The significance of conservative forces is that, if a force is a conservative force, then it can have an associated quantity defined for it called the potential energy function.  In particular, the change in potential energy, DPE, is defined to be negative the work done by the conservative force.Since there are two conservative forces with which we are currently familiar, that means that there are two potential energies with which we must become familiar: the gravitational potential energy and the spring potential energy.   We summarize these two potential energies below. But first, we note that, since work is just the transfer of energy, it must follow that energy and work have the same MKS unit – namely, joules (J).

Gravitational Potential Energy (PE_g):

Since only the change in potential energy is defined, we must always give a reference position with respect to which the change in potential energy is given. As far as gravity is concerned, we need only worry about the change in height of an object above the earth’s surface. (This form of the gravitational potential energy is valid only at the earth’s surface.)  In particular, choose some position above the earth’s surface to be the y = 0 position.  (This is usually taken to be at ground level, or at the lowest position of the object under consideration.)  The gravitational potential energy of an object of mass m at a position yabove the y = 0 reference position is then given by

PE_g = mgy .

(The position y is taken to be negative if it is below the y = 0 position, so the potential energy will also be negative there.)

Spring Potential Energy (PE_sp):

As with the gravitational potential energy above, we must define a reference position in order to give the potential energy of a spring.  With one end of the spring fixed (to a wall, ceiling, etc.), the position of the other end of the spring is measured relative to the unstretched  and uncompressed position of the spring. That is, if we use x to denote the position of the end of the spring, then the x = 0 position is when the spring is in its relaxed state (at its so-called equilibrium length).  Relative to this position, the spring potential energy is given by

PE_sp = (½) k x² ,

where k is the spring constant, and x is the distance of the end of the spring from its x = 0 equilibrium position. 

Note that, whereas the gravitational potential energy can be either positive or negative (above or below the y = 0 position), the spring potential energy is always positive (since it varies as x² ).