Energy

Learning Goals: Know where Mechanical Energy fits with respect to the Total Energy that includes other forms of energy including Heat, Internal Energy, Chemical, etc.

Be able to distinguish conservative from non-conservative interactions and forces.

Be able to calculate the work due to non-conservative forces, and the potential energy for conservative interactions.

Introduction to Energy

Total Energy

The world is awaking to the "energy crisis".  Most people realize that there won't be enough energy to meet the growing needs of expanding civilization without great financial or environmental costs.  An intelligent businessman recently sstated "...the hydrogen economy is the best solution because hydrogen can be made from water, which is plentiful".  He didn't understand the problem: hydrogen isn't scarce or expensive - the key commodity is the energy necessary to separate it from oxygen. Energy is a conserved physical quantity - it cannot be created from nothing.  Energy only can be transformed from  light to chemical energy to mechanical energy to gravitational energy, and back again.  

Energy can also be converted into heat, but heat can't be converted completely back into the other forms of energy - understanding this lies at the heart of Thermodynamics.  Because people did not understand that heat was a form of energy until well into the 19th century (they thought it was a sort of fluid called Phlogisten), they didn't understand the conservation of energy, and didn't know how to tell if a particular mechanical system would conserve mechanical energy (the sum of all kinetic energy and all potential energy due to springs, gravity, etc.).  Momentum was understood in Principia, published in 1684.  Energy was not understood until ~150 years later.   Now many courses ask you to learn these two subjects in subsequent weeks!

Types of Energy

The total energy of a system is simply the sum of the various kinds of energy it contains:
E^tot = K^inetic + U^potential + U^internal.+U^chemical + U^nuclear....
The systems we consider in this text consist of mechanical bodies each of which has kinetic energy due to both translation of its center of mass of and rotation about its center of mass.  Potential energy of each body will be due to gravity and material deformation (as in springs or flexible levers).  

Importantly, the energy of a system is a function of its state variables only.  It does depend on the prior history of the system.  Examples of state variables that determine various energies are shown here.  The first four are very important in this WIKItext:

State Variable    Energy

velocity    translational kinetic energy
angular velocity    rotational kinetic energy
position     gravitational potential energy
size (lenegth)    spring's potential energy

These contribute to Total Energy only:
temperature    internal energy    
chemicals    chemical energy
nuclear isotopes    nuclear energy

You may wonder why we haven't listed the mass as contributing to the total energy via E=mc^2.  That's because we're working in the non-relativistic limit in this WIKItext, and take the mass to be constant and unchanging.  In actuality, the various forms of energy listed above do contribute slightly to the mass - for example, the mass of two hydrogen atoms combining to form one hydrogen molecule is more than a part per billion more than the resulting molecule.  All other conventional chemical reactions yield less that a part per billion of their rest mass as energy.

Change of Total Energy: Model

For an isolated system such as we just discussed, the total energy is constant (conserved) - it can only change form.  For example, friction may reduce mechanical energy and increase internal (thermal) energy.  A chemical reaction might heat the entire system, increasing its internal energy.  An explosion might convert chemical energy into kinetic energy of the fragments.  If the system contains a heat engine, some thermal energy could be turned back into mechanical energy, etc..

To change the total energy of a system, one must pass energy through the boundaries of the system.  This can be accomplished in several ways, even ignoring adding mass or electromagnetic radiation.  The total energy of a system can be increased by heating the system or by doing mechanical work on the system.  These possibilities are encompassed by the First Law of Thermodynamics, which is really the law of conservation of energy for a system that is interacting with the environment.  We now present the vocabulary and the model for "Total Energy", even though it won't be used in Introductory Mechanics unless we consider statistical mechanics and thermodynamics of ideal gasses.

Model: First Law of Thermodynamics

Our usual model stuff
E_f = E+i + Q_

 
This equation is called the First Law of Thermodynamics.  Note that the subscript on the energy before and after is are single letters - this stresses that energy is a property of the system that depends only on it's state at a given time, taken to be the final and initial conditions of the system.  Heat and Work, on the other hand, are processes and therefore depend on the details of the path taken to the final state from the initial state. 

SAQs.  If a system is taken from an initial set of conditions to a final set of conditions,
1.    The amount of heat added is independent of the path
2.    The amount of work needed is independent of the path
3.    The sum of the heat energy added and the amount of work added is independent of the path
4.    Both the amount of heat added and the amount of work needed are independent of the path

Mechanical Energy

Mechanical Energy Conserved Touchstone Example

INtroducgion - Refer back to falling ball, do it with energy    

Note use of mgh for inclined plane
Use double inclined plane with one bend in middle - still energy cons works.
Use arbitrary shape and mention work? - NOT YET

Model: Mechanical Energy and Non-Conservative Work

In this WIKItext, as in most introductory mechanics books, the only form of energy that we will be greatly concerned with is mechanical energy.  Mechanical Energy is the sum of kinetic energy and (mechanical) potential energy but but excludes thermal energy, chemical energy, etc even though these may change the mechanical motion of the system.  Thus the mechanical energy of an isolated (from external influences) system can change due to purely internal interactions that are non-conservative. For example, friction between two objects which are both inside the system can reduce the kinetic energy (and increase the thermal energy), all without any external interactions being present.

Non-conservative forces, by definition, do not conserve mechanical energy. (Of course they do conserve total energy.)  They have the property that the work they do depends on the path of the system.  Generally it is easiest to recognize them because they dissipate (or create) mechanical energy.  For example, friction is non-conservative because if you do work to move a heavy box on a level floor, you must do more work to move it back to its original position.  In contrast if you compress a spring, you can recover the energy your work put into the spring by letting the spring do work when you release it.  This allows you to use a potential energy to represent the spring energy and then you don't have to calculate the work due to the spring force.  

When you are thinking conceptually about mechanics problems, it is important to determine whether each interaction and its associated force(s) is conservative or non-conservative.  Here are some pointers:

Conservative: The only conservative interactions we consider in mechanics are gravity (both uniform and universal), the linear force of a spring (Hooke's Law), and occasionally the electrostatic force.  Like these forces, conservative forces are generally functions of position only (so the work doesn't depend on the direction of motion).

Non-conservative: These obviously must transfer mechanical energy to/from some other form of energy.  Often they are dissipative, turning mechanical energy into heat.  Friction, air turbulence, interactions that bend or cut sheet metal, etc.

WARNING: in this WIKI, "energy" generally means "mechanical energy" - shorthand common to mechanics books.