8.3:
Conservation of Mechanical Energy
Consider a tire swinging like a pendulum back and forth under the influence of gravitational force. The pendulum-Earth system is assumed to be an isolated system.
As the pendulum swings, the system's energy varies back and forth between gravitational potential energy and kinetic energy, while the system's mechanical energy—the sum of the two—remains constant.
When the pendulum is at its lowest position, it's at its maximum speed, and all the energy is kinetic energy. At its highest position, where the pendulum momentarily stops, the speed is zero, and all the energy is potential energy.
At the intermediate positions, half the energy is kinetic energy, and half is potential energy.
In any isolated system, the law of conservation of mechanical energy states that the total mechanical energy of the system at any instant remains constant. The subscripts refer to different time instants during an energy transfer process.
8.3:
Conservation of Mechanical Energy
The mechanical energy E of a system is the sum of its potential energy U and the kinetic energy K of the objects within it. What happens to this mechanical energy when only conservative forces cause energy transfers within the system—that is, when frictional and drag forces do not act on the objects in the system? Also assume that the system is isolated from its environment; in other words no external force from an object outside the system causes energy changes inside the system.
When a conservative force does work W on an object within a system, that force transfers energy between kinetic energy K of the object and potential energy U of the system. In an isolated system, where only conservative forces cause energy changes, the kinetic energy and potential energy can change; however, their sum, the mechanical energy E of the system, cannot change. This result is called the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy allows us to solve problems that would be difficult to solve using only Newton's laws. When the mechanical energy of a system is conserved, the sum of the kinetic energy and potential energy at one instant can be related to their sum at another instant without finding the work done by the forces involved. A great advantage of using the conservation of energy instead of Newton's laws of motion is that it is possible to go from the initial state to the final state without considering all the intermediate motion.
Suggested Reading
- Walker, J., Halliday, D., Resnick, R. (2007), Fundamentals of Physics, 10th Edition, John Wiley & Sons, Inc.; section 8.2; pages 184-186.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/8-3-conservation-of-energy section 8.3; pages 372-377.