition of potential energy
The change in potential energy of a system is the negative of the work done by forces internal to the system.

10. Multiparticle energy principle

11. Multiparticle energy principle
Net change of mechanical energy
(机械能) inside system
Work done on the system by external forces

12. Multiparticle energy principle
Work
SYSTEM
Net change of energy inside the system

13. A ball of mass 0.1 kg is dropped from rest near the surface of the Earth.
It travels downward 2 m, speeding up.
System: Ball
What is the work done by the surroundings?
0 J
+1.96 J
-1.96 J
2 m

14. A ball of mass 0.1 kg is dropped from rest near the surface of the Earth.
It travels downward 2 m, speeding up.
System: Ball + Earth
What is the work done by the surroundings?
0 J
+1.96 J
-1.96 J
2 m

15. A ball of mass 0.1 kg is dropped from rest near the surface of the Earth.
It travels downward 2 m, speeding up.
System: Ball + Earth
Work done by surroundings = 0.
However, did the kinetic energy of the system change?
K increased (2) K decreased (3) no change
2 m

16. Gravitational potential energy
for (object + Earth) system, close to the Earth’s surface y

17. Gravitational potential energy
where r is the distance between the two objects
Binary star system

18. Gravitational potential energy
U = 0 at r→∞
Even though U is negative (负), the change in U can be either positive (正) or negative.
Only changes in energy are important, not the absolute value.

19. Remember this?
What is the least speed for a projectile launched vertically upward from the Earth’s surface, such that it will continue into space and never return to Earth?
Assume that gravity is the only force felt after launch.

20. System: rocket + Earth
Surroundings: none
Initial state: Rocket at Earth’s surface, with initial speed vi.
Final state: Rocket at infinite distance from the Earth, with zero speed. r

21. Energy principle:
There is no work done on the system (W = 0), so r

22. The mechanical energy of the system is conserved.
When:
no work is done on a system, and
no friction is present, then:
The mechanical energy of the system is conserved.
机械能守恒

23. Potential energy of a spring (弹簧)
The force generated by a spring, stretched a distance s, is:
where k is a constant.
The potential energy of a stretched spring is:

24. Potential energy of a spring (弹簧)
This is only true when s is not very large.
Of course, when s approaches infinity (∞), the spring will break!

25. Example: Bungee jumping
Dave (m = 75 kg) jumps off a bridge with a bungee cord (k = 50 N/m) tied to his feet. He falls for 15 meters before the cord begins to stretch.
How far will he fall before he stops?

26. Cord starts stretching
System: Dave + cord + Earth
Surroundings: none
Example: y h s
15 m
y = 0
Initial
Initial state: Dave on top of bridge, at rest
Final state: Cord stretched, Dave hanging under the bridge, at rest
Cord starts stretching
Final

27. Cord starts stretching
Energy principle:
No work done on system, so
Example: y h s
15 m
Initial
Cord starts stretching
y = 0
Final