Summarizing Conservative Forces and Potential Energy

Summary

• A conservative force is one for which work depends only on the starting and ending points of a motion, not on the path taken.
• We can define potential energy \(\left(\text{PE}\right)\) for any conservative force, just as we defined \({\text{PE}}_{g}\) for the gravitational force.
• The potential energy of a spring is \({\text{PE}}_{s}=\frac{1}{2}{\text{kx}}^{2}\), where \(k\) is the spring’s force constant and \(x\) is the displacement from its undeformed position.
• Mechanical energy is defined to be \(\text{KE}+\text{PE}\) for a conservative force.
• When only conservative forces act on and within a system, the total mechanical energy is constant. In equation form,

where i and f denote initial and final values. This is known as the conservation of mechanical energy.

Glossary

conservative force

a force that does the same work for any given initial and final configuration, regardless of the path followed

potential energy

energy due to position, shape, or configuration

potential energy of a spring

the stored energy of a spring as a function of its displacement; when Hooke’s law applies, it is given by the expression \(\frac{1}{2}{\text{kx}}^{2}\) where \(x\) is the distance the spring is compressed or extended and \(k\) is the spring constant

conservation of mechanical energy

the rule that the sum of the kinetic energies and potential energies remains constant if only conservative forces act on and within a system

mechanical energy

the sum of kinetic energy and potential energy