Fall 2011 PHYS 172: Modern Mechanics Lecture 9 – The Energy Principle Read 6.1 – 6.7

TODAY The Energy Principle Energy is Conserved Rest Energy and Kinetic Energy Work =

What Is Energy? Energy is just some number you can compute that obeys a conservation law. It’s helpful book-keeping. (See Feynman Lectures on Physics, Vol. 1, Sec. 4-1) Why Introduce Energy? say something about initial and final states of a system predict whether some process CAN occur Without following the details of a complicated process, we can: Basically, for some problems, using energy is much simpler than using the momentum principle. Often, however, we’ll use both principles in combination.

The Energy Principle “effect” “cause” (due to interactions) Compare With Momentum Principle: “effect”“cause” (due to interactions) NOTE: Momentum is a vector. Energy is a scalar (no direction). * We mean the Work done ON a System by a Force in the Surroundings*

Energy is A Conserved Quantity Compare with Conservation of Momentum: Reminder: Momentum is a vector. Energy is a scalar (no direction). Energy is a useful thing to consider because energy can’t be destroyed: it can only change forms.

Energy of a Single Particle System Experimentally, this is the expression that works in a conservation law

Rest Energy A particle at rest has some energy just because it exists! Rest energy = mc 2.

Kinetic Energy Excess energy above rest energy (mc 2 ) is due to motion (ignoring internal degrees of freedom): mc 2

Kinetic Energy What?? What happened to ? At low velocities, Use a Taylor expansion (power series):

Energy And Momentum Use ALWAYS TRUE. EVERY REFERENCE FRAME.

It takes twice as much energy to raise a heavy box a distance 2h, compared to lifting it only h. Thus, it is distance that seems to matter, not time. Energy and Force Does this match your real-world experience?

It takes twice as much energy to raise a heavy box a distance 2h, compared to lifting it only h. Thus, it is distance that seems to matter, not time. In more general vector notation, this is: * We mean the Work done ON a System by a Force in the Surroundings* Energy and Force WORK due to mechanical energy transfer:

Force at an angle y x Only x -component of force “works”! Definition of work Force component along the path of motion

14 Work as a dot product x y Dot product:

Sun does work on comet, speeding it up. Earth does no work on the Moon. Circular orbit Sign of Work

Game: What’s the Sign of the Work? * We mean the Work done ON a System by a Force in the Surroundings*

What Causes The Energy of A System to Change? Reminder: Momentum Principle Interactions: some P surroundings transforms into P system. Forces transfer momentum from one object to another. Energy can be transferred from the surroundings to a system in two ways: Energy transfer by heat (next chapter) Energy transfer by work set = 0 for now

Example You hold a ball of mass 0.5 kg at rest in your hand and throw it forward so that it leaves your hand at speed 20 m/s. How much work did you do on the ball? v f = 20 m/s v i = 0 m/s Rest energy did not change

Change of identity – change of rest energy Neutron decay An electron-volt (eV) unit: 1 eV = 1.6  J 1 MeV = 10 6 eV 1 Volt How much kinetic energy do the products have? =0 Mass of products is smaller than mass of reactant! Mass is converted into kinetic energy >0

WHAT WE DID TODAY The Energy Principle Energy is Conserved Rest Energy and Kinetic Energy Work =