Force and Position Relationships To look into this more carefully, let's reexamine some important graphs for a vibrating spring. Notice that the position and acceleration/force graphs are 180º outofphase: when the spring's displacement from its equilibrium is UP, the restoring force and acceleration are DOWN and viceversa. Our next graphs draw our attention to the spring's displacement, energy modes and restoring force. Notice that
 when the spring is either in a state of maximum extension or compression its potential energy is also a maximum
 when the spring's displacement is DOWN the restoring force is UP
 when the potential energy function has a negative slope, the restoring force is positive and viceversa
 when the restoring force is zero, the potential energy is zero

at any point in the cycle, the total energy is constant, U + K = U_{max} = K_{max}
Force Functions Our next step will be to show that a function representing the instantaneous values of the restoring force can be expressed as the negative of the derivative of our potential energy function Remember our two relationships involving work The work done by a conservative force decreases an object's potential energy while it is increasing its kinetic energy
Defining the initial potential energy U_{o} = 0, gives us
Using the calculus, we see that our desired expression of the instantaneous restoring force being equal to the negative derivative of the potential energy function. Let's practice this relationship with an example. 