Potential Energy

Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other.

Potential energy exists whenever an object which has mass has a position within a force field.

Potential Energy of a System

The work done by an external agent on the system of the book and the Earth as the book is lifted from a height ya to a height yb is equal to mgyb = mgya.

While the book was at the highest point, the energy of the system had

the potential to become kinetic energy, but did not do so until the book was allowed to

fall. Thus, we call the energy storage mechanism before we release the book potential

energy. We will find that a potential energy can only be associated with specific types of

forces. In this particular case, we are discussing gravitational potential energy.

we can identify the quantity mgy as the gravitational potential energy Ug :

Ug ≡ mgy                 

Using our definition of gravitational potential energy, Equation 8.1 can now be rewritten as

W = ∆Ug                 

The Isolated System-Conservation of Mechanical Energy

As the book falls from yb to ya, the work done by the gravitational force on the book is :

W on book=(mg) . ∆r = (- mgĵ) . [(yb-ya)ĵ] = mgyb-mgya        

The work done on the book is equal to the change in the kinetic energy of the book:

W on book = ∆K book

    It can be written as:

∆Kbook = mgyb-mgya

We define the sum of kinetic and potential energies as mechanical energy:

 E mech = K + Ug

   we can write the general form of the definition for mechanical energy without a subscript on U:

 E mech = K + U

Isolated System

An isolated system is one for which there are no energy transfers across the boundary. The energy in such a system is conserved—the sum of the kinetic and potential energies remains constant.

Elastic Potential Energy

The elastic potential energy function associated with the block–spring system is defined by:

The elastic potential energy of the system can be thought of as the energy stored in the deformed spring (one that is either compressed or stretched from its equilibrium position).

Conservative Forces

Conservative forces have these two equivalent properties:

1. The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle.

2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one in which the beginning and end points are identical.)

Nonconservative Forces

Nonconservative forces acting within a system cause a change in the mechanical energy Emech of the system.

8.4      Changes in Mechanical  for Nonconservative  Forces

Ø  if the forces acting on objects within a system are conservative, then the mechanical energy of the system is conserved.

Ø  if some of the forces acting on objects within the system are not conservative, then the mechanical energy of the system changes.

Consider the book sliding across the surface in the preceding section. As the book
moves through a distance d, the only force that does work on it is the force of kinetic friction. This force causes a decrease in the kinetic energy of the book. This decrease was calculated in Chapter 7, leading to 7.20, which we repeat here:

∆K   = -f_kd

                                      Changes kinetic energy = amount by which the       mechanical energy  of the changes because of the force of kinetic friction

o   In general, if a friction force acts within a system,

o   if the book moves on an incline that is not frictionless, there is a change in both the kinetic energy and the gravitational potential energy of the book–Earth system.

>> where ∆U is the change in all forms of potential energy

v  Change in mechanical energy of a system due to friction within the system

8.5  Relationship Between Conservative Forces and Potential Energy

The work done on a member of a system by a conservative force between the members does not depend on the path taken by the moving member. The work depends only on the initial and final coordinates.

A potential energy function U such that the work done by a conservative

force equals the decrease in the potential energy of the system,

The work done by a conservative force acting between members of a system equals the negative of the change in the potential energy associated with that force when the configurationof the system changes, where the change in the potential energy is defined as   

                                            

                                                     

 

o   If the point of application of the force undergoes an infinitesimal displacement
dx, we can express the infinitesimal change in the potential energy of the system dU as

o   Therefore, the conservative force is related to the potential energy function through

o   The x component of a conservative force acting on an object within a system equals the negative derivative of the potential energy of the system with respect to x

8.6 Energy Diagrams and Equilibrium of a System

The motion of a sy stem can often be understood qualitatively through a graph of its potential energy versus the position of a member of the system.