Force and Position Relationships To look into this more carefully, let's re-examine some important graphs for a vibrating spring. Notice that the position and acceleration/force graphs are 180º out-of-phase: when the spring's displacement from its equilibrium is UP, the restoring force and acceleration are DOWN and vice-versa. 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 vice-versa
- when the restoring force is zero, the potential energy is zero
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at any point in the cycle, the total energy is constant, U + K = Umax = Kmax
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 Uo = 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. |