It has previously been mentioned that there is a relationship between work and mechanical energy change. Whenever work is done upon an object by an external force (or nonconservative force), there will be a change in the total mechanical energy of the object. If only internal forces are doing work (no work done by external forces), then there is no change in the total amount of mechanical energy. The total mechanical energy is said to be conserved. In this part of Lesson 2, we will further explore the quantitative relationship between work and mechanical energy in situations in which there are no external forces doing work.
The quantitative relationship between work and the two forms of mechanical energy is expressed by the following equation:
<span>KEi + PEi + Wext = KEf + PEf</span>
The equation illustrates that the total mechanical energy (KE + PE) of the object is changed as a result of work done by external forces. There are a host of other situations in which the only forces doing work are internal or conservative forces. In such situations, the total mechanical energy of the object is not changed. The external work term cancels from the above equation and mechanical energy is conserved. The previous equation is simplified to the following form:
<span>KEi + PEi = KEf + PEf</span>
In these situations, the sum of the kinetic and potential energy is everywhere the same. As the potential energy is increased due to the stretch/compression of a spring or an increase in its height above the earth, the kinetic energy is decreased due to the object slowing down. As the potential energy is decreased due to the return of a spring to its rest position or a decrease in height above the earth, the kinetic energy is increased due to the object speeding up. We would say that energy is transformed or changes its form from kinetic energy to potential energy (or vice versa); yet the total amount present is conserved - i.e., always the same.
