In physics, the use of the term mechanical energy usually involves three types of energies: potential gravitational energy, kinetic energy, and elastic potential energy. Although potential energy is often represented by the expression PE, in this lesson we will use the variable U; similarly, kinetic energy will be represented by the variable K.

 

In the absence of non-conservative, or dissipative forces, these energies obey the law of conservation of energy, or ΔU + ΔK = 0. That is, when a system is only acting under the influence of conservative forces its total energy content never changes, the energy just converts between forms.

 

 

Let's practice with an example of a problem using conservation of energy and a potential energy graph.

 

 

 

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

 

 

 

 

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.

 

0 £ x < 1   1 < x < 3   3 < x £ 6

segment U(x) 0 ≤ x < 1 1 < x < 3 3 < x ≤ 6 4 - 3.5x ½(x - 2)2 ½x - 1

 

On a potential energy graph, when the function's derivative is equal to zero, then the net force acting on the system is equal to zero. When an object is located at one of these positions or in one of these regions it is said to be in a state of equilibrium: stable, unstable, dynamic, and static (or neutral).

 

On the following diagram,

• x₃ and x₅ are points of stable equilibrium - bowls. If the system is slightly displaced to either side the forces on either side will return the object back to these positions.
• x₄ is a position of unstable equilibrium - a crest or peak. If the object is displaced ever so slightly from this position, the internal forces on either side will act to encourage further displacement instead of returning it back to x₄.
• x₆ is a position of either dynamic or static (neutral) equilibrium - a plateau. Since there is no net force acting on the object it must either possess only potential energy and be at rest or, it also possesses kinetic energy and must be moving at a constant velocity.