Question

define potential energ, and derive the equation

Answer 1

Derivation for the equation of Potenstial Energy:
Let the work done on the object against gravity = W
                 Work done, W = force × displacement
                 Work done, W  = mg × h
                 Work done, W = mgh

Since workdone on the object is equal to mgh, an energy equal  to mgh units is gained by the object . This is the potential energy (Ep) of the object.
                              Ep = mgh

Answer 2

The extension produced x is directly proportional to the force applied to extend or compress the spring. F=kx were k is spring constant. The work done to stretch or compress the spring is stored as its PE W= integral 0 to x (Fdx) =0 to x (kxdx) =1/2 kx^2