Chapter 6 Work and Energy

0) Conservation laws Newton’s laws (with kinematic equations) allow a complete and continuous description of motion, but requires –knowledge of the force at all times –detailed calculations which may be prohibitive Conservation laws allow intermediate details to be ignored by realizing some quantities are conserved Conservation of momentum, which follows directly from Newton’s laws and will be considered later, was discovered first (Descartes). Momentum (mv) was considered an essential quality of motion.

0) Conservation laws The idea that we should “pay” for producing motion is absent in Newton’s laws and momentum. Huygens never cared for the idea that momentum was an essential quality of motion: A bomb could explode and yet the total momentum remains zero. What kind of essential quality of motion remains zero when bodies are being hurtled through the air? Huygens found the quantity mv 2 was conserved in billiard ball collisions In 1807 Thomas Young associated the term energy with mv 2

1) Energy It is important to realize that in physics today, we have no knowledge of what energy is. Richard Feynman As we cannot give a general definition of energy, the principle of the conservation of energy simply signifies that there is something which remains constant. Well, whatever new notions of the world future experiments may give us, we know beforehand that there will be something which remains constant and which we will be able to call energy. Henri Poincaré ( )

1) Energy Energy is some quantity, which takes many forms, associated with a system which we know intuitively is conserved. Or else we could make a perpetual motion machine. Feynman writes that the conservation of energy is the statement that perpetual motion is not possible. State of a system in relation to fundamental forces Force is the agent of change; energy is measure of change

2) Work a) Work along a straight line: F || s Work is the change in the energy of a system resulting from the application of a force acting over a distance. F m s Units: 1 N m = 1 joule = 1 J Work done by force F, is If the displacement s is zero, no work is done.

2) Work (b) Force and displacement not parallel F m s  The change in the system is the same as if a parallel force F s = F cos  acted over a distance s, so the work done by F is

2) Work (c) Negative work If a force acts opposite to the direction of motion (like friction), it does negative work. It takes energy out of the system. f m s v v = 0 

3) Kinetic Energy Work produces a change in energy. Work on an otherwise free object produces motion. The energy associated with motion is called kinetic energy, KE. Work - energy theorem: If as a result of work on an object (system) the only change is its resulting motion:

3) Kinetic Energy F m s vfvf vivi Using gives for the work done by F: Define Kinetic Energy: so that

3) Kinetic energy Work-energy theorem qualifications: –Force is the net force on the object (system) –No internal changes to object (system) e. g. Lifting a rock produces no kinetic energy because there is no net force on the rock (on average). e.g. Compressing a spring produces no kinetic energy because there are internal changes to the system

4) Conservative forces Work done by a conservative force is independent of path Work done by a conservative force in a closed loop is zero –e.g. gravity, spring, electromagnetic all fundamental forces any force determined by the configuration of the system m mg m m e.g. Work done by gravity m mg

5) Potential Energy Work produces a change in energy Work against a conservative force (lifting a rock, compressing a spring), can be converted to motion (kinetic energy) by releasing the system; the system has a potential to do work. Work on a system of interacting particles can produce a change in the configuration (compressed spring). Energy associated with the configuration of a system is called potential energy, PE. If, as a result of work (W) on a system (against conservative forces), the only change is its configuration:

5) Potential Energy If, as a result of work (W) on a system (against conservative forces), the only change is its configuration: Work done by conservative forces in a system produces a negative change in the potential energy.

6) Gravitational potential energy (near earth) Consider an object lifted by an external force to a height h. If it is at rest before and after, then the only change is the configuration of the earth-object system F = mg m h The change in the potential energy is the work done: The position of zero PE is arbitrary; only changes are interesting. Define PE = 0 for h = 0. Then, (Work done by gravity is -mgh)

7) Conservation of Mechanical Energy Mechanical energy: E = PE + KE If all forces in an isolated system are conservative, mechanical energy is constant Work-energy: W =  KE If all work is done by conservative forces: W = -  PE Therefore, -  PE =  KE, or