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

Same problem, different approach:

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?

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

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

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

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

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

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.

Multiparticle energy principle

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

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

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

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

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

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

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

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.

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.

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

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

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

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:

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!

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?

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

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