Potential Energy

For Conservative Forces like potential energy, the energy itself represents the state of the system, not an individual object. For example, a block high above the earth has a gravitational potential energy related to the block-earth system, and not the block itself. Changes in the system, like the block falling to the ground, represent a change in potential energy, or a Work:

For a state initial and a state final that we define,

We take the inverse to find the force:

Conservation of Mechanical Energy

For an isolated system for only conservative forces:

In other words:

In an isolated system in which only conservative forces act, the total mechanical energy remains constant.

Using this along with the Work-Energy Theorem, we can use this to analyze conservative systems where we usually used Newton’s Laws. In a more applicable form:

For instances with rotation, we can separate kinetic into and translational and rotational kinetic energy:

At an equilibrium point of a system we consider that if we place the object (or in general the objects) there with zero Kinetic energy ,the object will stay there . The fact that the object stays there means that it will not change its position , thus the Potential energy will remain the same .

Using we can know the future behavior of the particle. We can find using energy:

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An important concept to know is Equillibrium, a state where when you change the position of the particle the change potential energy remains at zero.

At an equilibrium point of a system we consider that if we place the object (or in general the objects) there with zero Kinetic energy, the object will stay there. The fact that the object stays there means that it will not change its position, thus the Potential energy will remain the same.